*> \brief \b DBBCSD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBBCSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbbcsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbbcsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbbcsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
* THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
* V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
* B22D, B22E, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
* INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B11D( * ), B11E( * ), B12D( * ), B12E( * ),
* $ B21D( * ), B21E( * ), B22D( * ), B22E( * ),
* $ PHI( * ), THETA( * ), WORK( * )
* DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
* $ V2T( LDV2T, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBBCSD computes the CS decomposition of an orthogonal matrix in
*> bidiagonal-block form,
*>
*>
*> [ B11 | B12 0 0 ]
*> [ 0 | 0 -I 0 ]
*> X = [----------------]
*> [ B21 | B22 0 0 ]
*> [ 0 | 0 0 I ]
*>
*> [ C | -S 0 0 ]
*> [ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**T
*> = [---------] [---------------] [---------] .
*> [ | U2 ] [ S | C 0 0 ] [ | V2 ]
*> [ 0 | 0 0 I ]
*>
*> X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
*> than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
*> transposed and/or permuted. This can be done in constant time using
*> the TRANS and SIGNS options. See DORCSD for details.)
*>
*> The bidiagonal matrices B11, B12, B21, and B22 are represented
*> implicitly by angles THETA(1:Q) and PHI(1:Q-1).
*>
*> The orthogonal matrices U1, U2, V1T, and V2T are input/output.
*> The input matrices are pre- or post-multiplied by the appropriate
*> singular vector matrices.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU1
*> \verbatim
*> JOBU1 is CHARACTER
*> = 'Y': U1 is updated;
*> otherwise: U1 is not updated.
*> \endverbatim
*>
*> \param[in] JOBU2
*> \verbatim
*> JOBU2 is CHARACTER
*> = 'Y': U2 is updated;
*> otherwise: U2 is not updated.
*> \endverbatim
*>
*> \param[in] JOBV1T
*> \verbatim
*> JOBV1T is CHARACTER
*> = 'Y': V1T is updated;
*> otherwise: V1T is not updated.
*> \endverbatim
*>
*> \param[in] JOBV2T
*> \verbatim
*> JOBV2T is CHARACTER
*> = 'Y': V2T is updated;
*> otherwise: V2T is not updated.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER
*> = 'T': X, U1, U2, V1T, and V2T are stored in row-major
*> order;
*> otherwise: X, U1, U2, V1T, and V2T are stored in column-
*> major order.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows and columns in X, the orthogonal matrix in
*> bidiagonal-block form.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows in the top-left block of X. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is INTEGER
*> The number of columns in the top-left block of X.
*> 0 <= Q <= MIN(P,M-P,M-Q).
*> \endverbatim
*>
*> \param[in,out] THETA
*> \verbatim
*> THETA is DOUBLE PRECISION array, dimension (Q)
*> On entry, the angles THETA(1),...,THETA(Q) that, along with
*> PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
*> form. On exit, the angles whose cosines and sines define the
*> diagonal blocks in the CS decomposition.
*> \endverbatim
*>
*> \param[in,out] PHI
*> \verbatim
*> PHI is DOUBLE PRECISION array, dimension (Q-1)
*> The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
*> THETA(Q), define the matrix in bidiagonal-block form.
*> \endverbatim
*>
*> \param[in,out] U1
*> \verbatim
*> U1 is DOUBLE PRECISION array, dimension (LDU1,P)
*> On entry, a P-by-P matrix. On exit, U1 is postmultiplied
*> by the left singular vector matrix common to [ B11 ; 0 ] and
*> [ B12 0 0 ; 0 -I 0 0 ].
*> \endverbatim
*>
*> \param[in] LDU1
*> \verbatim
*> LDU1 is INTEGER
*> The leading dimension of the array U1, LDU1 >= MAX(1,P).
*> \endverbatim
*>
*> \param[in,out] U2
*> \verbatim
*> U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
*> On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
*> postmultiplied by the left singular vector matrix common to
*> [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*> LDU2 is INTEGER
*> The leading dimension of the array U2, LDU2 >= MAX(1,M-P).
*> \endverbatim
*>
*> \param[in,out] V1T
*> \verbatim
*> V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
*> On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
*> by the transpose of the right singular vector
*> matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
*> \endverbatim
*>
*> \param[in] LDV1T
*> \verbatim
*> LDV1T is INTEGER
*> The leading dimension of the array V1T, LDV1T >= MAX(1,Q).
*> \endverbatim
*>
*> \param[in,out] V2T
*> \verbatim
*> V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
*> On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
*> premultiplied by the transpose of the right
*> singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
*> [ B22 0 0 ; 0 0 I ].
*> \endverbatim
*>
*> \param[in] LDV2T
*> \verbatim
*> LDV2T is INTEGER
*> The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).
*> \endverbatim
*>
*> \param[out] B11D
*> \verbatim
*> B11D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B11D contains the cosines of THETA(1),
*> ..., THETA(Q). If DBBCSD fails to converge, then B11D
*> contains the diagonal of the partially reduced top-left
*> block.
*> \endverbatim
*>
*> \param[out] B11E
*> \verbatim
*> B11E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B11E contains zeros. If DBBCSD fails
*> to converge, then B11E contains the superdiagonal of the
*> partially reduced top-left block.
*> \endverbatim
*>
*> \param[out] B12D
*> \verbatim
*> B12D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B12D contains the negative sines of
*> THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*> B12D contains the diagonal of the partially reduced top-right
*> block.
*> \endverbatim
*>
*> \param[out] B12E
*> \verbatim
*> B12E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B12E contains zeros. If DBBCSD fails
*> to converge, then B12E contains the subdiagonal of the
*> partially reduced top-right block.
*> \endverbatim
*>
*> \param[out] B21D
*> \verbatim
*> B21D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B21D contains the negative sines of
*> THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*> B21D contains the diagonal of the partially reduced bottom-left
*> block.
*> \endverbatim
*>
*> \param[out] B21E
*> \verbatim
*> B21E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B21E contains zeros. If DBBCSD fails
*> to converge, then B21E contains the subdiagonal of the
*> partially reduced bottom-left block.
*> \endverbatim
*>
*> \param[out] B22D
*> \verbatim
*> B22D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B22D contains the negative sines of
*> THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*> B22D contains the diagonal of the partially reduced bottom-right
*> block.
*> \endverbatim
*>
*> \param[out] B22E
*> \verbatim
*> B22E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B22E contains zeros. If DBBCSD fails
*> to converge, then B22E contains the subdiagonal of the
*> partially reduced bottom-right block.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= MAX(1,8*Q).
*>
*> If LWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the WORK array,
*> returns this value as the first entry of the work array, and
*> no error message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if DBBCSD did not converge, INFO specifies the number
*> of nonzero entries in PHI, and B11D, B11E, etc.,
*> contain the partially reduced matrix.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> TOLMUL DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
*> TOLMUL controls the convergence criterion of the QR loop.
*> Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
*> are within TOLMUL*EPS of either bound.
*> \endverbatim
*
*> \par References:
* ================
*>
*> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*> Algorithms, 50(1):33-65, 2009.
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup bbcsd
*
* =====================================================================
SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P,
$ Q,
$ THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
$ V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
$ B22D, B22E, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
* ..
* .. Array Arguments ..
DOUBLE PRECISION B11D( * ), B11E( * ), B12D( * ), B12E( * ),
$ B21D( * ), B21E( * ), B22D( * ), B22E( * ),
$ PHI( * ), THETA( * ), WORK( * )
DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
$ V2T( LDV2T, * )
* ..
*
* ===================================================================
*
* .. Parameters ..
INTEGER MAXITR
PARAMETER ( MAXITR = 6 )
DOUBLE PRECISION HUNDRED, MEIGHTH, ONE, TEN, ZERO
PARAMETER ( HUNDRED = 100.0D0, MEIGHTH = -0.125D0,
$ ONE = 1.0D0, TEN = 10.0D0, ZERO = 0.0D0 )
DOUBLE PRECISION NEGONE
PARAMETER ( NEGONE = -1.0D0 )
DOUBLE PRECISION PIOVER2
PARAMETER ( PIOVER2 = 1.57079632679489661923132169163975144210D0 )
* ..
* .. Local Scalars ..
LOGICAL COLMAJOR, LQUERY, RESTART11, RESTART12,
$ RESTART21, RESTART22, WANTU1, WANTU2, WANTV1T,
$ WANTV2T
INTEGER I, IMIN, IMAX, ITER, IU1CS, IU1SN, IU2CS,
$ IU2SN, IV1TCS, IV1TSN, IV2TCS, IV2TSN, J,
$ LWORKMIN, LWORKOPT, MAXIT, MINI
DOUBLE PRECISION B11BULGE, B12BULGE, B21BULGE, B22BULGE, DUMMY,
$ EPS, MU, NU, R, SIGMA11, SIGMA21,
$ TEMP, THETAMAX, THETAMIN, THRESH, TOL, TOLMUL,
$ UNFL, X1, X2, Y1, Y2
*
* .. External Subroutines ..
EXTERNAL DLASR, DSCAL, DSWAP, DLARTGP, DLARTGS,
$ DLAS2,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
LOGICAL LSAME
EXTERNAL LSAME, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, ATAN2, COS, MAX, MIN, SIN, SQRT
* ..
* .. Executable Statements ..
*
* Test input arguments
*
INFO = 0
LQUERY = LWORK .EQ. -1
WANTU1 = LSAME( JOBU1, 'Y' )
WANTU2 = LSAME( JOBU2, 'Y' )
WANTV1T = LSAME( JOBV1T, 'Y' )
WANTV2T = LSAME( JOBV2T, 'Y' )
COLMAJOR = .NOT. LSAME( TRANS, 'T' )
*
IF( M .LT. 0 ) THEN
INFO = -6
ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
INFO = -7
ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
INFO = -8
ELSE IF( Q .GT. P .OR. Q .GT. M-P .OR. Q .GT. M-Q ) THEN
INFO = -8
ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
INFO = -12
ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
INFO = -14
ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
INFO = -16
ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
INFO = -18
END IF
*
* Quick return if Q = 0
*
IF( INFO .EQ. 0 .AND. Q .EQ. 0 ) THEN
LWORKMIN = 1
WORK(1) = LWORKMIN
RETURN
END IF
*
* Compute workspace
*
IF( INFO .EQ. 0 ) THEN
IU1CS = 1
IU1SN = IU1CS + Q
IU2CS = IU1SN + Q
IU2SN = IU2CS + Q
IV1TCS = IU2SN + Q
IV1TSN = IV1TCS + Q
IV2TCS = IV1TSN + Q
IV2TSN = IV2TCS + Q
LWORKOPT = IV2TSN + Q - 1
LWORKMIN = LWORKOPT
WORK(1) = LWORKOPT
IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
INFO = -28
END IF
END IF
*
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'DBBCSD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'Epsilon' )
UNFL = DLAMCH( 'Safe minimum' )
TOLMUL = MAX( TEN, MIN( HUNDRED, EPS**MEIGHTH ) )
TOL = TOLMUL*EPS
THRESH = MAX( TOL, MAXITR*Q*Q*UNFL )
*
* Test for negligible sines or cosines
*
DO I = 1, Q
IF( THETA(I) .LT. THRESH ) THEN
THETA(I) = ZERO
ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
THETA(I) = PIOVER2
END IF
END DO
DO I = 1, Q-1
IF( PHI(I) .LT. THRESH ) THEN
PHI(I) = ZERO
ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
PHI(I) = PIOVER2
END IF
END DO
*
* Initial deflation
*
IMAX = Q
DO WHILE( IMAX .GT. 1 )
IF( PHI(IMAX-1) .NE. ZERO ) THEN
EXIT
END IF
IMAX = IMAX - 1
END DO
IMIN = IMAX - 1
IF ( IMIN .GT. 1 ) THEN
DO WHILE( PHI(IMIN-1) .NE. ZERO )
IMIN = IMIN - 1
IF ( IMIN .LE. 1 ) EXIT
END DO
END IF
*
* Initialize iteration counter
*
MAXIT = MAXITR*Q*Q
ITER = 0
*
* Begin main iteration loop
*
DO WHILE( IMAX .GT. 1 )
*
* Compute the matrix entries
*
B11D(IMIN) = COS( THETA(IMIN) )
B21D(IMIN) = -SIN( THETA(IMIN) )
DO I = IMIN, IMAX - 1
B11E(I) = -SIN( THETA(I) ) * SIN( PHI(I) )
B11D(I+1) = COS( THETA(I+1) ) * COS( PHI(I) )
B12D(I) = SIN( THETA(I) ) * COS( PHI(I) )
B12E(I) = COS( THETA(I+1) ) * SIN( PHI(I) )
B21E(I) = -COS( THETA(I) ) * SIN( PHI(I) )
B21D(I+1) = -SIN( THETA(I+1) ) * COS( PHI(I) )
B22D(I) = COS( THETA(I) ) * COS( PHI(I) )
B22E(I) = -SIN( THETA(I+1) ) * SIN( PHI(I) )
END DO
B12D(IMAX) = SIN( THETA(IMAX) )
B22D(IMAX) = COS( THETA(IMAX) )
*
* Abort if not converging; otherwise, increment ITER
*
IF( ITER .GT. MAXIT ) THEN
INFO = 0
DO I = 1, Q
IF( PHI(I) .NE. ZERO )
$ INFO = INFO + 1
END DO
RETURN
END IF
*
ITER = ITER + IMAX - IMIN
*
* Compute shifts
*
THETAMAX = THETA(IMIN)
THETAMIN = THETA(IMIN)
DO I = IMIN+1, IMAX
IF( THETA(I) > THETAMAX )
$ THETAMAX = THETA(I)
IF( THETA(I) < THETAMIN )
$ THETAMIN = THETA(I)
END DO
*
IF( THETAMAX .GT. PIOVER2 - THRESH ) THEN
*
* Zero on diagonals of B11 and B22; induce deflation with a
* zero shift
*
MU = ZERO
NU = ONE
*
ELSE IF( THETAMIN .LT. THRESH ) THEN
*
* Zero on diagonals of B12 and B22; induce deflation with a
* zero shift
*
MU = ONE
NU = ZERO
*
ELSE
*
* Compute shifts for B11 and B21 and use the lesser
*
CALL DLAS2( B11D(IMAX-1), B11E(IMAX-1), B11D(IMAX),
$ SIGMA11,
$ DUMMY )
CALL DLAS2( B21D(IMAX-1), B21E(IMAX-1), B21D(IMAX),
$ SIGMA21,
$ DUMMY )
*
IF( SIGMA11 .LE. SIGMA21 ) THEN
MU = SIGMA11
NU = SQRT( ONE - MU**2 )
IF( MU .LT. THRESH ) THEN
MU = ZERO
NU = ONE
END IF
ELSE
NU = SIGMA21
MU = SQRT( 1.0 - NU**2 )
IF( NU .LT. THRESH ) THEN
MU = ONE
NU = ZERO
END IF
END IF
END IF
*
* Rotate to produce bulges in B11 and B21
*
IF( MU .LE. NU ) THEN
CALL DLARTGS( B11D(IMIN), B11E(IMIN), MU,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
ELSE
CALL DLARTGS( B21D(IMIN), B21E(IMIN), NU,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
END IF
*
TEMP = WORK(IV1TCS+IMIN-1)*B11D(IMIN) +
$ WORK(IV1TSN+IMIN-1)*B11E(IMIN)
B11E(IMIN) = WORK(IV1TCS+IMIN-1)*B11E(IMIN) -
$ WORK(IV1TSN+IMIN-1)*B11D(IMIN)
B11D(IMIN) = TEMP
B11BULGE = WORK(IV1TSN+IMIN-1)*B11D(IMIN+1)
B11D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B11D(IMIN+1)
TEMP = WORK(IV1TCS+IMIN-1)*B21D(IMIN) +
$ WORK(IV1TSN+IMIN-1)*B21E(IMIN)
B21E(IMIN) = WORK(IV1TCS+IMIN-1)*B21E(IMIN) -
$ WORK(IV1TSN+IMIN-1)*B21D(IMIN)
B21D(IMIN) = TEMP
B21BULGE = WORK(IV1TSN+IMIN-1)*B21D(IMIN+1)
B21D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B21D(IMIN+1)
*
* Compute THETA(IMIN)
*
THETA( IMIN ) = ATAN2( SQRT( B21D(IMIN)**2+B21BULGE**2 ),
$ SQRT( B11D(IMIN)**2+B11BULGE**2 ) )
*
* Chase the bulges in B11(IMIN+1,IMIN) and B21(IMIN+1,IMIN)
*
IF( B11D(IMIN)**2+B11BULGE**2 .GT. THRESH**2 ) THEN
CALL DLARTGP( B11BULGE, B11D(IMIN), WORK(IU1SN+IMIN-1),
$ WORK(IU1CS+IMIN-1), R )
ELSE IF( MU .LE. NU ) THEN
CALL DLARTGS( B11E( IMIN ), B11D( IMIN + 1 ), MU,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
ELSE
CALL DLARTGS( B12D( IMIN ), B12E( IMIN ), NU,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
END IF
IF( B21D(IMIN)**2+B21BULGE**2 .GT. THRESH**2 ) THEN
CALL DLARTGP( B21BULGE, B21D(IMIN), WORK(IU2SN+IMIN-1),
$ WORK(IU2CS+IMIN-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B21E( IMIN ), B21D( IMIN + 1 ), NU,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
ELSE
CALL DLARTGS( B22D(IMIN), B22E(IMIN), MU,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
END IF
WORK(IU2CS+IMIN-1) = -WORK(IU2CS+IMIN-1)
WORK(IU2SN+IMIN-1) = -WORK(IU2SN+IMIN-1)
*
TEMP = WORK(IU1CS+IMIN-1)*B11E(IMIN) +
$ WORK(IU1SN+IMIN-1)*B11D(IMIN+1)
B11D(IMIN+1) = WORK(IU1CS+IMIN-1)*B11D(IMIN+1) -
$ WORK(IU1SN+IMIN-1)*B11E(IMIN)
B11E(IMIN) = TEMP
IF( IMAX .GT. IMIN+1 ) THEN
B11BULGE = WORK(IU1SN+IMIN-1)*B11E(IMIN+1)
B11E(IMIN+1) = WORK(IU1CS+IMIN-1)*B11E(IMIN+1)
END IF
TEMP = WORK(IU1CS+IMIN-1)*B12D(IMIN) +
$ WORK(IU1SN+IMIN-1)*B12E(IMIN)
B12E(IMIN) = WORK(IU1CS+IMIN-1)*B12E(IMIN) -
$ WORK(IU1SN+IMIN-1)*B12D(IMIN)
B12D(IMIN) = TEMP
B12BULGE = WORK(IU1SN+IMIN-1)*B12D(IMIN+1)
B12D(IMIN+1) = WORK(IU1CS+IMIN-1)*B12D(IMIN+1)
TEMP = WORK(IU2CS+IMIN-1)*B21E(IMIN) +
$ WORK(IU2SN+IMIN-1)*B21D(IMIN+1)
B21D(IMIN+1) = WORK(IU2CS+IMIN-1)*B21D(IMIN+1) -
$ WORK(IU2SN+IMIN-1)*B21E(IMIN)
B21E(IMIN) = TEMP
IF( IMAX .GT. IMIN+1 ) THEN
B21BULGE = WORK(IU2SN+IMIN-1)*B21E(IMIN+1)
B21E(IMIN+1) = WORK(IU2CS+IMIN-1)*B21E(IMIN+1)
END IF
TEMP = WORK(IU2CS+IMIN-1)*B22D(IMIN) +
$ WORK(IU2SN+IMIN-1)*B22E(IMIN)
B22E(IMIN) = WORK(IU2CS+IMIN-1)*B22E(IMIN) -
$ WORK(IU2SN+IMIN-1)*B22D(IMIN)
B22D(IMIN) = TEMP
B22BULGE = WORK(IU2SN+IMIN-1)*B22D(IMIN+1)
B22D(IMIN+1) = WORK(IU2CS+IMIN-1)*B22D(IMIN+1)
*
* Inner loop: chase bulges from B11(IMIN,IMIN+2),
* B12(IMIN,IMIN+1), B21(IMIN,IMIN+2), and B22(IMIN,IMIN+1) to
* bottom-right
*
DO I = IMIN+1, IMAX-1
*
* Compute PHI(I-1)
*
X1 = SIN(THETA(I-1))*B11E(I-1) + COS(THETA(I-1))*B21E(I-1)
X2 = SIN(THETA(I-1))*B11BULGE + COS(THETA(I-1))*B21BULGE
Y1 = SIN(THETA(I-1))*B12D(I-1) + COS(THETA(I-1))*B22D(I-1)
Y2 = SIN(THETA(I-1))*B12BULGE + COS(THETA(I-1))*B22BULGE
*
PHI(I-1) = ATAN2( SQRT(X1**2+X2**2), SQRT(Y1**2+Y2**2) )
*
* Determine if there are bulges to chase or if a new direct
* summand has been reached
*
RESTART11 = B11E(I-1)**2 + B11BULGE**2 .LE. THRESH**2
RESTART21 = B21E(I-1)**2 + B21BULGE**2 .LE. THRESH**2
RESTART12 = B12D(I-1)**2 + B12BULGE**2 .LE. THRESH**2
RESTART22 = B22D(I-1)**2 + B22BULGE**2 .LE. THRESH**2
*
* If possible, chase bulges from B11(I-1,I+1), B12(I-1,I),
* B21(I-1,I+1), and B22(I-1,I). If necessary, restart bulge-
* chasing by applying the original shift again.
*
IF( .NOT. RESTART11 .AND. .NOT. RESTART21 ) THEN
CALL DLARTGP( X2, X1, WORK(IV1TSN+I-1),
$ WORK(IV1TCS+I-1),
$ R )
ELSE IF( .NOT. RESTART11 .AND. RESTART21 ) THEN
CALL DLARTGP( B11BULGE, B11E(I-1), WORK(IV1TSN+I-1),
$ WORK(IV1TCS+I-1), R )
ELSE IF( RESTART11 .AND. .NOT. RESTART21 ) THEN
CALL DLARTGP( B21BULGE, B21E(I-1), WORK(IV1TSN+I-1),
$ WORK(IV1TCS+I-1), R )
ELSE IF( MU .LE. NU ) THEN
CALL DLARTGS( B11D(I), B11E(I), MU, WORK(IV1TCS+I-1),
$ WORK(IV1TSN+I-1) )
ELSE
CALL DLARTGS( B21D(I), B21E(I), NU, WORK(IV1TCS+I-1),
$ WORK(IV1TSN+I-1) )
END IF
WORK(IV1TCS+I-1) = -WORK(IV1TCS+I-1)
WORK(IV1TSN+I-1) = -WORK(IV1TSN+I-1)
IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( Y2, Y1, WORK(IV2TSN+I-1-1),
$ WORK(IV2TCS+I-1-1), R )
ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
CALL DLARTGP( B12BULGE, B12D(I-1), WORK(IV2TSN+I-1-1),
$ WORK(IV2TCS+I-1-1), R )
ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( B22BULGE, B22D(I-1), WORK(IV2TSN+I-1-1),
$ WORK(IV2TCS+I-1-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B12E(I-1), B12D(I), NU,
$ WORK(IV2TCS+I-1-1),
$ WORK(IV2TSN+I-1-1) )
ELSE
CALL DLARTGS( B22E(I-1), B22D(I), MU,
$ WORK(IV2TCS+I-1-1),
$ WORK(IV2TSN+I-1-1) )
END IF
*
TEMP = WORK(IV1TCS+I-1)*B11D(I) + WORK(IV1TSN+I-1)*B11E(I)
B11E(I) = WORK(IV1TCS+I-1)*B11E(I) -
$ WORK(IV1TSN+I-1)*B11D(I)
B11D(I) = TEMP
B11BULGE = WORK(IV1TSN+I-1)*B11D(I+1)
B11D(I+1) = WORK(IV1TCS+I-1)*B11D(I+1)
TEMP = WORK(IV1TCS+I-1)*B21D(I) + WORK(IV1TSN+I-1)*B21E(I)
B21E(I) = WORK(IV1TCS+I-1)*B21E(I) -
$ WORK(IV1TSN+I-1)*B21D(I)
B21D(I) = TEMP
B21BULGE = WORK(IV1TSN+I-1)*B21D(I+1)
B21D(I+1) = WORK(IV1TCS+I-1)*B21D(I+1)
TEMP = WORK(IV2TCS+I-1-1)*B12E(I-1) +
$ WORK(IV2TSN+I-1-1)*B12D(I)
B12D(I) = WORK(IV2TCS+I-1-1)*B12D(I) -
$ WORK(IV2TSN+I-1-1)*B12E(I-1)
B12E(I-1) = TEMP
B12BULGE = WORK(IV2TSN+I-1-1)*B12E(I)
B12E(I) = WORK(IV2TCS+I-1-1)*B12E(I)
TEMP = WORK(IV2TCS+I-1-1)*B22E(I-1) +
$ WORK(IV2TSN+I-1-1)*B22D(I)
B22D(I) = WORK(IV2TCS+I-1-1)*B22D(I) -
$ WORK(IV2TSN+I-1-1)*B22E(I-1)
B22E(I-1) = TEMP
B22BULGE = WORK(IV2TSN+I-1-1)*B22E(I)
B22E(I) = WORK(IV2TCS+I-1-1)*B22E(I)
*
* Compute THETA(I)
*
X1 = COS(PHI(I-1))*B11D(I) + SIN(PHI(I-1))*B12E(I-1)
X2 = COS(PHI(I-1))*B11BULGE + SIN(PHI(I-1))*B12BULGE
Y1 = COS(PHI(I-1))*B21D(I) + SIN(PHI(I-1))*B22E(I-1)
Y2 = COS(PHI(I-1))*B21BULGE + SIN(PHI(I-1))*B22BULGE
*
THETA(I) = ATAN2( SQRT(Y1**2+Y2**2), SQRT(X1**2+X2**2) )
*
* Determine if there are bulges to chase or if a new direct
* summand has been reached
*
RESTART11 = B11D(I)**2 + B11BULGE**2 .LE. THRESH**2
RESTART12 = B12E(I-1)**2 + B12BULGE**2 .LE. THRESH**2
RESTART21 = B21D(I)**2 + B21BULGE**2 .LE. THRESH**2
RESTART22 = B22E(I-1)**2 + B22BULGE**2 .LE. THRESH**2
*
* If possible, chase bulges from B11(I+1,I), B12(I+1,I-1),
* B21(I+1,I), and B22(I+1,I-1). If necessary, restart bulge-
* chasing by applying the original shift again.
*
IF( .NOT. RESTART11 .AND. .NOT. RESTART12 ) THEN
CALL DLARTGP( X2, X1, WORK(IU1SN+I-1),
$ WORK(IU1CS+I-1),
$ R )
ELSE IF( .NOT. RESTART11 .AND. RESTART12 ) THEN
CALL DLARTGP( B11BULGE, B11D(I), WORK(IU1SN+I-1),
$ WORK(IU1CS+I-1), R )
ELSE IF( RESTART11 .AND. .NOT. RESTART12 ) THEN
CALL DLARTGP( B12BULGE, B12E(I-1), WORK(IU1SN+I-1),
$ WORK(IU1CS+I-1), R )
ELSE IF( MU .LE. NU ) THEN
CALL DLARTGS( B11E(I), B11D(I+1), MU, WORK(IU1CS+I-1),
$ WORK(IU1SN+I-1) )
ELSE
CALL DLARTGS( B12D(I), B12E(I), NU, WORK(IU1CS+I-1),
$ WORK(IU1SN+I-1) )
END IF
IF( .NOT. RESTART21 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( Y2, Y1, WORK(IU2SN+I-1),
$ WORK(IU2CS+I-1),
$ R )
ELSE IF( .NOT. RESTART21 .AND. RESTART22 ) THEN
CALL DLARTGP( B21BULGE, B21D(I), WORK(IU2SN+I-1),
$ WORK(IU2CS+I-1), R )
ELSE IF( RESTART21 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( B22BULGE, B22E(I-1), WORK(IU2SN+I-1),
$ WORK(IU2CS+I-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B21E(I), B21E(I+1), NU, WORK(IU2CS+I-1),
$ WORK(IU2SN+I-1) )
ELSE
CALL DLARTGS( B22D(I), B22E(I), MU, WORK(IU2CS+I-1),
$ WORK(IU2SN+I-1) )
END IF
WORK(IU2CS+I-1) = -WORK(IU2CS+I-1)
WORK(IU2SN+I-1) = -WORK(IU2SN+I-1)
*
TEMP = WORK(IU1CS+I-1)*B11E(I) + WORK(IU1SN+I-1)*B11D(I+1)
B11D(I+1) = WORK(IU1CS+I-1)*B11D(I+1) -
$ WORK(IU1SN+I-1)*B11E(I)
B11E(I) = TEMP
IF( I .LT. IMAX - 1 ) THEN
B11BULGE = WORK(IU1SN+I-1)*B11E(I+1)
B11E(I+1) = WORK(IU1CS+I-1)*B11E(I+1)
END IF
TEMP = WORK(IU2CS+I-1)*B21E(I) + WORK(IU2SN+I-1)*B21D(I+1)
B21D(I+1) = WORK(IU2CS+I-1)*B21D(I+1) -
$ WORK(IU2SN+I-1)*B21E(I)
B21E(I) = TEMP
IF( I .LT. IMAX - 1 ) THEN
B21BULGE = WORK(IU2SN+I-1)*B21E(I+1)
B21E(I+1) = WORK(IU2CS+I-1)*B21E(I+1)
END IF
TEMP = WORK(IU1CS+I-1)*B12D(I) + WORK(IU1SN+I-1)*B12E(I)
B12E(I) = WORK(IU1CS+I-1)*B12E(I) - WORK(IU1SN+I-1)*B12D(I)
B12D(I) = TEMP
B12BULGE = WORK(IU1SN+I-1)*B12D(I+1)
B12D(I+1) = WORK(IU1CS+I-1)*B12D(I+1)
TEMP = WORK(IU2CS+I-1)*B22D(I) + WORK(IU2SN+I-1)*B22E(I)
B22E(I) = WORK(IU2CS+I-1)*B22E(I) - WORK(IU2SN+I-1)*B22D(I)
B22D(I) = TEMP
B22BULGE = WORK(IU2SN+I-1)*B22D(I+1)
B22D(I+1) = WORK(IU2CS+I-1)*B22D(I+1)
*
END DO
*
* Compute PHI(IMAX-1)
*
X1 = SIN(THETA(IMAX-1))*B11E(IMAX-1) +
$ COS(THETA(IMAX-1))*B21E(IMAX-1)
Y1 = SIN(THETA(IMAX-1))*B12D(IMAX-1) +
$ COS(THETA(IMAX-1))*B22D(IMAX-1)
Y2 = SIN(THETA(IMAX-1))*B12BULGE + COS(THETA(IMAX-1))*B22BULGE
*
PHI(IMAX-1) = ATAN2( ABS(X1), SQRT(Y1**2+Y2**2) )
*
* Chase bulges from B12(IMAX-1,IMAX) and B22(IMAX-1,IMAX)
*
RESTART12 = B12D(IMAX-1)**2 + B12BULGE**2 .LE. THRESH**2
RESTART22 = B22D(IMAX-1)**2 + B22BULGE**2 .LE. THRESH**2
*
IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( Y2, Y1, WORK(IV2TSN+IMAX-1-1),
$ WORK(IV2TCS+IMAX-1-1), R )
ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
CALL DLARTGP( B12BULGE, B12D(IMAX-1),
$ WORK(IV2TSN+IMAX-1-1),
$ WORK(IV2TCS+IMAX-1-1), R )
ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( B22BULGE, B22D(IMAX-1),
$ WORK(IV2TSN+IMAX-1-1),
$ WORK(IV2TCS+IMAX-1-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B12E(IMAX-1), B12D(IMAX), NU,
$ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
ELSE
CALL DLARTGS( B22E(IMAX-1), B22D(IMAX), MU,
$ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
END IF
*
TEMP = WORK(IV2TCS+IMAX-1-1)*B12E(IMAX-1) +
$ WORK(IV2TSN+IMAX-1-1)*B12D(IMAX)
B12D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B12D(IMAX) -
$ WORK(IV2TSN+IMAX-1-1)*B12E(IMAX-1)
B12E(IMAX-1) = TEMP
TEMP = WORK(IV2TCS+IMAX-1-1)*B22E(IMAX-1) +
$ WORK(IV2TSN+IMAX-1-1)*B22D(IMAX)
B22D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B22D(IMAX) -
$ WORK(IV2TSN+IMAX-1-1)*B22E(IMAX-1)
B22E(IMAX-1) = TEMP
*
* Update singular vectors
*
IF( WANTU1 ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'R', 'V', 'F', P, IMAX-IMIN+1,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
$ U1(1,IMIN), LDU1 )
ELSE
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, P,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
$ U1(IMIN,1), LDU1 )
END IF
END IF
IF( WANTU2 ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'R', 'V', 'F', M-P, IMAX-IMIN+1,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
$ U2(1,IMIN), LDU2 )
ELSE
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-P,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
$ U2(IMIN,1), LDU2 )
END IF
END IF
IF( WANTV1T ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, Q,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
$ V1T(IMIN,1), LDV1T )
ELSE
CALL DLASR( 'R', 'V', 'F', Q, IMAX-IMIN+1,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
$ V1T(1,IMIN), LDV1T )
END IF
END IF
IF( WANTV2T ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-Q,
$ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
$ V2T(IMIN,1), LDV2T )
ELSE
CALL DLASR( 'R', 'V', 'F', M-Q, IMAX-IMIN+1,
$ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
$ V2T(1,IMIN), LDV2T )
END IF
END IF
*
* Fix signs on B11(IMAX-1,IMAX) and B21(IMAX-1,IMAX)
*
IF( B11E(IMAX-1)+B21E(IMAX-1) .GT. 0 ) THEN
B11D(IMAX) = -B11D(IMAX)
B21D(IMAX) = -B21D(IMAX)
IF( WANTV1T ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( Q, NEGONE, V1T(IMAX,1), LDV1T )
ELSE
CALL DSCAL( Q, NEGONE, V1T(1,IMAX), 1 )
END IF
END IF
END IF
*
* Compute THETA(IMAX)
*
X1 = COS(PHI(IMAX-1))*B11D(IMAX) +
$ SIN(PHI(IMAX-1))*B12E(IMAX-1)
Y1 = COS(PHI(IMAX-1))*B21D(IMAX) +
$ SIN(PHI(IMAX-1))*B22E(IMAX-1)
*
THETA(IMAX) = ATAN2( ABS(Y1), ABS(X1) )
*
* Fix signs on B11(IMAX,IMAX), B12(IMAX,IMAX-1), B21(IMAX,IMAX),
* and B22(IMAX,IMAX-1)
*
IF( B11D(IMAX)+B12E(IMAX-1) .LT. 0 ) THEN
B12D(IMAX) = -B12D(IMAX)
IF( WANTU1 ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( P, NEGONE, U1(1,IMAX), 1 )
ELSE
CALL DSCAL( P, NEGONE, U1(IMAX,1), LDU1 )
END IF
END IF
END IF
IF( B21D(IMAX)+B22E(IMAX-1) .GT. 0 ) THEN
B22D(IMAX) = -B22D(IMAX)
IF( WANTU2 ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( M-P, NEGONE, U2(1,IMAX), 1 )
ELSE
CALL DSCAL( M-P, NEGONE, U2(IMAX,1), LDU2 )
END IF
END IF
END IF
*
* Fix signs on B12(IMAX,IMAX) and B22(IMAX,IMAX)
*
IF( B12D(IMAX)+B22D(IMAX) .LT. 0 ) THEN
IF( WANTV2T ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( M-Q, NEGONE, V2T(IMAX,1), LDV2T )
ELSE
CALL DSCAL( M-Q, NEGONE, V2T(1,IMAX), 1 )
END IF
END IF
END IF
*
* Test for negligible sines or cosines
*
DO I = IMIN, IMAX
IF( THETA(I) .LT. THRESH ) THEN
THETA(I) = ZERO
ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
THETA(I) = PIOVER2
END IF
END DO
DO I = IMIN, IMAX-1
IF( PHI(I) .LT. THRESH ) THEN
PHI(I) = ZERO
ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
PHI(I) = PIOVER2
END IF
END DO
*
* Deflate
*
IF (IMAX .GT. 1) THEN
DO WHILE( PHI(IMAX-1) .EQ. ZERO )
IMAX = IMAX - 1
IF (IMAX .LE. 1) EXIT
END DO
END IF
IF( IMIN .GT. IMAX - 1 )
$ IMIN = IMAX - 1
IF (IMIN .GT. 1) THEN
DO WHILE (PHI(IMIN-1) .NE. ZERO)
IMIN = IMIN - 1
IF (IMIN .LE. 1) EXIT
END DO
END IF
*
* Repeat main iteration loop
*
END DO
*
* Postprocessing: order THETA from least to greatest
*
DO I = 1, Q
*
MINI = I
THETAMIN = THETA(I)
DO J = I+1, Q
IF( THETA(J) .LT. THETAMIN ) THEN
MINI = J
THETAMIN = THETA(J)
END IF
END DO
*
IF( MINI .NE. I ) THEN
THETA(MINI) = THETA(I)
THETA(I) = THETAMIN
IF( COLMAJOR ) THEN
IF( WANTU1 )
$ CALL DSWAP( P, U1(1,I), 1, U1(1,MINI), 1 )
IF( WANTU2 )
$ CALL DSWAP( M-P, U2(1,I), 1, U2(1,MINI), 1 )
IF( WANTV1T )
$ CALL DSWAP( Q, V1T(I,1), LDV1T, V1T(MINI,1),
$ LDV1T )
IF( WANTV2T )
$ CALL DSWAP( M-Q, V2T(I,1), LDV2T, V2T(MINI,1),
$ LDV2T )
ELSE
IF( WANTU1 )
$ CALL DSWAP( P, U1(I,1), LDU1, U1(MINI,1), LDU1 )
IF( WANTU2 )
$ CALL DSWAP( M-P, U2(I,1), LDU2, U2(MINI,1), LDU2 )
IF( WANTV1T )
$ CALL DSWAP( Q, V1T(1,I), 1, V1T(1,MINI), 1 )
IF( WANTV2T )
$ CALL DSWAP( M-Q, V2T(1,I), 1, V2T(1,MINI), 1 )
END IF
END IF
*
END DO
*
RETURN
*
* End of DBBCSD
*
END
*> \brief \b DBDSDC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSDC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbdsdc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbdsdc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbdsdc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPQ, UPLO
* INTEGER INFO, LDU, LDVT, N
* ..
* .. Array Arguments ..
* INTEGER IQ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), Q( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBDSDC computes the singular value decomposition (SVD) of a real
*> N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT,
*> using a divide and conquer method, where S is a diagonal matrix
*> with non-negative diagonal elements (the singular values of B), and
*> U and VT are orthogonal matrices of left and right singular vectors,
*> respectively. DBDSDC can be used to compute all singular values,
*> and optionally, singular vectors or singular vectors in compact form.
*>
*> The code currently calls DLASDQ if singular values only are desired.
*> However, it can be slightly modified to compute singular values
*> using the divide and conquer method.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': B is upper bidiagonal.
*> = 'L': B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> Specifies whether singular vectors are to be computed
*> as follows:
*> = 'N': Compute singular values only;
*> = 'P': Compute singular values and compute singular
*> vectors in compact form;
*> = 'I': Compute singular values and singular vectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the bidiagonal matrix B.
*> On exit, if INFO=0, the singular values of B.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the elements of E contain the offdiagonal
*> elements of the bidiagonal matrix whose SVD is desired.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,N)
*> If COMPQ = 'I', then:
*> On exit, if INFO = 0, U contains the left singular vectors
*> of the bidiagonal matrix.
*> For other values of COMPQ, U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= 1.
*> If singular vectors are desired, then LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
*> If COMPQ = 'I', then:
*> On exit, if INFO = 0, VT**T contains the right singular
*> vectors of the bidiagonal matrix.
*> For other values of COMPQ, VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= 1.
*> If singular vectors are desired, then LDVT >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ)
*> If COMPQ = 'P', then:
*> On exit, if INFO = 0, Q and IQ contain the left
*> and right singular vectors in a compact form,
*> requiring O(N log N) space instead of 2*N**2.
*> In particular, Q contains all the DOUBLE PRECISION data in
*> LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
*> words of memory, where SMLSIZ is returned by ILAENV and
*> is equal to the maximum size of the subproblems at the
*> bottom of the computation tree (usually about 25).
*> For other values of COMPQ, Q is not referenced.
*> \endverbatim
*>
*> \param[out] IQ
*> \verbatim
*> IQ is INTEGER array, dimension (LDIQ)
*> If COMPQ = 'P', then:
*> On exit, if INFO = 0, Q and IQ contain the left
*> and right singular vectors in a compact form,
*> requiring O(N log N) space instead of 2*N**2.
*> In particular, IQ contains all INTEGER data in
*> LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
*> words of memory, where SMLSIZ is returned by ILAENV and
*> is equal to the maximum size of the subproblems at the
*> bottom of the computation tree (usually about 25).
*> For other values of COMPQ, IQ is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> If COMPQ = 'N' then LWORK >= (4 * N).
*> If COMPQ = 'P' then LWORK >= (6 * N).
*> If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute a singular value.
*> The update process of divide and conquer failed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup bdsdc
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q,
$ IQ,
$ WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPQ, UPLO
INTEGER INFO, LDU, LDVT, N
* ..
* .. Array Arguments ..
INTEGER IQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), Q( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
* Changed dimension statement in comment describing E from (N) to
* (N-1). Sven, 17 Feb 05.
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
INTEGER DIFL, DIFR, GIVCOL, GIVNUM, GIVPTR, I, IC,
$ ICOMPQ, IERR, II, IS, IU, IUPLO, IVT, J, K, KK,
$ MLVL, NM1, NSIZE, PERM, POLES, QSTART, SMLSIZ,
$ SMLSZP, SQRE, START, WSTART, Z
DOUBLE PRECISION CS, EPS, ORGNRM, P, R, SN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, ILAENV, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLARTG, DLASCL, DLASD0, DLASDA,
$ DLASDQ,
$ DLASET, DLASR, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, SIGN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IUPLO = 0
IF( LSAME( UPLO, 'U' ) )
$ IUPLO = 1
IF( LSAME( UPLO, 'L' ) )
$ IUPLO = 2
IF( LSAME( COMPQ, 'N' ) ) THEN
ICOMPQ = 0
ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ICOMPQ = 2
ELSE
ICOMPQ = -1
END IF
IF( IUPLO.EQ.0 ) THEN
INFO = -1
ELSE IF( ICOMPQ.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ( LDU.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDU.LT.
$ N ) ) ) THEN
INFO = -7
ELSE IF( ( LDVT.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDVT.LT.
$ N ) ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DBDSDC', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
SMLSIZ = ILAENV( 9, 'DBDSDC', ' ', 0, 0, 0, 0 )
IF( N.EQ.1 ) THEN
IF( ICOMPQ.EQ.1 ) THEN
Q( 1 ) = SIGN( ONE, D( 1 ) )
Q( 1+SMLSIZ*N ) = ONE
ELSE IF( ICOMPQ.EQ.2 ) THEN
U( 1, 1 ) = SIGN( ONE, D( 1 ) )
VT( 1, 1 ) = ONE
END IF
D( 1 ) = ABS( D( 1 ) )
RETURN
END IF
NM1 = N - 1
*
* If matrix lower bidiagonal, rotate to be upper bidiagonal
* by applying Givens rotations on the left
*
WSTART = 1
QSTART = 3
IF( ICOMPQ.EQ.1 ) THEN
CALL DCOPY( N, D, 1, Q( 1 ), 1 )
CALL DCOPY( N-1, E, 1, Q( N+1 ), 1 )
END IF
IF( IUPLO.EQ.2 ) THEN
QSTART = 5
IF( ICOMPQ .EQ. 2 ) WSTART = 2*N - 1
DO 10 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
IF( ICOMPQ.EQ.1 ) THEN
Q( I+2*N ) = CS
Q( I+3*N ) = SN
ELSE IF( ICOMPQ.EQ.2 ) THEN
WORK( I ) = CS
WORK( NM1+I ) = -SN
END IF
10 CONTINUE
END IF
*
* If ICOMPQ = 0, use DLASDQ to compute the singular values.
*
IF( ICOMPQ.EQ.0 ) THEN
* Ignore WSTART, instead using WORK( 1 ), since the two vectors
* for CS and -SN above are added only if ICOMPQ == 2,
* and adding them exceeds documented WORK size of 4*n.
CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
$ LDU, WORK( 1 ), INFO )
GO TO 40
END IF
*
* If N is smaller than the minimum divide size SMLSIZ, then solve
* the problem with another solver.
*
IF( N.LE.SMLSIZ ) THEN
IF( ICOMPQ.EQ.2 ) THEN
CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
CALL DLASDQ( 'U', 0, N, N, N, 0, D, E, VT, LDVT, U, LDU,
$ U,
$ LDU, WORK( WSTART ), INFO )
ELSE IF( ICOMPQ.EQ.1 ) THEN
IU = 1
IVT = IU + N
CALL DLASET( 'A', N, N, ZERO, ONE,
$ Q( IU+( QSTART-1 )*N ),
$ N )
CALL DLASET( 'A', N, N, ZERO, ONE,
$ Q( IVT+( QSTART-1 )*N ),
$ N )
CALL DLASDQ( 'U', 0, N, N, N, 0, D, E,
$ Q( IVT+( QSTART-1 )*N ), N,
$ Q( IU+( QSTART-1 )*N ), N,
$ Q( IU+( QSTART-1 )*N ), N, WORK( WSTART ),
$ INFO )
END IF
GO TO 40
END IF
*
IF( ICOMPQ.EQ.2 ) THEN
CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
END IF
*
* Scale.
*
ORGNRM = DLANST( 'M', N, D, E )
IF( ORGNRM.EQ.ZERO )
$ RETURN
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, IERR )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, IERR )
*
EPS = (0.9D+0)*DLAMCH( 'Epsilon' )
*
MLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
SMLSZP = SMLSIZ + 1
*
IF( ICOMPQ.EQ.1 ) THEN
IU = 1
IVT = 1 + SMLSIZ
DIFL = IVT + SMLSZP
DIFR = DIFL + MLVL
Z = DIFR + MLVL*2
IC = Z + MLVL
IS = IC + 1
POLES = IS + 1
GIVNUM = POLES + 2*MLVL
*
K = 1
GIVPTR = 2
PERM = 3
GIVCOL = PERM + MLVL
END IF
*
DO 20 I = 1, N
IF( ABS( D( I ) ).LT.EPS ) THEN
D( I ) = SIGN( EPS, D( I ) )
END IF
20 CONTINUE
*
START = 1
SQRE = 0
*
DO 30 I = 1, NM1
IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
*
* Subproblem found. First determine its size and then
* apply divide and conquer on it.
*
IF( I.LT.NM1 ) THEN
*
* A subproblem with E(I) small for I < NM1.
*
NSIZE = I - START + 1
ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
*
* A subproblem with E(NM1) not too small but I = NM1.
*
NSIZE = N - START + 1
ELSE
*
* A subproblem with E(NM1) small. This implies an
* 1-by-1 subproblem at D(N). Solve this 1-by-1 problem
* first.
*
NSIZE = I - START + 1
IF( ICOMPQ.EQ.2 ) THEN
U( N, N ) = SIGN( ONE, D( N ) )
VT( N, N ) = ONE
ELSE IF( ICOMPQ.EQ.1 ) THEN
Q( N+( QSTART-1 )*N ) = SIGN( ONE, D( N ) )
Q( N+( SMLSIZ+QSTART-1 )*N ) = ONE
END IF
D( N ) = ABS( D( N ) )
END IF
IF( ICOMPQ.EQ.2 ) THEN
CALL DLASD0( NSIZE, SQRE, D( START ), E( START ),
$ U( START, START ), LDU, VT( START, START ),
$ LDVT, SMLSIZ, IWORK, WORK( WSTART ), INFO )
ELSE
CALL DLASDA( ICOMPQ, SMLSIZ, NSIZE, SQRE, D( START ),
$ E( START ), Q( START+( IU+QSTART-2 )*N ), N,
$ Q( START+( IVT+QSTART-2 )*N ),
$ IQ( START+K*N ), Q( START+( DIFL+QSTART-2 )*
$ N ), Q( START+( DIFR+QSTART-2 )*N ),
$ Q( START+( Z+QSTART-2 )*N ),
$ Q( START+( POLES+QSTART-2 )*N ),
$ IQ( START+GIVPTR*N ), IQ( START+GIVCOL*N ),
$ N, IQ( START+PERM*N ),
$ Q( START+( GIVNUM+QSTART-2 )*N ),
$ Q( START+( IC+QSTART-2 )*N ),
$ Q( START+( IS+QSTART-2 )*N ),
$ WORK( WSTART ), IWORK, INFO )
END IF
IF( INFO.NE.0 ) THEN
RETURN
END IF
START = I + 1
END IF
30 CONTINUE
*
* Unscale
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, IERR )
40 CONTINUE
*
* Use Selection Sort to minimize swaps of singular vectors
*
DO 60 II = 2, N
I = II - 1
KK = I
P = D( I )
DO 50 J = II, N
IF( D( J ).GT.P ) THEN
KK = J
P = D( J )
END IF
50 CONTINUE
IF( KK.NE.I ) THEN
D( KK ) = D( I )
D( I ) = P
IF( ICOMPQ.EQ.1 ) THEN
IQ( I ) = KK
ELSE IF( ICOMPQ.EQ.2 ) THEN
CALL DSWAP( N, U( 1, I ), 1, U( 1, KK ), 1 )
CALL DSWAP( N, VT( I, 1 ), LDVT, VT( KK, 1 ), LDVT )
END IF
ELSE IF( ICOMPQ.EQ.1 ) THEN
IQ( I ) = I
END IF
60 CONTINUE
*
* If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO
*
IF( ICOMPQ.EQ.1 ) THEN
IF( IUPLO.EQ.1 ) THEN
IQ( N ) = 1
ELSE
IQ( N ) = 0
END IF
END IF
*
* If B is lower bidiagonal, update U by those Givens rotations
* which rotated B to be upper bidiagonal
*
IF( ( IUPLO.EQ.2 ) .AND. ( ICOMPQ.EQ.2 ) )
$ CALL DLASR( 'L', 'V', 'B', N, N, WORK( 1 ), WORK( N ), U,
$ LDU )
*
RETURN
*
* End of DBDSDC
*
END
*> \brief \b DBDSQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbdsqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
* LDU, C, LDC, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBDSQR computes the singular values and, optionally, the right and/or
*> left singular vectors from the singular value decomposition (SVD) of
*> a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
*> zero-shift QR algorithm. The SVD of B has the form
*>
*> B = Q * S * P**T
*>
*> where S is the diagonal matrix of singular values, Q is an orthogonal
*> matrix of left singular vectors, and P is an orthogonal matrix of
*> right singular vectors. If left singular vectors are requested, this
*> subroutine actually returns U*Q instead of Q, and, if right singular
*> vectors are requested, this subroutine returns P**T*VT instead of
*> P**T, for given real input matrices U and VT. When U and VT are the
*> orthogonal matrices that reduce a general matrix A to bidiagonal
*> form: A = U*B*VT, as computed by DGEBRD, then
*>
*> A = (U*Q) * S * (P**T*VT)
*>
*> is the SVD of A. Optionally, the subroutine may also compute Q**T*C
*> for a given real input matrix C.
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices With
*> Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
*> LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
*> no. 5, pp. 873-912, Sept 1990) and
*> "Accurate singular values and differential qd algorithms," by
*> B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
*> Department, University of California at Berkeley, July 1992
*> for a detailed description of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': B is upper bidiagonal;
*> = 'L': B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix B. N >= 0.
*> \endverbatim
*>
*> \param[in] NCVT
*> \verbatim
*> NCVT is INTEGER
*> The number of columns of the matrix VT. NCVT >= 0.
*> \endverbatim
*>
*> \param[in] NRU
*> \verbatim
*> NRU is INTEGER
*> The number of rows of the matrix U. NRU >= 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*> NCC is INTEGER
*> The number of columns of the matrix C. NCC >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the bidiagonal matrix B.
*> On exit, if INFO=0, the singular values of B in decreasing
*> order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the N-1 offdiagonal elements of the bidiagonal
*> matrix B.
*> On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
*> will contain the diagonal and superdiagonal elements of a
*> bidiagonal matrix orthogonally equivalent to the one given
*> as input.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
*> On entry, an N-by-NCVT matrix VT.
*> On exit, VT is overwritten by P**T * VT.
*> Not referenced if NCVT = 0.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT.
*> LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU, N)
*> On entry, an NRU-by-N matrix U.
*> On exit, U is overwritten by U * Q.
*> Not referenced if NRU = 0.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,NRU).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC, NCC)
*> On entry, an N-by-NCC matrix C.
*> On exit, C is overwritten by Q**T * C.
*> Not referenced if NCC = 0.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C.
*> LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> LWORK = 4*N, if NCVT = NRU = NCC = 0, and
*> LWORK = 4*(N-1), otherwise
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: If INFO = -i, the i-th argument had an illegal value
*> > 0:
*> if NCVT = NRU = NCC = 0,
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 30*N
*> iterations (in inner while loop)
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> else NCVT = NRU = NCC = 0,
*> the algorithm did not converge; D and E contain the
*> elements of a bidiagonal matrix which is orthogonally
*> similar to the input matrix B; if INFO = i, i
*> elements of E have not converged to zero.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
*> TOLMUL controls the convergence criterion of the QR loop.
*> If it is positive, TOLMUL*EPS is the desired relative
*> precision in the computed singular values.
*> If it is negative, abs(TOLMUL*EPS*sigma_max) is the
*> desired absolute accuracy in the computed singular
*> values (corresponds to relative accuracy
*> abs(TOLMUL*EPS) in the largest singular value.
*> abs(TOLMUL) should be between 1 and 1/EPS, and preferably
*> between 10 (for fast convergence) and .1/EPS
*> (for there to be some accuracy in the results).
*> Default is to lose at either one eighth or 2 of the
*> available decimal digits in each computed singular value
*> (whichever is smaller).
*>
*> MAXITR INTEGER, default = 6
*> MAXITR controls the maximum number of passes of the
*> algorithm through its inner loop. The algorithms stops
*> (and so fails to converge) if the number of passes
*> through the inner loop exceeds MAXITR*N**2.
*>
*> \endverbatim
*
*> \par Note:
* ===========
*>
*> \verbatim
*> Bug report from Cezary Dendek.
*> On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
*> removed since it can overflow pretty easily (for N larger or equal
*> than 18,919). We instead use MAXITDIVN = MAXITR*N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup bdsqr
*
* =====================================================================
SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
$ LDU, C, LDC, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION NEGONE
PARAMETER ( NEGONE = -1.0D0 )
DOUBLE PRECISION HNDRTH
PARAMETER ( HNDRTH = 0.01D0 )
DOUBLE PRECISION TEN
PARAMETER ( TEN = 10.0D0 )
DOUBLE PRECISION HNDRD
PARAMETER ( HNDRD = 100.0D0 )
DOUBLE PRECISION MEIGTH
PARAMETER ( MEIGTH = -0.125D0 )
INTEGER MAXITR
PARAMETER ( MAXITR = 6 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, ROTATE
INTEGER I, IDIR, ISUB, ITER, ITERDIVN, J, LL, LLL, M,
$ MAXITDIVN, NM1, NM12, NM13, OLDLL, OLDM
DOUBLE PRECISION ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU,
$ OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL,
$ SINR, SLL, SMAX, SMIN, SMINOA,
$ SN, THRESH, TOL, TOLMUL, UNFL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLARTG, DLAS2, DLASQ1, DLASR, DLASV2,
$ DROT,
$ DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NCVT.LT.0 ) THEN
INFO = -3
ELSE IF( NRU.LT.0 ) THEN
INFO = -4
ELSE IF( NCC.LT.0 ) THEN
INFO = -5
ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR.
$ ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN
INFO = -9
ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN
INFO = -11
ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR.
$ ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DBDSQR', -INFO )
RETURN
END IF
IF( N.EQ.0 )
$ RETURN
IF( N.EQ.1 )
$ GO TO 160
*
* ROTATE is true if any singular vectors desired, false otherwise
*
ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 )
*
* If no singular vectors desired, use qd algorithm
*
IF( .NOT.ROTATE ) THEN
CALL DLASQ1( N, D, E, WORK, INFO )
*
* If INFO equals 2, dqds didn't finish, try to finish
*
IF( INFO .NE. 2 ) RETURN
INFO = 0
END IF
*
NM1 = N - 1
NM12 = NM1 + NM1
NM13 = NM12 + NM1
IDIR = 0
*
* Get machine constants
*
EPS = DLAMCH( 'Epsilon' )
UNFL = DLAMCH( 'Safe minimum' )
*
* If matrix lower bidiagonal, rotate to be upper bidiagonal
* by applying Givens rotations on the left
*
IF( LOWER ) THEN
DO 10 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
WORK( I ) = CS
WORK( NM1+I ) = SN
10 CONTINUE
*
* Update singular vectors if desired
*
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), WORK( N ),
$ U,
$ LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), WORK( N ),
$ C,
$ LDC )
END IF
*
* Compute singular values to relative accuracy TOL
* (By setting TOL to be negative, algorithm will compute
* singular values to absolute accuracy ABS(TOL)*norm(input matrix))
*
TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) )
TOL = TOLMUL*EPS
*
* Compute approximate maximum, minimum singular values
*
SMAX = ZERO
DO 20 I = 1, N
SMAX = MAX( SMAX, ABS( D( I ) ) )
20 CONTINUE
DO 30 I = 1, N - 1
SMAX = MAX( SMAX, ABS( E( I ) ) )
30 CONTINUE
SMIN = ZERO
IF( TOL.GE.ZERO ) THEN
*
* Relative accuracy desired
*
SMINOA = ABS( D( 1 ) )
IF( SMINOA.EQ.ZERO )
$ GO TO 50
MU = SMINOA
DO 40 I = 2, N
MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) )
SMINOA = MIN( SMINOA, MU )
IF( SMINOA.EQ.ZERO )
$ GO TO 50
40 CONTINUE
50 CONTINUE
SMINOA = SMINOA / SQRT( DBLE( N ) )
THRESH = MAX( TOL*SMINOA, MAXITR*(N*(N*UNFL)) )
ELSE
*
* Absolute accuracy desired
*
THRESH = MAX( ABS( TOL )*SMAX, MAXITR*(N*(N*UNFL)) )
END IF
*
* Prepare for main iteration loop for the singular values
* (MAXIT is the maximum number of passes through the inner
* loop permitted before nonconvergence signalled.)
*
MAXITDIVN = MAXITR*N
ITERDIVN = 0
ITER = -1
OLDLL = -1
OLDM = -1
*
* M points to last element of unconverged part of matrix
*
M = N
*
* Begin main iteration loop
*
60 CONTINUE
*
* Check for convergence or exceeding iteration count
*
IF( M.LE.1 )
$ GO TO 160
*
IF( ITER.GE.N ) THEN
ITER = ITER - N
ITERDIVN = ITERDIVN + 1
IF( ITERDIVN.GE.MAXITDIVN )
$ GO TO 200
END IF
*
* Find diagonal block of matrix to work on
*
IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH )
$ D( M ) = ZERO
SMAX = ABS( D( M ) )
DO 70 LLL = 1, M - 1
LL = M - LLL
ABSS = ABS( D( LL ) )
ABSE = ABS( E( LL ) )
IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH )
$ D( LL ) = ZERO
IF( ABSE.LE.THRESH )
$ GO TO 80
SMAX = MAX( SMAX, ABSS, ABSE )
70 CONTINUE
LL = 0
GO TO 90
80 CONTINUE
E( LL ) = ZERO
*
* Matrix splits since E(LL) = 0
*
IF( LL.EQ.M-1 ) THEN
*
* Convergence of bottom singular value, return to top of loop
*
M = M - 1
GO TO 60
END IF
90 CONTINUE
LL = LL + 1
*
* E(LL) through E(M-1) are nonzero, E(LL-1) is zero
*
IF( LL.EQ.M-1 ) THEN
*
* 2 by 2 block, handle separately
*
CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR,
$ COSR, SINL, COSL )
D( M-1 ) = SIGMX
E( M-1 ) = ZERO
D( M ) = SIGMN
*
* Compute singular vectors, if desired
*
IF( NCVT.GT.0 )
$ CALL DROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT,
$ COSR,
$ SINR )
IF( NRU.GT.0 )
$ CALL DROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL,
$ SINL )
IF( NCC.GT.0 )
$ CALL DROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL,
$ SINL )
M = M - 2
GO TO 60
END IF
*
* If working on new submatrix, choose shift direction
* (from larger end diagonal element towards smaller)
*
IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN
IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN
*
* Chase bulge from top (big end) to bottom (small end)
*
IDIR = 1
ELSE
*
* Chase bulge from bottom (big end) to top (small end)
*
IDIR = 2
END IF
END IF
*
* Apply convergence tests
*
IF( IDIR.EQ.1 ) THEN
*
* Run convergence test in forward direction
* First apply standard test to bottom of matrix
*
IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR.
$ ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN
E( M-1 ) = ZERO
GO TO 60
END IF
*
IF( TOL.GE.ZERO ) THEN
*
* If relative accuracy desired,
* apply convergence criterion forward
*
MU = ABS( D( LL ) )
SMIN = MU
DO 100 LLL = LL, M - 1
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
E( LLL ) = ZERO
GO TO 60
END IF
MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
SMIN = MIN( SMIN, MU )
100 CONTINUE
END IF
*
ELSE
*
* Run convergence test in backward direction
* First apply standard test to top of matrix
*
IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR.
$ ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN
E( LL ) = ZERO
GO TO 60
END IF
*
IF( TOL.GE.ZERO ) THEN
*
* If relative accuracy desired,
* apply convergence criterion backward
*
MU = ABS( D( M ) )
SMIN = MU
DO 110 LLL = M - 1, LL, -1
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
E( LLL ) = ZERO
GO TO 60
END IF
MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
SMIN = MIN( SMIN, MU )
110 CONTINUE
END IF
END IF
OLDLL = LL
OLDM = M
*
* Compute shift. First, test if shifting would ruin relative
* accuracy, and if so set the shift to zero.
*
IF( TOL.GE.ZERO .AND. N*TOL*( SMIN / SMAX ).LE.
$ MAX( EPS, HNDRTH*TOL ) ) THEN
*
* Use a zero shift to avoid loss of relative accuracy
*
SHIFT = ZERO
ELSE
*
* Compute the shift from 2-by-2 block at end of matrix
*
IF( IDIR.EQ.1 ) THEN
SLL = ABS( D( LL ) )
CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R )
ELSE
SLL = ABS( D( M ) )
CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R )
END IF
*
* Test if shift negligible, and if so set to zero
*
IF( SLL.GT.ZERO ) THEN
IF( ( SHIFT / SLL )**2.LT.EPS )
$ SHIFT = ZERO
END IF
END IF
*
* Increment iteration count
*
ITER = ITER + M - LL
*
* If SHIFT = 0, do simplified QR iteration
*
IF( SHIFT.EQ.ZERO ) THEN
IF( IDIR.EQ.1 ) THEN
*
* Chase bulge from top to bottom
* Save cosines and sines for later singular vector updates
*
CS = ONE
OLDCS = ONE
DO 120 I = LL, M - 1
CALL DLARTG( D( I )*CS, E( I ), CS, SN, R )
IF( I.GT.LL )
$ E( I-1 ) = OLDSN*R
CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN,
$ D( I ) )
WORK( I-LL+1 ) = CS
WORK( I-LL+1+NM1 ) = SN
WORK( I-LL+1+NM12 ) = OLDCS
WORK( I-LL+1+NM13 ) = OLDSN
120 CONTINUE
H = D( M )*CS
D( M ) = H*OLDCS
E( M-1 ) = H*OLDSN
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
$ WORK( N ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1,
$ WORK( NM12+1 ),
$ WORK( NM13+1 ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC,
$ WORK( NM12+1 ),
$ WORK( NM13+1 ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( M-1 ) ).LE.THRESH )
$ E( M-1 ) = ZERO
*
ELSE
*
* Chase bulge from bottom to top
* Save cosines and sines for later singular vector updates
*
CS = ONE
OLDCS = ONE
DO 130 I = M, LL + 1, -1
CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R )
IF( I.LT.M )
$ E( I ) = OLDSN*R
CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN,
$ D( I ) )
WORK( I-LL ) = CS
WORK( I-LL+NM1 ) = -SN
WORK( I-LL+NM12 ) = OLDCS
WORK( I-LL+NM13 ) = -OLDSN
130 CONTINUE
H = D( LL )*CS
D( LL ) = H*OLDCS
E( LL ) = H*OLDSN
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT,
$ WORK( NM12+1 ),
$ WORK( NM13+1 ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
$ WORK( N ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
$ WORK( N ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( LL ) ).LE.THRESH )
$ E( LL ) = ZERO
END IF
ELSE
*
* Use nonzero shift
*
IF( IDIR.EQ.1 ) THEN
*
* Chase bulge from top to bottom
* Save cosines and sines for later singular vector updates
*
F = ( ABS( D( LL ) )-SHIFT )*
$ ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) )
G = E( LL )
DO 140 I = LL, M - 1
CALL DLARTG( F, G, COSR, SINR, R )
IF( I.GT.LL )
$ E( I-1 ) = R
F = COSR*D( I ) + SINR*E( I )
E( I ) = COSR*E( I ) - SINR*D( I )
G = SINR*D( I+1 )
D( I+1 ) = COSR*D( I+1 )
CALL DLARTG( F, G, COSL, SINL, R )
D( I ) = R
F = COSL*E( I ) + SINL*D( I+1 )
D( I+1 ) = COSL*D( I+1 ) - SINL*E( I )
IF( I.LT.M-1 ) THEN
G = SINL*E( I+1 )
E( I+1 ) = COSL*E( I+1 )
END IF
WORK( I-LL+1 ) = COSR
WORK( I-LL+1+NM1 ) = SINR
WORK( I-LL+1+NM12 ) = COSL
WORK( I-LL+1+NM13 ) = SINL
140 CONTINUE
E( M-1 ) = F
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
$ WORK( N ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1,
$ WORK( NM12+1 ),
$ WORK( NM13+1 ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC,
$ WORK( NM12+1 ),
$ WORK( NM13+1 ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( M-1 ) ).LE.THRESH )
$ E( M-1 ) = ZERO
*
ELSE
*
* Chase bulge from bottom to top
* Save cosines and sines for later singular vector updates
*
F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT /
$ D( M ) )
G = E( M-1 )
DO 150 I = M, LL + 1, -1
CALL DLARTG( F, G, COSR, SINR, R )
IF( I.LT.M )
$ E( I ) = R
F = COSR*D( I ) + SINR*E( I-1 )
E( I-1 ) = COSR*E( I-1 ) - SINR*D( I )
G = SINR*D( I-1 )
D( I-1 ) = COSR*D( I-1 )
CALL DLARTG( F, G, COSL, SINL, R )
D( I ) = R
F = COSL*E( I-1 ) + SINL*D( I-1 )
D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 )
IF( I.GT.LL+1 ) THEN
G = SINL*E( I-2 )
E( I-2 ) = COSL*E( I-2 )
END IF
WORK( I-LL ) = COSR
WORK( I-LL+NM1 ) = -SINR
WORK( I-LL+NM12 ) = COSL
WORK( I-LL+NM13 ) = -SINL
150 CONTINUE
E( LL ) = F
*
* Test convergence
*
IF( ABS( E( LL ) ).LE.THRESH )
$ E( LL ) = ZERO
*
* Update singular vectors if desired
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT,
$ WORK( NM12+1 ),
$ WORK( NM13+1 ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
$ WORK( N ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
$ WORK( N ), C( LL, 1 ), LDC )
END IF
END IF
*
* QR iteration finished, go back and check convergence
*
GO TO 60
*
* All singular values converged, so make them positive
*
160 CONTINUE
DO 170 I = 1, N
IF( D( I ).EQ.ZERO ) THEN
*
* Avoid -ZERO
*
D( I ) = ZERO
END IF
IF( D( I ).LT.ZERO ) THEN
D( I ) = -D( I )
*
* Change sign of singular vectors, if desired
*
IF( NCVT.GT.0 )
$ CALL DSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT )
END IF
170 CONTINUE
*
* Sort the singular values into decreasing order (insertion sort on
* singular values, but only one transposition per singular vector)
*
DO 190 I = 1, N - 1
*
* Scan for smallest D(I)
*
ISUB = 1
SMIN = D( 1 )
DO 180 J = 2, N + 1 - I
IF( D( J ).LE.SMIN ) THEN
ISUB = J
SMIN = D( J )
END IF
180 CONTINUE
IF( ISUB.NE.N+1-I ) THEN
*
* Swap singular values and vectors
*
D( ISUB ) = D( N+1-I )
D( N+1-I ) = SMIN
IF( NCVT.GT.0 )
$ CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ),
$ LDVT )
IF( NRU.GT.0 )
$ CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 )
IF( NCC.GT.0 )
$ CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ),
$ LDC )
END IF
190 CONTINUE
GO TO 220
*
* Maximum number of iterations exceeded, failure to converge
*
200 CONTINUE
INFO = 0
DO 210 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
210 CONTINUE
220 CONTINUE
RETURN
*
* End of DBDSQR
*
END
*> \brief \b DDISNA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DDISNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ddisna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ddisna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ddisna.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB
* INTEGER INFO, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), SEP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DDISNA computes the reciprocal condition numbers for the eigenvectors
*> of a real symmetric or complex Hermitian matrix or for the left or
*> right singular vectors of a general m-by-n matrix. The reciprocal
*> condition number is the 'gap' between the corresponding eigenvalue or
*> singular value and the nearest other one.
*>
*> The bound on the error, measured by angle in radians, in the I-th
*> computed vector is given by
*>
*> DLAMCH( 'E' ) * ( ANORM / SEP( I ) )
*>
*> where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed
*> to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of
*> the error bound.
*>
*> DDISNA may also be used to compute error bounds for eigenvectors of
*> the generalized symmetric definite eigenproblem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies for which problem the reciprocal condition numbers
*> should be computed:
*> = 'E': the eigenvectors of a symmetric/Hermitian matrix;
*> = 'L': the left singular vectors of a general matrix;
*> = 'R': the right singular vectors of a general matrix.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> If JOB = 'L' or 'R', the number of columns of the matrix,
*> in which case N >= 0. Ignored if JOB = 'E'.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
*> dimension (min(M,N)) if JOB = 'L' or 'R'
*> The eigenvalues (if JOB = 'E') or singular values (if JOB =
*> 'L' or 'R') of the matrix, in either increasing or decreasing
*> order. If singular values, they must be non-negative.
*> \endverbatim
*>
*> \param[out] SEP
*> \verbatim
*> SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
*> dimension (min(M,N)) if JOB = 'L' or 'R'
*> The reciprocal condition numbers of the vectors.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup disna
*
* =====================================================================
SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), SEP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL DECR, EIGEN, INCR, LEFT, RIGHT, SING
INTEGER I, K
DOUBLE PRECISION ANORM, EPS, NEWGAP, OLDGAP, SAFMIN, THRESH
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
EIGEN = LSAME( JOB, 'E' )
LEFT = LSAME( JOB, 'L' )
RIGHT = LSAME( JOB, 'R' )
SING = LEFT .OR. RIGHT
IF( EIGEN ) THEN
K = M
ELSE IF( SING ) THEN
K = MIN( M, N )
END IF
IF( .NOT.EIGEN .AND. .NOT.SING ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( K.LT.0 ) THEN
INFO = -3
ELSE
INCR = .TRUE.
DECR = .TRUE.
DO 10 I = 1, K - 1
IF( INCR )
$ INCR = INCR .AND. D( I ).LE.D( I+1 )
IF( DECR )
$ DECR = DECR .AND. D( I ).GE.D( I+1 )
10 CONTINUE
IF( SING .AND. K.GT.0 ) THEN
IF( INCR )
$ INCR = INCR .AND. ZERO.LE.D( 1 )
IF( DECR )
$ DECR = DECR .AND. D( K ).GE.ZERO
END IF
IF( .NOT.( INCR .OR. DECR ) )
$ INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DDISNA', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 )
$ RETURN
*
* Compute reciprocal condition numbers
*
IF( K.EQ.1 ) THEN
SEP( 1 ) = DLAMCH( 'O' )
ELSE
OLDGAP = ABS( D( 2 )-D( 1 ) )
SEP( 1 ) = OLDGAP
DO 20 I = 2, K - 1
NEWGAP = ABS( D( I+1 )-D( I ) )
SEP( I ) = MIN( OLDGAP, NEWGAP )
OLDGAP = NEWGAP
20 CONTINUE
SEP( K ) = OLDGAP
END IF
IF( SING ) THEN
IF( ( LEFT .AND. M.GT.N ) .OR. ( RIGHT .AND. M.LT.N ) ) THEN
IF( INCR )
$ SEP( 1 ) = MIN( SEP( 1 ), D( 1 ) )
IF( DECR )
$ SEP( K ) = MIN( SEP( K ), D( K ) )
END IF
END IF
*
* Ensure that reciprocal condition numbers are not less than
* threshold, in order to limit the size of the error bound
*
EPS = DLAMCH( 'E' )
SAFMIN = DLAMCH( 'S' )
ANORM = MAX( ABS( D( 1 ) ), ABS( D( K ) ) )
IF( ANORM.EQ.ZERO ) THEN
THRESH = EPS
ELSE
THRESH = MAX( EPS*ANORM, SAFMIN )
END IF
DO 30 I = 1, K
SEP( I ) = MAX( SEP( I ), THRESH )
30 CONTINUE
*
RETURN
*
* End of DDISNA
*
END
*> \brief \b DGBBRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBBRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbbrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbbrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbbrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
* LDQ, PT, LDPT, C, LDC, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER VECT
* INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
* $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBBRD reduces a real general m-by-n band matrix A to upper
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> The routine computes B, and optionally forms Q or P**T, or computes
*> Q**T*C for a given matrix C.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> Specifies whether or not the matrices Q and P**T are to be
*> formed.
*> = 'N': do not form Q or P**T;
*> = 'Q': form Q only;
*> = 'P': form P**T only;
*> = 'B': form both.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*> NCC is INTEGER
*> The number of columns of the matrix C. NCC >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals of the matrix A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals of the matrix A. KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the m-by-n band matrix A, stored in rows 1 to
*> KL+KU+1. The j-th column of A is stored in the j-th column of
*> the array AB as follows:
*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
*> On exit, A is overwritten by values generated during the
*> reduction.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array A. LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (min(M,N))
*> The diagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*> The superdiagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,M)
*> If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
*> If VECT = 'N' or 'P', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] PT
*> \verbatim
*> PT is DOUBLE PRECISION array, dimension (LDPT,N)
*> If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
*> If VECT = 'N' or 'Q', the array PT is not referenced.
*> \endverbatim
*>
*> \param[in] LDPT
*> \verbatim
*> LDPT is INTEGER
*> The leading dimension of the array PT.
*> LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,NCC)
*> On entry, an m-by-ncc matrix C.
*> On exit, C is overwritten by Q**T*C.
*> C is not referenced if NCC = 0.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C.
*> LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*max(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbbrd
*
* =====================================================================
SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
$ LDQ, PT, LDPT, C, LDC, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER VECT
INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
$ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL WANTB, WANTC, WANTPT, WANTQ
INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
$ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
DOUBLE PRECISION RA, RB, RC, RS
* ..
* .. External Subroutines ..
EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTB = LSAME( VECT, 'B' )
WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
WANTC = NCC.GT.0
KLU1 = KL + KU + 1
INFO = 0
IF( .NOT.WANTQ .AND.
$ .NOT.WANTPT .AND.
$ .NOT.LSAME( VECT, 'N' ) )
$ THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NCC.LT.0 ) THEN
INFO = -4
ELSE IF( KL.LT.0 ) THEN
INFO = -5
ELSE IF( KU.LT.0 ) THEN
INFO = -6
ELSE IF( LDAB.LT.KLU1 ) THEN
INFO = -8
ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
INFO = -12
ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBBRD', -INFO )
RETURN
END IF
*
* Initialize Q and P**T to the unit matrix, if needed
*
IF( WANTQ )
$ CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
IF( WANTPT )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
*
* Quick return if possible.
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
MINMN = MIN( M, N )
*
IF( KL+KU.GT.1 ) THEN
*
* Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
* first to lower bidiagonal form and then transform to upper
* bidiagonal
*
IF( KU.GT.0 ) THEN
ML0 = 1
MU0 = 2
ELSE
ML0 = 2
MU0 = 1
END IF
*
* Wherever possible, plane rotations are generated and applied in
* vector operations of length NR over the index set J1:J2:KLU1.
*
* The sines of the plane rotations are stored in WORK(1:max(m,n))
* and the cosines in WORK(max(m,n)+1:2*max(m,n)).
*
MN = MAX( M, N )
KLM = MIN( M-1, KL )
KUN = MIN( N-1, KU )
KB = KLM + KUN
KB1 = KB + 1
INCA = KB1*LDAB
NR = 0
J1 = KLM + 2
J2 = 1 - KUN
*
DO 90 I = 1, MINMN
*
* Reduce i-th column and i-th row of matrix to bidiagonal form
*
ML = KLM + 1
MU = KUN + 1
DO 80 KK = 1, KB
J1 = J1 + KB
J2 = J2 + KB
*
* generate plane rotations to annihilate nonzero elements
* which have been created below the band
*
IF( NR.GT.0 )
$ CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
$ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
*
* apply plane rotations from the left
*
DO 10 L = 1, KB
IF( J2-KLM+L-1.GT.N ) THEN
NRT = NR - 1
ELSE
NRT = NR
END IF
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ),
$ INCA,
$ AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
$ WORK( MN+J1 ), WORK( J1 ), KB1 )
10 CONTINUE
*
IF( ML.GT.ML0 ) THEN
IF( ML.LE.M-I+1 ) THEN
*
* generate plane rotation to annihilate a(i+ml-1,i)
* within the band, and apply rotation from the left
*
CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
$ WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
$ RA )
AB( KU+ML-1, I ) = RA
IF( I.LT.N )
$ CALL DROT( MIN( KU+ML-2, N-I ),
$ AB( KU+ML-2, I+1 ), LDAB-1,
$ AB( KU+ML-1, I+1 ), LDAB-1,
$ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
END IF
NR = NR + 1
J1 = J1 - KB1
END IF
*
IF( WANTQ ) THEN
*
* accumulate product of plane rotations in Q
*
DO 20 J = J1, J2, KB1
CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
$ WORK( MN+J ), WORK( J ) )
20 CONTINUE
END IF
*
IF( WANTC ) THEN
*
* apply plane rotations to C
*
DO 30 J = J1, J2, KB1
CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ),
$ LDC,
$ WORK( MN+J ), WORK( J ) )
30 CONTINUE
END IF
*
IF( J2+KUN.GT.N ) THEN
*
* adjust J2 to keep within the bounds of the matrix
*
NR = NR - 1
J2 = J2 - KB1
END IF
*
DO 40 J = J1, J2, KB1
*
* create nonzero element a(j-1,j+ku) above the band
* and store it in WORK(n+1:2*n)
*
WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
40 CONTINUE
*
* generate plane rotations to annihilate nonzero elements
* which have been generated above the band
*
IF( NR.GT.0 )
$ CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
$ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
$ KB1 )
*
* apply plane rotations from the right
*
DO 50 L = 1, KB
IF( J2+L-1.GT.M ) THEN
NRT = NR - 1
ELSE
NRT = NR
END IF
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
$ AB( L, J1+KUN ), INCA,
$ WORK( MN+J1+KUN ), WORK( J1+KUN ),
$ KB1 )
50 CONTINUE
*
IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
IF( MU.LE.N-I+1 ) THEN
*
* generate plane rotation to annihilate a(i,i+mu-1)
* within the band, and apply rotation from the right
*
CALL DLARTG( AB( KU-MU+3, I+MU-2 ),
$ AB( KU-MU+2, I+MU-1 ),
$ WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
$ RA )
AB( KU-MU+3, I+MU-2 ) = RA
CALL DROT( MIN( KL+MU-2, M-I ),
$ AB( KU-MU+4, I+MU-2 ), 1,
$ AB( KU-MU+3, I+MU-1 ), 1,
$ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
END IF
NR = NR + 1
J1 = J1 - KB1
END IF
*
IF( WANTPT ) THEN
*
* accumulate product of plane rotations in P**T
*
DO 60 J = J1, J2, KB1
CALL DROT( N, PT( J+KUN-1, 1 ), LDPT,
$ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
$ WORK( J+KUN ) )
60 CONTINUE
END IF
*
IF( J2+KB.GT.M ) THEN
*
* adjust J2 to keep within the bounds of the matrix
*
NR = NR - 1
J2 = J2 - KB1
END IF
*
DO 70 J = J1, J2, KB1
*
* create nonzero element a(j+kl+ku,j+ku-1) below the
* band and store it in WORK(1:n)
*
WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
70 CONTINUE
*
IF( ML.GT.ML0 ) THEN
ML = ML - 1
ELSE
MU = MU - 1
END IF
80 CONTINUE
90 CONTINUE
END IF
*
IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
*
* A has been reduced to lower bidiagonal form
*
* Transform lower bidiagonal form to upper bidiagonal by applying
* plane rotations from the left, storing diagonal elements in D
* and off-diagonal elements in E
*
DO 100 I = 1, MIN( M-1, N )
CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
D( I ) = RA
IF( I.LT.N ) THEN
E( I ) = RS*AB( 1, I+1 )
AB( 1, I+1 ) = RC*AB( 1, I+1 )
END IF
IF( WANTQ )
$ CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
IF( WANTC )
$ CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
$ RS )
100 CONTINUE
IF( M.LE.N )
$ D( M ) = AB( 1, M )
ELSE IF( KU.GT.0 ) THEN
*
* A has been reduced to upper bidiagonal form
*
IF( M.LT.N ) THEN
*
* Annihilate a(m,m+1) by applying plane rotations from the
* right, storing diagonal elements in D and off-diagonal
* elements in E
*
RB = AB( KU, M+1 )
DO 110 I = M, 1, -1
CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA )
D( I ) = RA
IF( I.GT.1 ) THEN
RB = -RS*AB( KU, I )
E( I-1 ) = RC*AB( KU, I )
END IF
IF( WANTPT )
$ CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
$ RC, RS )
110 CONTINUE
ELSE
*
* Copy off-diagonal elements to E and diagonal elements to D
*
DO 120 I = 1, MINMN - 1
E( I ) = AB( KU, I+1 )
120 CONTINUE
DO 130 I = 1, MINMN
D( I ) = AB( KU+1, I )
130 CONTINUE
END IF
ELSE
*
* A is diagonal. Set elements of E to zero and copy diagonal
* elements to D.
*
DO 140 I = 1, MINMN - 1
E( I ) = ZERO
140 CONTINUE
DO 150 I = 1, MINMN
D( I ) = AB( 1, I )
150 CONTINUE
END IF
RETURN
*
* End of DGBBRD
*
END
*> \brief \b DGBCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER INFO, KL, KU, LDAB, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBCON estimates the reciprocal of the condition number of a real
*> general band matrix A, in either the 1-norm or the infinity-norm,
*> using the LU factorization computed by DGBTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as
*> RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies whether the 1-norm condition number or the
*> infinity-norm condition number is required:
*> = '1' or 'O': 1-norm;
*> = 'I': Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> Details of the LU factorization of the band matrix A, as
*> computed by DGBTRF. U is stored as an upper triangular band
*> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
*> the multipliers used during the factorization are stored in
*> rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= N, row i of the matrix was
*> interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> If NORM = '1' or 'O', the 1-norm of the original matrix A.
*> If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbcon
*
* =====================================================================
SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM,
$ RCOND,
$ WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER INFO, KL, KU, LDAB, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LNOTI, ONENRM
CHARACTER NORMIN
INTEGER IX, J, JP, KASE, KASE1, KD, LM
DOUBLE PRECISION AINVNM, SCALE, SMLNUM, T
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLACN2, DLATBS, DRSCL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN
INFO = -6
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
SMLNUM = DLAMCH( 'Safe minimum' )
*
* Estimate the norm of inv(A).
*
AINVNM = ZERO
NORMIN = 'N'
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KD = KL + KU + 1
LNOTI = KL.GT.0
KASE = 0
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(L).
*
IF( LNOTI ) THEN
DO 20 J = 1, N - 1
LM = MIN( KL, N-J )
JP = IPIV( J )
T = WORK( JP )
IF( JP.NE.J ) THEN
WORK( JP ) = WORK( J )
WORK( J ) = T
END IF
CALL DAXPY( LM, -T, AB( KD+1, J ), 1, WORK( J+1 ),
$ 1 )
20 CONTINUE
END IF
*
* Multiply by inv(U).
*
CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
$ INFO )
ELSE
*
* Multiply by inv(U**T).
*
CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
$ KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
$ INFO )
*
* Multiply by inv(L**T).
*
IF( LNOTI ) THEN
DO 30 J = N - 1, 1, -1
LM = MIN( KL, N-J )
WORK( J ) = WORK( J ) - DDOT( LM, AB( KD+1, J ), 1,
$ WORK( J+1 ), 1 )
JP = IPIV( J )
IF( JP.NE.J ) THEN
T = WORK( JP )
WORK( JP ) = WORK( J )
WORK( J ) = T
END IF
30 CONTINUE
END IF
END IF
*
* Divide X by 1/SCALE if doing so will not cause overflow.
*
NORMIN = 'Y'
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 40
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
40 CONTINUE
RETURN
*
* End of DGBCON
*
END
*> \brief \b DGBEQU
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
* AMAX, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, KL, KU, LDAB, M, N
* DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBEQU computes row and column scalings intended to equilibrate an
*> M-by-N band matrix A and reduce its condition number. R returns the
*> row scale factors and C the column scale factors, chosen to try to
*> make the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
*>
*> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
*> number and BIGNUM = largest safe number. Use of these scaling
*> factors is not guaranteed to reduce the condition number of A but
*> works well in practice.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The band matrix A, stored in rows 1 to KL+KU+1. The j-th
*> column of A is stored in the j-th column of the array AB as
*> follows:
*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (M)
*> If INFO = 0, or INFO > M, R contains the row scale factors
*> for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*> ROWCND is DOUBLE PRECISION
*> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
*> AMAX is neither too large nor too small, it is not worth
*> scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*> COLCND is DOUBLE PRECISION
*> If INFO = 0, COLCND contains the ratio of the smallest
*> C(i) to the largest C(i). If COLCND >= 0.1, it is not
*> worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= M: the i-th row of A is exactly zero
*> > M: the (i-M)-th column of A is exactly zero
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbequ
*
* =====================================================================
SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
$ COLCND,
$ AMAX, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, KD
DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBEQU', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
ROWCND = ONE
COLCND = ONE
AMAX = ZERO
RETURN
END IF
*
* Get machine constants.
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
* Compute row scale factors.
*
DO 10 I = 1, M
R( I ) = ZERO
10 CONTINUE
*
* Find the maximum element in each row.
*
KD = KU + 1
DO 30 J = 1, N
DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
20 CONTINUE
30 CONTINUE
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 40 I = 1, M
RCMAX = MAX( RCMAX, R( I ) )
RCMIN = MIN( RCMIN, R( I ) )
40 CONTINUE
AMAX = RCMAX
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 50 I = 1, M
IF( R( I ).EQ.ZERO ) THEN
INFO = I
RETURN
END IF
50 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 60 I = 1, M
R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
60 CONTINUE
*
* Compute ROWCND = min(R(I)) / max(R(I))
*
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
* Compute column scale factors
*
DO 70 J = 1, N
C( J ) = ZERO
70 CONTINUE
*
* Find the maximum element in each column,
* assuming the row scaling computed above.
*
KD = KU + 1
DO 90 J = 1, N
DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
80 CONTINUE
90 CONTINUE
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 100 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
100 CONTINUE
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 110 J = 1, N
IF( C( J ).EQ.ZERO ) THEN
INFO = M + J
RETURN
END IF
110 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 120 J = 1, N
C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
120 CONTINUE
*
* Compute COLCND = min(C(J)) / max(C(J))
*
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
RETURN
*
* End of DGBEQU
*
END
*> \brief \b DGBEQUB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
* AMAX, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, KL, KU, LDAB, M, N
* DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBEQUB computes row and column scalings intended to equilibrate an
*> M-by-N matrix A and reduce its condition number. R returns the row
*> scale factors and C the column scale factors, chosen to try to make
*> the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
*> the radix.
*>
*> R(i) and C(j) are restricted to be a power of the radix between
*> SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
*> of these scaling factors is not guaranteed to reduce the condition
*> number of A but works well in practice.
*>
*> This routine differs from DGEEQU by restricting the scaling factors
*> to a power of the radix. Barring over- and underflow, scaling by
*> these factors introduces no additional rounding errors. However, the
*> scaled entries' magnitudes are no longer approximately 1 but lie
*> between sqrt(radix) and 1/sqrt(radix).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array A. LDAB >= max(1,M).
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (M)
*> If INFO = 0 or INFO > M, R contains the row scale factors
*> for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*> ROWCND is DOUBLE PRECISION
*> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
*> AMAX is neither too large nor too small, it is not worth
*> scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*> COLCND is DOUBLE PRECISION
*> If INFO = 0, COLCND contains the ratio of the smallest
*> C(i) to the largest C(i). If COLCND >= 0.1, it is not
*> worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= M: the i-th row of A is exactly zero
*> > M: the (i-M)-th column of A is exactly zero
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbequb
*
* =====================================================================
SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
$ COLCND,
$ AMAX, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, KD
DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, LOG
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBEQUB', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
ROWCND = ONE
COLCND = ONE
AMAX = ZERO
RETURN
END IF
*
* Get machine constants. Assume SMLNUM is a power of the radix.
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
RADIX = DLAMCH( 'B' )
LOGRDX = LOG(RADIX)
*
* Compute row scale factors.
*
DO 10 I = 1, M
R( I ) = ZERO
10 CONTINUE
*
* Find the maximum element in each row.
*
KD = KU + 1
DO 30 J = 1, N
DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
20 CONTINUE
30 CONTINUE
DO I = 1, M
IF( R( I ).GT.ZERO ) THEN
R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
END IF
END DO
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 40 I = 1, M
RCMAX = MAX( RCMAX, R( I ) )
RCMIN = MIN( RCMIN, R( I ) )
40 CONTINUE
AMAX = RCMAX
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 50 I = 1, M
IF( R( I ).EQ.ZERO ) THEN
INFO = I
RETURN
END IF
50 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 60 I = 1, M
R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
60 CONTINUE
*
* Compute ROWCND = min(R(I)) / max(R(I)).
*
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
* Compute column scale factors.
*
DO 70 J = 1, N
C( J ) = ZERO
70 CONTINUE
*
* Find the maximum element in each column,
* assuming the row scaling computed above.
*
DO 90 J = 1, N
DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
80 CONTINUE
IF( C( J ).GT.ZERO ) THEN
C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
END IF
90 CONTINUE
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 100 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
100 CONTINUE
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 110 J = 1, N
IF( C( J ).EQ.ZERO ) THEN
INFO = M + J
RETURN
END IF
110 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 120 J = 1, N
C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
120 CONTINUE
*
* Compute COLCND = min(C(J)) / max(C(J)).
*
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
RETURN
*
* End of DGBEQUB
*
END
*> \brief \b DGBRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
* IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is banded, and provides
*> error bounds and backward error estimates for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The original band matrix A, stored in rows 1 to KL+KU+1.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[in] AFB
*> \verbatim
*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*> Details of the LU factorization of the band matrix A, as
*> computed by DGBTRF. U is stored as an upper triangular band
*> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
*> the multipliers used during the factorization are stored in
*> rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*> LDAFB is INTEGER
*> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices from DGBTRF; for 1<=i<=N, row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DGBTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbrfs
*
* =====================================================================
SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
$ LDAFB,
$ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
CHARACTER TRANST
INTEGER COUNT, I, J, K, KASE, KK, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGBMV, DGBTRS, DLACN2,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
INFO = -7
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
INFO = -9
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = MIN( KL+KU+2, N+1 )
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - op(A) * X,
* where op(A) = A, A**T, or A**H, depending on TRANS.
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ),
$ 1,
$ ONE, WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(op(A))*abs(X) + abs(B).
*
IF( NOTRAN ) THEN
DO 50 K = 1, N
KK = KU + 1 - K
XK = ABS( X( K, J ) )
DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
40 CONTINUE
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
KK = KU + 1 - K
DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
60 CONTINUE
WORK( K ) = WORK( K ) + S
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
$ WORK( N+1 ), N, INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(op(A)))*
* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(A)) is the inverse of op(A)
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(op(A)) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**T).
*
CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
$ WORK( N+1 ), N, INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( N+I )*WORK( I )
110 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( N+I )*WORK( I )
120 CONTINUE
CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
$ WORK( N+1 ), N, INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DGBRFS
*
END
*> \brief <b> DGBSV computes the solution to system of linear equations A * X = B for GB matrices</b> (simple driver)
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBSV computes the solution to a real system of linear equations
*> A * X = B, where A is a band matrix of order N with KL subdiagonals
*> and KU superdiagonals, and X and B are N-by-NRHS matrices.
*>
*> The LU decomposition with partial pivoting and row interchanges is
*> used to factor A as A = L * U, where L is a product of permutation
*> and unit lower triangular matrices with KL subdiagonals, and U is
*> upper triangular with KL+KU superdiagonals. The factored form of A
*> is then used to solve the system of equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the matrix A in band storage, in rows KL+1 to
*> 2*KL+KU+1; rows 1 to KL of the array need not be set.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
*> On exit, details of the factorization: U is stored as an
*> upper triangular band matrix with KL+KU superdiagonals in
*> rows 1 to KL+KU+1, and the multipliers used during the
*> factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices that define the permutation matrix P;
*> row i of the matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and the solution has not been computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbsv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> M = N = 6, KL = 2, KU = 1:
*>
*> On entry: On exit:
*>
*> * * * + + + * * * u14 u25 u36
*> * * + + + + * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
*> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
*>
*> Array elements marked * are not used by the routine; elements marked
*> + need not be set on entry, but are required by the routine to store
*> elements of U because of fill-in resulting from the row interchanges.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. External Subroutines ..
EXTERNAL DGBTRF, DGBTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( KL.LT.0 ) THEN
INFO = -2
ELSE IF( KU.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( N, 1 ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBSV ', -INFO )
RETURN
END IF
*
* Compute the LU factorization of the band matrix A.
*
CALL DGBTRF( N, N, KL, KU, AB, LDAB, IPIV, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DGBTRS( 'No transpose', N, KL, KU, NRHS, AB, LDAB,
$ IPIV,
$ B, LDB, INFO )
END IF
RETURN
*
* End of DGBSV
*
END
*> \brief <b> DGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
* RCOND, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, TRANS
* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
* $ BERR( * ), C( * ), FERR( * ), R( * ),
* $ WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBSVX uses the LU factorization to compute the solution to a real
*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
*> where A is a band matrix of order N with KL subdiagonals and KU
*> superdiagonals, and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed by this subroutine:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*> Whether or not the system will be equilibrated depends on the
*> scaling of the matrix A, but if equilibration is used, A is
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
*> or diag(C)*B (if TRANS = 'T' or 'C').
*>
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
*> matrix A (after equilibration if FACT = 'E') as
*> A = L * U,
*> where L is a product of permutation and unit lower triangular
*> matrices with KL subdiagonals, and U is upper triangular with
*> KL+KU superdiagonals.
*>
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
*> returns with INFO = i. Otherwise, the factored form of A is used
*> to estimate the condition number of the matrix A. If the
*> reciprocal of the condition number is less than machine precision,
*> INFO = N+1 is returned as a warning, but the routine still goes on
*> to solve for X and compute error bounds as described below.
*>
*> 4. The system of equations is solved for X using the factored form
*> of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
*> that it solves the original system before equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of the matrix A is
*> supplied on entry, and if not, whether the matrix A should be
*> equilibrated before it is factored.
*> = 'F': On entry, AFB and IPIV contain the factored form of
*> A. If EQUED is not 'N', the matrix A has been
*> equilibrated with scaling factors given by R and C.
*> AB, AFB, and IPIV are not modified.
*> = 'N': The matrix A will be copied to AFB and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> copied to AFB and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations.
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
*>
*> If FACT = 'F' and EQUED is not 'N', then A must have been
*> equilibrated by the scaling factors in R and/or C. AB is not
*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
*> EQUED = 'N' on exit.
*>
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
*> EQUED = 'R': A := diag(R) * A
*> EQUED = 'C': A := A * diag(C)
*> EQUED = 'B': A := diag(R) * A * diag(C).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[in,out] AFB
*> \verbatim
*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*> If FACT = 'F', then AFB is an input argument and on entry
*> contains details of the LU factorization of the band matrix
*> A, as computed by DGBTRF. U is stored as an upper triangular
*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
*> and the multipliers used during the factorization are stored
*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
*> the factored form of the equilibrated matrix A.
*>
*> If FACT = 'N', then AFB is an output argument and on exit
*> returns details of the LU factorization of A.
*>
*> If FACT = 'E', then AFB is an output argument and on exit
*> returns details of the LU factorization of the equilibrated
*> matrix A (see the description of AB for the form of the
*> equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*> LDAFB is INTEGER
*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains the pivot indices from the factorization A = L*U
*> as computed by DGBTRF; row i of the matrix was interchanged
*> with row IPIV(i).
*>
*> If FACT = 'N', then IPIV is an output argument and on exit
*> contains the pivot indices from the factorization A = L*U
*> of the original matrix A.
*>
*> If FACT = 'E', then IPIV is an output argument and on exit
*> contains the pivot indices from the factorization A = L*U
*> of the equilibrated matrix A.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration (always true if FACT = 'N').
*> = 'R': Row equilibration, i.e., A has been premultiplied by
*> diag(R).
*> = 'C': Column equilibration, i.e., A has been postmultiplied
*> by diag(C).
*> = 'B': Both row and column equilibration, i.e., A has been
*> replaced by diag(R) * A * diag(C).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (N)
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
*> is not accessed. R is an input argument if FACT = 'F';
*> otherwise, R is an output argument. If FACT = 'F' and
*> EQUED = 'R' or 'B', each element of R must be positive.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
*> is not accessed. C is an input argument if FACT = 'F';
*> otherwise, C is an output argument. If FACT = 'F' and
*> EQUED = 'C' or 'B', each element of C must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit,
*> if EQUED = 'N', B is not modified;
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
*> diag(R)*B;
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
*> overwritten by diag(C)*B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
*> to the original system of equations. Note that A and B are
*> modified on exit if EQUED .ne. 'N', and the solution to the
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
*> and EQUED = 'R' or 'B'.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A after equilibration (if done). If RCOND is less than the
*> machine precision (in particular, if RCOND = 0), the matrix
*> is singular to working precision. This condition is
*> indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,3*N))
*> On exit, WORK(1) contains the reciprocal pivot growth
*> factor norm(A)/norm(U). The "max absolute element" norm is
*> used. If WORK(1) is much less than 1, then the stability
*> of the LU factorization of the (equilibrated) matrix A
*> could be poor. This also means that the solution X, condition
*> estimator RCOND, and forward error bound FERR could be
*> unreliable. If factorization fails with 0<INFO<=N, then
*> WORK(1) contains the reciprocal pivot growth factor for the
*> leading INFO columns of A.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, so the solution and error bounds
*> could not be computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbsvx
*
* =====================================================================
SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
$ RCOND, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ BERR( * ), C( * ), FERR( * ), R( * ),
$ WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
CHARACTER NORM
INTEGER I, INFEQU, J, J1, J2
DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
$ ROWCND, RPVGRW, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGB, DLANTB
EXTERNAL LSAME, DLAMCH, DLANGB, DLANTB
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF,
$ DGBTRS,
$ DLACPY, DLAQGB, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
NOTRAN = LSAME( TRANS, 'N' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
ROWEQU = .FALSE.
COLEQU = .FALSE.
ELSE
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND.
$ .NOT.EQUIL .AND.
$ .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KL.LT.0 ) THEN
INFO = -4
ELSE IF( KU.LT.0 ) THEN
INFO = -5
ELSE IF( NRHS.LT.0 ) THEN
INFO = -6
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
INFO = -8
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
INFO = -10
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -12
ELSE
IF( ROWEQU ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 10 J = 1, N
RCMIN = MIN( RCMIN, R( J ) )
RCMAX = MAX( RCMAX, R( J ) )
10 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -13
ELSE IF( N.GT.0 ) THEN
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
ROWCND = ONE
END IF
END IF
IF( COLEQU .AND. INFO.EQ.0 ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 20 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
20 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -14
ELSE IF( N.GT.0 ) THEN
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
COLCND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -16
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -18
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
$ AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND,
$ COLCND,
$ AMAX, EQUED )
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
END IF
END IF
*
* Scale the right hand side.
*
IF( NOTRAN ) THEN
IF( ROWEQU ) THEN
DO 40 J = 1, NRHS
DO 30 I = 1, N
B( I, J ) = R( I )*B( I, J )
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( COLEQU ) THEN
DO 60 J = 1, NRHS
DO 50 I = 1, N
B( I, J ) = C( I )*B( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the LU factorization of the band matrix A.
*
DO 70 J = 1, N
J1 = MAX( J-KU, 1 )
J2 = MIN( J+KL, N )
CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
$ AFB( KL+KU+1-J+J1, J ), 1 )
70 CONTINUE
*
CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 ) THEN
*
* Compute the reciprocal pivot growth factor of the
* leading rank-deficient INFO columns of A.
*
ANORM = ZERO
DO 90 J = 1, INFO
DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
80 CONTINUE
90 CONTINUE
RPVGRW = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1,
$ KL+KU ),
$ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
$ WORK )
IF( RPVGRW.EQ.ZERO ) THEN
RPVGRW = ONE
ELSE
RPVGRW = ANORM / RPVGRW
END IF
WORK( 1 ) = RPVGRW
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A and the
* reciprocal pivot growth factor RPVGRW.
*
IF( NOTRAN ) THEN
NORM = '1'
ELSE
NORM = 'I'
END IF
ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
IF( RPVGRW.EQ.ZERO ) THEN
RPVGRW = ONE
ELSE
RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
END IF
*
* Compute the reciprocal of the condition number of A.
*
CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
$ WORK, IWORK, INFO )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
$ INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
$ IPIV,
$ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( NOTRAN ) THEN
IF( COLEQU ) THEN
DO 110 J = 1, NRHS
DO 100 I = 1, N
X( I, J ) = C( I )*X( I, J )
100 CONTINUE
110 CONTINUE
DO 120 J = 1, NRHS
FERR( J ) = FERR( J ) / COLCND
120 CONTINUE
END IF
ELSE IF( ROWEQU ) THEN
DO 140 J = 1, NRHS
DO 130 I = 1, N
X( I, J ) = R( I )*X( I, J )
130 CONTINUE
140 CONTINUE
DO 150 J = 1, NRHS
FERR( J ) = FERR( J ) / ROWCND
150 CONTINUE
END IF
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
WORK( 1 ) = RPVGRW
RETURN
*
* End of DGBSVX
*
END
*> \brief \b DGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbtf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbtf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbtf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBTF2 computes an LU factorization of a real m-by-n band matrix A
*> using partial pivoting with row interchanges.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the matrix A in band storage, in rows KL+1 to
*> 2*KL+KU+1; rows 1 to KL of the array need not be set.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
*>
*> On exit, details of the factorization: U is stored as an
*> upper triangular band matrix with KL+KU superdiagonals in
*> rows 1 to KL+KU+1, and the multipliers used during the
*> factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (min(M,N))
*> The pivot indices; for 1 <= i <= min(M,N), row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and division by zero will occur if it is used
*> to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbtf2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> M = N = 6, KL = 2, KU = 1:
*>
*> On entry: On exit:
*>
*> * * * + + + * * * u14 u25 u36
*> * * + + + + * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
*> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
*>
*> Array elements marked * are not used by the routine; elements marked
*> + need not be set on entry, but are required by the routine to store
*> elements of U, because of fill-in resulting from the row
*> interchanges.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, JP, JU, KM, KV
* ..
* .. External Functions ..
INTEGER IDAMAX
EXTERNAL IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DGER, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* KV is the number of superdiagonals in the factor U, allowing for
* fill-in.
*
KV = KU + KL
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KL+KV+1 ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Gaussian elimination with partial pivoting
*
* Set fill-in elements in columns KU+2 to KV to zero.
*
DO 20 J = KU + 2, MIN( KV, N )
DO 10 I = KV - J + 2, KL
AB( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* JU is the index of the last column affected by the current stage
* of the factorization.
*
JU = 1
*
DO 40 J = 1, MIN( M, N )
*
* Set fill-in elements in column J+KV to zero.
*
IF( J+KV.LE.N ) THEN
DO 30 I = 1, KL
AB( I, J+KV ) = ZERO
30 CONTINUE
END IF
*
* Find pivot and test for singularity. KM is the number of
* subdiagonal elements in the current column.
*
KM = MIN( KL, M-J )
JP = IDAMAX( KM+1, AB( KV+1, J ), 1 )
IPIV( J ) = JP + J - 1
IF( AB( KV+JP, J ).NE.ZERO ) THEN
JU = MAX( JU, MIN( J+KU+JP-1, N ) )
*
* Apply interchange to columns J to JU.
*
IF( JP.NE.1 )
$ CALL DSWAP( JU-J+1, AB( KV+JP, J ), LDAB-1,
$ AB( KV+1, J ), LDAB-1 )
*
IF( KM.GT.0 ) THEN
*
* Compute multipliers.
*
CALL DSCAL( KM, ONE / AB( KV+1, J ), AB( KV+2, J ),
$ 1 )
*
* Update trailing submatrix within the band.
*
IF( JU.GT.J )
$ CALL DGER( KM, JU-J, -ONE, AB( KV+2, J ), 1,
$ AB( KV, J+1 ), LDAB-1, AB( KV+1, J+1 ),
$ LDAB-1 )
END IF
ELSE
*
* If pivot is zero, set INFO to the index of the pivot
* unless a zero pivot has already been found.
*
IF( INFO.EQ.0 )
$ INFO = J
END IF
40 CONTINUE
RETURN
*
* End of DGBTF2
*
END
*> \brief \b DGBTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbtrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbtrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbtrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBTRF computes an LU factorization of a real m-by-n band matrix A
*> using partial pivoting with row interchanges.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the matrix A in band storage, in rows KL+1 to
*> 2*KL+KU+1; rows 1 to KL of the array need not be set.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
*>
*> On exit, details of the factorization: U is stored as an
*> upper triangular band matrix with KL+KU superdiagonals in
*> rows 1 to KL+KU+1, and the multipliers used during the
*> factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (min(M,N))
*> The pivot indices; for 1 <= i <= min(M,N), row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and division by zero will occur if it is used
*> to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbtrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> M = N = 6, KL = 2, KU = 1:
*>
*> On entry: On exit:
*>
*> * * * + + + * * * u14 u25 u36
*> * * + + + + * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
*> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
*>
*> Array elements marked * are not used by the routine; elements marked
*> + need not be set on entry, but are required by the routine to store
*> elements of U because of fill-in resulting from the row interchanges.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
INTEGER NBMAX, LDWORK
PARAMETER ( NBMAX = 64, LDWORK = NBMAX+1 )
* ..
* .. Local Scalars ..
INTEGER I, I2, I3, II, IP, J, J2, J3, JB, JJ, JM, JP,
$ JU, K2, KM, KV, NB, NW
DOUBLE PRECISION TEMP
* ..
* .. Local Arrays ..
DOUBLE PRECISION WORK13( LDWORK, NBMAX ),
$ WORK31( LDWORK, NBMAX )
* ..
* .. External Functions ..
INTEGER IDAMAX, ILAENV
EXTERNAL IDAMAX, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGBTF2, DGEMM, DGER, DLASWP,
$ DSCAL,
$ DSWAP, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* KV is the number of superdiagonals in the factor U, allowing for
* fill-in
*
KV = KU + KL
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KL+KV+1 ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment
*
NB = ILAENV( 1, 'DGBTRF', ' ', M, N, KL, KU )
*
* The block size must not exceed the limit set by the size of the
* local arrays WORK13 and WORK31.
*
NB = MIN( NB, NBMAX )
*
IF( NB.LE.1 .OR. NB.GT.KL ) THEN
*
* Use unblocked code
*
CALL DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
ELSE
*
* Use blocked code
*
* Zero the superdiagonal elements of the work array WORK13
*
DO 20 J = 1, NB
DO 10 I = 1, J - 1
WORK13( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* Zero the subdiagonal elements of the work array WORK31
*
DO 40 J = 1, NB
DO 30 I = J + 1, NB
WORK31( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
*
* Gaussian elimination with partial pivoting
*
* Set fill-in elements in columns KU+2 to KV to zero
*
DO 60 J = KU + 2, MIN( KV, N )
DO 50 I = KV - J + 2, KL
AB( I, J ) = ZERO
50 CONTINUE
60 CONTINUE
*
* JU is the index of the last column affected by the current
* stage of the factorization
*
JU = 1
*
DO 180 J = 1, MIN( M, N ), NB
JB = MIN( NB, MIN( M, N )-J+1 )
*
* The active part of the matrix is partitioned
*
* A11 A12 A13
* A21 A22 A23
* A31 A32 A33
*
* Here A11, A21 and A31 denote the current block of JB columns
* which is about to be factorized. The number of rows in the
* partitioning are JB, I2, I3 respectively, and the numbers
* of columns are JB, J2, J3. The superdiagonal elements of A13
* and the subdiagonal elements of A31 lie outside the band.
*
I2 = MIN( KL-JB, M-J-JB+1 )
I3 = MIN( JB, M-J-KL+1 )
*
* J2 and J3 are computed after JU has been updated.
*
* Factorize the current block of JB columns
*
DO 80 JJ = J, J + JB - 1
*
* Set fill-in elements in column JJ+KV to zero
*
IF( JJ+KV.LE.N ) THEN
DO 70 I = 1, KL
AB( I, JJ+KV ) = ZERO
70 CONTINUE
END IF
*
* Find pivot and test for singularity. KM is the number of
* subdiagonal elements in the current column.
*
KM = MIN( KL, M-JJ )
JP = IDAMAX( KM+1, AB( KV+1, JJ ), 1 )
IPIV( JJ ) = JP + JJ - J
IF( AB( KV+JP, JJ ).NE.ZERO ) THEN
JU = MAX( JU, MIN( JJ+KU+JP-1, N ) )
IF( JP.NE.1 ) THEN
*
* Apply interchange to columns J to J+JB-1
*
IF( JP+JJ-1.LT.J+KL ) THEN
*
CALL DSWAP( JB, AB( KV+1+JJ-J, J ), LDAB-1,
$ AB( KV+JP+JJ-J, J ), LDAB-1 )
ELSE
*
* The interchange affects columns J to JJ-1 of A31
* which are stored in the work array WORK31
*
CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
$ WORK31( JP+JJ-J-KL, 1 ), LDWORK )
CALL DSWAP( J+JB-JJ, AB( KV+1, JJ ), LDAB-1,
$ AB( KV+JP, JJ ), LDAB-1 )
END IF
END IF
*
* Compute multipliers
*
CALL DSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2,
$ JJ ),
$ 1 )
*
* Update trailing submatrix within the band and within
* the current block. JM is the index of the last column
* which needs to be updated.
*
JM = MIN( JU, J+JB-1 )
IF( JM.GT.JJ )
$ CALL DGER( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1,
$ AB( KV, JJ+1 ), LDAB-1,
$ AB( KV+1, JJ+1 ), LDAB-1 )
ELSE
*
* If pivot is zero, set INFO to the index of the pivot
* unless a zero pivot has already been found.
*
IF( INFO.EQ.0 )
$ INFO = JJ
END IF
*
* Copy current column of A31 into the work array WORK31
*
NW = MIN( JJ-J+1, I3 )
IF( NW.GT.0 )
$ CALL DCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1,
$ WORK31( 1, JJ-J+1 ), 1 )
80 CONTINUE
IF( J+JB.LE.N ) THEN
*
* Apply the row interchanges to the other blocks.
*
J2 = MIN( JU-J+1, KV ) - JB
J3 = MAX( 0, JU-J-KV+1 )
*
* Use DLASWP to apply the row interchanges to A12, A22, and
* A32.
*
CALL DLASWP( J2, AB( KV+1-JB, J+JB ), LDAB-1, 1, JB,
$ IPIV( J ), 1 )
*
* Adjust the pivot indices.
*
DO 90 I = J, J + JB - 1
IPIV( I ) = IPIV( I ) + J - 1
90 CONTINUE
*
* Apply the row interchanges to A13, A23, and A33
* columnwise.
*
K2 = J - 1 + JB + J2
DO 110 I = 1, J3
JJ = K2 + I
DO 100 II = J + I - 1, J + JB - 1
IP = IPIV( II )
IF( IP.NE.II ) THEN
TEMP = AB( KV+1+II-JJ, JJ )
AB( KV+1+II-JJ, JJ ) = AB( KV+1+IP-JJ, JJ )
AB( KV+1+IP-JJ, JJ ) = TEMP
END IF
100 CONTINUE
110 CONTINUE
*
* Update the relevant part of the trailing submatrix
*
IF( J2.GT.0 ) THEN
*
* Update A12
*
CALL DTRSM( 'Left', 'Lower', 'No transpose',
$ 'Unit',
$ JB, J2, ONE, AB( KV+1, J ), LDAB-1,
$ AB( KV+1-JB, J+JB ), LDAB-1 )
*
IF( I2.GT.0 ) THEN
*
* Update A22
*
CALL DGEMM( 'No transpose', 'No transpose', I2,
$ J2,
$ JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
$ AB( KV+1-JB, J+JB ), LDAB-1, ONE,
$ AB( KV+1, J+JB ), LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Update A32
*
CALL DGEMM( 'No transpose', 'No transpose', I3,
$ J2,
$ JB, -ONE, WORK31, LDWORK,
$ AB( KV+1-JB, J+JB ), LDAB-1, ONE,
$ AB( KV+KL+1-JB, J+JB ), LDAB-1 )
END IF
END IF
*
IF( J3.GT.0 ) THEN
*
* Copy the lower triangle of A13 into the work array
* WORK13
*
DO 130 JJ = 1, J3
DO 120 II = JJ, JB
WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 )
120 CONTINUE
130 CONTINUE
*
* Update A13 in the work array
*
CALL DTRSM( 'Left', 'Lower', 'No transpose',
$ 'Unit',
$ JB, J3, ONE, AB( KV+1, J ), LDAB-1,
$ WORK13, LDWORK )
*
IF( I2.GT.0 ) THEN
*
* Update A23
*
CALL DGEMM( 'No transpose', 'No transpose', I2,
$ J3,
$ JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
$ WORK13, LDWORK, ONE, AB( 1+JB, J+KV ),
$ LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Update A33
*
CALL DGEMM( 'No transpose', 'No transpose', I3,
$ J3,
$ JB, -ONE, WORK31, LDWORK, WORK13,
$ LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 )
END IF
*
* Copy the lower triangle of A13 back into place
*
DO 150 JJ = 1, J3
DO 140 II = JJ, JB
AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ )
140 CONTINUE
150 CONTINUE
END IF
ELSE
*
* Adjust the pivot indices.
*
DO 160 I = J, J + JB - 1
IPIV( I ) = IPIV( I ) + J - 1
160 CONTINUE
END IF
*
* Partially undo the interchanges in the current block to
* restore the upper triangular form of A31 and copy the upper
* triangle of A31 back into place
*
DO 170 JJ = J + JB - 1, J, -1
JP = IPIV( JJ ) - JJ + 1
IF( JP.NE.1 ) THEN
*
* Apply interchange to columns J to JJ-1
*
IF( JP+JJ-1.LT.J+KL ) THEN
*
* The interchange does not affect A31
*
CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
$ AB( KV+JP+JJ-J, J ), LDAB-1 )
ELSE
*
* The interchange does affect A31
*
CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
$ WORK31( JP+JJ-J-KL, 1 ), LDWORK )
END IF
END IF
*
* Copy the current column of A31 back into place
*
NW = MIN( I3, JJ-J+1 )
IF( NW.GT.0 )
$ CALL DCOPY( NW, WORK31( 1, JJ-J+1 ), 1,
$ AB( KV+KL+1-JJ+J, JJ ), 1 )
170 CONTINUE
180 CONTINUE
END IF
*
RETURN
*
* End of DGBTRF
*
END
*> \brief \b DGBTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBTRS solves a system of linear equations
*> A * X = B or A**T * X = B
*> with a general band matrix A using the LU factorization computed
*> by DGBTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations.
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T* X = B (Transpose)
*> = 'C': A**T* X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> Details of the LU factorization of the band matrix A, as
*> computed by DGBTRF. U is stored as an upper triangular band
*> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
*> the multipliers used during the factorization are stored in
*> rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= N, row i of the matrix was
*> interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbtrs
*
* =====================================================================
SUBROUTINE DGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B,
$ LDB,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LNOTI, NOTRAN
INTEGER I, J, KD, L, LM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER, DSWAP, DTBSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.( 2*KL+KU+1 ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
KD = KU + KL + 1
LNOTI = KL.GT.0
*
IF( NOTRAN ) THEN
*
* Solve A*X = B.
*
* Solve L*X = B, overwriting B with X.
*
* L is represented as a product of permutations and unit lower
* triangular matrices L = P(1) * L(1) * ... * P(n-1) * L(n-1),
* where each transformation L(i) is a rank-one modification of
* the identity matrix.
*
IF( LNOTI ) THEN
DO 10 J = 1, N - 1
LM = MIN( KL, N-J )
L = IPIV( J )
IF( L.NE.J )
$ CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
CALL DGER( LM, NRHS, -ONE, AB( KD+1, J ), 1, B( J,
$ 1 ),
$ LDB, B( J+1, 1 ), LDB )
10 CONTINUE
END IF
*
DO 20 I = 1, NRHS
*
* Solve U*X = B, overwriting B with X.
*
CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N,
$ KL+KU,
$ AB, LDAB, B( 1, I ), 1 )
20 CONTINUE
*
ELSE
*
* Solve A**T*X = B.
*
DO 30 I = 1, NRHS
*
* Solve U**T*X = B, overwriting B with X.
*
CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KL+KU,
$ AB,
$ LDAB, B( 1, I ), 1 )
30 CONTINUE
*
* Solve L**T*X = B, overwriting B with X.
*
IF( LNOTI ) THEN
DO 40 J = N - 1, 1, -1
LM = MIN( KL, N-J )
CALL DGEMV( 'Transpose', LM, NRHS, -ONE, B( J+1, 1 ),
$ LDB, AB( KD+1, J ), 1, ONE, B( J, 1 ), LDB )
L = IPIV( J )
IF( L.NE.J )
$ CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
40 CONTINUE
END IF
END IF
RETURN
*
* End of DGBTRS
*
END
*> \brief \b DGEBAK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBAK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebak.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebak.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebak.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB, SIDE
* INTEGER IHI, ILO, INFO, LDV, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION SCALE( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBAK forms the right or left eigenvectors of a real general matrix
*> by backward transformation on the computed eigenvectors of the
*> balanced matrix output by DGEBAL.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies the type of backward transformation required:
*> = 'N': do nothing, return immediately;
*> = 'P': do backward transformation for permutation only;
*> = 'S': do backward transformation for scaling only;
*> = 'B': do backward transformations for both permutation and
*> scaling.
*> JOB must be the same as the argument JOB supplied to DGEBAL.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': V contains right eigenvectors;
*> = 'L': V contains left eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrix V. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> The integers ILO and IHI determined by DGEBAL.
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutation and scaling factors, as returned
*> by DGEBAL.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of columns of the matrix V. M >= 0.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,M)
*> On entry, the matrix of right or left eigenvectors to be
*> transformed, as returned by DHSEIN or DTREVC.
*> On exit, V is overwritten by the transformed eigenvectors.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebak
*
* =====================================================================
SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOB, SIDE
INTEGER IHI, ILO, INFO, LDV, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION SCALE( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFTV, RIGHTV
INTEGER I, II, K
DOUBLE PRECISION S
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Decode and Test the input parameters
*
RIGHTV = LSAME( SIDE, 'R' )
LEFTV = LSAME( SIDE, 'L' )
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND.
$ .NOT.LSAME( JOB, 'P' ) .AND.
$ .NOT.LSAME( JOB, 'S' ) .AND.
$ .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -7
ELSE IF( LDV.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEBAK', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( M.EQ.0 )
$ RETURN
IF( LSAME( JOB, 'N' ) )
$ RETURN
*
IF( ILO.EQ.IHI )
$ GO TO 30
*
* Backward balance
*
IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
IF( RIGHTV ) THEN
DO 10 I = ILO, IHI
S = SCALE( I )
CALL DSCAL( M, S, V( I, 1 ), LDV )
10 CONTINUE
END IF
*
IF( LEFTV ) THEN
DO 20 I = ILO, IHI
S = ONE / SCALE( I )
CALL DSCAL( M, S, V( I, 1 ), LDV )
20 CONTINUE
END IF
*
END IF
*
* Backward permutation
*
* For I = ILO-1 step -1 until 1,
* IHI+1 step 1 until N do --
*
30 CONTINUE
IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF( RIGHTV ) THEN
DO 40 II = 1, N
I = II
IF( I.GE.ILO .AND. I.LE.IHI )
$ GO TO 40
IF( I.LT.ILO )
$ I = ILO - II
K = INT( SCALE( I ) )
IF( K.EQ.I )
$ GO TO 40
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
40 CONTINUE
END IF
*
IF( LEFTV ) THEN
DO 50 II = 1, N
I = II
IF( I.GE.ILO .AND. I.LE.IHI )
$ GO TO 50
IF( I.LT.ILO )
$ I = ILO - II
K = INT( SCALE( I ) )
IF( K.EQ.I )
$ GO TO 50
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
50 CONTINUE
END IF
END IF
*
RETURN
*
* End of DGEBAK
*
END
*> \brief \b DGEBAL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBAL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebal.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebal.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebal.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB
* INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), SCALE( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBAL balances a general real matrix A. This involves, first,
*> permuting A by a similarity transformation to isolate eigenvalues
*> in the first 1 to ILO-1 and last IHI+1 to N elements on the
*> diagonal; and second, applying a diagonal similarity transformation
*> to rows and columns ILO to IHI to make the rows and columns as
*> close in norm as possible. Both steps are optional.
*>
*> Balancing may reduce the 1-norm of the matrix, and improve the
*> accuracy of the computed eigenvalues and/or eigenvectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies the operations to be performed on A:
*> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
*> for i = 1,...,N;
*> = 'P': permute only;
*> = 'S': scale only;
*> = 'B': both permute and scale.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the input matrix A.
*> On exit, A is overwritten by the balanced matrix.
*> If JOB = 'N', A is not referenced.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*> \param[out] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI are set to integers such that on exit
*> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
*> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and scaling factors applied to
*> A. If P(j) is the index of the row and column interchanged
*> with row and column j and D(j) is the scaling factor
*> applied to row and column j, then
*> SCALE(j) = P(j) for j = 1,...,ILO-1
*> = D(j) for j = ILO,...,IHI
*> = P(j) for j = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebal
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The permutations consist of row and column interchanges which put
*> the matrix in the form
*>
*> ( T1 X Y )
*> P A P = ( 0 B Z )
*> ( 0 0 T2 )
*>
*> where T1 and T2 are upper triangular matrices whose eigenvalues lie
*> along the diagonal. The column indices ILO and IHI mark the starting
*> and ending columns of the submatrix B. Balancing consists of applying
*> a diagonal similarity transformation inv(D) * B * D to make the
*> 1-norms of each row of B and its corresponding column nearly equal.
*> The output matrix is
*>
*> ( T1 X*D Y )
*> ( 0 inv(D)*B*D inv(D)*Z ).
*> ( 0 0 T2 )
*>
*> Information about the permutations P and the diagonal matrix D is
*> returned in the vector SCALE.
*>
*> This subroutine is based on the EISPACK routine BALANC.
*>
*> Modified by Tzu-Yi Chen, Computer Science Division, University of
*> California at Berkeley, USA
*>
*> Refactored by Evert Provoost, Department of Computer Science,
*> KU Leuven, Belgium
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SCALE( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION SCLFAC
PARAMETER ( SCLFAC = 2.0D+0 )
DOUBLE PRECISION FACTOR
PARAMETER ( FACTOR = 0.95D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOCONV, CANSWAP
INTEGER I, ICA, IRA, J, K, L
DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
$ SFMIN2
* ..
* .. External Functions ..
LOGICAL DISNAN, LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DNRM2
EXTERNAL DISNAN, LSAME, IDAMAX, DLAMCH,
$ DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* Test the input parameters
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND.
$ .NOT.LSAME( JOB, 'P' ) .AND.
$ .NOT.LSAME( JOB, 'S' ) .AND.
$ .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEBAL', -INFO )
RETURN
END IF
*
* Quick returns.
*
IF( N.EQ.0 ) THEN
ILO = 1
IHI = 0
RETURN
END IF
*
IF( LSAME( JOB, 'N' ) ) THEN
DO I = 1, N
SCALE( I ) = ONE
END DO
ILO = 1
IHI = N
RETURN
END IF
*
* Permutation to isolate eigenvalues if possible.
*
K = 1
L = N
*
IF( .NOT.LSAME( JOB, 'S' ) ) THEN
*
* Row and column exchange.
*
NOCONV = .TRUE.
DO WHILE( NOCONV )
*
* Search for rows isolating an eigenvalue and push them down.
*
NOCONV = .FALSE.
DO I = L, 1, -1
CANSWAP = .TRUE.
DO J = 1, L
IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
CANSWAP = .FALSE.
EXIT
END IF
END DO
*
IF( CANSWAP ) THEN
SCALE( L ) = I
IF( I.NE.L ) THEN
CALL DSWAP( L, A( 1, I ), 1, A( 1, L ), 1 )
CALL DSWAP( N-K+1, A( I, K ), LDA, A( L, K ),
$ LDA )
END IF
NOCONV = .TRUE.
*
IF( L.EQ.1 ) THEN
ILO = 1
IHI = 1
RETURN
END IF
*
L = L - 1
END IF
END DO
*
END DO
NOCONV = .TRUE.
DO WHILE( NOCONV )
*
* Search for columns isolating an eigenvalue and push them left.
*
NOCONV = .FALSE.
DO J = K, L
CANSWAP = .TRUE.
DO I = K, L
IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
CANSWAP = .FALSE.
EXIT
END IF
END DO
*
IF( CANSWAP ) THEN
SCALE( K ) = J
IF( J.NE.K ) THEN
CALL DSWAP( L, A( 1, J ), 1, A( 1, K ), 1 )
CALL DSWAP( N-K+1, A( J, K ), LDA, A( K, K ),
$ LDA )
END IF
NOCONV = .TRUE.
*
K = K + 1
END IF
END DO
*
END DO
*
END IF
*
* Initialize SCALE for non-permuted submatrix.
*
DO I = K, L
SCALE( I ) = ONE
END DO
*
* If we only had to permute, we are done.
*
IF( LSAME( JOB, 'P' ) ) THEN
ILO = K
IHI = L
RETURN
END IF
*
* Balance the submatrix in rows K to L.
*
* Iterative loop for norm reduction.
*
SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
SFMAX1 = ONE / SFMIN1
SFMIN2 = SFMIN1*SCLFAC
SFMAX2 = ONE / SFMIN2
*
NOCONV = .TRUE.
DO WHILE( NOCONV )
NOCONV = .FALSE.
*
DO I = K, L
*
C = DNRM2( L-K+1, A( K, I ), 1 )
R = DNRM2( L-K+1, A( I, K ), LDA )
ICA = IDAMAX( L, A( 1, I ), 1 )
CA = ABS( A( ICA, I ) )
IRA = IDAMAX( N-K+1, A( I, K ), LDA )
RA = ABS( A( I, IRA+K-1 ) )
*
* Guard against zero C or R due to underflow.
*
IF( C.EQ.ZERO .OR. R.EQ.ZERO ) CYCLE
*
* Exit if NaN to avoid infinite loop
*
IF( DISNAN( C+CA+R+RA ) ) THEN
INFO = -3
CALL XERBLA( 'DGEBAL', -INFO )
RETURN
END IF
*
G = R / SCLFAC
F = ONE
S = C + R
*
DO WHILE( C.LT.G .AND. MAX( F, C, CA ).LT.SFMAX2 .AND.
$ MIN( R, G, RA ).GT.SFMIN2 )
F = F*SCLFAC
C = C*SCLFAC
CA = CA*SCLFAC
R = R / SCLFAC
G = G / SCLFAC
RA = RA / SCLFAC
END DO
*
G = C / SCLFAC
*
DO WHILE( G.GE.R .AND. MAX( R, RA ).LT.SFMAX2 .AND.
$ MIN( F, C, G, CA ).GT.SFMIN2 )
F = F / SCLFAC
C = C / SCLFAC
G = G / SCLFAC
CA = CA / SCLFAC
R = R*SCLFAC
RA = RA*SCLFAC
END DO
*
* Now balance.
*
IF( ( C+R ).GE.FACTOR*S ) CYCLE
IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
IF( F*SCALE( I ).LE.SFMIN1 ) CYCLE
END IF
IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
IF( SCALE( I ).GE.SFMAX1 / F ) CYCLE
END IF
G = ONE / F
SCALE( I ) = SCALE( I )*F
NOCONV = .TRUE.
*
CALL DSCAL( N-K+1, G, A( I, K ), LDA )
CALL DSCAL( L, F, A( 1, I ), 1 )
*
END DO
*
END DO
*
ILO = K
IHI = L
*
RETURN
*
* End of DGEBAL
*
END
*> \brief \b DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBD2 reduces a real general m by n matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n general matrix to be reduced.
*> On exit,
*> if m >= n, the diagonal and the first superdiagonal are
*> overwritten with the upper bidiagonal matrix B; the
*> elements below the diagonal, with the array TAUQ, represent
*> the orthogonal matrix Q as a product of elementary
*> reflectors, and the elements above the first superdiagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors;
*> if m < n, the diagonal and the first subdiagonal are
*> overwritten with the lower bidiagonal matrix B; the
*> elements below the first subdiagonal, with the array TAUQ,
*> represent the orthogonal matrix Q as a product of
*> elementary reflectors, and the elements above the diagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (min(M,N))
*> The diagonal elements of the bidiagonal matrix B:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*> The off-diagonal elements of the bidiagonal matrix B:
*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebd2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> If m >= n,
*>
*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n,
*>
*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The contents of A on exit are illustrated by the following examples:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
*> ( v1 v2 v3 v4 v5 )
*>
*> where d and e denote diagonal and off-diagonal elements of B, vi
*> denotes an element of the vector defining H(i), and ui an element of
*> the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.LT.0 ) THEN
CALL XERBLA( 'DGEBD2', -INFO )
RETURN
END IF
*
IF( M.GE.N ) THEN
*
* Reduce to upper bidiagonal form
*
DO 10 I = 1, N
*
* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAUQ( I ) )
D( I ) = A( I, I )
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
IF( I.LT.N )
$ CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1,
$ TAUQ( I ),
$ A( I, I+1 ), LDA, WORK )
*
IF( I.LT.N ) THEN
*
* Generate elementary reflector G(i) to annihilate
* A(i,i+2:n)
*
CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
$ LDA, TAUP( I ) )
E( I ) = A( I, I+1 )
*
* Apply G(i) to A(i+1:m,i+1:n) from the right
*
CALL DLARF1F( 'Right', M-I, N-I, A( I, I+1 ), LDA,
$ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
ELSE
TAUP( I ) = ZERO
END IF
10 CONTINUE
ELSE
*
* Reduce to lower bidiagonal form
*
DO 20 I = 1, M
*
* Generate elementary reflector G(i) to annihilate A(i,i+1:n)
*
CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ),
$ LDA,
$ TAUP( I ) )
D( I ) = A( I, I )
*
* Apply G(i) to A(i+1:m,i:n) from the right
*
IF( I.LT.M )
$ CALL DLARF1F( 'Right', M-I, N-I+1, A( I, I ), LDA,
$ TAUP( I ), A( I+1, I ), LDA, WORK )
*
IF( I.LT.M ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:m,i)
*
CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ),
$ 1,
$ TAUQ( I ) )
E( I ) = A( I+1, I )
*
* Apply H(i) to A(i+1:m,i+1:n) from the left
*
CALL DLARF1F( 'Left', M-I, N-I, A( I+1, I ), 1,
$ TAUQ( I ),
$ A( I+1, I+1 ), LDA, WORK )
ELSE
TAUQ( I ) = ZERO
END IF
20 CONTINUE
END IF
RETURN
*
* End of DGEBD2
*
END
*> \brief \b DGEBRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBRD reduces a general real M-by-N matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N general matrix to be reduced.
*> On exit,
*> if m >= n, the diagonal and the first superdiagonal are
*> overwritten with the upper bidiagonal matrix B; the
*> elements below the diagonal, with the array TAUQ, represent
*> the orthogonal matrix Q as a product of elementary
*> reflectors, and the elements above the first superdiagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors;
*> if m < n, the diagonal and the first subdiagonal are
*> overwritten with the lower bidiagonal matrix B; the
*> elements below the first subdiagonal, with the array TAUQ,
*> represent the orthogonal matrix Q as a product of
*> elementary reflectors, and the elements above the diagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (min(M,N))
*> The diagonal elements of the bidiagonal matrix B:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*> The off-diagonal elements of the bidiagonal matrix B:
*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK.
*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
*> For optimum performance LWORK >= (M+N)*NB, where NB
*> is the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> If m >= n,
*>
*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n,
*>
*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The contents of A on exit are illustrated by the following examples:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
*> ( v1 v2 v3 v4 v5 )
*>
*> where d and e denote diagonal and off-diagonal elements of B, vi
*> denotes an element of the vector defining H(i), and ui an element of
*> the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKMIN, LWKOPT,
$ MINMN, NB, NBMIN, NX, WS
* ..
* .. External Subroutines ..
EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
MINMN = MIN( M, N )
IF( MINMN.EQ.0 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
LWKMIN = MAX( M, N )
NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
LWKOPT = ( M+N )*NB
ENDIF
WORK( 1 ) = DBLE( LWKOPT )
*
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
IF( INFO.LT.0 ) THEN
CALL XERBLA( 'DGEBRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( MINMN.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
WS = MAX( M, N )
LDWRKX = M
LDWRKY = N
*
IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
*
* Set the crossover point NX.
*
NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
*
* Determine when to switch from blocked to unblocked code.
*
IF( NX.LT.MINMN ) THEN
WS = LWKOPT
IF( LWORK.LT.WS ) THEN
*
* Not enough work space for the optimal NB, consider using
* a smaller block size.
*
NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
IF( LWORK.GE.( M+N )*NBMIN ) THEN
NB = LWORK / ( M+N )
ELSE
NB = 1
NX = MINMN
END IF
END IF
END IF
ELSE
NX = MINMN
END IF
*
DO 30 I = 1, MINMN - NX, NB
*
* Reduce rows and columns i:i+nb-1 to bidiagonal form and return
* the matrices X and Y which are needed to update the unreduced
* part of the matrix
*
CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ),
$ E( I ),
$ TAUQ( I ), TAUP( I ), WORK, LDWRKX,
$ WORK( LDWRKX*NB+1 ), LDWRKY )
*
* Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
* of the form A := A - V*Y**T - X*U**T
*
CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
$ NB, -ONE, A( I+NB, I ), LDA,
$ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
$ A( I+NB, I+NB ), LDA )
CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1,
$ N-I-NB+1,
$ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
$ ONE, A( I+NB, I+NB ), LDA )
*
* Copy diagonal and off-diagonal elements of B back into A
*
IF( M.GE.N ) THEN
DO 10 J = I, I + NB - 1
A( J, J ) = D( J )
A( J, J+1 ) = E( J )
10 CONTINUE
ELSE
DO 20 J = I, I + NB - 1
A( J, J ) = D( J )
A( J+1, J ) = E( J )
20 CONTINUE
END IF
30 CONTINUE
*
* Use unblocked code to reduce the remainder of the matrix
*
CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAUQ( I ), TAUP( I ), WORK, IINFO )
WORK( 1 ) = WS
RETURN
*
* End of DGEBRD
*
END
*> \brief \b DGECON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGECON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgecon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgecon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgecon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGECON( NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER INFO, LDA, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGECON estimates the reciprocal of the condition number of a general
*> real matrix A, in either the 1-norm or the infinity-norm, using
*> the LU factorization computed by DGETRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as
*> RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies whether the 1-norm condition number or the
*> infinity-norm condition number is required:
*> = '1' or 'O': 1-norm;
*> = 'I': Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The factors L and U from the factorization A = P*L*U
*> as computed by DGETRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> If NORM = '1' or 'O', the 1-norm of the original matrix A.
*> If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> NaNs are illegal values for ANORM, and they propagate to
*> the output parameter RCOND.
*> Infinity is illegal for ANORM, and it propagates to the output
*> parameter RCOND as 0.
*> = 1: if RCOND = NaN, or
*> RCOND = Inf, or
*> the computed norm of the inverse of A is 0.
*> In the latter, RCOND = 0 is returned.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gecon
*
* =====================================================================
SUBROUTINE DGECON( NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER INFO, LDA, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ONENRM
CHARACTER NORMIN
INTEGER IX, KASE, KASE1
DOUBLE PRECISION AINVNM, SCALE, SL, SMLNUM, SU, HUGEVAL
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLATRS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
HUGEVAL = DLAMCH( 'Overflow' )
*
* Test the input parameters.
*
INFO = 0
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGECON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
ELSE IF( DISNAN( ANORM ) ) THEN
RCOND = ANORM
INFO = -5
RETURN
ELSE IF( ANORM.GT.HUGEVAL ) THEN
INFO = -5
RETURN
END IF
*
SMLNUM = DLAMCH( 'Safe minimum' )
*
* Estimate the norm of inv(A).
*
AINVNM = ZERO
NORMIN = 'N'
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KASE = 0
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(L).
*
CALL DLATRS( 'Lower', 'No transpose', 'Unit', NORMIN, N,
$ A,
$ LDA, WORK, SL, WORK( 2*N+1 ), INFO )
*
* Multiply by inv(U).
*
CALL DLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ A, LDA, WORK, SU, WORK( 3*N+1 ), INFO )
ELSE
*
* Multiply by inv(U**T).
*
CALL DLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
$ A,
$ LDA, WORK, SU, WORK( 3*N+1 ), INFO )
*
* Multiply by inv(L**T).
*
CALL DLATRS( 'Lower', 'Transpose', 'Unit', NORMIN, N, A,
$ LDA, WORK, SL, WORK( 2*N+1 ), INFO )
END IF
*
* Divide X by 1/(SL*SU) if doing so will not cause overflow.
*
SCALE = SL*SU
NORMIN = 'Y'
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO ) THEN
RCOND = ( ONE / AINVNM ) / ANORM
ELSE
INFO = 1
RETURN
END IF
*
* Check for NaNs and Infs
*
IF( DISNAN( RCOND ) .OR. RCOND.GT.HUGEVAL )
$ INFO = 1
*
20 CONTINUE
RETURN
*
* End of DGECON
*
END
*> \brief \b DGEEQU
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEEQU computes row and column scalings intended to equilibrate an
*> M-by-N matrix A and reduce its condition number. R returns the row
*> scale factors and C the column scale factors, chosen to try to make
*> the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
*>
*> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
*> number and BIGNUM = largest safe number. Use of these scaling
*> factors is not guaranteed to reduce the condition number of A but
*> works well in practice.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The M-by-N matrix whose equilibration factors are
*> to be computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (M)
*> If INFO = 0 or INFO > M, R contains the row scale factors
*> for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*> ROWCND is DOUBLE PRECISION
*> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
*> AMAX is neither too large nor too small, it is not worth
*> scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*> COLCND is DOUBLE PRECISION
*> If INFO = 0, COLCND contains the ratio of the smallest
*> C(i) to the largest C(i). If COLCND >= 0.1, it is not
*> worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= M: the i-th row of A is exactly zero
*> > M: the (i-M)-th column of A is exactly zero
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geequ
*
* =====================================================================
SUBROUTINE DGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEQU', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
ROWCND = ONE
COLCND = ONE
AMAX = ZERO
RETURN
END IF
*
* Get machine constants.
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
* Compute row scale factors.
*
DO 10 I = 1, M
R( I ) = ZERO
10 CONTINUE
*
* Find the maximum element in each row.
*
DO 30 J = 1, N
DO 20 I = 1, M
R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
20 CONTINUE
30 CONTINUE
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 40 I = 1, M
RCMAX = MAX( RCMAX, R( I ) )
RCMIN = MIN( RCMIN, R( I ) )
40 CONTINUE
AMAX = RCMAX
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 50 I = 1, M
IF( R( I ).EQ.ZERO ) THEN
INFO = I
RETURN
END IF
50 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 60 I = 1, M
R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
60 CONTINUE
*
* Compute ROWCND = min(R(I)) / max(R(I))
*
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
* Compute column scale factors
*
DO 70 J = 1, N
C( J ) = ZERO
70 CONTINUE
*
* Find the maximum element in each column,
* assuming the row scaling computed above.
*
DO 90 J = 1, N
DO 80 I = 1, M
C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
80 CONTINUE
90 CONTINUE
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 100 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
100 CONTINUE
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 110 J = 1, N
IF( C( J ).EQ.ZERO ) THEN
INFO = M + J
RETURN
END IF
110 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 120 J = 1, N
C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
120 CONTINUE
*
* Compute COLCND = min(C(J)) / max(C(J))
*
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
RETURN
*
* End of DGEEQU
*
END
*> \brief \b DGEEQUB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEEQUB computes row and column scalings intended to equilibrate an
*> M-by-N matrix A and reduce its condition number. R returns the row
*> scale factors and C the column scale factors, chosen to try to make
*> the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
*> the radix.
*>
*> R(i) and C(j) are restricted to be a power of the radix between
*> SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
*> of these scaling factors is not guaranteed to reduce the condition
*> number of A but works well in practice.
*>
*> This routine differs from DGEEQU by restricting the scaling factors
*> to a power of the radix. Barring over- and underflow, scaling by
*> these factors introduces no additional rounding errors. However, the
*> scaled entries' magnitudes are no longer approximately 1 but lie
*> between sqrt(radix) and 1/sqrt(radix).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The M-by-N matrix whose equilibration factors are
*> to be computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (M)
*> If INFO = 0 or INFO > M, R contains the row scale factors
*> for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*> ROWCND is DOUBLE PRECISION
*> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
*> AMAX is neither too large nor too small, it is not worth
*> scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*> COLCND is DOUBLE PRECISION
*> If INFO = 0, COLCND contains the ratio of the smallest
*> C(i) to the largest C(i). If COLCND >= 0.1, it is not
*> worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= M: the i-th row of A is exactly zero
*> > M: the (i-M)-th column of A is exactly zero
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geequb
*
* =====================================================================
SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, LOG
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEQUB', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
ROWCND = ONE
COLCND = ONE
AMAX = ZERO
RETURN
END IF
*
* Get machine constants. Assume SMLNUM is a power of the radix.
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
RADIX = DLAMCH( 'B' )
LOGRDX = LOG( RADIX )
*
* Compute row scale factors.
*
DO 10 I = 1, M
R( I ) = ZERO
10 CONTINUE
*
* Find the maximum element in each row.
*
DO 30 J = 1, N
DO 20 I = 1, M
R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
20 CONTINUE
30 CONTINUE
DO I = 1, M
IF( R( I ).GT.ZERO ) THEN
R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
END IF
END DO
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 40 I = 1, M
RCMAX = MAX( RCMAX, R( I ) )
RCMIN = MIN( RCMIN, R( I ) )
40 CONTINUE
AMAX = RCMAX
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 50 I = 1, M
IF( R( I ).EQ.ZERO ) THEN
INFO = I
RETURN
END IF
50 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 60 I = 1, M
R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
60 CONTINUE
*
* Compute ROWCND = min(R(I)) / max(R(I)).
*
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
* Compute column scale factors
*
DO 70 J = 1, N
C( J ) = ZERO
70 CONTINUE
*
* Find the maximum element in each column,
* assuming the row scaling computed above.
*
DO 90 J = 1, N
DO 80 I = 1, M
C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
80 CONTINUE
IF( C( J ).GT.ZERO ) THEN
C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
END IF
90 CONTINUE
*
* Find the maximum and minimum scale factors.
*
RCMIN = BIGNUM
RCMAX = ZERO
DO 100 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
100 CONTINUE
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 110 J = 1, N
IF( C( J ).EQ.ZERO ) THEN
INFO = M + J
RETURN
END IF
110 CONTINUE
ELSE
*
* Invert the scale factors.
*
DO 120 J = 1, N
C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
120 CONTINUE
*
* Compute COLCND = min(C(J)) / max(C(J)).
*
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
END IF
*
RETURN
*
* End of DGEEQUB
*
END
*> \brief <b> DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEES + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgees.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgees.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgees.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
* VS, LDVS, WORK, LWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVS, SORT
* INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
* $ WR( * )
* ..
* .. Function Arguments ..
* LOGICAL SELECT
* EXTERNAL SELECT
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEES computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues, the real Schur form T, and, optionally, the matrix of
*> Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
*>
*> Optionally, it also orders the eigenvalues on the diagonal of the
*> real Schur form so that selected eigenvalues are at the top left.
*> The leading columns of Z then form an orthonormal basis for the
*> invariant subspace corresponding to the selected eigenvalues.
*>
*> A matrix is in real Schur form if it is upper quasi-triangular with
*> 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
*> form
*> [ a b ]
*> [ c a ]
*>
*> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVS
*> \verbatim
*> JOBVS is CHARACTER*1
*> = 'N': Schur vectors are not computed;
*> = 'V': Schur vectors are computed.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*> SORT is CHARACTER*1
*> Specifies whether or not to order the eigenvalues on the
*> diagonal of the Schur form.
*> = 'N': Eigenvalues are not ordered;
*> = 'S': Eigenvalues are ordered (see SELECT).
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
*> SELECT must be declared EXTERNAL in the calling subroutine.
*> If SORT = 'S', SELECT is used to select eigenvalues to sort
*> to the top left of the Schur form.
*> If SORT = 'N', SELECT is not referenced.
*> An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
*> SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
*> conjugate pair of eigenvalues is selected, then both complex
*> eigenvalues are selected.
*> Note that a selected complex eigenvalue may no longer
*> satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
*> ordering may change the value of complex eigenvalues
*> (especially if the eigenvalue is ill-conditioned); in this
*> case INFO is set to N+2 (see INFO below).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N matrix A.
*> On exit, A has been overwritten by its real Schur form T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*> SDIM is INTEGER
*> If SORT = 'N', SDIM = 0.
*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*> for which SELECT is true. (Complex conjugate
*> pairs for which SELECT is true for either
*> eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> WR and WI contain the real and imaginary parts,
*> respectively, of the computed eigenvalues in the same order
*> that they appear on the diagonal of the output Schur form T.
*> Complex conjugate pairs of eigenvalues will appear
*> consecutively with the eigenvalue having the positive
*> imaginary part first.
*> \endverbatim
*>
*> \param[out] VS
*> \verbatim
*> VS is DOUBLE PRECISION array, dimension (LDVS,N)
*> If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
*> vectors.
*> If JOBVS = 'N', VS is not referenced.
*> \endverbatim
*>
*> \param[in] LDVS
*> \verbatim
*> LDVS is INTEGER
*> The leading dimension of the array VS. LDVS >= 1; if
*> JOBVS = 'V', LDVS >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,3*N).
*> For good performance, LWORK must generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, and i is
*> <= N: the QR algorithm failed to compute all the
*> eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
*> contain those eigenvalues which have converged; if
*> JOBVS = 'V', VS contains the matrix which reduces A
*> to its partially converged Schur form.
*> = N+1: the eigenvalues could not be reordered because some
*> eigenvalues were too close to separate (the problem
*> is very ill-conditioned);
*> = N+2: after reordering, roundoff changed values of some
*> complex eigenvalues so that leading eigenvalues in
*> the Schur form no longer satisfy SELECT=.TRUE. This
*> could also be caused by underflow due to scaling.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gees
*
* =====================================================================
SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
$ VS, LDVS, WORK, LWORK, BWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVS, SORT
INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
$ WR( * )
* ..
* .. Function Arguments ..
LOGICAL SELECT
EXTERNAL SELECT
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTST,
$ WANTVS
INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
$ IHI, ILO, INXT, IP, ITAU, IWRK, MAXWRK, MINWRK
DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR,
$ DLACPY,
$ DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
WANTVS = LSAME( JOBVS, 'V' )
WANTST = LSAME( SORT, 'S' )
IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTST ) .AND.
$ ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
INFO = -11
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by DHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
MINWRK = 3*N
*
CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS,
$ LDVS,
$ WORK, -1, IEVAL )
HSWORK = INT( WORK( 1 ) )
*
IF( .NOT.WANTVS ) THEN
MAXWRK = MAX( MAXWRK, N + HSWORK )
ELSE
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
MAXWRK = MAX( MAXWRK, N + HSWORK )
END IF
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEES ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SDIM = 0
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
SCALEA = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
SCALEA = .TRUE.
CSCALE = SMLNUM
ELSE IF( ANRM.GT.BIGNUM ) THEN
SCALEA = .TRUE.
CSCALE = BIGNUM
END IF
IF( SCALEA )
$ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Permute the matrix to make it more nearly triangular
* (Workspace: need N)
*
IBAL = 1
CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = N + IBAL
IWRK = N + ITAU
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVS ) THEN
*
* Copy Householder vectors to VS
*
CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS )
*
* Generate orthogonal matrix in VS
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ),
$ WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
END IF
*
SDIM = 0
*
* Perform QR iteration, accumulating Schur vectors in VS if desired
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS,
$ WORK( IWRK ), LWORK-IWRK+1, IEVAL )
IF( IEVAL.GT.0 )
$ INFO = IEVAL
*
* Sort eigenvalues if desired
*
IF( WANTST .AND. INFO.EQ.0 ) THEN
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR )
END IF
DO 10 I = 1, N
BWORK( I ) = SELECT( WR( I ), WI( I ) )
10 CONTINUE
*
* Reorder eigenvalues and transform Schur vectors
* (Workspace: none needed)
*
CALL DTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI,
$ SDIM, S, SEP, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
$ ICOND )
IF( ICOND.GT.0 )
$ INFO = N + ICOND
END IF
*
IF( WANTVS ) THEN
*
* Undo balancing
* (Workspace: need N)
*
CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS,
$ LDVS,
$ IERR )
END IF
*
IF( SCALEA ) THEN
*
* Undo scaling for the Schur form of A
*
CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
CALL DCOPY( N, A, LDA+1, WR, 1 )
IF( CSCALE.EQ.SMLNUM ) THEN
*
* If scaling back towards underflow, adjust WI if an
* offdiagonal element of a 2-by-2 block in the Schur form
* underflows.
*
IF( IEVAL.GT.0 ) THEN
I1 = IEVAL + 1
I2 = IHI - 1
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI,
$ MAX( ILO-1, 1 ), IERR )
ELSE IF( WANTST ) THEN
I1 = 1
I2 = N - 1
ELSE
I1 = ILO
I2 = IHI - 1
END IF
INXT = I1 - 1
DO 20 I = I1, I2
IF( I.LT.INXT )
$ GO TO 20
IF( WI( I ).EQ.ZERO ) THEN
INXT = I + 1
ELSE
IF( A( I+1, I ).EQ.ZERO ) THEN
WI( I ) = ZERO
WI( I+1 ) = ZERO
ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ.
$ ZERO ) THEN
WI( I ) = ZERO
WI( I+1 ) = ZERO
IF( I.GT.1 )
$ CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ),
$ 1 )
IF( N.GT.I+1 )
$ CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
$ A( I+1, I+2 ), LDA )
IF( WANTVS ) THEN
CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ),
$ 1 )
END IF
A( I, I+1 ) = A( I+1, I )
A( I+1, I ) = ZERO
END IF
INXT = I + 2
END IF
20 CONTINUE
END IF
*
* Undo scaling for the imaginary part of the eigenvalues
*
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1,
$ WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR )
END IF
*
IF( WANTST .AND. INFO.EQ.0 ) THEN
*
* Check if reordering successful
*
LASTSL = .TRUE.
LST2SL = .TRUE.
SDIM = 0
IP = 0
DO 30 I = 1, N
CURSL = SELECT( WR( I ), WI( I ) )
IF( WI( I ).EQ.ZERO ) THEN
IF( CURSL )
$ SDIM = SDIM + 1
IP = 0
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
ELSE
IF( IP.EQ.1 ) THEN
*
* Last eigenvalue of conjugate pair
*
CURSL = CURSL .OR. LASTSL
LASTSL = CURSL
IF( CURSL )
$ SDIM = SDIM + 2
IP = -1
IF( CURSL .AND. .NOT.LST2SL )
$ INFO = N + 2
ELSE
*
* First eigenvalue of conjugate pair
*
IP = 1
END IF
END IF
LST2SL = LASTSL
LASTSL = CURSL
30 CONTINUE
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGEES
*
END
*> \brief <b> DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEESX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeesx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeesx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeesx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
* WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
* IWORK, LIWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVS, SENSE, SORT
* INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
* DOUBLE PRECISION RCONDE, RCONDV
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
* $ WR( * )
* ..
* .. Function Arguments ..
* LOGICAL SELECT
* EXTERNAL SELECT
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEESX computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues, the real Schur form T, and, optionally, the matrix of
*> Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
*>
*> Optionally, it also orders the eigenvalues on the diagonal of the
*> real Schur form so that selected eigenvalues are at the top left;
*> computes a reciprocal condition number for the average of the
*> selected eigenvalues (RCONDE); and computes a reciprocal condition
*> number for the right invariant subspace corresponding to the
*> selected eigenvalues (RCONDV). The leading columns of Z form an
*> orthonormal basis for this invariant subspace.
*>
*> For further explanation of the reciprocal condition numbers RCONDE
*> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
*> these quantities are called s and sep respectively).
*>
*> A real matrix is in real Schur form if it is upper quasi-triangular
*> with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
*> the form
*> [ a b ]
*> [ c a ]
*>
*> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVS
*> \verbatim
*> JOBVS is CHARACTER*1
*> = 'N': Schur vectors are not computed;
*> = 'V': Schur vectors are computed.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*> SORT is CHARACTER*1
*> Specifies whether or not to order the eigenvalues on the
*> diagonal of the Schur form.
*> = 'N': Eigenvalues are not ordered;
*> = 'S': Eigenvalues are ordered (see SELECT).
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
*> SELECT must be declared EXTERNAL in the calling subroutine.
*> If SORT = 'S', SELECT is used to select eigenvalues to sort
*> to the top left of the Schur form.
*> If SORT = 'N', SELECT is not referenced.
*> An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
*> SELECT(WR(j),WI(j)) is true; i.e., if either one of a
*> complex conjugate pair of eigenvalues is selected, then both
*> are. Note that a selected complex eigenvalue may no longer
*> satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
*> ordering may change the value of complex eigenvalues
*> (especially if the eigenvalue is ill-conditioned); in this
*> case INFO may be set to N+3 (see INFO below).
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*> SENSE is CHARACTER*1
*> Determines which reciprocal condition numbers are computed.
*> = 'N': None are computed;
*> = 'E': Computed for average of selected eigenvalues only;
*> = 'V': Computed for selected right invariant subspace only;
*> = 'B': Computed for both.
*> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the N-by-N matrix A.
*> On exit, A is overwritten by its real Schur form T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*> SDIM is INTEGER
*> If SORT = 'N', SDIM = 0.
*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*> for which SELECT is true. (Complex conjugate
*> pairs for which SELECT is true for either
*> eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> WR and WI contain the real and imaginary parts, respectively,
*> of the computed eigenvalues, in the same order that they
*> appear on the diagonal of the output Schur form T. Complex
*> conjugate pairs of eigenvalues appear consecutively with the
*> eigenvalue having the positive imaginary part first.
*> \endverbatim
*>
*> \param[out] VS
*> \verbatim
*> VS is DOUBLE PRECISION array, dimension (LDVS,N)
*> If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
*> vectors.
*> If JOBVS = 'N', VS is not referenced.
*> \endverbatim
*>
*> \param[in] LDVS
*> \verbatim
*> LDVS is INTEGER
*> The leading dimension of the array VS. LDVS >= 1, and if
*> JOBVS = 'V', LDVS >= N.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*> RCONDE is DOUBLE PRECISION
*> If SENSE = 'E' or 'B', RCONDE contains the reciprocal
*> condition number for the average of the selected eigenvalues.
*> Not referenced if SENSE = 'N' or 'V'.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*> RCONDV is DOUBLE PRECISION
*> If SENSE = 'V' or 'B', RCONDV contains the reciprocal
*> condition number for the selected right invariant subspace.
*> Not referenced if SENSE = 'N' or 'E'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,3*N).
*> Also, if SENSE = 'E' or 'V' or 'B',
*> LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
*> selected eigenvalues computed by this routine. Note that
*> N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
*> returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
*> 'B' this may not be large enough.
*> For good performance, LWORK must generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates upper bounds on the optimal sizes of the
*> arrays WORK and IWORK, returns these values as the first
*> entries of the WORK and IWORK arrays, and no error messages
*> related to LWORK or LIWORK are issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
*> Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
*> only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
*> may not be large enough.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates upper bounds on the optimal sizes of
*> the arrays WORK and IWORK, returns these values as the first
*> entries of the WORK and IWORK arrays, and no error messages
*> related to LWORK or LIWORK are issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, and i is
*> <= N: the QR algorithm failed to compute all the
*> eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
*> contain those eigenvalues which have converged; if
*> JOBVS = 'V', VS contains the transformation which
*> reduces A to its partially converged Schur form.
*> = N+1: the eigenvalues could not be reordered because some
*> eigenvalues were too close to separate (the problem
*> is very ill-conditioned);
*> = N+2: after reordering, roundoff changed values of some
*> complex eigenvalues so that leading eigenvalues in
*> the Schur form no longer satisfy SELECT=.TRUE. This
*> could also be caused by underflow due to scaling.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geesx
*
* =====================================================================
SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
$ WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
$ IWORK, LIWORK, BWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVS, SENSE, SORT
INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
DOUBLE PRECISION RCONDE, RCONDV
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
$ WR( * )
* ..
* .. Function Arguments ..
LOGICAL SELECT
EXTERNAL SELECT
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTSB,
$ WANTSE, WANTSN, WANTST, WANTSV, WANTVS
INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
$ IHI, ILO, INXT, IP, ITAU, IWRK, LIWRK, LWRK,
$ MAXWRK, MINWRK
DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR,
$ DLACPY,
$ DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
WANTVS = LSAME( JOBVS, 'V' )
WANTST = LSAME( SORT, 'S' )
WANTSN = LSAME( SENSE, 'N' )
WANTSE = LSAME( SENSE, 'E' )
WANTSV = LSAME( SENSE, 'V' )
WANTSB = LSAME( SENSE, 'B' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTST ) .AND.
$ ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
$ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
INFO = -12
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "RWorkspace:" describe the
* minimal amount of real workspace needed at that point in the
* code, as well as the preferred amount for good performance.
* IWorkspace refers to integer workspace.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by DHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.
* If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
* depends on SDIM, which is computed by the routine DTRSEN later
* in the code.)
*
IF( INFO.EQ.0 ) THEN
LIWRK = 1
IF( N.EQ.0 ) THEN
MINWRK = 1
LWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
MINWRK = 3*N
*
CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS,
$ LDVS,
$ WORK, -1, IEVAL )
HSWORK = INT( WORK( 1 ) )
*
IF( .NOT.WANTVS ) THEN
MAXWRK = MAX( MAXWRK, N + HSWORK )
ELSE
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
MAXWRK = MAX( MAXWRK, N + HSWORK )
END IF
LWRK = MAXWRK
IF( .NOT.WANTSN )
$ LWRK = MAX( LWRK, N + ( N*N )/2 )
IF( WANTSV .OR. WANTSB )
$ LIWRK = ( N*N )/4
END IF
IWORK( 1 ) = LIWRK
WORK( 1 ) = LWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -16
ELSE IF( LIWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEESX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SDIM = 0
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
SCALEA = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
SCALEA = .TRUE.
CSCALE = SMLNUM
ELSE IF( ANRM.GT.BIGNUM ) THEN
SCALEA = .TRUE.
CSCALE = BIGNUM
END IF
IF( SCALEA )
$ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Permute the matrix to make it more nearly triangular
* (RWorkspace: need N)
*
IBAL = 1
CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (RWorkspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = N + IBAL
IWRK = N + ITAU
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVS ) THEN
*
* Copy Householder vectors to VS
*
CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS )
*
* Generate orthogonal matrix in VS
* (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ),
$ WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
END IF
*
SDIM = 0
*
* Perform QR iteration, accumulating Schur vectors in VS if desired
* (RWorkspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS,
$ WORK( IWRK ), LWORK-IWRK+1, IEVAL )
IF( IEVAL.GT.0 )
$ INFO = IEVAL
*
* Sort eigenvalues if desired
*
IF( WANTST .AND. INFO.EQ.0 ) THEN
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR )
END IF
DO 10 I = 1, N
BWORK( I ) = SELECT( WR( I ), WI( I ) )
10 CONTINUE
*
* Reorder eigenvalues, transform Schur vectors, and compute
* reciprocal condition numbers
* (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM)
* otherwise, need N )
* (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM)
* otherwise, need 0 )
*
CALL DTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, WR,
$ WI,
$ SDIM, RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
$ IWORK, LIWORK, ICOND )
IF( .NOT.WANTSN )
$ MAXWRK = MAX( MAXWRK, N+2*SDIM*( N-SDIM ) )
IF( ICOND.EQ.-15 ) THEN
*
* Not enough real workspace
*
INFO = -16
ELSE IF( ICOND.EQ.-17 ) THEN
*
* Not enough integer workspace
*
INFO = -18
ELSE IF( ICOND.GT.0 ) THEN
*
* DTRSEN failed to reorder or to restore standard Schur form
*
INFO = ICOND + N
END IF
END IF
*
IF( WANTVS ) THEN
*
* Undo balancing
* (RWorkspace: need N)
*
CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS,
$ LDVS,
$ IERR )
END IF
*
IF( SCALEA ) THEN
*
* Undo scaling for the Schur form of A
*
CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
CALL DCOPY( N, A, LDA+1, WR, 1 )
IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN
DUM( 1 ) = RCONDV
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1,
$ IERR )
RCONDV = DUM( 1 )
END IF
IF( CSCALE.EQ.SMLNUM ) THEN
*
* If scaling back towards underflow, adjust WI if an
* offdiagonal element of a 2-by-2 block in the Schur form
* underflows.
*
IF( IEVAL.GT.0 ) THEN
I1 = IEVAL + 1
I2 = IHI - 1
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
$ IERR )
ELSE IF( WANTST ) THEN
I1 = 1
I2 = N - 1
ELSE
I1 = ILO
I2 = IHI - 1
END IF
INXT = I1 - 1
DO 20 I = I1, I2
IF( I.LT.INXT )
$ GO TO 20
IF( WI( I ).EQ.ZERO ) THEN
INXT = I + 1
ELSE
IF( A( I+1, I ).EQ.ZERO ) THEN
WI( I ) = ZERO
WI( I+1 ) = ZERO
ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ.
$ ZERO ) THEN
WI( I ) = ZERO
WI( I+1 ) = ZERO
IF( I.GT.1 )
$ CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ),
$ 1 )
IF( N.GT.I+1 )
$ CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
$ A( I+1, I+2 ), LDA )
IF( WANTVS ) THEN
CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ),
$ 1 )
END IF
A( I, I+1 ) = A( I+1, I )
A( I+1, I ) = ZERO
END IF
INXT = I + 2
END IF
20 CONTINUE
END IF
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1,
$ WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR )
END IF
*
IF( WANTST .AND. INFO.EQ.0 ) THEN
*
* Check if reordering successful
*
LASTSL = .TRUE.
LST2SL = .TRUE.
SDIM = 0
IP = 0
DO 30 I = 1, N
CURSL = SELECT( WR( I ), WI( I ) )
IF( WI( I ).EQ.ZERO ) THEN
IF( CURSL )
$ SDIM = SDIM + 1
IP = 0
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
ELSE
IF( IP.EQ.1 ) THEN
*
* Last eigenvalue of conjugate pair
*
CURSL = CURSL .OR. LASTSL
LASTSL = CURSL
IF( CURSL )
$ SDIM = SDIM + 2
IP = -1
IF( CURSL .AND. .NOT.LST2SL )
$ INFO = N + 2
ELSE
*
* First eigenvalue of conjugate pair
*
IP = 1
END IF
END IF
LST2SL = LASTSL
LASTSL = CURSL
30 CONTINUE
END IF
*
WORK( 1 ) = MAXWRK
IF( WANTSV .OR. WANTSB ) THEN
IWORK( 1 ) = MAX( 1, SDIM*( N-SDIM ) )
ELSE
IWORK( 1 ) = 1
END IF
*
RETURN
*
* End of DGEESX
*
END
*> \brief <b> DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeev.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
* LDVR, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WI( * ), WORK( * ), WR( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEEV computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> The right eigenvector v(j) of A satisfies
*> A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*> u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate-transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': left eigenvectors of A are not computed;
*> = 'V': left eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': right eigenvectors of A are not computed;
*> = 'V': right eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N matrix A.
*> On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> WR and WI contain the real and imaginary parts,
*> respectively, of the computed eigenvalues. Complex
*> conjugate pairs of eigenvalues appear consecutively
*> with the eigenvalue having the positive imaginary part
*> first.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
*> after another in the columns of VL, in the same order
*> as their eigenvalues.
*> If JOBVL = 'N', VL is not referenced.
*> If the j-th eigenvalue is real, then u(j) = VL(:,j),
*> the j-th column of VL.
*> If the j-th and (j+1)-st eigenvalues form a complex
*> conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1; if
*> JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
*> after another in the columns of VR, in the same order
*> as their eigenvalues.
*> If JOBVR = 'N', VR is not referenced.
*> If the j-th eigenvalue is real, then v(j) = VR(:,j),
*> the j-th column of VR.
*> If the j-th and (j+1)-st eigenvalues form a complex
*> conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
*> v(j+1) = VR(:,j) - i*VR(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1; if
*> JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,3*N), and
*> if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good
*> performance, LWORK must generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, the QR algorithm failed to compute all the
*> eigenvalues, and no eigenvectors have been computed;
*> elements i+1:N of WR and WI contain eigenvalues which
*> have converged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*
* @precisions fortran d -> s
*
*> \ingroup geev
*
* =====================================================================
SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL,
$ VR,
$ LDVR, WORK, LWORK, INFO )
implicit none
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WI( * ), WORK( * ), WR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SCALEA, WANTVL, WANTVR
CHARACTER SIDE
INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
$ LWORK_TREVC, MAXWRK, MINWRK, NOUT
DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
$ SN
* ..
* .. Local Arrays ..
LOGICAL SELECT( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY,
$ DLARTG,
$ DLASCL, DORGHR, DROT, DSCAL, DTREVC3, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX, ILAENV
DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE,
$ DLAPY2,
$ DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
WANTVL = LSAME( JOBVL, 'V' )
WANTVR = LSAME( JOBVR, 'V' )
IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTVR ) .AND.
$ ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -9
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -11
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by DHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
IF( WANTVL ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL,
$ LDVL,
$ WORK, -1, INFO )
HSWORK = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA,
$ VL, LDVL, VR, LDVR, N, NOUT,
$ WORK, -1, IERR )
LWORK_TREVC = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE IF( WANTVR ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR,
$ LDVR,
$ WORK, -1, INFO )
HSWORK = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA,
$ VL, LDVL, VR, LDVR, N, NOUT,
$ WORK, -1, IERR )
LWORK_TREVC = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE
MINWRK = 3*N
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
$ LDVR,
$ WORK, -1, INFO )
HSWORK = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
SCALEA = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
SCALEA = .TRUE.
CSCALE = SMLNUM
ELSE IF( ANRM.GT.BIGNUM ) THEN
SCALEA = .TRUE.
CSCALE = BIGNUM
END IF
IF( SCALEA )
$ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Balance the matrix
* (Workspace: need N)
*
IBAL = 1
CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = IBAL + N
IWRK = ITAU + N
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVL ) THEN
*
* Want left eigenvectors
* Copy Householder vectors to VL
*
SIDE = 'L'
CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ),
$ WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL,
$ LDVL,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
IF( WANTVR ) THEN
*
* Want left and right eigenvectors
* Copy Schur vectors to VR
*
SIDE = 'B'
CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
* Want right eigenvectors
* Copy Householder vectors to VR
*
SIDE = 'R'
CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ),
$ WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR,
$ LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR,
$ LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
* If INFO .NE. 0 from DHSEQR, then quit
*
IF( INFO.NE.0 )
$ GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
* Compute left and/or right eigenvectors
* (Workspace: need 4*N, prefer N + N + 2*N*NB)
*
CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR,
$ LDVR,
$ N, NOUT, WORK( IWRK ), LWORK-IWRK+1, IERR )
END IF
*
IF( WANTVL ) THEN
*
* Undo balancing of left eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL,
$ LDVL,
$ IERR )
*
* Normalize left eigenvectors and make largest component real
*
DO 20 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
$ DNRM2( N, VL( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
DO 10 K = 1, N
WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2
10 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
VL( K, I+1 ) = ZERO
END IF
20 CONTINUE
END IF
*
IF( WANTVR ) THEN
*
* Undo balancing of right eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR,
$ LDVR,
$ IERR )
*
* Normalize right eigenvectors and make largest component real
*
DO 40 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
$ DNRM2( N, VR( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
DO 30 K = 1, N
WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2
30 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
VR( K, I+1 ) = ZERO
END IF
40 CONTINUE
END IF
*
* Undo scaling if necessary
*
50 CONTINUE
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
$ WR( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
$ WI( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
IF( INFO.GT.0 ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
$ IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGEEV
*
END
*> \brief <b> DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
* VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
* RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER BALANC, JOBVL, JOBVR, SENSE
* INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
* DOUBLE PRECISION ABNRM
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
* $ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WI( * ), WORK( * ), WR( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> Optionally also, it computes a balancing transformation to improve
*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
*> (RCONDE), and reciprocal condition numbers for the right
*> eigenvectors (RCONDV).
*>
*> The right eigenvector v(j) of A satisfies
*> A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*> u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate-transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*>
*> Balancing a matrix means permuting the rows and columns to make it
*> more nearly upper triangular, and applying a diagonal similarity
*> transformation D * A * D**(-1), where D is a diagonal matrix, to
*> make its rows and columns closer in norm and the condition numbers
*> of its eigenvalues and eigenvectors smaller. The computed
*> reciprocal condition numbers correspond to the balanced matrix.
*> Permuting rows and columns will not change the condition numbers
*> (in exact arithmetic) but diagonal scaling will. For further
*> explanation of balancing, see section 4.10.2 of the LAPACK
*> Users' Guide.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] BALANC
*> \verbatim
*> BALANC is CHARACTER*1
*> Indicates how the input matrix should be diagonally scaled
*> and/or permuted to improve the conditioning of its
*> eigenvalues.
*> = 'N': Do not diagonally scale or permute;
*> = 'P': Perform permutations to make the matrix more nearly
*> upper triangular. Do not diagonally scale;
*> = 'S': Diagonally scale the matrix, i.e. replace A by
*> D*A*D**(-1), where D is a diagonal matrix chosen
*> to make the rows and columns of A more equal in
*> norm. Do not permute;
*> = 'B': Both diagonally scale and permute A.
*>
*> Computed reciprocal condition numbers will be for the matrix
*> after balancing and/or permuting. Permuting does not change
*> condition numbers (in exact arithmetic), but balancing does.
*> \endverbatim
*>
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': left eigenvectors of A are not computed;
*> = 'V': left eigenvectors of A are computed.
*> If SENSE = 'E' or 'B', JOBVL must = 'V'.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': right eigenvectors of A are not computed;
*> = 'V': right eigenvectors of A are computed.
*> If SENSE = 'E' or 'B', JOBVR must = 'V'.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*> SENSE is CHARACTER*1
*> Determines which reciprocal condition numbers are computed.
*> = 'N': None are computed;
*> = 'E': Computed for eigenvalues only;
*> = 'V': Computed for right eigenvectors only;
*> = 'B': Computed for eigenvalues and right eigenvectors.
*>
*> If SENSE = 'E' or 'B', both left and right eigenvectors
*> must also be computed (JOBVL = 'V' and JOBVR = 'V').
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N matrix A.
*> On exit, A has been overwritten. If JOBVL = 'V' or
*> JOBVR = 'V', A contains the real Schur form of the balanced
*> version of the input matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> WR and WI contain the real and imaginary parts,
*> respectively, of the computed eigenvalues. Complex
*> conjugate pairs of eigenvalues will appear consecutively
*> with the eigenvalue having the positive imaginary part
*> first.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
*> after another in the columns of VL, in the same order
*> as their eigenvalues.
*> If JOBVL = 'N', VL is not referenced.
*> If the j-th eigenvalue is real, then u(j) = VL(:,j),
*> the j-th column of VL.
*> If the j-th and (j+1)-st eigenvalues form a complex
*> conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1; if
*> JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
*> after another in the columns of VR, in the same order
*> as their eigenvalues.
*> If JOBVR = 'N', VR is not referenced.
*> If the j-th eigenvalue is real, then v(j) = VR(:,j),
*> the j-th column of VR.
*> If the j-th and (j+1)-st eigenvalues form a complex
*> conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
*> v(j+1) = VR(:,j) - i*VR(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1, and if
*> JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI are integer values determined when A was
*> balanced. The balanced A(i,j) = 0 if I > J and
*> J = 1,...,ILO-1 or I = IHI+1,...,N.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and scaling factors applied
*> when balancing A. If P(j) is the index of the row and column
*> interchanged with row and column j, and D(j) is the scaling
*> factor applied to row and column j, then
*> SCALE(J) = P(J), for J = 1,...,ILO-1
*> = D(J), for J = ILO,...,IHI
*> = P(J) for J = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] ABNRM
*> \verbatim
*> ABNRM is DOUBLE PRECISION
*> The one-norm of the balanced matrix (the maximum
*> of the sum of absolute values of elements of any column).
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*> RCONDE is DOUBLE PRECISION array, dimension (N)
*> RCONDE(j) is the reciprocal condition number of the j-th
*> eigenvalue.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*> RCONDV is DOUBLE PRECISION array, dimension (N)
*> RCONDV(j) is the reciprocal condition number of the j-th
*> right eigenvector.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. If SENSE = 'N' or 'E',
*> LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
*> LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6).
*> For good performance, LWORK must generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (2*N-2)
*> If SENSE = 'N' or 'E', not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, the QR algorithm failed to compute all the
*> eigenvalues, and no eigenvectors or condition numbers
*> have been computed; elements 1:ILO-1 and i+1:N of WR
*> and WI contain eigenvalues which have converged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*
* @precisions fortran d -> s
*
*> \ingroup geevx
*
* =====================================================================
SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR,
$ WI,
$ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
$ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
implicit none
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
DOUBLE PRECISION ABNRM
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
$ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
$ WI( * ), WORK( * ), WR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
$ WNTSNN, WNTSNV
CHARACTER JOB, SIDE
INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K,
$ LWORK_TREVC, MAXWRK, MINWRK, NOUT
DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
$ SN
* ..
* .. Local Arrays ..
LOGICAL SELECT( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY,
$ DLARTG,
$ DLASCL, DORGHR, DROT, DSCAL, DTREVC3, DTRSNA,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX, ILAENV
DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE,
$ DLAPY2,
$ DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
WANTVL = LSAME( JOBVL, 'V' )
WANTVR = LSAME( JOBVR, 'V' )
WNTSNN = LSAME( SENSE, 'N' )
WNTSNE = LSAME( SENSE, 'E' )
WNTSNV = LSAME( SENSE, 'V' )
WNTSNB = LSAME( SENSE, 'B' )
IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' )
$ .OR.
$ LSAME( BALANC, 'P' ) .OR.
$ LSAME( BALANC, 'B' ) ) )
$ THEN
INFO = -1
ELSE IF( ( .NOT.WANTVL ) .AND.
$ ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -2
ELSE IF( ( .NOT.WANTVR ) .AND.
$ ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -3
ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
$ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
$ WANTVR ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -13
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by DHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
*
IF( WANTVL ) THEN
CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA,
$ VL, LDVL, VR, LDVR,
$ N, NOUT, WORK, -1, IERR )
LWORK_TREVC = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL,
$ LDVL,
$ WORK, -1, INFO )
ELSE IF( WANTVR ) THEN
CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA,
$ VL, LDVL, VR, LDVR,
$ N, NOUT, WORK, -1, IERR )
LWORK_TREVC = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR,
$ LDVR,
$ WORK, -1, INFO )
ELSE
IF( WNTSNN ) THEN
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
$ LDVR, WORK, -1, INFO )
ELSE
CALL DHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
$ LDVR, WORK, -1, INFO )
END IF
END IF
HSWORK = INT( WORK(1) )
*
IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
MINWRK = 2*N
IF( .NOT.WNTSNN )
$ MINWRK = MAX( MINWRK, N*N+6*N )
MAXWRK = MAX( MAXWRK, HSWORK )
IF( .NOT.WNTSNN )
$ MAXWRK = MAX( MAXWRK, N*N + 6*N )
ELSE
MINWRK = 3*N
IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
$ MINWRK = MAX( MINWRK, N*N + 6*N )
MAXWRK = MAX( MAXWRK, HSWORK )
MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR',
$ ' ', N, 1, N, -1 ) )
IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
$ MAXWRK = MAX( MAXWRK, N*N + 6*N )
MAXWRK = MAX( MAXWRK, 3*N )
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -21
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ICOND = 0
ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
SCALEA = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
SCALEA = .TRUE.
CSCALE = SMLNUM
ELSE IF( ANRM.GT.BIGNUM ) THEN
SCALEA = .TRUE.
CSCALE = BIGNUM
END IF
IF( SCALEA )
$ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Balance the matrix and compute ABNRM
*
CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
ABNRM = DLANGE( '1', N, N, A, LDA, DUM )
IF( SCALEA ) THEN
DUM( 1 ) = ABNRM
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
ABNRM = DUM( 1 )
END IF
*
* Reduce to upper Hessenberg form
* (Workspace: need 2*N, prefer N+N*NB)
*
ITAU = 1
IWRK = ITAU + N
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVL ) THEN
*
* Want left eigenvectors
* Copy Householder vectors to VL
*
SIDE = 'L'
CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
* Generate orthogonal matrix in VL
* (Workspace: need 2*N-1, prefer N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ),
$ WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need 1, prefer HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL,
$ LDVL,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
IF( WANTVR ) THEN
*
* Want left and right eigenvectors
* Copy Schur vectors to VR
*
SIDE = 'B'
CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
* Want right eigenvectors
* Copy Householder vectors to VR
*
SIDE = 'R'
CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
* Generate orthogonal matrix in VR
* (Workspace: need 2*N-1, prefer N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ),
$ WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need 1, prefer HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR,
$ LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
* Compute eigenvalues only
* If condition numbers desired, compute Schur form
*
IF( WNTSNN ) THEN
JOB = 'E'
ELSE
JOB = 'S'
END IF
*
* (Workspace: need 1, prefer HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR,
$ LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
* If INFO .NE. 0 from DHSEQR, then quit
*
IF( INFO.NE.0 )
$ GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
* Compute left and/or right eigenvectors
* (Workspace: need 3*N, prefer N + 2*N*NB)
*
CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR,
$ LDVR,
$ N, NOUT, WORK( IWRK ), LWORK-IWRK+1, IERR )
END IF
*
* Compute condition numbers if desired
* (Workspace: need N*N+6*N unless SENSE = 'E')
*
IF( .NOT.WNTSNN ) THEN
CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR,
$ LDVR,
$ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
$ ICOND )
END IF
*
IF( WANTVL ) THEN
*
* Undo balancing of left eigenvectors
*
CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
$ IERR )
*
* Normalize left eigenvectors and make largest component real
*
DO 20 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
$ DNRM2( N, VL( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
DO 10 K = 1, N
WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
10 CONTINUE
K = IDAMAX( N, WORK, 1 )
CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
VL( K, I+1 ) = ZERO
END IF
20 CONTINUE
END IF
*
IF( WANTVR ) THEN
*
* Undo balancing of right eigenvectors
*
CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
$ IERR )
*
* Normalize right eigenvectors and make largest component real
*
DO 40 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
$ DNRM2( N, VR( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
DO 30 K = 1, N
WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
30 CONTINUE
K = IDAMAX( N, WORK, 1 )
CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
VR( K, I+1 ) = ZERO
END IF
40 CONTINUE
END IF
*
* Undo scaling if necessary
*
50 CONTINUE
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
$ WR( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
$ WI( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
IF( INFO.EQ.0 ) THEN
IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
$ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
$ IERR )
ELSE
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
$ IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGEEVX
*
END
*> \brief <b> DGEGS computes the eigenvalues, real Schur form, and, optionally, the left and/or right Schur vectors of a real matrix pair (A,B)</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
* ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
* $ VSR( LDVSR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGES.
*>
*> DGEGS computes the eigenvalues, real Schur form, and, optionally,
*> left and or/right Schur vectors of a real matrix pair (A,B).
*> Given two square matrices A and B, the generalized real Schur
*> factorization has the form
*>
*> A = Q*S*Z**T, B = Q*T*Z**T
*>
*> where Q and Z are orthogonal matrices, T is upper triangular, and S
*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
*> of eigenvalues of (A,B). The columns of Q are the left Schur vectors
*> and the columns of Z are the right Schur vectors.
*>
*> If only the eigenvalues of (A,B) are needed, the driver routine
*> DGEGV should be used instead. See DGEGV for a description of the
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
*> (GNEP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors (returned in VSL).
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors (returned in VSR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the matrix A.
*> On exit, the upper quasi-triangular matrix S from the
*> generalized real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the matrix B.
*> On exit, the upper triangular matrix T from the generalized
*> real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue
*> of GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> The imaginary parts of each scalar alpha defining an
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
*> eigenvalue is real; if positive, then the j-th and (j+1)-st
*> eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> The scalars beta that define the eigenvalues of GNEP.
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
*> pair (A,B), in one of the forms lambda = alpha/beta or
*> mu = beta/alpha. Since either lambda or mu may overflow,
*> they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >=1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,4*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
*> The optimal LWORK is 2*N + N*(NB+1).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. (A,B) are not in Schur
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*> be correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from DGGBAL
*> =N+2: error return from DGEQRF
*> =N+3: error return from DORMQR
*> =N+4: error return from DORGQR
*> =N+5: error return from DGGHRD
*> =N+6: error return from DHGEQZ (other than failed
*> iteration)
*> =N+7: error return from DGGBAK (computing VSL)
*> =N+8: error return from DGGBAK (computing VSR)
*> =N+9: error return from DLASCL (various places)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEeigen
*
* =====================================================================
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
$ LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SAFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
* Test the input arguments
*
LWKMIN = MAX( 4*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -14
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = 2*N + N*( NB+1 )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEGS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
SAFMIN = DLAMCH( 'S' )
SMLNUM = N*SAFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (2*N words -- "work..." not actually used)
* left_permutation, right_permutation, work...
*
ILEFT = 1
IRIGHT = N + 1
IWORK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 10
END IF
*
* Reduce B to triangular form, and initialize VSL and/or VSR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWORK
IWORK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 10
END IF
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 10
END IF
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 10
END IF
END IF
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 10
END IF
*
* Perform QZ algorithm, computing Schur vectors if desired
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 10
END IF
*
* Apply permutation to VSL and VSR
*
IF( ILVSL ) THEN
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 10
END IF
END IF
IF( ILVSR ) THEN
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 10
END IF
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
10 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEGS
*
END
*> \brief <b> DGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGEV.
*>
*> DGEGV computes the eigenvalues and, optionally, the left and/or right
*> eigenvectors of a real matrix pair (A,B).
*> Given two square matrices A and B,
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
*> that
*>
*> A*x = lambda*B*x.
*>
*> An alternate form is to find the eigenvalues mu and corresponding
*> eigenvectors y such that
*>
*> mu*A*y = B*y.
*>
*> These two forms are equivalent with mu = 1/lambda and x = y if
*> neither lambda nor mu is zero. In order to deal with the case that
*> lambda or mu is zero or small, two values alpha and beta are returned
*> for each eigenvalue, such that lambda = alpha/beta and
*> mu = beta/alpha.
*>
*> The vectors x and y in the above equations are right eigenvectors of
*> the matrix pair (A,B). Vectors u and v satisfying
*>
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
*>
*> are left eigenvectors of (A,B).
*>
*> Note: this routine performs "full balancing" on A and B
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors (returned
*> in VL).
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors (returned
*> in VR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the matrix A.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
*> contains the real Schur form of A from the generalized Schur
*> factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only the diagonal
*> blocks from the Schur form will be correct. See DGGHRD and
*> DHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the matrix B.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
*> upper triangular matrix obtained from B in the generalized
*> Schur factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only those elements of
*> B corresponding to the diagonal blocks from the Schur form of
*> A will be correct. See DGGHRD and DHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue of
*> GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> The imaginary parts of each scalar alpha defining an
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
*> eigenvalue is real; if positive, then the j-th and
*> (j+1)-st eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> The scalars beta that define the eigenvalues of GNEP.
*>
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
*> pair (A,B), in one of the forms lambda = alpha/beta or
*> mu = beta/alpha. Since either lambda or mu may overflow,
*> they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
*> in the columns of VL, in the same order as their eigenvalues.
*> If the j-th eigenvalue is real, then u(j) = VL(:,j).
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
*> pair, then
*> u(j) = VL(:,j) + i*VL(:,j+1)
*> and
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
*>
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
*> in the columns of VR, in the same order as their eigenvalues.
*> If the j-th eigenvalue is real, then x(j) = VR(:,j).
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
*> pair, then
*> x(j) = VR(:,j) + i*VR(:,j+1)
*> and
*> x(j+1) = VR(:,j) - i*VR(:,j+1).
*>
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvalues
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,8*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
*> The optimal LWORK is:
*> 2*N + MAX( 6*N, N*(NB+1) ).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*> should be correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from DGGBAL
*> =N+2: error return from DGEQRF
*> =N+3: error return from DORMQR
*> =N+4: error return from DORGQR
*> =N+5: error return from DGGHRD
*> =N+6: error return from DHGEQZ (other than failed
*> iteration)
*> =N+7: error return from DTGEVC
*> =N+8: error return from DGGBAK (computing VL)
*> =N+9: error return from DGGBAK (computing VR)
*> =N+10: error return from DLASCL (various calls)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEeigen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Balancing
*> ---------
*>
*> This driver calls DGGBAL to both permute and scale rows and columns
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
*> and PL*B*R will be upper triangular except for the diagonal blocks
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
*> possible. The diagonal scaling matrices DL and DR are chosen so
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
*> one (except for the elements that start out zero.)
*>
*> After the eigenvalues and eigenvectors of the balanced matrices
*> have been computed, DGGBAK transforms the eigenvectors back to what
*> they would have been (in perfect arithmetic) if they had not been
*> balanced.
*>
*> Contents of A and B on Exit
*> -------- -- - --- - -- ----
*>
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
*> both), then on exit the arrays A and B will contain the real Schur
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
*> are computed, then only the diagonal blocks will be correct.
*>
*> [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
*> by Golub & van Loan, pub. by Johns Hopkins U. Press.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
$ SALFAI, SALFAR, SBETA, SCALE, TEMP
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
* Test the input arguments
*
LWKMIN = MAX( 8*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -14
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEGV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
SAFMIN = DLAMCH( 'S' )
SAFMIN = SAFMIN + SAFMIN
SAFMAX = ONE / SAFMIN
ONEPLS = ONE + ( 4*EPS )
*
* Scale A
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ANRM1 = ANRM
ANRM2 = ONE
IF( ANRM.LT.ONE ) THEN
IF( SAFMAX*ANRM.LT.ONE ) THEN
ANRM1 = SAFMIN
ANRM2 = SAFMAX*ANRM
END IF
END IF
*
IF( ANRM.GT.ZERO ) THEN
CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Scale B
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
BNRM1 = BNRM
BNRM2 = ONE
IF( BNRM.LT.ONE ) THEN
IF( SAFMAX*BNRM.LT.ONE ) THEN
BNRM1 = SAFMIN
BNRM2 = SAFMAX*BNRM
END IF
END IF
*
IF( BNRM.GT.ZERO ) THEN
CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (8*N words -- "work" requires 6*N words)
* left_permutation, right_permutation, work...
*
ILEFT = 1
IRIGHT = N + 1
IWORK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 120
END IF
*
* Reduce B to triangular form, and initialize VL and/or VR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = IWORK
IWORK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 120
END IF
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 120
END IF
*
IF( ILVL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 120
END IF
END IF
*
IF( ILVR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IINFO )
ELSE
CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
END IF
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 120
END IF
*
* Perform QZ algorithm
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 120
END IF
*
IF( ILV ) THEN
*
* Compute Eigenvectors (DTGEVC requires 6*N words of workspace)
*
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 120
END IF
*
* Undo balancing on VL and VR, rescale
*
IF( ILVL ) THEN
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VL, LDVL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 120
END IF
DO 50 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 50
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
10 CONTINUE
ELSE
DO 20 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
$ ABS( VL( JR, JC+1 ) ) )
20 CONTINUE
END IF
IF( TEMP.LT.SAFMIN )
$ GO TO 50
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 30 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
30 CONTINUE
ELSE
DO 40 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
40 CONTINUE
END IF
50 CONTINUE
END IF
IF( ILVR ) THEN
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VR, LDVR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
GO TO 120
END IF
DO 100 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 100
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 60 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
60 CONTINUE
ELSE
DO 70 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
$ ABS( VR( JR, JC+1 ) ) )
70 CONTINUE
END IF
IF( TEMP.LT.SAFMIN )
$ GO TO 100
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 80 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
80 CONTINUE
ELSE
DO 90 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
90 CONTINUE
END IF
100 CONTINUE
END IF
*
* End of eigenvector calculation
*
END IF
*
* Undo scaling in alpha, beta
*
* Note: this does not give the alpha and beta for the unscaled
* problem.
*
* Un-scaling is limited to avoid underflow in alpha and beta
* if they are significant.
*
DO 110 JC = 1, N
ABSAR = ABS( ALPHAR( JC ) )
ABSAI = ABS( ALPHAI( JC ) )
ABSB = ABS( BETA( JC ) )
SALFAR = ANRM*ALPHAR( JC )
SALFAI = ANRM*ALPHAI( JC )
SBETA = BNRM*BETA( JC )
ILIMIT = .FALSE.
SCALE = ONE
*
* Check for significant underflow in ALPHAI
*
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
*
ELSE IF( SALFAI.EQ.ZERO ) THEN
*
* If insignificant underflow in ALPHAI, then make the
* conjugate eigenvalue real.
*
IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
ALPHAI( JC-1 ) = ZERO
ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
ALPHAI( JC+1 ) = ZERO
END IF
END IF
*
* Check for significant underflow in ALPHAR
*
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
END IF
*
* Check for significant underflow in BETA
*
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
$ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
END IF
*
* Check for possible overflow when limiting scaling
*
IF( ILIMIT ) THEN
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
$ ABS( SBETA ) )
IF( TEMP.GT.ONE )
$ SCALE = SCALE / TEMP
IF( SCALE.LT.ONE )
$ ILIMIT = .FALSE.
END IF
*
* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
*
IF( ILIMIT ) THEN
SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
SBETA = ( SCALE*BETA( JC ) )*BNRM
END IF
ALPHAR( JC ) = SALFAR
ALPHAI( JC ) = SALFAI
BETA( JC ) = SBETA
110 CONTINUE
*
120 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEGV
*
END
*> \brief \b DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEHD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgehd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgehd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgehd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
*> an orthogonal similarity transformation: Q**T * A * Q = H .
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> It is assumed that A is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*> set by a previous call to DGEBAL; otherwise they should be
*> set to 1 and N respectively. See Further Details.
*> 1 <= ILO <= IHI <= max(1,N).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the n by n general matrix to be reduced.
*> On exit, the upper triangle and the first subdiagonal of A
*> are overwritten with the upper Hessenberg matrix H, and the
*> elements below the first subdiagonal, with the array TAU,
*> represent the orthogonal matrix Q as a product of elementary
*> reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gehd2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of (ihi-ilo) elementary
*> reflectors
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
*> exit in A(i+2:ihi,i), and tau in TAU(i).
*>
*> The contents of A are illustrated by the following example, with
*> n = 7, ilo = 2 and ihi = 6:
*>
*> on entry, on exit,
*>
*> ( a a a a a a a ) ( a a h h h h a )
*> ( a a a a a a ) ( a h h h h a )
*> ( a a a a a a ) ( h h h h h h )
*> ( a a a a a a ) ( v2 h h h h h )
*> ( a a a a a a ) ( v2 v3 h h h h )
*> ( a a a a a a ) ( v2 v3 v4 h h h )
*> ( a ) ( a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEHD2', -INFO )
RETURN
END IF
*
DO 10 I = ILO, IHI - 1
*
* Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
*
CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAU( I ) )
*
* Apply H(i) to A(1:ihi,i+1:ihi) from the right
*
CALL DLARF1F( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
$ A( 1, I+1 ), LDA, WORK )
*
* Apply H(i) to A(i+1:ihi,i+1:n) from the left
*
CALL DLARF1F( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
$ A( I+1, I+1 ), LDA, WORK )
*
10 CONTINUE
*
RETURN
*
* End of DGEHD2
*
END
*> \brief \b DGEHRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEHRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgehrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgehrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgehrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEHRD reduces a real general matrix A to upper Hessenberg form H by
*> an orthogonal similarity transformation: Q**T * A * Q = H .
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> It is assumed that A is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*> set by a previous call to DGEBAL; otherwise they should be
*> set to 1 and N respectively. See Further Details.
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N general matrix to be reduced.
*> On exit, the upper triangle and the first subdiagonal of A
*> are overwritten with the upper Hessenberg matrix H, and the
*> elements below the first subdiagonal, with the array TAU,
*> represent the orthogonal matrix Q as a product of elementary
*> reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
*> zero.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= max(1,N).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gehrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of (ihi-ilo) elementary
*> reflectors
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
*> exit in A(i+2:ihi,i), and tau in TAU(i).
*>
*> The contents of A are illustrated by the following example, with
*> n = 7, ilo = 2 and ihi = 6:
*>
*> on entry, on exit,
*>
*> ( a a a a a a a ) ( a a h h h h a )
*> ( a a a a a a ) ( a h h h h a )
*> ( a a a a a a ) ( h h h h h h )
*> ( a a a a a a ) ( v2 h h h h h )
*> ( a a a a a a ) ( v2 v3 h h h h )
*> ( a a a a a a ) ( v2 v3 v4 h h h )
*> ( a ) ( a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*>
*> This file is a slight modification of LAPACK-3.0's DGEHRD
*> subroutine incorporating improvements proposed by Quintana-Orti and
*> Van de Geijn (2006). (See DLAHR2.)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0,
$ ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB,
$ NBMIN, NH, NX
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB,
$ DTRMM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
*
NH = IHI - ILO + 1
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
IF( NH.LE.1 ) THEN
LWKOPT = 1
ELSE
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI,
$ -1 ) )
LWKOPT = N*NB + TSIZE
ENDIF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEHRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
*
DO 10 I = 1, ILO - 1
TAU( I ) = ZERO
10 CONTINUE
DO 20 I = MAX( 1, IHI ), N - 1
TAU( I ) = ZERO
20 CONTINUE
*
* Quick return if possible
*
IF( NH.LE.1 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Determine the block size
*
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
NBMIN = 2
IF( NB.GT.1 .AND. NB.LT.NH ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code)
*
NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
IF( NX.LT.NH ) THEN
*
* Determine if workspace is large enough for blocked code
*
IF( LWORK.LT.LWKOPT ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code
*
NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI,
$ -1 ) )
IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN
NB = (LWORK-TSIZE) / N
ELSE
NB = 1
END IF
END IF
END IF
END IF
LDWORK = N
*
IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
*
* Use unblocked code below
*
I = ILO
*
ELSE
*
* Use blocked code
*
IWT = 1 + N*NB
DO 40 I = ILO, IHI - 1 - NX, NB
IB = MIN( NB, IHI-I )
*
* Reduce columns i:i+ib-1 to Hessenberg form, returning the
* matrices V and T of the block reflector H = I - V*T*V**T
* which performs the reduction, and also the matrix Y = A*V*T
*
CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ),
$ WORK( IWT ), LDT, WORK, LDWORK )
*
* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
* right, computing A := A - Y * V**T. V(i+ib,ib-1) must be set
* to 1
*
EI = A( I+IB, I+IB-1 )
A( I+IB, I+IB-1 ) = ONE
CALL DGEMM( 'No transpose', 'Transpose',
$ IHI, IHI-I-IB+1,
$ IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
$ A( 1, I+IB ), LDA )
A( I+IB, I+IB-1 ) = EI
*
* Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
* right
*
CALL DTRMM( 'Right', 'Lower', 'Transpose',
$ 'Unit', I, IB-1,
$ ONE, A( I+1, I ), LDA, WORK, LDWORK )
DO 30 J = 0, IB-2
CALL DAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
$ A( 1, I+J+1 ), 1 )
30 CONTINUE
*
* Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
* left
*
CALL DLARFB( 'Left', 'Transpose', 'Forward',
$ 'Columnwise',
$ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA,
$ WORK( IWT ), LDT, A( I+1, I+IB ), LDA,
$ WORK, LDWORK )
40 CONTINUE
END IF
*
* Use unblocked code to reduce the rest of the matrix
*
CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEHRD
*
END
*> \brief \b DGEJSV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEJSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgejsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgejsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgejsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
* M, N, A, LDA, SVA, U, LDU, V, LDV,
* WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* IMPLICIT NONE
* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
* $ WORK( LWORK )
* INTEGER IWORK( * )
* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEJSV computes the singular value decomposition (SVD) of a real M-by-N
*> matrix [A], where M >= N. The SVD of [A] is written as
*>
*> [A] = [U] * [SIGMA] * [V]^t,
*>
*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
*> diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
*> [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
*> the singular values of [A]. The columns of [U] and [V] are the left and
*> the right singular vectors of [A], respectively. The matrices [U] and [V]
*> are computed and stored in the arrays U and V, respectively. The diagonal
*> of [SIGMA] is computed and stored in the array SVA.
*> DGEJSV can sometimes compute tiny singular values and their singular vectors much
*> more accurately than other SVD routines, see below under Further Details.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBA
*> \verbatim
*> JOBA is CHARACTER*1
*> Specifies the level of accuracy:
*> = 'C': This option works well (high relative accuracy) if A = B * D,
*> with well-conditioned B and arbitrary diagonal matrix D.
*> The accuracy cannot be spoiled by COLUMN scaling. The
*> accuracy of the computed output depends on the condition of
*> B, and the procedure aims at the best theoretical accuracy.
*> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
*> bounded by f(M,N)*epsilon* cond(B), independent of D.
*> The input matrix is preprocessed with the QRF with column
*> pivoting. This initial preprocessing and preconditioning by
*> a rank revealing QR factorization is common for all values of
*> JOBA. Additional actions are specified as follows:
*> = 'E': Computation as with 'C' with an additional estimate of the
*> condition number of B. It provides a realistic error bound.
*> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
*> D1, D2, and well-conditioned matrix C, this option gives
*> higher accuracy than the 'C' option. If the structure of the
*> input matrix is not known, and relative accuracy is
*> desirable, then this option is advisable. The input matrix A
*> is preprocessed with QR factorization with FULL (row and
*> column) pivoting.
*> = 'G': Computation as with 'F' with an additional estimate of the
*> condition number of B, where A=D*B. If A has heavily weighted
*> rows, then using this condition number gives too pessimistic
*> error bound.
*> = 'A': Small singular values are the noise and the matrix is treated
*> as numerically rank deficient. The error in the computed
*> singular values is bounded by f(m,n)*epsilon*||A||.
*> The computed SVD A = U * S * V^t restores A up to
*> f(m,n)*epsilon*||A||.
*> This gives the procedure the licence to discard (set to zero)
*> all singular values below N*epsilon*||A||.
*> = 'R': Similar as in 'A'. Rank revealing property of the initial
*> QR factorization is used do reveal (using triangular factor)
*> a gap sigma_{r+1} < epsilon * sigma_r in which case the
*> numerical RANK is declared to be r. The SVD is computed with
*> absolute error bounds, but more accurately than with 'A'.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> Specifies whether to compute the columns of U:
*> = 'U': N columns of U are returned in the array U.
*> = 'F': full set of M left sing. vectors is returned in the array U.
*> = 'W': U may be used as workspace of length M*N. See the description
*> of U.
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether to compute the matrix V:
*> = 'V': N columns of V are returned in the array V; Jacobi rotations
*> are not explicitly accumulated.
*> = 'J': N columns of V are returned in the array V, but they are
*> computed as the product of Jacobi rotations. This option is
*> allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
*> = 'W': V may be used as workspace of length N*N. See the description
*> of V.
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBR
*> \verbatim
*> JOBR is CHARACTER*1
*> Specifies the RANGE for the singular values. Issues the licence to
*> set to zero small positive singular values if they are outside
*> specified range. If A .NE. 0 is scaled so that the largest singular
*> value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
*> the licence to kill columns of A whose norm in c*A is less than
*> DSQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN,
*> where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
*> = 'N': Do not kill small columns of c*A. This option assumes that
*> BLAS and QR factorizations and triangular solvers are
*> implemented to work in that range. If the condition of A
*> is greater than BIG, use DGESVJ.
*> = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]
*> (roughly, as described above). This option is recommended.
*> ~~~~~~~~~~~~~~~~~~~~~~~~~~~
*> For computing the singular values in the FULL range [SFMIN,BIG]
*> use DGESVJ.
*> \endverbatim
*>
*> \param[in] JOBT
*> \verbatim
*> JOBT is CHARACTER*1
*> If the matrix is square then the procedure may determine to use
*> transposed A if A^t seems to be better with respect to convergence.
*> If the matrix is not square, JOBT is ignored. This is subject to
*> changes in the future.
*> The decision is based on two values of entropy over the adjoint
*> orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
*> = 'T': transpose if entropy test indicates possibly faster
*> convergence of Jacobi process if A^t is taken as input. If A is
*> replaced with A^t, then the row pivoting is included automatically.
*> = 'N': do not speculate.
*> This option can be used to compute only the singular values, or the
*> full SVD (U, SIGMA and V). For only one set of singular vectors
*> (U or V), the caller should provide both U and V, as one of the
*> matrices is used as workspace if the matrix A is transposed.
*> The implementer can easily remove this constraint and make the
*> code more complicated. See the descriptions of U and V.
*> \endverbatim
*>
*> \param[in] JOBP
*> \verbatim
*> JOBP is CHARACTER*1
*> Issues the licence to introduce structured perturbations to drown
*> denormalized numbers. This licence should be active if the
*> denormals are poorly implemented, causing slow computation,
*> especially in cases of fast convergence (!). For details see [1,2].
*> For the sake of simplicity, this perturbations are included only
*> when the full SVD or only the singular values are requested. The
*> implementer/user can easily add the perturbation for the cases of
*> computing one set of singular vectors.
*> = 'P': introduce perturbation
*> = 'N': do not perturb
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A. M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] SVA
*> \verbatim
*> SVA is DOUBLE PRECISION array, dimension (N)
*> On exit,
*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
*> computation SVA contains Euclidean column norms of the
*> iterated matrices in the array A.
*> - For WORK(1) .NE. WORK(2): The singular values of A are
*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
*> sigma_max(A) overflows or if small singular values have been
*> saved from underflow by scaling the input matrix A.
*> - If JOBR='R' then some of the singular values may be returned
*> as exact zeros obtained by "set to zero" because they are
*> below the numerical rank threshold or are denormalized numbers.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension ( LDU, N ) or ( LDU, M )
*> If JOBU = 'U', then U contains on exit the M-by-N matrix of
*> the left singular vectors.
*> If JOBU = 'F', then U contains on exit the M-by-M matrix of
*> the left singular vectors, including an ONB
*> of the orthogonal complement of the Range(A).
*> If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N),
*> then U is used as workspace if the procedure
*> replaces A with A^t. In that case, [V] is computed
*> in U as left singular vectors of A^t and then
*> copied back to the V array. This 'W' option is just
*> a reminder to the caller that in this case U is
*> reserved as workspace of length N*N.
*> If JOBU = 'N' U is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U, LDU >= 1.
*> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension ( LDV, N )
*> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
*> the right singular vectors;
*> If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N),
*> then V is used as workspace if the procedure
*> replaces A with A^t. In that case, [U] is computed
*> in V as right singular vectors of A^t and then
*> copied back to the U array. This 'W' option is just
*> a reminder to the caller that in this case V is
*> reserved as workspace of length N*N.
*> If JOBV = 'N' V is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V, LDV >= 1.
*> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(7,LWORK))
*> On exit, if N > 0 .AND. M > 0 (else not referenced),
*> WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
*> that SCALE*SVA(1:N) are the computed singular values
*> of A. (See the description of SVA().)
*> WORK(2) = See the description of WORK(1).
*> WORK(3) = SCONDA is an estimate for the condition number of
*> column equilibrated A. (If JOBA = 'E' or 'G')
*> SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
*> It is computed using DPOCON. It holds
*> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
*> where R is the triangular factor from the QRF of A.
*> However, if R is truncated and the numerical rank is
*> determined to be strictly smaller than N, SCONDA is
*> returned as -1, thus indicating that the smallest
*> singular values might be lost.
*>
*> If full SVD is needed, the following two condition numbers are
*> useful for the analysis of the algorithm. They are provided for
*> a developer/implementer who is familiar with the details of
*> the method.
*>
*> WORK(4) = an estimate of the scaled condition number of the
*> triangular factor in the first QR factorization.
*> WORK(5) = an estimate of the scaled condition number of the
*> triangular factor in the second QR factorization.
*> The following two parameters are computed if JOBT = 'T'.
*> They are provided for a developer/implementer who is familiar
*> with the details of the method.
*>
*> WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
*> of diag(A^t*A) / Trace(A^t*A) taken as point in the
*> probability simplex.
*> WORK(7) = the entropy of A*A^t.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> Length of WORK to confirm proper allocation of work space.
*> LWORK depends on the job:
*>
*> If only SIGMA is needed (JOBU = 'N', JOBV = 'N') and
*> -> .. no scaled condition estimate required (JOBE = 'N'):
*> LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
*> ->> For optimal performance (blocked code) the optimal value
*> is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
*> block size for DGEQP3 and DGEQRF.
*> In general, optimal LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
*> -> .. an estimate of the scaled condition number of A is
*> required (JOBA='E', 'G'). In this case, LWORK is the maximum
*> of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
*> ->> For optimal performance (blocked code) the optimal value
*> is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
*> In general, the optimal length LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
*> N+N*N+LWORK(DPOCON),7).
*>
*> If SIGMA and the right singular vectors are needed (JOBV = 'V'),
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
*> -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
*> where NB is the optimal block size for DGEQP3, DGEQRF, DGELQF,
*> DORMLQ. In general, the optimal length LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
*> N+LWORK(DGELQF), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
*>
*> If SIGMA and the left singular vectors are needed
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
*> -> For optimal performance:
*> if JOBU = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
*> if JOBU = 'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
*> where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
*> In general, the optimal length LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
*> 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
*> Here LWORK(DORMQR) equals N*NB (for JOBU = 'U') or
*> M*NB (for JOBU = 'F').
*>
*> If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
*> -> if JOBV = 'V'
*> the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
*> -> if JOBV = 'J' the minimal requirement is
*> LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
*> -> For optimal performance, LWORK should be additionally
*> larger than N+M*NB, where NB is the optimal block size
*> for DORMQR.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(3,M+3*N)).
*> On exit,
*> IWORK(1) = the numerical rank determined after the initial
*> QR factorization with pivoting. See the descriptions
*> of JOBA and JOBR.
*> IWORK(2) = the number of the computed nonzero singular values
*> IWORK(3) = if nonzero, a warning message:
*> If IWORK(3) = 1 then some of the column norms of A
*> were denormalized floats. The requested high accuracy
*> is not warranted by the data.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> < 0: if INFO = -i, then the i-th argument had an illegal value.
*> = 0: successful exit;
*> > 0: DGEJSV did not converge in the maximal allowed number
*> of sweeps. The computed values may be inaccurate.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gejsv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3,
*> DGEQRF, and DGELQF as preprocessors and preconditioners. Optionally, an
*> additional row pivoting can be used as a preprocessor, which in some
*> cases results in much higher accuracy. An example is matrix A with the
*> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
*> diagonal matrices and C is well-conditioned matrix. In that case, complete
*> pivoting in the first QR factorizations provides accuracy dependent on the
*> condition number of C, and independent of D1, D2. Such higher accuracy is
*> not completely understood theoretically, but it works well in practice.
*> Further, if A can be written as A = B*D, with well-conditioned B and some
*> diagonal D, then the high accuracy is guaranteed, both theoretically and
*> in software, independent of D. For more details see [1], [2].
*> The computational range for the singular values can be the full range
*> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
*> & LAPACK routines called by DGEJSV are implemented to work in that range.
*> If that is not the case, then the restriction for safe computation with
*> the singular values in the range of normalized IEEE numbers is that the
*> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
*> overflow. This code (DGEJSV) is best used in this restricted range,
*> meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
*> returned as zeros. See JOBR for details on this.
*> Further, this implementation is somewhat slower than the one described
*> in [1,2] due to replacement of some non-LAPACK components, and because
*> the choice of some tuning parameters in the iterative part (DGESVJ) is
*> left to the implementer on a particular machine.
*> The rank revealing QR factorization (in this code: DGEQP3) should be
*> implemented as in [3]. We have a new version of DGEQP3 under development
*> that is more robust than the current one in LAPACK, with a cleaner cut in
*> rank deficient cases. It will be available in the SIGMA library [4].
*> If M is much larger than N, it is obvious that the initial QRF with
*> column pivoting can be preprocessed by the QRF without pivoting. That
*> well known trick is not used in DGEJSV because in some cases heavy row
*> weighting can be treated with complete pivoting. The overhead in cases
*> M much larger than N is then only due to pivoting, but the benefits in
*> terms of accuracy have prevailed. The implementer/user can incorporate
*> this extra QRF step easily. The implementer can also improve data movement
*> (matrix transpose, matrix copy, matrix transposed copy) - this
*> implementation of DGEJSV uses only the simplest, naive data movement.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
*> LAPACK Working note 169.
*> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
*> LAPACK Working note 170.
*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
*> factorization software - a case study.
*> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
*> LAPACK Working note 176.
*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*> QSVD, (H,K)-SVD computations.
*> Department of Mathematics, University of Zagreb, 2008.
*> \endverbatim
*
*> \par Bugs, examples and comments:
* =================================
*>
*> Please report all bugs and send interesting examples and/or comments to
*> drmac@math.hr. Thank you.
*>
* =====================================================================
SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
$ M, N, A, LDA, SVA, U, LDU, V, LDV,
$ WORK, LWORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
IMPLICIT NONE
INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
$ WORK( LWORK )
INTEGER IWORK( * )
CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
* ..
*
* ===========================================================================
*
* .. Local Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
$ CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN, MAXPRJ, SCALEM,
$ SCONDA, SFMIN, SMALL, TEMP1, USCAL1, USCAL2, XSC
INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LSVEC,
$ L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,
$ NOSCAL, ROWPIV, RSVEC, TRANSP
* ..
* .. Intrinsic Functions ..
INTRINSIC DABS, DLOG, MAX, MIN, DBLE, IDNINT, DSIGN, DSQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DNRM2
INTEGER IDAMAX
LOGICAL LSAME
EXTERNAL IDAMAX, LSAME, DLAMCH, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGELQF, DGEQP3, DGEQRF, DLACPY,
$ DLASCL,
$ DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ,
$ DORMQR, DPOCON, DSCAL, DSWAP, DTRSM, XERBLA
*
EXTERNAL DGESVJ
* ..
*
* Test the input arguments
*
LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
JRACC = LSAME( JOBV, 'J' )
RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
L2RANK = LSAME( JOBA, 'R' )
L2ABER = LSAME( JOBA, 'A' )
ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
L2TRAN = LSAME( JOBT, 'T' )
L2KILL = LSAME( JOBR, 'R' )
DEFR = LSAME( JOBR, 'N' )
L2PERT = LSAME( JOBP, 'P' )
*
IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
$ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
INFO = - 1
ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
$ LSAME( JOBU, 'W' )) ) THEN
INFO = - 2
ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
$ LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN
INFO = - 3
ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
INFO = - 4
ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN
INFO = - 5
ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
INFO = - 6
ELSE IF ( M .LT. 0 ) THEN
INFO = - 7
ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
INFO = - 8
ELSE IF ( LDA .LT. M ) THEN
INFO = - 10
ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
INFO = - 13
ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
INFO = - 15
ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.
& (LWORK .LT. MAX(7,4*N+1,2*M+N))) .OR.
& (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND.
& (LWORK .LT. MAX(7,4*N+N*N,2*M+N))) .OR.
& (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
& .OR.
& (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
& .OR.
& (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND.
& (LWORK.LT.MAX(2*M+N,6*N+2*N*N)))
& .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND.
& LWORK.LT.MAX(2*M+N,4*N+N*N,2*N+N*N+6)))
& THEN
INFO = - 17
ELSE
* #:)
INFO = 0
END IF
*
IF ( INFO .NE. 0 ) THEN
* #:(
CALL XERBLA( 'DGEJSV', - INFO )
RETURN
END IF
*
* Quick return for void matrix (Y3K safe)
* #:)
IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
IWORK(1:3) = 0
WORK(1:7) = 0
RETURN
ENDIF
*
* Determine whether the matrix U should be M x N or M x M
*
IF ( LSVEC ) THEN
N1 = N
IF ( LSAME( JOBU, 'F' ) ) N1 = M
END IF
*
* Set numerical parameters
*
*! NOTE: Make sure DLAMCH() does not fail on the target architecture.
*
EPSLN = DLAMCH('Epsilon')
SFMIN = DLAMCH('SafeMinimum')
SMALL = SFMIN / EPSLN
BIG = DLAMCH('O')
* BIG = ONE / SFMIN
*
* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
*
*(!) If necessary, scale SVA() to protect the largest norm from
* overflow. It is possible that this scaling pushes the smallest
* column norm left from the underflow threshold (extreme case).
*
SCALEM = ONE / DSQRT(DBLE(M)*DBLE(N))
NOSCAL = .TRUE.
GOSCAL = .TRUE.
DO 1874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL DLASSQ( M, A(1,p), 1, AAPP, AAQQ )
IF ( AAPP .GT. BIG ) THEN
INFO = - 9
CALL XERBLA( 'DGEJSV', -INFO )
RETURN
END IF
AAQQ = DSQRT(AAQQ)
IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN
SVA(p) = AAPP * AAQQ
ELSE
NOSCAL = .FALSE.
SVA(p) = AAPP * ( AAQQ * SCALEM )
IF ( GOSCAL ) THEN
GOSCAL = .FALSE.
CALL DSCAL( p-1, SCALEM, SVA, 1 )
END IF
END IF
1874 CONTINUE
*
IF ( NOSCAL ) SCALEM = ONE
*
AAPP = ZERO
AAQQ = BIG
DO 4781 p = 1, N
AAPP = MAX( AAPP, SVA(p) )
IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
4781 CONTINUE
*
* Quick return for zero M x N matrix
* #:)
IF ( AAPP .EQ. ZERO ) THEN
IF ( LSVEC ) CALL DLASET( 'G', M, N1, ZERO, ONE, U, LDU )
IF ( RSVEC ) CALL DLASET( 'G', N, N, ZERO, ONE, V, LDV )
WORK(1) = ONE
WORK(2) = ONE
IF ( ERREST ) WORK(3) = ONE
IF ( LSVEC .AND. RSVEC ) THEN
WORK(4) = ONE
WORK(5) = ONE
END IF
IF ( L2TRAN ) THEN
WORK(6) = ZERO
WORK(7) = ZERO
END IF
IWORK(1) = 0
IWORK(2) = 0
IWORK(3) = 0
RETURN
END IF
*
* Issue warning if denormalized column norms detected. Override the
* high relative accuracy request. Issue licence to kill columns
* (set them to zero) whose norm is less than sigma_max / BIG (roughly).
* #:(
WARNING = 0
IF ( AAQQ .LE. SFMIN ) THEN
L2RANK = .TRUE.
L2KILL = .TRUE.
WARNING = 1
END IF
*
* Quick return for one-column matrix
* #:)
IF ( N .EQ. 1 ) THEN
*
IF ( LSVEC ) THEN
CALL DLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
CALL DLACPY( 'A', M, 1, A, LDA, U, LDU )
* computing all M left singular vectors of the M x 1 matrix
IF ( N1 .NE. N ) THEN
CALL DGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,
$ IERR )
CALL DORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,
$ IERR )
CALL DCOPY( M, A(1,1), 1, U(1,1), 1 )
END IF
END IF
IF ( RSVEC ) THEN
V(1,1) = ONE
END IF
IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
SVA(1) = SVA(1) / SCALEM
SCALEM = ONE
END IF
WORK(1) = ONE / SCALEM
WORK(2) = ONE
IF ( SVA(1) .NE. ZERO ) THEN
IWORK(1) = 1
IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
IWORK(2) = 1
ELSE
IWORK(2) = 0
END IF
ELSE
IWORK(1) = 0
IWORK(2) = 0
END IF
IWORK(3) = 0
IF ( ERREST ) WORK(3) = ONE
IF ( LSVEC .AND. RSVEC ) THEN
WORK(4) = ONE
WORK(5) = ONE
END IF
IF ( L2TRAN ) THEN
WORK(6) = ZERO
WORK(7) = ZERO
END IF
RETURN
*
END IF
*
TRANSP = .FALSE.
L2TRAN = L2TRAN .AND. ( M .EQ. N )
*
AATMAX = -ONE
AATMIN = BIG
IF ( ROWPIV .OR. L2TRAN ) THEN
*
* Compute the row norms, needed to determine row pivoting sequence
* (in the case of heavily row weighted A, row pivoting is strongly
* advised) and to collect information needed to compare the
* structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).
*
IF ( L2TRAN ) THEN
DO 1950 p = 1, M
XSC = ZERO
TEMP1 = ONE
CALL DLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
* DLASSQ gets both the ell_2 and the ell_infinity norm
* in one pass through the vector
WORK(M+N+p) = XSC * SCALEM
WORK(N+p) = XSC * (SCALEM*DSQRT(TEMP1))
AATMAX = MAX( AATMAX, WORK(N+p) )
IF (WORK(N+p) .NE. ZERO) AATMIN = MIN(AATMIN,WORK(N+p))
1950 CONTINUE
ELSE
DO 1904 p = 1, M
WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) )
AATMAX = MAX( AATMAX, WORK(M+N+p) )
AATMIN = MIN( AATMIN, WORK(M+N+p) )
1904 CONTINUE
END IF
*
END IF
*
* For square matrix A try to determine whether A^t would be better
* input for the preconditioned Jacobi SVD, with faster convergence.
* The decision is based on an O(N) function of the vector of column
* and row norms of A, based on the Shannon entropy. This should give
* the right choice in most cases when the difference actually matters.
* It may fail and pick the slower converging side.
*
ENTRA = ZERO
ENTRAT = ZERO
IF ( L2TRAN ) THEN
*
XSC = ZERO
TEMP1 = ONE
CALL DLASSQ( N, SVA, 1, XSC, TEMP1 )
TEMP1 = ONE / TEMP1
*
ENTRA = ZERO
DO 1113 p = 1, N
BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1
IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1)
1113 CONTINUE
ENTRA = - ENTRA / DLOG(DBLE(N))
*
* Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.
* It is derived from the diagonal of A^t * A. Do the same with the
* diagonal of A * A^t, compute the entropy of the corresponding
* probability distribution. Note that A * A^t and A^t * A have the
* same trace.
*
ENTRAT = ZERO
DO 1114 p = N+1, N+M
BIG1 = ( ( WORK(p) / XSC )**2 ) * TEMP1
IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)
1114 CONTINUE
ENTRAT = - ENTRAT / DLOG(DBLE(M))
*
* Analyze the entropies and decide A or A^t. Smaller entropy
* usually means better input for the algorithm.
*
TRANSP = ( ENTRAT .LT. ENTRA )
*
* If A^t is better than A, transpose A.
*
IF ( TRANSP ) THEN
* In an optimal implementation, this trivial transpose
* should be replaced with faster transpose.
DO 1115 p = 1, N - 1
DO 1116 q = p + 1, N
TEMP1 = A(q,p)
A(q,p) = A(p,q)
A(p,q) = TEMP1
1116 CONTINUE
1115 CONTINUE
DO 1117 p = 1, N
WORK(M+N+p) = SVA(p)
SVA(p) = WORK(N+p)
1117 CONTINUE
TEMP1 = AAPP
AAPP = AATMAX
AATMAX = TEMP1
TEMP1 = AAQQ
AAQQ = AATMIN
AATMIN = TEMP1
KILL = LSVEC
LSVEC = RSVEC
RSVEC = KILL
IF ( LSVEC ) N1 = N
*
ROWPIV = .TRUE.
END IF
*
END IF
* END IF L2TRAN
*
* Scale the matrix so that its maximal singular value remains less
* than DSQRT(BIG) -- the matrix is scaled so that its maximal column
* has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep
* DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and
* BLAS routines that, in some implementations, are not capable of
* working in the full interval [SFMIN,BIG] and that they may provoke
* overflows in the intermediate results. If the singular values spread
* from SFMIN to BIG, then DGESVJ will compute them. So, in that case,
* one should use DGESVJ instead of DGEJSV.
*
BIG1 = DSQRT( BIG )
TEMP1 = DSQRT( BIG / DBLE(N) )
*
CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
AAQQ = ( AAQQ / AAPP ) * TEMP1
ELSE
AAQQ = ( AAQQ * TEMP1 ) / AAPP
END IF
TEMP1 = TEMP1 * SCALEM
CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
*
* To undo scaling at the end of this procedure, multiply the
* computed singular values with USCAL2 / USCAL1.
*
USCAL1 = TEMP1
USCAL2 = AAPP
*
IF ( L2KILL ) THEN
* L2KILL enforces computation of nonzero singular values in
* the restricted range of condition number of the initial A,
* sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN).
XSC = DSQRT( SFMIN )
ELSE
XSC = SMALL
*
* Now, if the condition number of A is too big,
* sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN,
* as a precaution measure, the full SVD is computed using DGESVJ
* with accumulated Jacobi rotations. This provides numerically
* more robust computation, at the cost of slightly increased run
* time. Depending on the concrete implementation of BLAS and LAPACK
* (i.e. how they behave in presence of extreme ill-conditioning) the
* implementor may decide to remove this switch.
IF ( ( AAQQ.LT.DSQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
JRACC = .TRUE.
END IF
*
END IF
IF ( AAQQ .LT. XSC ) THEN
DO 700 p = 1, N
IF ( SVA(p) .LT. XSC ) THEN
CALL DLASET( 'A', M, 1, ZERO, ZERO, A(1,p), LDA )
SVA(p) = ZERO
END IF
700 CONTINUE
END IF
*
* Preconditioning using QR factorization with pivoting
*
IF ( ROWPIV ) THEN
* Optional row permutation (Bjoerck row pivoting):
* A result by Cox and Higham shows that the Bjoerck's
* row pivoting combined with standard column pivoting
* has similar effect as Powell-Reid complete pivoting.
* The ell-infinity norms of A are made nonincreasing.
DO 1952 p = 1, M - 1
q = IDAMAX( M-p+1, WORK(M+N+p), 1 ) + p - 1
IWORK(2*N+p) = q
IF ( p .NE. q ) THEN
TEMP1 = WORK(M+N+p)
WORK(M+N+p) = WORK(M+N+q)
WORK(M+N+q) = TEMP1
END IF
1952 CONTINUE
CALL DLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 )
END IF
*
* End of the preparation phase (scaling, optional sorting and
* transposing, optional flushing of small columns).
*
* Preconditioning
*
* If the full SVD is needed, the right singular vectors are computed
* from a matrix equation, and for that we need theoretical analysis
* of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF.
* In all other cases the first RR QRF can be chosen by other criteria
* (eg speed by replacing global with restricted window pivoting, such
* as in SGEQPX from TOMS # 782). Good results will be obtained using
* SGEQPX with properly (!) chosen numerical parameters.
* Any improvement of DGEQP3 improves overall performance of DGEJSV.
*
* A * P1 = Q1 * [ R1^t 0]^t:
DO 1963 p = 1, N
* .. all columns are free columns
IWORK(p) = 0
1963 CONTINUE
CALL DGEQP3( M,N,A,LDA, IWORK,WORK, WORK(N+1),LWORK-N, IERR )
*
* The upper triangular matrix R1 from the first QRF is inspected for
* rank deficiency and possibilities for deflation, or possible
* ill-conditioning. Depending on the user specified flag L2RANK,
* the procedure explores possibilities to reduce the numerical
* rank by inspecting the computed upper triangular factor. If
* L2RANK or L2ABER are up, then DGEJSV will compute the SVD of
* A + dA, where ||dA|| <= f(M,N)*EPSLN.
*
NR = 1
IF ( L2ABER ) THEN
* Standard absolute error bound suffices. All sigma_i with
* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
* aggressive enforcement of lower numerical rank by introducing a
* backward error of the order of N*EPSLN*||A||.
TEMP1 = DSQRT(DBLE(N))*EPSLN
DO 3001 p = 2, N
IF ( DABS(A(p,p)) .GE. (TEMP1*DABS(A(1,1))) ) THEN
NR = NR + 1
ELSE
GO TO 3002
END IF
3001 CONTINUE
3002 CONTINUE
ELSE IF ( L2RANK ) THEN
* .. similarly as above, only slightly more gentle (less aggressive).
* Sudden drop on the diagonal of R1 is used as the criterion for
* close-to-rank-deficient.
TEMP1 = DSQRT(SFMIN)
DO 3401 p = 2, N
IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR.
$ ( DABS(A(p,p)) .LT. SMALL ) .OR.
$ ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
NR = NR + 1
3401 CONTINUE
3402 CONTINUE
*
ELSE
* The goal is high relative accuracy. However, if the matrix
* has high scaled condition number the relative accuracy is in
* general not feasible. Later on, a condition number estimator
* will be deployed to estimate the scaled condition number.
* Here we just remove the underflowed part of the triangular
* factor. This prevents the situation in which the code is
* working hard to get the accuracy not warranted by the data.
TEMP1 = DSQRT(SFMIN)
DO 3301 p = 2, N
IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR.
$ ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
NR = NR + 1
3301 CONTINUE
3302 CONTINUE
*
END IF
*
ALMORT = .FALSE.
IF ( NR .EQ. N ) THEN
MAXPRJ = ONE
DO 3051 p = 2, N
TEMP1 = DABS(A(p,p)) / SVA(IWORK(p))
MAXPRJ = MIN( MAXPRJ, TEMP1 )
3051 CONTINUE
IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.
END IF
*
*
SCONDA = - ONE
CONDR1 = - ONE
CONDR2 = - ONE
*
IF ( ERREST ) THEN
IF ( N .EQ. NR ) THEN
IF ( RSVEC ) THEN
* .. V is available as workspace
CALL DLACPY( 'U', N, N, A, LDA, V, LDV )
DO 3053 p = 1, N
TEMP1 = SVA(IWORK(p))
CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 )
3053 CONTINUE
CALL DPOCON( 'U', N, V, LDV, ONE, TEMP1,
$ WORK(N+1), IWORK(2*N+M+1), IERR )
ELSE IF ( LSVEC ) THEN
* .. U is available as workspace
CALL DLACPY( 'U', N, N, A, LDA, U, LDU )
DO 3054 p = 1, N
TEMP1 = SVA(IWORK(p))
CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 )
3054 CONTINUE
CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1,
$ WORK(N+1), IWORK(2*N+M+1), IERR )
ELSE
CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N )
DO 3052 p = 1, N
TEMP1 = SVA(IWORK(p))
CALL DSCAL( p, ONE/TEMP1, WORK(N+(p-1)*N+1), 1 )
3052 CONTINUE
* .. the columns of R are scaled to have unit Euclidean lengths.
CALL DPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,
$ WORK(N+N*N+1), IWORK(2*N+M+1), IERR )
END IF
SCONDA = ONE / DSQRT(TEMP1)
* SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
ELSE
SCONDA = - ONE
END IF
END IF
*
L2PERT = L2PERT .AND. ( DABS( A(1,1)/A(NR,NR) ) .GT. DSQRT(BIG1) )
* If there is no violent scaling, artificial perturbation is not needed.
*
* Phase 3:
*
IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
*
* Singular Values only
*
* .. transpose A(1:NR,1:N)
DO 1946 p = 1, MIN( N-1, NR )
CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
1946 CONTINUE
*
* The following two DO-loops introduce small relative perturbation
* into the strict upper triangle of the lower triangular matrix.
* Small entries below the main diagonal are also changed.
* This modification is useful if the computing environment does not
* provide/allow FLUSH TO ZERO underflow, for it prevents many
* annoying denormalized numbers in case of strongly scaled matrices.
* The perturbation is structured so that it does not introduce any
* new perturbation of the singular values, and it does not destroy
* the job done by the preconditioner.
* The licence for this perturbation is in the variable L2PERT, which
* should be .FALSE. if FLUSH TO ZERO underflow is active.
*
IF ( .NOT. ALMORT ) THEN
*
IF ( L2PERT ) THEN
* XSC = DSQRT(SMALL)
XSC = EPSLN / DBLE(N)
DO 4947 q = 1, NR
TEMP1 = XSC*DABS(A(q,q))
DO 4949 p = 1, N
IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
$ .OR. ( p .LT. q ) )
$ A(p,q) = DSIGN( TEMP1, A(p,q) )
4949 CONTINUE
4947 CONTINUE
ELSE
CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, A(1,2),LDA )
END IF
*
* .. second preconditioning using the QR factorization
*
CALL DGEQRF( N,NR, A,LDA, WORK, WORK(N+1),LWORK-N, IERR )
*
* .. and transpose upper to lower triangular
DO 1948 p = 1, NR - 1
CALL DCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
1948 CONTINUE
*
END IF
*
* Row-cyclic Jacobi SVD algorithm with column pivoting
*
* .. again some perturbation (a "background noise") is added
* to drown denormals
IF ( L2PERT ) THEN
* XSC = DSQRT(SMALL)
XSC = EPSLN / DBLE(N)
DO 1947 q = 1, NR
TEMP1 = XSC*DABS(A(q,q))
DO 1949 p = 1, NR
IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
$ .OR. ( p .LT. q ) )
$ A(p,q) = DSIGN( TEMP1, A(p,q) )
1949 CONTINUE
1947 CONTINUE
ELSE
CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, A(1,2),
$ LDA )
END IF
*
* .. and one-sided Jacobi rotations are started on a lower
* triangular matrix (plus perturbation which is ignored in
* the part which destroys triangular form (confusing?!))
*
CALL DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,
$ N, V, LDV, WORK, LWORK, INFO )
*
SCALEM = WORK(1)
NUMRANK = IDNINT(WORK(2))
*
*
ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN
*
* -> Singular Values and Right Singular Vectors <-
*
IF ( ALMORT ) THEN
*
* .. in this case NR equals N
DO 1998 p = 1, NR
CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
1998 CONTINUE
CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2),
$ LDV )
*
CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,
$ WORK, LWORK, INFO )
SCALEM = WORK(1)
NUMRANK = IDNINT(WORK(2))
ELSE
*
* .. two more QR factorizations ( one QRF is not enough, two require
* accumulated product of Jacobi rotations, three are perfect )
*
CALL DLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1),
$ LDA )
CALL DGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N,
$ IERR)
CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV )
CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2),
$ LDV )
CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
DO 8998 p = 1, NR
CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
8998 CONTINUE
CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2),
$ LDV )
*
CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,
$ LDU, WORK(N+1), LWORK, INFO )
SCALEM = WORK(N+1)
NUMRANK = IDNINT(WORK(N+2))
IF ( NR .LT. N ) THEN
CALL DLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1),
$ LDV )
CALL DLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1),
$ LDV )
CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1),
$ LDV )
END IF
*
CALL DORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,
$ V, LDV, WORK(N+1), LWORK-N, IERR )
*
END IF
*
DO 8991 p = 1, N
CALL DCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
8991 CONTINUE
CALL DLACPY( 'All', N, N, A, LDA, V, LDV )
*
IF ( TRANSP ) THEN
CALL DLACPY( 'All', N, N, V, LDV, U, LDU )
END IF
*
ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
*
* .. Singular Values and Left Singular Vectors ..
*
* .. second preconditioning step to avoid need to accumulate
* Jacobi rotations in the Jacobi iterations.
DO 1965 p = 1, NR
CALL DCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
1965 CONTINUE
CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
*
CALL DGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
*
DO 1967 p = 1, NR - 1
CALL DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
1967 CONTINUE
CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
*
CALL DGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
$ LDA, WORK(N+1), LWORK-N, INFO )
SCALEM = WORK(N+1)
NUMRANK = IDNINT(WORK(N+2))
*
IF ( NR .LT. M ) THEN
CALL DLASET( 'A', M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU )
IF ( NR .LT. N1 ) THEN
CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1),
$ LDU )
CALL DLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),
$ LDU )
END IF
END IF
*
CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
*
IF ( ROWPIV )
$ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
DO 1974 p = 1, N1
XSC = ONE / DNRM2( M, U(1,p), 1 )
CALL DSCAL( M, XSC, U(1,p), 1 )
1974 CONTINUE
*
IF ( TRANSP ) THEN
CALL DLACPY( 'All', N, N, U, LDU, V, LDV )
END IF
*
ELSE
*
* .. Full SVD ..
*
IF ( .NOT. JRACC ) THEN
*
IF ( .NOT. ALMORT ) THEN
*
* Second Preconditioning Step (QRF [with pivoting])
* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
* equivalent to an LQF CALL. Since in many libraries the QRF
* seems to be better optimized than the LQF, we do explicit
* transpose and use the QRF. This is subject to changes in an
* optimized implementation of DGEJSV.
*
DO 1968 p = 1, NR
CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
1968 CONTINUE
*
* .. the following two loops perturb small entries to avoid
* denormals in the second QR factorization, where they are
* as good as zeros. This is done to avoid painfully slow
* computation with denormals. The relative size of the perturbation
* is a parameter that can be changed by the implementer.
* This perturbation device will be obsolete on machines with
* properly implemented arithmetic.
* To switch it off, set L2PERT=.FALSE. To remove it from the
* code, remove the action under L2PERT=.TRUE., leave the ELSE part.
* The following two loops should be blocked and fused with the
* transposed copy above.
*
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL)
DO 2969 q = 1, NR
TEMP1 = XSC*DABS( V(q,q) )
DO 2968 p = 1, N
IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
$ .OR. ( p .LT. q ) )
$ V(p,q) = DSIGN( TEMP1, V(p,q) )
IF ( p .LT. q ) V(p,q) = - V(p,q)
2968 CONTINUE
2969 CONTINUE
ELSE
CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2),
$ LDV )
END IF
*
* Estimate the row scaled condition number of R1
* (If R1 is rectangular, N > NR, then the condition number
* of the leading NR x NR submatrix is estimated.)
*
CALL DLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR )
DO 3950 p = 1, NR
TEMP1 = DNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1)
CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)
3950 CONTINUE
CALL DPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,
$ WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)
CONDR1 = ONE / DSQRT(TEMP1)
* .. here need a second opinion on the condition number
* .. then assume worst case scenario
* R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)
* more conservative <=> CONDR1 .LT. DSQRT(DBLE(N))
*
COND_OK = DSQRT(DBLE(NR))
*[TP] COND_OK is a tuning parameter.
IF ( CONDR1 .LT. COND_OK ) THEN
* .. the second QRF without pivoting. Note: in an optimized
* implementation, this QRF should be implemented as the QRF
* of a lower triangular matrix.
* R1^t = Q2 * R2
CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
*
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL)/EPSLN
DO 3959 p = 2, NR
DO 3958 q = 1, p - 1
TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
IF ( DABS(V(q,p)) .LE. TEMP1 )
$ V(q,p) = DSIGN( TEMP1, V(q,p) )
3958 CONTINUE
3959 CONTINUE
END IF
*
IF ( NR .NE. N )
$ CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
* .. save ...
*
* .. this transposed copy should be better than naive
DO 1969 p = 1, NR - 1
CALL DCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
1969 CONTINUE
*
CONDR2 = CONDR1
*
ELSE
*
* .. ill-conditioned case: second QRF with pivoting
* Note that windowed pivoting would be equally good
* numerically, and more run-time efficient. So, in
* an optimal implementation, the next call to DGEQP3
* should be replaced with eg. CALL SGEQPX (ACM TOMS #782)
* with properly (carefully) chosen parameters.
*
* R1^t * P2 = Q2 * R2
DO 3003 p = 1, NR
IWORK(N+p) = 0
3003 CONTINUE
CALL DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),
$ WORK(2*N+1), LWORK-2*N, IERR )
** CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
** $ LWORK-2*N, IERR )
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL)
DO 3969 p = 2, NR
DO 3968 q = 1, p - 1
TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
IF ( DABS(V(q,p)) .LE. TEMP1 )
$ V(q,p) = DSIGN( TEMP1, V(q,p) )
3968 CONTINUE
3969 CONTINUE
END IF
*
CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
*
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL)
DO 8970 p = 2, NR
DO 8971 q = 1, p - 1
TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
V(p,q) = - DSIGN( TEMP1, V(q,p) )
8971 CONTINUE
8970 CONTINUE
ELSE
CALL DLASET( 'L',NR-1,NR-1,ZERO,ZERO,V(2,1),LDV )
END IF
* Now, compute R2 = L3 * Q3, the LQ factorization.
CALL DGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),
$ WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
* .. and estimate the condition number
CALL DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )
DO 4950 p = 1, NR
TEMP1 = DNRM2( p, WORK(2*N+N*NR+NR+p), NR )
CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )
4950 CONTINUE
CALL DPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
$ WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )
CONDR2 = ONE / DSQRT(TEMP1)
*
IF ( CONDR2 .GE. COND_OK ) THEN
* .. save the Householder vectors used for Q3
* (this overwrites the copy of R2, as it will not be
* needed in this branch, but it does not overwrite the
* Huseholder vectors of Q2.).
CALL DLACPY( 'U', NR, NR, V, LDV, WORK(2*N+1), N )
* .. and the rest of the information on Q3 is in
* WORK(2*N+N*NR+1:2*N+N*NR+N)
END IF
*
END IF
*
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL)
DO 4968 q = 2, NR
TEMP1 = XSC * V(q,q)
DO 4969 p = 1, q - 1
* V(p,q) = - DSIGN( TEMP1, V(q,p) )
V(p,q) = - DSIGN( TEMP1, V(p,q) )
4969 CONTINUE
4968 CONTINUE
ELSE
CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, V(1,2), LDV )
END IF
*
* Second preconditioning finished; continue with Jacobi SVD
* The input matrix is lower triangular.
*
* Recover the right singular vectors as solution of a well
* conditioned triangular matrix equation.
*
IF ( CONDR1 .LT. COND_OK ) THEN
*
CALL DGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,
$ LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )
SCALEM = WORK(2*N+N*NR+NR+1)
NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
DO 3970 p = 1, NR
CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )
CALL DSCAL( NR, SVA(p), V(1,p), 1 )
3970 CONTINUE
* .. pick the right matrix equation and solve it
*
IF ( NR .EQ. N ) THEN
* :)) .. best case, R1 is inverted. The solution of this matrix
* equation is Q2*V2 = the product of the Jacobi rotations
* used in DGESVJ, premultiplied with the orthogonal matrix
* from the second QR factorization.
CALL DTRSM( 'L','U','N','N', NR,NR,ONE, A,LDA, V,
$ LDV )
ELSE
* .. R1 is well conditioned, but non-square. Transpose(R2)
* is inverted to get the product of the Jacobi rotations
* used in DGESVJ. The Q-factor from the second QR
* factorization is then built in explicitly.
CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),
$ N,V,LDV)
IF ( NR .LT. N ) THEN
CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)
CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)
CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),
$ LDV)
END IF
CALL DORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
END IF
*
ELSE IF ( CONDR2 .LT. COND_OK ) THEN
*
* :) .. the input matrix A is very likely a relative of
* the Kahan matrix :)
* The matrix R2 is inverted. The solution of the matrix equation
* is Q3^T*V3 = the product of the Jacobi rotations (applied to
* the lower triangular L3 from the LQ factorization of
* R2=L3*Q3), pre-multiplied with the transposed Q3.
CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR,
$ U,
$ LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
SCALEM = WORK(2*N+N*NR+NR+1)
NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
DO 3870 p = 1, NR
CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )
CALL DSCAL( NR, SVA(p), U(1,p), 1 )
3870 CONTINUE
CALL DTRSM('L','U','N','N',NR,NR,ONE,WORK(2*N+1),N,U,
$ LDU)
* .. apply the permutation from the second QR factorization
DO 873 q = 1, NR
DO 872 p = 1, NR
WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
872 CONTINUE
DO 874 p = 1, NR
U(p,q) = WORK(2*N+N*NR+NR+p)
874 CONTINUE
873 CONTINUE
IF ( NR .LT. N ) THEN
CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),
$ LDV )
END IF
CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
ELSE
* Last line of defense.
* #:( This is a rather pathological case: no scaled condition
* improvement after two pivoted QR factorizations. Other
* possibility is that the rank revealing QR factorization
* or the condition estimator has failed, or the COND_OK
* is set very close to ONE (which is unnecessary). Normally,
* this branch should never be executed, but in rare cases of
* failure of the RRQR or condition estimator, the last line of
* defense ensures that DGEJSV completes the task.
* Compute the full SVD of L3 using DGESVJ with explicit
* accumulation of Jacobi rotations.
CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR,
$ U,
$ LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
SCALEM = WORK(2*N+N*NR+NR+1)
NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
IF ( NR .LT. N ) THEN
CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),
$ LDV )
END IF
CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
*
CALL DORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,
$ WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),
$ LWORK-2*N-N*NR-NR, IERR )
DO 773 q = 1, NR
DO 772 p = 1, NR
WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
772 CONTINUE
DO 774 p = 1, NR
U(p,q) = WORK(2*N+N*NR+NR+p)
774 CONTINUE
773 CONTINUE
*
END IF
*
* Permute the rows of V using the (column) permutation from the
* first QRF. Also, scale the columns to make them unit in
* Euclidean norm. This applies to all cases.
*
TEMP1 = DSQRT(DBLE(N)) * EPSLN
DO 1972 q = 1, N
DO 972 p = 1, N
WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
972 CONTINUE
DO 973 p = 1, N
V(p,q) = WORK(2*N+N*NR+NR+p)
973 CONTINUE
XSC = ONE / DNRM2( N, V(1,q), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL DSCAL( N, XSC, V(1,q), 1 )
1972 CONTINUE
* At this moment, V contains the right singular vectors of A.
* Next, assemble the left singular vector matrix U (M x N).
IF ( NR .LT. M ) THEN
CALL DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1),
$ LDU )
IF ( NR .LT. N1 ) THEN
CALL DLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU)
CALL DLASET('A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),
$ LDU)
END IF
END IF
*
* The Q matrix from the first QRF is built into the left singular
* matrix U. This applies to all cases.
*
CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
* The columns of U are normalized. The cost is O(M*N) flops.
TEMP1 = DSQRT(DBLE(M)) * EPSLN
DO 1973 p = 1, NR
XSC = ONE / DNRM2( M, U(1,p), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL DSCAL( M, XSC, U(1,p), 1 )
1973 CONTINUE
*
* If the initial QRF is computed with row pivoting, the left
* singular vectors must be adjusted.
*
IF ( ROWPIV )
$ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
ELSE
*
* .. the initial matrix A has almost orthogonal columns and
* the second QRF is not needed
*
CALL DLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N )
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL)
DO 5970 p = 2, N
TEMP1 = XSC * WORK( N + (p-1)*N + p )
DO 5971 q = 1, p - 1
WORK(N+(q-1)*N+p)=-DSIGN(TEMP1,WORK(N+(p-1)*N+q))
5971 CONTINUE
5970 CONTINUE
ELSE
CALL DLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N )
END IF
*
CALL DGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,
$ N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )
*
SCALEM = WORK(N+N*N+1)
NUMRANK = IDNINT(WORK(N+N*N+2))
DO 6970 p = 1, N
CALL DCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 )
CALL DSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 )
6970 CONTINUE
*
CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
$ ONE, A, LDA, WORK(N+1), N )
DO 6972 p = 1, N
CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )
6972 CONTINUE
TEMP1 = DSQRT(DBLE(N))*EPSLN
DO 6971 p = 1, N
XSC = ONE / DNRM2( N, V(1,p), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL DSCAL( N, XSC, V(1,p), 1 )
6971 CONTINUE
*
* Assemble the left singular vector matrix U (M x N).
*
IF ( N .LT. M ) THEN
CALL DLASET( 'A', M-N, N, ZERO, ZERO, U(N+1,1), LDU )
IF ( N .LT. N1 ) THEN
CALL DLASET( 'A',N, N1-N, ZERO, ZERO, U(1,N+1),
$ LDU )
CALL DLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),
$ LDU )
END IF
END IF
CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
TEMP1 = DSQRT(DBLE(M))*EPSLN
DO 6973 p = 1, N1
XSC = ONE / DNRM2( M, U(1,p), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL DSCAL( M, XSC, U(1,p), 1 )
6973 CONTINUE
*
IF ( ROWPIV )
$ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
END IF
*
* end of the >> almost orthogonal case << in the full SVD
*
ELSE
*
* This branch deploys a preconditioned Jacobi SVD with explicitly
* accumulated rotations. It is included as optional, mainly for
* experimental purposes. It does perform well, and can also be used.
* In this implementation, this branch will be automatically activated
* if the condition number sigma_max(A) / sigma_min(A) is predicted
* to be greater than the overflow threshold. This is because the
* a posteriori computation of the singular vectors assumes robust
* implementation of BLAS and some LAPACK procedures, capable of working
* in presence of extreme values. Since that is not always the case, ...
*
DO 7968 p = 1, NR
CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
7968 CONTINUE
*
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL/EPSLN)
DO 5969 q = 1, NR
TEMP1 = XSC*DABS( V(q,q) )
DO 5968 p = 1, N
IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
$ .OR. ( p .LT. q ) )
$ V(p,q) = DSIGN( TEMP1, V(p,q) )
IF ( p .LT. q ) V(p,q) = - V(p,q)
5968 CONTINUE
5969 CONTINUE
ELSE
CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
END IF
CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )
*
DO 7969 p = 1, NR
CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
7969 CONTINUE
IF ( L2PERT ) THEN
XSC = DSQRT(SMALL/EPSLN)
DO 9970 q = 2, NR
DO 9971 p = 1, q - 1
TEMP1 = XSC * MIN(DABS(U(p,p)),DABS(U(q,q)))
U(p,q) = - DSIGN( TEMP1, U(q,p) )
9971 CONTINUE
9970 CONTINUE
ELSE
CALL DLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
END IF
CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA,
$ N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )
SCALEM = WORK(2*N+N*NR+1)
NUMRANK = IDNINT(WORK(2*N+N*NR+2))
IF ( NR .LT. N ) THEN
CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
END IF
CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
*
* Permute the rows of V using the (column) permutation from the
* first QRF. Also, scale the columns to make them unit in
* Euclidean norm. This applies to all cases.
*
TEMP1 = DSQRT(DBLE(N)) * EPSLN
DO 7972 q = 1, N
DO 8972 p = 1, N
WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
8972 CONTINUE
DO 8973 p = 1, N
V(p,q) = WORK(2*N+N*NR+NR+p)
8973 CONTINUE
XSC = ONE / DNRM2( N, V(1,q), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL DSCAL( N, XSC, V(1,q), 1 )
7972 CONTINUE
*
* At this moment, V contains the right singular vectors of A.
* Next, assemble the left singular vector matrix U (M x N).
*
IF ( NR .LT. M ) THEN
CALL DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )
IF ( NR .LT. N1 ) THEN
CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1),
$ LDU )
CALL DLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),
$ LDU )
END IF
END IF
*
CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
*
IF ( ROWPIV )
$ CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
*
END IF
IF ( TRANSP ) THEN
* .. swap U and V because the procedure worked on A^t
DO 6974 p = 1, N
CALL DSWAP( N, U(1,p), 1, V(1,p), 1 )
6974 CONTINUE
END IF
*
END IF
* end of the full SVD
*
* Undo scaling, if necessary (and possible)
*
IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N,
$ IERR )
USCAL1 = ONE
USCAL2 = ONE
END IF
*
IF ( NR .LT. N ) THEN
DO 3004 p = NR+1, N
SVA(p) = ZERO
3004 CONTINUE
END IF
*
WORK(1) = USCAL2 * SCALEM
WORK(2) = USCAL1
IF ( ERREST ) WORK(3) = SCONDA
IF ( LSVEC .AND. RSVEC ) THEN
WORK(4) = CONDR1
WORK(5) = CONDR2
END IF
IF ( L2TRAN ) THEN
WORK(6) = ENTRA
WORK(7) = ENTRAT
END IF
*
IWORK(1) = NR
IWORK(2) = NUMRANK
IWORK(3) = WARNING
*
RETURN
* ..
* .. END OF DGEJSV
* ..
END
*
*> \brief \b DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELQ2 computes an LQ factorization of a real m-by-n matrix A:
*>
*> A = ( L 0 ) * Q
*>
*> where:
*>
*> Q is a n-by-n orthogonal matrix;
*> L is a lower-triangular m-by-m matrix;
*> 0 is a m-by-(n-m) zero matrix, if m < n.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, the elements on and below the diagonal of the array
*> contain the m by min(m,n) lower trapezoidal matrix L (L is
*> lower triangular if m <= n); the elements above the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelq2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELQ2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = 1, K
*
* Generate elementary reflector H(i) to annihilate A(i,i+1:n)
*
CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
$ TAU( I ) )
IF( I.LT.M ) THEN
*
* Apply H(i) to A(i+1:m,i:n) from the right
*
CALL DLARF1F( 'Right', M-I, N-I+1, A( I, I ), LDA,
$ TAU( I ),
$ A( I+1, I ), LDA, WORK )
END IF
10 CONTINUE
RETURN
*
* End of DGELQ2
*
END
*> \brief \b DGELQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELQF computes an LQ factorization of a real M-by-N matrix A:
*>
*> A = ( L 0 ) * Q
*>
*> where:
*>
*> Q is a N-by-N orthogonal matrix;
*> L is a lower-triangular M-by-M matrix;
*> 0 is a M-by-(N-M) zero matrix, if M < N.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and below the diagonal of the array
*> contain the m-by-min(m,n) lower trapezoidal matrix L (L is
*> lower triangular if m <= n); the elements above the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
*> For optimum performance LWORK >= M*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelqf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DGELQ2, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
K = MIN( M, N )
NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( .NOT.LQUERY ) THEN
IF( LWORK.LE.0 .OR. ( N.GT.0 .AND. LWORK.LT.MAX( 1, M ) ) )
$ INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
IF( K.EQ.0 ) THEN
LWKOPT = 1
ELSE
LWKOPT = M*NB
END IF
WORK( 1 ) = LWKOPT
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGELQF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGELQF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially
*
DO 10 I = 1, K - NX, NB
IB = MIN( K-I+1, NB )
*
* Compute the LQ factorization of the current block
* A(i:i+ib-1,i:n)
*
CALL DGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
IF( I+IB.LE.M ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I,
$ I ),
$ LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(i+ib:m,i:n) from the right
*
CALL DLARFB( 'Right', 'No transpose', 'Forward',
$ 'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ),
$ LDA, WORK, LDWORK, A( I+IB, I ), LDA,
$ WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
ELSE
I = 1
END IF
*
* Use unblocked code to factor the last or only block.
*
IF( I.LE.K )
$ CALL DGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGELQF
*
END
*> \brief <b> DGELS solves overdetermined or underdetermined systems for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgels.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgels.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgels.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELS solves overdetermined or underdetermined real linear systems
*> involving an M-by-N matrix A, or its transpose, using a QR or LQ
*> factorization of A.
*>
*> It is assumed that A has full rank, and only a rudimentary protection
*> against rank-deficient matrices is provided. This subroutine only detects
*> exact rank-deficiency, where a diagonal element of the triangular factor
*> of A is exactly zero.
*>
*> It is conceivable for one (or more) of the diagonal elements of the triangular
*> factor of A to be subnormally tiny numbers without this subroutine signalling
*> an error. The solutions computed for such almost-rank-deficient matrices may
*> be less accurate due to a loss of numerical precision.
*>
*> The following options are provided:
*>
*> 1. If TRANS = 'N' and m >= n: find the least squares solution of
*> an overdetermined system, i.e., solve the least squares problem
*> minimize || B - A*X ||.
*>
*> 2. If TRANS = 'N' and m < n: find the minimum norm solution of
*> an underdetermined system A * X = B.
*>
*> 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
*> an underdetermined system A**T * X = B.
*>
*> 4. If TRANS = 'T' and m < n: find the least squares solution of
*> an overdetermined system, i.e., solve the least squares problem
*> minimize || B - A**T * X ||.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': the linear system involves A;
*> = 'T': the linear system involves A**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of
*> columns of the matrices B and X. NRHS >=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit,
*> if M >= N, A is overwritten by details of its QR
*> factorization as returned by DGEQRF;
*> if M < N, A is overwritten by details of its LQ
*> factorization as returned by DGELQF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the matrix B of right hand side vectors, stored
*> columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
*> if TRANS = 'T'.
*> On exit, if INFO = 0, B is overwritten by the solution
*> vectors, stored columnwise:
*> if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
*> squares solution vectors; the residual sum of squares for the
*> solution in each column is given by the sum of squares of
*> elements N+1 to M in that column;
*> if TRANS = 'N' and m < n, rows 1 to N of B contain the
*> minimum norm solution vectors;
*> if TRANS = 'T' and m >= n, rows 1 to M of B contain the
*> minimum norm solution vectors;
*> if TRANS = 'T' and m < n, rows 1 to M of B contain the
*> least squares solution vectors; the residual sum of squares
*> for the solution in each column is given by the sum of
*> squares of elements M+1 to N in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= MAX(1,M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= max( 1, MN + max( MN, NRHS ) ).
*> For optimal performance,
*> LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
*> where MN = min(M,N) and NB is the optimum block size.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element of the
*> triangular factor of A is exactly zero, so that A does not have
*> full rank; the least squares solution could not be
*> computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gels
*
* =====================================================================
SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK,
$ LWORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, TPSD
INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE
DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM
* ..
* .. Local Arrays ..
DOUBLE PRECISION RWORK( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DGELQF, DGEQRF, DLASCL, DLASET, DORMLQ,
$ DORMQR,
$ DTRTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
MN = MIN( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.( LSAME( TRANS, 'N' ) .OR.
$ LSAME( TRANS, 'T' ) ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
$ THEN
INFO = -10
END IF
*
* Figure out optimal block size
*
IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
*
TPSD = .TRUE.
IF( LSAME( TRANS, 'N' ) )
$ TPSD = .FALSE.
*
IF( M.GE.N ) THEN
NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
IF( TPSD ) THEN
NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LN', M, NRHS, N,
$ -1 ) )
ELSE
NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N,
$ -1 ) )
END IF
ELSE
NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
IF( TPSD ) THEN
NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M,
$ -1 ) )
ELSE
NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LN', N, NRHS, M,
$ -1 ) )
END IF
END IF
*
WSIZE = MAX( 1, MN+MAX( MN, NRHS )*NB )
WORK( 1 ) = DBLE( WSIZE )
*
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N, NRHS ).EQ.0 ) THEN
CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
RETURN
END IF
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
*
* Scale A, B if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, RWORK )
IASCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
IASCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
IASCL = 2
ELSE IF( ANRM.EQ.ZERO ) THEN
*
* Matrix all zero. Return zero solution.
*
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
GO TO 50
END IF
*
BROW = M
IF( TPSD )
$ BROW = N
BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
$ INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
$ INFO )
IBSCL = 2
END IF
*
IF( M.GE.N ) THEN
*
* compute QR factorization of A
*
CALL DGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ),
$ LWORK-MN,
$ INFO )
*
* workspace at least N, optimally N*NB
*
IF( .NOT.TPSD ) THEN
*
* Least-Squares Problem min || A * X - B ||
*
* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
CALL DORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
* B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
*
CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N,
$ NRHS,
$ A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) THEN
RETURN
END IF
*
SCLLEN = N
*
ELSE
*
* Underdetermined system of equations A**T * X = B
*
* B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
*
CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
$ A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) THEN
RETURN
END IF
*
* B(N+1:M,1:NRHS) = ZERO
*
DO 20 J = 1, NRHS
DO 10 I = N + 1, M
B( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
*
CALL DORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
SCLLEN = M
*
END IF
*
ELSE
*
* Compute LQ factorization of A
*
CALL DGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ),
$ LWORK-MN,
$ INFO )
*
* workspace at least M, optimally M*NB.
*
IF( .NOT.TPSD ) THEN
*
* underdetermined system of equations A * X = B
*
* B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
*
CALL DTRTRS( 'Lower', 'No transpose', 'Non-unit', M,
$ NRHS,
$ A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) THEN
RETURN
END IF
*
* B(M+1:N,1:NRHS) = 0
*
DO 40 J = 1, NRHS
DO 30 I = M + 1, N
B( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
*
* B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
*
CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
SCLLEN = N
*
ELSE
*
* overdetermined system min || A**T * X - B ||
*
* B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
*
CALL DORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
* B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
*
CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
$ A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) THEN
RETURN
END IF
*
SCLLEN = M
*
END IF
*
END IF
*
* Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
$ INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
$ INFO )
END IF
*
50 CONTINUE
WORK( 1 ) = DBLE( WSIZE )
*
RETURN
*
* End of DGELS
*
END
*> \brief <b> DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
* WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELSD computes the minimum-norm solution to a real linear least
*> squares problem:
*> minimize 2-norm(| b - A*x |)
*> using the singular value decomposition (SVD) of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*>
*> The problem is solved in three steps:
*> (1) Reduce the coefficient matrix A to bidiagonal form with
*> Householder transformations, reducing the original problem
*> into a "bidiagonal least squares problem" (BLS)
*> (2) Solve the BLS using a divide and conquer approach.
*> (3) Apply back all the Householder transformations to solve
*> the original least squares problem.
*>
*> The effective rank of A is determined by treating as zero those
*> singular values which are less than RCOND times the largest singular
*> value.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, A has been destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the M-by-NRHS right hand side matrix B.
*> On exit, B is overwritten by the N-by-NRHS solution
*> matrix X. If m >= n and RANK = n, the residual
*> sum-of-squares for the solution in the i-th column is given
*> by the sum of squares of elements n+1:m in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,max(M,N)).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (min(M,N))
*> The singular values of A in decreasing order.
*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> RCOND is used to determine the effective rank of A.
*> Singular values S(i) <= RCOND*S(1) are treated as zero.
*> If RCOND < 0, machine precision is used instead.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The effective rank of A, i.e., the number of singular values
*> which are greater than RCOND*S(1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK must be at least 1.
*> The exact minimum amount of workspace needed depends on M,
*> N and NRHS. As long as LWORK is at least
*> 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
*> if M is greater than or equal to N or
*> 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
*> if M is less than N, the code will execute correctly.
*> SMLSIZ is returned by ILAENV and is equal to the maximum
*> size of the subproblems at the bottom of the computation
*> tree (usually about 25), and
*> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
*> where MINMN = MIN( M,N ).
*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: the algorithm for computing the SVD failed to converge;
*> if INFO = i, i off-diagonal elements of an intermediate
*> bidiagonal form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelsd
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
*> California at Berkeley, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*
* =====================================================================
SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
$ WORK, LWORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
$ LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
$ MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL DGEBRD, DGELQF, DGEQRF, DLACPY,
$ DLALSD,
$ DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, INT, LOG, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
MINMN = MIN( M, N )
MAXMN = MAX( M, N )
MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
INFO = -7
END IF
*
SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
*
* Compute workspace.
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.)
*
MINWRK = 1
LIWORK = 1
MINMN = MAX( 1, MINMN )
NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
$ LOG( TWO ) ) + 1, 0 )
*
IF( INFO.EQ.0 ) THEN
MAXWRK = 1
LIWORK = 3*MINMN*NLVL + 11*MINMN
MM = M
IF( M.GE.N .AND. M.GE.MNTHR ) THEN
*
* Path 1a - overdetermined, with many more rows than columns.
*
MM = N
MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
$ -1, -1 ) )
MAXWRK = MAX( MAXWRK, N+NRHS*
$ ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
END IF
IF( M.GE.N ) THEN
*
* Path 1 - overdetermined or exactly determined.
*
MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
$ ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
MAXWRK = MAX( MAXWRK, 3*N+NRHS*
$ ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
$ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
END IF
IF( N.GT.M ) THEN
WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
IF( N.GE.MNTHR ) THEN
*
* Path 2a - underdetermined, with many more columns
* than rows.
*
MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
$ -1 )
MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
$ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M,
$ -1 ) )
MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
$ ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M,
$ -1 ) )
IF( NRHS.GT.1 ) THEN
MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
ELSE
MAXWRK = MAX( MAXWRK, M*M+2*M )
END IF
MAXWRK = MAX( MAXWRK, M+NRHS*
$ ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
! XXX: Ensure the Path 2a case below is triggered. The workspace
! calculation should use queries for all routines eventually.
MAXWRK = MAX( MAXWRK,
$ 4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
ELSE
*
* Path 2 - remaining underdetermined cases.
*
MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
$ -1, -1 )
MAXWRK = MAX( MAXWRK, 3*M+NRHS*
$ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N,
$ -1 ) )
MAXWRK = MAX( MAXWRK, 3*M+M*
$ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M,
$ -1 ) )
MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
END IF
MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
END IF
MINWRK = MIN( MINWRK, MAXWRK )
WORK( 1 ) = MAXWRK
IWORK( 1 ) = LIWORK
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELSD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
GO TO 10
END IF
*
* Quick return if possible.
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RANK = 0
RETURN
END IF
*
* Get machine parameters.
*
EPS = DLAMCH( 'P' )
SFMIN = DLAMCH( 'S' )
SMLNUM = SFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max entry outside range [SMLNUM,BIGNUM].
*
ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
IASCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM.
*
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
IASCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM.
*
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
IASCL = 2
ELSE IF( ANRM.EQ.ZERO ) THEN
*
* Matrix all zero. Return zero solution.
*
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
RANK = 0
GO TO 10
END IF
*
* Scale B if max entry outside range [SMLNUM,BIGNUM].
*
BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM.
*
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM.
*
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 2
END IF
*
* If M < N make sure certain entries of B are zero.
*
IF( M.LT.N )
$ CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
*
* Overdetermined case.
*
IF( M.GE.N ) THEN
*
* Path 1 - overdetermined or exactly determined.
*
MM = M
IF( M.GE.MNTHR ) THEN
*
* Path 1a - overdetermined, with many more rows than columns.
*
MM = N
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R.
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, INFO )
*
* Multiply B by transpose(Q).
* (Workspace: need N+NRHS, prefer N+NRHS*NB)
*
CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ),
$ B,
$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
* Zero out below R.
*
IF( N.GT.1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
END IF
END IF
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in A.
* (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
*
CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors of R.
* (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA,
$ WORK( ITAUQ ),
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
* Solve the bidiagonal least squares problem.
*
CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
$ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
IF( INFO.NE.0 ) THEN
GO TO 10
END IF
*
* Multiply B by right bidiagonalizing vectors of R.
*
CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA,
$ WORK( ITAUP ),
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
$ MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
*
* Path 2a - underdetermined, with many more columns than rows
* and sufficient workspace for an efficient algorithm.
*
LDWORK = M
IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
$ M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
ITAU = 1
NWORK = M + 1
*
* Compute A=L*Q.
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, INFO )
IL = NWORK
*
* Copy L to WORK(IL), zeroing out above its diagonal.
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
$ LDWORK )
IE = IL + LDWORK*M
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL).
* (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK-NWORK+1, INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors of L.
* (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
$ WORK( ITAUQ ), B, LDB, WORK( NWORK ),
$ LWORK-NWORK+1, INFO )
*
* Solve the bidiagonal least squares problem.
*
CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
$ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
IF( INFO.NE.0 ) THEN
GO TO 10
END IF
*
* Multiply B by right bidiagonalizing vectors of L.
*
CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
$ WORK( ITAUP ), B, LDB, WORK( NWORK ),
$ LWORK-NWORK+1, INFO )
*
* Zero out below first M rows of B.
*
CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
NWORK = ITAU + M
*
* Multiply transpose(Q) by B.
* (Workspace: need M+NRHS, prefer M+NRHS*NB)
*
CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
ELSE
*
* Path 2 - remaining underdetermined cases.
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize A.
* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors.
* (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
$ WORK( ITAUQ ),
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
* Solve the bidiagonal least squares problem.
*
CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
$ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
IF( INFO.NE.0 ) THEN
GO TO 10
END IF
*
* Multiply B by right bidiagonalizing vectors of A.
*
CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA,
$ WORK( ITAUP ),
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
END IF
*
* Undo scaling.
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
END IF
*
10 CONTINUE
WORK( 1 ) = MAXWRK
IWORK( 1 ) = LIWORK
RETURN
*
* End of DGELSD
*
END
*> \brief <b> DGELSS solves overdetermined or underdetermined systems for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelss.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelss.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelss.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELSS computes the minimum norm solution to a real linear least
*> squares problem:
*>
*> Minimize 2-norm(| b - A*x |).
*>
*> using the singular value decomposition (SVD) of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix
*> X.
*>
*> The effective rank of A is determined by treating as zero those
*> singular values which are less than RCOND times the largest singular
*> value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the first min(m,n) rows of A are overwritten with
*> its right singular vectors, stored rowwise.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the M-by-NRHS right hand side matrix B.
*> On exit, B is overwritten by the N-by-NRHS solution
*> matrix X. If m >= n and RANK = n, the residual
*> sum-of-squares for the solution in the i-th column is given
*> by the sum of squares of elements n+1:m in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,max(M,N)).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (min(M,N))
*> The singular values of A in decreasing order.
*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> RCOND is used to determine the effective rank of A.
*> Singular values S(i) <= RCOND*S(1) are treated as zero.
*> If RCOND < 0, machine precision is used instead.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The effective rank of A, i.e., the number of singular values
*> which are greater than RCOND*S(1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1, and also:
*> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: the algorithm for computing the SVD failed to converge;
*> if INFO = i, i off-diagonal elements of an intermediate
*> bidiagonal form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelss
*
* =====================================================================
SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
$ WORK, LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL,
$ ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN,
$ MAXWRK, MINMN, MINWRK, MM, MNTHR
INTEGER LWORK_DGEQRF, LWORK_DORMQR, LWORK_DGEBRD,
$ LWORK_DORMBR, LWORK_DORGBR, LWORK_DORMLQ,
$ LWORK_DGELQF
DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSQR, DCOPY, DGEBRD, DGELQF, DGEMM,
$ DGEMV,
$ DGEQRF, DLACPY, DLASCL, DLASET, DORGBR,
$ DORMBR, DORMLQ, DORMQR, DRSCL, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
MINMN = MIN( M, N )
MAXMN = MAX( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
INFO = -7
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.)
*
IF( INFO.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
IF( MINMN.GT.0 ) THEN
MM = M
MNTHR = ILAENV( 6, 'DGELSS', ' ', M, N, NRHS, -1 )
IF( M.GE.N .AND. M.GE.MNTHR ) THEN
*
* Path 1a - overdetermined, with many more rows than
* columns
*
* Compute space needed for DGEQRF
CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, INFO )
LWORK_DGEQRF = INT( DUM(1) )
* Compute space needed for DORMQR
CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, DUM(1), B,
$ LDB, DUM(1), -1, INFO )
LWORK_DORMQR = INT( DUM(1) )
MM = N
MAXWRK = MAX( MAXWRK, N + LWORK_DGEQRF )
MAXWRK = MAX( MAXWRK, N + LWORK_DORMQR )
END IF
IF( M.GE.N ) THEN
*
* Path 1 - overdetermined or exactly determined
*
* Compute workspace needed for DBDSQR
*
BDSPAC = MAX( 1, 5*N )
* Compute space needed for DGEBRD
CALL DGEBRD( MM, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, INFO )
LWORK_DGEBRD = INT( DUM(1) )
* Compute space needed for DORMBR
CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA,
$ DUM(1),
$ B, LDB, DUM(1), -1, INFO )
LWORK_DORMBR = INT( DUM(1) )
* Compute space needed for DORGBR
CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, INFO )
LWORK_DORGBR = INT( DUM(1) )
* Compute total workspace needed
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORMBR )
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, BDSPAC )
MAXWRK = MAX( MAXWRK, N*NRHS )
MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC )
MAXWRK = MAX( MINWRK, MAXWRK )
END IF
IF( N.GT.M ) THEN
*
* Compute workspace needed for DBDSQR
*
BDSPAC = MAX( 1, 5*M )
MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC )
IF( N.GE.MNTHR ) THEN
*
* Path 2a - underdetermined, with many more columns
* than rows
*
* Compute space needed for DGELQF
CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1),
$ -1, INFO )
LWORK_DGELQF = INT( DUM(1) )
* Compute space needed for DGEBRD
CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, INFO )
LWORK_DGEBRD = INT( DUM(1) )
* Compute space needed for DORMBR
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
$ DUM(1), B, LDB, DUM(1), -1, INFO )
LWORK_DORMBR = INT( DUM(1) )
* Compute space needed for DORGBR
CALL DORGBR( 'P', M, M, M, A, LDA, DUM(1),
$ DUM(1), -1, INFO )
LWORK_DORGBR = INT( DUM(1) )
* Compute space needed for DORMLQ
CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, DUM(1),
$ B, LDB, DUM(1), -1, INFO )
LWORK_DORMLQ = INT( DUM(1) )
* Compute total workspace needed
MAXWRK = M + LWORK_DGELQF
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DGEBRD )
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORMBR )
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC )
IF( NRHS.GT.1 ) THEN
MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
ELSE
MAXWRK = MAX( MAXWRK, M*M + 2*M )
END IF
MAXWRK = MAX( MAXWRK, M + LWORK_DORMLQ )
ELSE
*
* Path 2 - underdetermined
*
* Compute space needed for DGEBRD
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, INFO )
LWORK_DGEBRD = INT( DUM(1) )
* Compute space needed for DORMBR
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, A, LDA,
$ DUM(1), B, LDB, DUM(1), -1, INFO )
LWORK_DORMBR = INT( DUM(1) )
* Compute space needed for DORGBR
CALL DORGBR( 'P', M, N, M, A, LDA, DUM(1),
$ DUM(1), -1, INFO )
LWORK_DORGBR = INT( DUM(1) )
MAXWRK = 3*M + LWORK_DGEBRD
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORMBR )
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, BDSPAC )
MAXWRK = MAX( MAXWRK, N*NRHS )
END IF
END IF
MAXWRK = MAX( MINWRK, MAXWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
$ INFO = -12
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELSS', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RANK = 0
RETURN
END IF
*
* Get machine parameters
*
EPS = DLAMCH( 'P' )
SFMIN = DLAMCH( 'S' )
SMLNUM = SFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
IASCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
IASCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
IASCL = 2
ELSE IF( ANRM.EQ.ZERO ) THEN
*
* Matrix all zero. Return zero solution.
*
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, MINMN )
RANK = 0
GO TO 70
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 2
END IF
*
* Overdetermined case
*
IF( M.GE.N ) THEN
*
* Path 1 - overdetermined or exactly determined
*
MM = M
IF( M.GE.MNTHR ) THEN
*
* Path 1a - overdetermined, with many more rows than columns
*
MM = N
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
*
* Multiply B by transpose(Q)
* (Workspace: need N+NRHS, prefer N+NRHS*NB)
*
CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ),
$ B,
$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
* Zero out below R
*
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
END IF
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in A
* (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
*
CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors of R
* (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA,
$ WORK( ITAUQ ),
$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
* Generate right bidiagonalizing vectors of R in A
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
IWORK = IE + N
*
* Perform bidiagonal QR iteration
* multiply B by transpose of left singular vectors
* compute right singular vectors in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
$ 1, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70
*
* Multiply B by reciprocals of singular values
*
THR = MAX( RCOND*S( 1 ), SFMIN )
IF( RCOND.LT.ZERO )
$ THR = MAX( EPS*S( 1 ), SFMIN )
RANK = 0
DO 10 I = 1, N
IF( S( I ).GT.THR ) THEN
CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
RANK = RANK + 1
ELSE
CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ),
$ LDB )
END IF
10 CONTINUE
*
* Multiply B by right singular vectors
* (Workspace: need N, prefer N*NRHS)
*
IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB,
$ ZERO,
$ WORK, LDB )
CALL DLACPY( 'G', N, NRHS, WORK, LDB, B, LDB )
ELSE IF( NRHS.GT.1 ) THEN
CHUNK = LWORK / N
DO 20 I = 1, NRHS, CHUNK
BL = MIN( NRHS-I+1, CHUNK )
CALL DGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1,
$ I ),
$ LDB, ZERO, WORK, N )
CALL DLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB )
20 CONTINUE
ELSE IF( NRHS.EQ.1 ) THEN
CALL DGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
CALL DCOPY( N, WORK, 1, B, 1 )
END IF
*
ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
$ MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN
*
* Path 2a - underdetermined, with many more columns than rows
* and sufficient workspace for an efficient algorithm
*
LDWORK = M
IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
$ M*LDA+M+M*NRHS ) )LDWORK = LDA
ITAU = 1
IWORK = M + 1
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
IL = IWORK
*
* Copy L to WORK(IL), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
$ LDWORK )
IE = IL + LDWORK*M
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
* (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors of L
* (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
$ WORK( ITAUQ ), B, LDB, WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
*
* Generate right bidiagonalizing vectors of R in WORK(IL)
* (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IL ), LDWORK,
$ WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
IWORK = IE + M
*
* Perform bidiagonal QR iteration,
* computing right singular vectors of L in WORK(IL) and
* multiplying B by transpose of left singular vectors
* (Workspace: need M*M+M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ),
$ LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70
*
* Multiply B by reciprocals of singular values
*
THR = MAX( RCOND*S( 1 ), SFMIN )
IF( RCOND.LT.ZERO )
$ THR = MAX( EPS*S( 1 ), SFMIN )
RANK = 0
DO 30 I = 1, M
IF( S( I ).GT.THR ) THEN
CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
RANK = RANK + 1
ELSE
CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ),
$ LDB )
END IF
30 CONTINUE
IWORK = IE
*
* Multiply B by right singular vectors of L in WORK(IL)
* (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS)
*
IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN
CALL DGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ),
$ LDWORK,
$ B, LDB, ZERO, WORK( IWORK ), LDB )
CALL DLACPY( 'G', M, NRHS, WORK( IWORK ), LDB, B, LDB )
ELSE IF( NRHS.GT.1 ) THEN
CHUNK = ( LWORK-IWORK+1 ) / M
DO 40 I = 1, NRHS, CHUNK
BL = MIN( NRHS-I+1, CHUNK )
CALL DGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ),
$ LDWORK,
$ B( 1, I ), LDB, ZERO, WORK( IWORK ), M )
CALL DLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ),
$ LDB )
40 CONTINUE
ELSE IF( NRHS.EQ.1 ) THEN
CALL DGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1,
$ 1 ),1, ZERO, WORK( IWORK ), 1 )
CALL DCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 )
END IF
*
* Zero out below first M rows of B
*
CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
IWORK = ITAU + M
*
* Multiply transpose(Q) by B
* (Workspace: need M+NRHS, prefer M+NRHS*NB)
*
CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
ELSE
*
* Path 2 - remaining underdetermined cases
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize A
* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors
* (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
$ WORK( ITAUQ ),
$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
* Generate right bidiagonalizing vectors in A
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
IWORK = IE + M
*
* Perform bidiagonal QR iteration,
* computing right singular vectors of A in A and
* multiplying B by transpose of left singular vectors
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
$ 1, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70
*
* Multiply B by reciprocals of singular values
*
THR = MAX( RCOND*S( 1 ), SFMIN )
IF( RCOND.LT.ZERO )
$ THR = MAX( EPS*S( 1 ), SFMIN )
RANK = 0
DO 50 I = 1, M
IF( S( I ).GT.THR ) THEN
CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
RANK = RANK + 1
ELSE
CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ),
$ LDB )
END IF
50 CONTINUE
*
* Multiply B by right singular vectors of A
* (Workspace: need N, prefer N*NRHS)
*
IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
CALL DGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB,
$ ZERO,
$ WORK, LDB )
CALL DLACPY( 'F', N, NRHS, WORK, LDB, B, LDB )
ELSE IF( NRHS.GT.1 ) THEN
CHUNK = LWORK / N
DO 60 I = 1, NRHS, CHUNK
BL = MIN( NRHS-I+1, CHUNK )
CALL DGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1,
$ I ),
$ LDB, ZERO, WORK, N )
CALL DLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB )
60 CONTINUE
ELSE IF( NRHS.EQ.1 ) THEN
CALL DGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
CALL DCOPY( N, WORK, 1, B, 1 )
END IF
END IF
*
* Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
END IF
*
70 CONTINUE
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGELSS
*
END
*> \brief <b> DGELSX solves overdetermined or underdetermined systems for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
* WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGELSY.
*>
*> DGELSX computes the minimum-norm solution to a real linear least
*> squares problem:
*> minimize || A * X - B ||
*> using a complete orthogonal factorization of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*>
*> The routine first computes a QR factorization with column pivoting:
*> A * P = Q * [ R11 R12 ]
*> [ 0 R22 ]
*> with R11 defined as the largest leading submatrix whose estimated
*> condition number is less than 1/RCOND. The order of R11, RANK,
*> is the effective rank of A.
*>
*> Then, R22 is considered to be negligible, and R12 is annihilated
*> by orthogonal transformations from the right, arriving at the
*> complete orthogonal factorization:
*> A * P = Q * [ T11 0 ] * Z
*> [ 0 0 ]
*> The minimum-norm solution is then
*> X = P * Z**T [ inv(T11)*Q1**T*B ]
*> [ 0 ]
*> where Q1 consists of the first RANK columns of Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of
*> columns of matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, A has been overwritten by details of its
*> complete orthogonal factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the M-by-NRHS right hand side matrix B.
*> On exit, the N-by-NRHS solution matrix X.
*> If m >= n and RANK = n, the residual sum-of-squares for
*> the solution in the i-th column is given by the sum of
*> squares of elements N+1:M in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,M,N).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(i) .ne. 0, the i-th column of A is an
*> initial column, otherwise it is a free column. Before
*> the QR factorization of A, all initial columns are
*> permuted to the leading positions; only the remaining
*> free columns are moved as a result of column pivoting
*> during the factorization.
*> On exit, if JPVT(i) = k, then the i-th column of A*P
*> was the k-th column of A.
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> RCOND is used to determine the effective rank of A, which
*> is defined as the order of the largest leading triangular
*> submatrix R11 in the QR factorization with pivoting of A,
*> whose estimated condition number < 1/RCOND.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The effective rank of A, i.e., the order of the submatrix
*> R11. This is the same as the order of the submatrix T11
*> in the complete orthogonal factorization of A.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEsolve
*
* =====================================================================
SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND,
$ RANK, WORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER IMAX, IMIN
PARAMETER ( IMAX = 1, IMIN = 2 )
DOUBLE PRECISION ZERO, ONE, DONE, NTDONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, DONE = ZERO,
$ NTDONE = ONE )
* ..
* .. Local Scalars ..
INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
$ SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DGEQPF, DLAIC1, DLASCL, DLASET, DLATZM, DORM2R,
$ DTRSM, DTZRQF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
MN = MIN( M, N )
ISMIN = MN + 1
ISMAX = 2*MN + 1
*
* Test the input arguments.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
INFO = -7
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELSX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N, NRHS ).EQ.0 ) THEN
RANK = 0
RETURN
END IF
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
*
* Scale A, B if max elements outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
IASCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
IASCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
IASCL = 2
ELSE IF( ANRM.EQ.ZERO ) THEN
*
* Matrix all zero. Return zero solution.
*
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
RANK = 0
GO TO 100
END IF
*
BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 2
END IF
*
* Compute QR factorization with column pivoting of A:
* A * P = Q * R
*
CALL DGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
$ INFO )
*
* workspace 3*N. Details of Householder rotations stored
* in WORK(1:MN).
*
* Determine RANK using incremental condition estimation
*
WORK( ISMIN ) = ONE
WORK( ISMAX ) = ONE
SMAX = ABS( A( 1, 1 ) )
SMIN = SMAX
IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
RANK = 0
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
GO TO 100
ELSE
RANK = 1
END IF
*
10 CONTINUE
IF( RANK.LT.MN ) THEN
I = RANK + 1
CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
$ A( I, I ), SMINPR, S1, C1 )
CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
$ A( I, I ), SMAXPR, S2, C2 )
*
IF( SMAXPR*RCOND.LE.SMINPR ) THEN
DO 20 I = 1, RANK
WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
20 CONTINUE
WORK( ISMIN+RANK ) = C1
WORK( ISMAX+RANK ) = C2
SMIN = SMINPR
SMAX = SMAXPR
RANK = RANK + 1
GO TO 10
END IF
END IF
*
* Logically partition R = [ R11 R12 ]
* [ 0 R22 ]
* where R11 = R(1:RANK,1:RANK)
*
* [R11,R12] = [ T11, 0 ] * Y
*
IF( RANK.LT.N )
$ CALL DTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
*
* Details of Householder rotations stored in WORK(MN+1:2*MN)
*
* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
CALL DORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA,
$ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), INFO )
*
* workspace NRHS
*
* B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
*
CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
$ NRHS, ONE, A, LDA, B, LDB )
*
DO 40 I = RANK + 1, N
DO 30 J = 1, NRHS
B( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
*
* B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
*
IF( RANK.LT.N ) THEN
DO 50 I = 1, RANK
CALL DLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
$ WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB,
$ WORK( 2*MN+1 ) )
50 CONTINUE
END IF
*
* workspace NRHS
*
* B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
*
DO 90 J = 1, NRHS
DO 60 I = 1, N
WORK( 2*MN+I ) = NTDONE
60 CONTINUE
DO 80 I = 1, N
IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
IF( JPVT( I ).NE.I ) THEN
K = I
T1 = B( K, J )
T2 = B( JPVT( K ), J )
70 CONTINUE
B( JPVT( K ), J ) = T1
WORK( 2*MN+K ) = DONE
T1 = T2
K = JPVT( K )
T2 = B( JPVT( K ), J )
IF( JPVT( K ).NE.I )
$ GO TO 70
B( I, J ) = T1
WORK( 2*MN+K ) = DONE
END IF
END IF
80 CONTINUE
90 CONTINUE
*
* Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
END IF
*
100 CONTINUE
*
RETURN
*
* End of DGELSX
*
END
*> \brief <b> DGELSY solves overdetermined or underdetermined systems for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsy.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELSY computes the minimum-norm solution to a real linear least
*> squares problem:
*> minimize || A * X - B ||
*> using a complete orthogonal factorization of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*>
*> The routine first computes a QR factorization with column pivoting:
*> A * P = Q * [ R11 R12 ]
*> [ 0 R22 ]
*> with R11 defined as the largest leading submatrix whose estimated
*> condition number is less than 1/RCOND. The order of R11, RANK,
*> is the effective rank of A.
*>
*> Then, R22 is considered to be negligible, and R12 is annihilated
*> by orthogonal transformations from the right, arriving at the
*> complete orthogonal factorization:
*> A * P = Q * [ T11 0 ] * Z
*> [ 0 0 ]
*> The minimum-norm solution is then
*> X = P * Z**T [ inv(T11)*Q1**T*B ]
*> [ 0 ]
*> where Q1 consists of the first RANK columns of Q.
*>
*> This routine is basically identical to the original xGELSX except
*> three differences:
*> o The call to the subroutine xGEQPF has been substituted by the
*> the call to the subroutine xGEQP3. This subroutine is a Blas-3
*> version of the QR factorization with column pivoting.
*> o Matrix B (the right hand side) is updated with Blas-3.
*> o The permutation of matrix B (the right hand side) is faster and
*> more simple.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of
*> columns of matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, A has been overwritten by details of its
*> complete orthogonal factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the M-by-NRHS right hand side matrix B.
*> On exit, the N-by-NRHS solution matrix X.
*> If M = 0 or N = 0, B is not referenced.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,M,N).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*> to the front of AP, otherwise column i is a free column.
*> On exit, if JPVT(i) = k, then the i-th column of AP
*> was the k-th column of A.
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> RCOND is used to determine the effective rank of A, which
*> is defined as the order of the largest leading triangular
*> submatrix R11 in the QR factorization with pivoting of A,
*> whose estimated condition number < 1/RCOND.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The effective rank of A, i.e., the order of the submatrix
*> R11. This is the same as the order of the submatrix T11
*> in the complete orthogonal factorization of A.
*> If NRHS = 0, RANK = 0 on output.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> The unblocked strategy requires that:
*> LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
*> where MN = min( M, N ).
*> The block algorithm requires that:
*> LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
*> where NB is an upper bound on the blocksize returned
*> by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
*> and DORMRZ.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: If INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelsy
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n
*> E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
*>
* =====================================================================
SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND,
$ RANK,
$ WORK, LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER IMAX, IMIN
PARAMETER ( IMAX = 1, IMIN = 2 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,
$ LWKOPT, MN, NB, NB1, NB2, NB3, NB4
DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
$ SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL ILAENV, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEQP3, DLAIC1, DLASCL,
$ DLASET,
$ DORMQR, DORMRZ, DTRSM, DTZRZF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
MN = MIN( M, N )
ISMIN = MN + 1
ISMAX = 2*MN + 1
*
* Test the input arguments.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
INFO = -7
END IF
*
* Figure out optimal block size
*
IF( INFO.EQ.0 ) THEN
IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, NRHS, -1 )
NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, NRHS, -1 )
NB = MAX( NB1, NB2, NB3, NB4 )
LWKMIN = MN + MAX( 2*MN, N + 1, MN + NRHS )
LWKOPT = MAX( LWKMIN,
$ MN + 2*N + NB*( N + 1 ), 2*MN + NB*NRHS )
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELSY', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
RANK = 0
RETURN
END IF
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
*
* Scale A, B if max entries outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
IASCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
IASCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
IASCL = 2
ELSE IF( ANRM.EQ.ZERO ) THEN
*
* Matrix all zero. Return zero solution.
*
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
RANK = 0
GO TO 70
END IF
*
BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
$ INFO )
IBSCL = 2
END IF
*
* Compute QR factorization with column pivoting of A:
* A * P = Q * R
*
CALL DGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
$ LWORK-MN, INFO )
WSIZE = MN + WORK( MN+1 )
*
* workspace: MN+2*N+NB*(N+1).
* Details of Householder rotations stored in WORK(1:MN).
*
* Determine RANK using incremental condition estimation
*
WORK( ISMIN ) = ONE
WORK( ISMAX ) = ONE
SMAX = ABS( A( 1, 1 ) )
SMIN = SMAX
IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
RANK = 0
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
GO TO 70
ELSE
RANK = 1
END IF
*
10 CONTINUE
IF( RANK.LT.MN ) THEN
I = RANK + 1
CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
$ A( I, I ), SMINPR, S1, C1 )
CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
$ A( I, I ), SMAXPR, S2, C2 )
*
IF( SMAXPR*RCOND.LE.SMINPR ) THEN
DO 20 I = 1, RANK
WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
20 CONTINUE
WORK( ISMIN+RANK ) = C1
WORK( ISMAX+RANK ) = C2
SMIN = SMINPR
SMAX = SMAXPR
RANK = RANK + 1
GO TO 10
END IF
END IF
*
* workspace: 3*MN.
*
* Logically partition R = [ R11 R12 ]
* [ 0 R22 ]
* where R11 = R(1:RANK,1:RANK)
*
* [R11,R12] = [ T11, 0 ] * Y
*
IF( RANK.LT.N )
$ CALL DTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
$ LWORK-2*MN, INFO )
*
* workspace: 2*MN.
* Details of Householder rotations stored in WORK(MN+1:2*MN)
*
* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
CALL DORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA,
$ WORK( 1 ),
$ B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
WSIZE = MAX( WSIZE, 2*MN+WORK( 2*MN+1 ) )
*
* workspace: 2*MN+NB*NRHS.
*
* B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
*
CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
$ NRHS, ONE, A, LDA, B, LDB )
*
DO 40 J = 1, NRHS
DO 30 I = RANK + 1, N
B( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
*
* B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
*
IF( RANK.LT.N ) THEN
CALL DORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A,
$ LDA, WORK( MN+1 ), B, LDB, WORK( 2*MN+1 ),
$ LWORK-2*MN, INFO )
END IF
*
* workspace: 2*MN+NRHS.
*
* B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
*
DO 60 J = 1, NRHS
DO 50 I = 1, N
WORK( JPVT( I ) ) = B( I, J )
50 CONTINUE
CALL DCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
60 CONTINUE
*
* workspace: N.
*
* Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
$ INFO )
CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
$ INFO )
END IF
*
70 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGELSY
*
END
*> \brief \b DGEMQRT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEMQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgemqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgemqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgemqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEMQRT( SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT,
* C, LDC, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDV, LDC, M, N, NB, LDT
* ..
* .. Array Arguments ..
* DOUBLE PRECISION V( LDV, * ), C( LDC, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEMQRT overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q C C Q
*> TRANS = 'T': Q**T C C Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of K
*> elementary reflectors:
*>
*> Q = H(1) H(2) . . . H(K) = I - V T V**T
*>
*> generated using the compact WY representation as returned by DGEQRT.
*>
*> Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'C': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The block size used for the storage of T. K >= NB >= 1.
*> This must be the same value of NB used to generate T
*> in DGEQRT.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQRT in the first K columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The upper triangular factors of the block reflectors
*> as returned by DGEQRT, stored as a NB-by-N matrix.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array. The dimension of
*> WORK is N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gemqrt
*
* =====================================================================
SUBROUTINE DGEMQRT( SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT,
$ C, LDC, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDV, LDC, M, N, NB, LDT
* ..
* .. Array Arguments ..
DOUBLE PRECISION V( LDV, * ), C( LDC, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* ..
* .. Local Scalars ..
LOGICAL LEFT, RIGHT, TRAN, NOTRAN
INTEGER I, IB, LDWORK, KF, Q
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DLARFB
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* .. Test the input arguments ..
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
RIGHT = LSAME( SIDE, 'R' )
TRAN = LSAME( TRANS, 'T' )
NOTRAN = LSAME( TRANS, 'N' )
*
IF( LEFT ) THEN
LDWORK = MAX( 1, N )
Q = M
ELSE IF ( RIGHT ) THEN
LDWORK = MAX( 1, M )
Q = N
END IF
IF( .NOT.LEFT .AND. .NOT.RIGHT ) THEN
INFO = -1
ELSE IF( .NOT.TRAN .AND. .NOT.NOTRAN ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.Q ) THEN
INFO = -5
ELSE IF( NB.LT.1 .OR. (NB.GT.K .AND. K.GT.0)) THEN
INFO = -6
ELSE IF( LDV.LT.MAX( 1, Q ) ) THEN
INFO = -8
ELSE IF( LDT.LT.NB ) THEN
INFO = -10
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -12
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEMQRT', -INFO )
RETURN
END IF
*
* .. Quick return if possible ..
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) RETURN
*
IF( LEFT .AND. TRAN ) THEN
*
DO I = 1, K, NB
IB = MIN( NB, K-I+1 )
CALL DLARFB( 'L', 'T', 'F', 'C', M-I+1, N, IB,
$ V( I, I ), LDV, T( 1, I ), LDT,
$ C( I, 1 ), LDC, WORK, LDWORK )
END DO
*
ELSE IF( RIGHT .AND. NOTRAN ) THEN
*
DO I = 1, K, NB
IB = MIN( NB, K-I+1 )
CALL DLARFB( 'R', 'N', 'F', 'C', M, N-I+1, IB,
$ V( I, I ), LDV, T( 1, I ), LDT,
$ C( 1, I ), LDC, WORK, LDWORK )
END DO
*
ELSE IF( LEFT .AND. NOTRAN ) THEN
*
KF = ((K-1)/NB)*NB+1
DO I = KF, 1, -NB
IB = MIN( NB, K-I+1 )
CALL DLARFB( 'L', 'N', 'F', 'C', M-I+1, N, IB,
$ V( I, I ), LDV, T( 1, I ), LDT,
$ C( I, 1 ), LDC, WORK, LDWORK )
END DO
*
ELSE IF( RIGHT .AND. TRAN ) THEN
*
KF = ((K-1)/NB)*NB+1
DO I = KF, 1, -NB
IB = MIN( NB, K-I+1 )
CALL DLARFB( 'R', 'T', 'F', 'C', M, N-I+1, IB,
$ V( I, I ), LDV, T( 1, I ), LDT,
$ C( 1, I ), LDC, WORK, LDWORK )
END DO
*
END IF
*
RETURN
*
* End of DGEMQRT
*
END
*> \brief \b DGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeql2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeql2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeql2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQL2 computes a QL factorization of a real m by n matrix A:
*> A = Q * L.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, if m >= n, the lower triangle of the subarray
*> A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
*> if m <= n, the elements on and below the (n-m)-th
*> superdiagonal contain the m by n lower trapezoidal matrix L;
*> the remaining elements, with the array TAU, represent the
*> orthogonal matrix Q as a product of elementary reflectors
*> (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geql2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
*> A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
* ..
* .. External Subroutines ..
EXTERNAL DLARF1L, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQL2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = K, 1, -1
*
* Generate elementary reflector H(i) to annihilate
* A(1:m-k+i-1,n-k+i)
*
CALL DLARFG( M-K+I, A( M-K+I, N-K+I ), A( 1, N-K+I ), 1,
$ TAU( I ) )
*
* Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left
*
CALL DLARF1L( 'Left', M-K+I, N-K+I-1, A( 1, N-K+I ), 1,
$ TAU( I ),
$ A, LDA, WORK )
10 CONTINUE
RETURN
*
* End of DGEQL2
*
END
*> \brief \b DGEQLF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQLF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqlf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqlf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqlf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQLF computes a QL factorization of a real M-by-N matrix A:
*> A = Q * L.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit,
*> if m >= n, the lower triangle of the subarray
*> A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
*> if m <= n, the elements on and below the (n-m)-th
*> superdiagonal contain the M-by-N lower trapezoidal matrix L;
*> the remaining elements, with the array TAU, represent the
*> orthogonal matrix Q as a product of elementary reflectors
*> (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqlf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
*> A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
$ MU, NB, NBMIN, NU, NX
* ..
* .. External Subroutines ..
EXTERNAL DGEQL2, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
*
IF( INFO.EQ.0 ) THEN
K = MIN( M, N )
IF( K.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = ILAENV( 1, 'DGEQLF', ' ', M, N, -1, -1 )
LWKOPT = N*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( .NOT.LQUERY ) THEN
IF( LWORK.LE.0 .OR. ( M.GT.0 .AND. LWORK.LT.MAX( 1, N ) ) )
$ INFO = -7
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQLF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 ) THEN
RETURN
END IF
*
NBMIN = 2
NX = 1
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGEQLF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGEQLF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially.
* The last kk columns are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
IB = MIN( K-I+1, NB )
*
* Compute the QL factorization of the current block
* A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1)
*
CALL DGEQL2( M-K+I+IB-1, IB, A( 1, N-K+I ), LDA,
$ TAU( I ),
$ WORK, IINFO )
IF( N-K+I.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
$ A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H**T to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
*
CALL DLARFB( 'Left', 'Transpose', 'Backward',
$ 'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
$ A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
$ WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
MU = M - K + I + NB - 1
NU = N - K + I + NB - 1
ELSE
MU = M
NU = N
END IF
*
* Use unblocked code to factor the last or only block
*
IF( MU.GT.0 .AND. NU.GT.0 )
$ CALL DGEQL2( MU, NU, A, LDA, TAU, WORK, IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGEQLF
*
END
*> \brief \b DGEQP3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQP3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqp3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqp3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqp3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQP3 computes a QR factorization with column pivoting of a
*> matrix A: A*P = Q*R using Level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the upper triangle of the array contains the
*> min(M,N)-by-N upper trapezoidal matrix R; the elements below
*> the diagonal, together with the array TAU, represent the
*> orthogonal matrix Q as a product of min(M,N) elementary
*> reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(J).ne.0, the J-th column of A is permuted
*> to the front of A*P (a leading column); if JPVT(J)=0,
*> the J-th column of A is a free column.
*> On exit, if JPVT(J)=K, then the J-th column of A*P was the
*> the K-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO=0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 3*N+1.
*> For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
*> is the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqp3
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real/complex vector
*> with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
*> A(i+1:m,i), and tau in TAU(i).
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*> X. Sun, Computer Science Dept., Duke University, USA
*>
* =====================================================================
SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER INB, INBMIN, IXOVER
PARAMETER ( INB = 1, INBMIN = 2, IXOVER = 3 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB,
$ NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DLAQP2, DLAQPS, DORMQR, DSWAP,
$ XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DNRM2
EXTERNAL ILAENV, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test input arguments
* ====================
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
*
IF( INFO.EQ.0 ) THEN
MINMN = MIN( M, N )
IF( MINMN.EQ.0 ) THEN
IWS = 1
LWKOPT = 1
ELSE
IWS = 3*N + 1
NB = ILAENV( INB, 'DGEQRF', ' ', M, N, -1, -1 )
LWKOPT = 2*N + ( N + 1 )*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQP3', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Move initial columns up front.
*
NFXD = 1
DO 10 J = 1, N
IF( JPVT( J ).NE.0 ) THEN
IF( J.NE.NFXD ) THEN
CALL DSWAP( M, A( 1, J ), 1, A( 1, NFXD ), 1 )
JPVT( J ) = JPVT( NFXD )
JPVT( NFXD ) = J
ELSE
JPVT( J ) = J
END IF
NFXD = NFXD + 1
ELSE
JPVT( J ) = J
END IF
10 CONTINUE
NFXD = NFXD - 1
*
* Factorize fixed columns
* =======================
*
* Compute the QR factorization of fixed columns and update
* remaining columns.
*
IF( NFXD.GT.0 ) THEN
NA = MIN( M, NFXD )
*CC CALL DGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
CALL DGEQRF( M, NA, A, LDA, TAU, WORK, LWORK, INFO )
IWS = MAX( IWS, INT( WORK( 1 ) ) )
IF( NA.LT.N ) THEN
*CC CALL DORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
*CC $ TAU, A( 1, NA+1 ), LDA, WORK, INFO )
CALL DORMQR( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
$ TAU,
$ A( 1, NA+1 ), LDA, WORK, LWORK, INFO )
IWS = MAX( IWS, INT( WORK( 1 ) ) )
END IF
END IF
*
* Factorize free columns
* ======================
*
IF( NFXD.LT.MINMN ) THEN
*
SM = M - NFXD
SN = N - NFXD
SMINMN = MINMN - NFXD
*
* Determine the block size.
*
NB = ILAENV( INB, 'DGEQRF', ' ', SM, SN, -1, -1 )
NBMIN = 2
NX = 0
*
IF( ( NB.GT.1 ) .AND. ( NB.LT.SMINMN ) ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( IXOVER, 'DGEQRF', ' ', SM, SN, -1,
$ -1 ) )
*
*
IF( NX.LT.SMINMN ) THEN
*
* Determine if workspace is large enough for blocked code.
*
MINWS = 2*SN + ( SN+1 )*NB
IWS = MAX( IWS, MINWS )
IF( LWORK.LT.MINWS ) THEN
*
* Not enough workspace to use optimal NB: Reduce NB and
* determine the minimum value of NB.
*
NB = ( LWORK-2*SN ) / ( SN+1 )
NBMIN = MAX( 2, ILAENV( INBMIN, 'DGEQRF', ' ', SM,
$ SN,
$ -1, -1 ) )
*
*
END IF
END IF
END IF
*
* Initialize partial column norms. The first N elements of work
* store the exact column norms.
*
DO 20 J = NFXD + 1, N
WORK( J ) = DNRM2( SM, A( NFXD+1, J ), 1 )
WORK( N+J ) = WORK( J )
20 CONTINUE
*
IF( ( NB.GE.NBMIN ) .AND. ( NB.LT.SMINMN ) .AND.
$ ( NX.LT.SMINMN ) ) THEN
*
* Use blocked code initially.
*
J = NFXD + 1
*
* Compute factorization: while loop.
*
*
TOPBMN = MINMN - NX
30 CONTINUE
IF( J.LE.TOPBMN ) THEN
JB = MIN( NB, TOPBMN-J+1 )
*
* Factorize JB columns among columns J:N.
*
CALL DLAQPS( M, N-J+1, J-1, JB, FJB, A( 1, J ), LDA,
$ JPVT( J ), TAU( J ), WORK( J ), WORK( N+J ),
$ WORK( 2*N+1 ), WORK( 2*N+JB+1 ), N-J+1 )
*
J = J + FJB
GO TO 30
END IF
ELSE
J = NFXD + 1
END IF
*
* Use unblocked code to factor the last or only block.
*
*
IF( J.LE.MINMN )
$ CALL DLAQP2( M, N-J+1, J-1, A( 1, J ), LDA, JPVT( J ),
$ TAU( J ), WORK( J ), WORK( N+J ),
$ WORK( 2*N+1 ) )
*
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DGEQP3
*
END
*> \brief \b DGEQPF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQPF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqpf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqpf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqpf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGEQP3.
*>
*> DGEQPF computes a QR factorization with column pivoting of a
*> real M-by-N matrix A: A*P = Q*R.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the upper triangle of the array contains the
*> min(M,N)-by-N upper triangular matrix R; the elements
*> below the diagonal, together with the array TAU,
*> represent the orthogonal matrix Q as a product of
*> min(m,n) elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*> to the front of A*P (a leading column); if JPVT(i) = 0,
*> the i-th column of A is a free column.
*> On exit, if JPVT(i) = k, then the i-th column of A*P
*> was the k-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(n)
*>
*> Each H(i) has the form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
*>
*> The matrix P is represented in jpvt as follows: If
*> jpvt(j) = i
*> then the jth column of P is the ith canonical unit vector.
*>
*> Partial column norm updating strategy modified by
*> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*> University of Zagreb, Croatia.
*> -- April 2011 --
*> For more details see LAPACK Working Note 176.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, ITEMP, J, MA, MN, PVT
DOUBLE PRECISION AII, TEMP, TEMP2, TOL3Z
* ..
* .. External Subroutines ..
EXTERNAL DGEQR2, DLARF, DLARFG, DORM2R, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DNRM2
EXTERNAL IDAMAX, DLAMCH, DNRM2
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQPF', -INFO )
RETURN
END IF
*
MN = MIN( M, N )
TOL3Z = SQRT(DLAMCH('Epsilon'))
*
* Move initial columns up front
*
ITEMP = 1
DO 10 I = 1, N
IF( JPVT( I ).NE.0 ) THEN
IF( I.NE.ITEMP ) THEN
CALL DSWAP( M, A( 1, I ), 1, A( 1, ITEMP ), 1 )
JPVT( I ) = JPVT( ITEMP )
JPVT( ITEMP ) = I
ELSE
JPVT( I ) = I
END IF
ITEMP = ITEMP + 1
ELSE
JPVT( I ) = I
END IF
10 CONTINUE
ITEMP = ITEMP - 1
*
* Compute the QR factorization and update remaining columns
*
IF( ITEMP.GT.0 ) THEN
MA = MIN( ITEMP, M )
CALL DGEQR2( M, MA, A, LDA, TAU, WORK, INFO )
IF( MA.LT.N ) THEN
CALL DORM2R( 'Left', 'Transpose', M, N-MA, MA, A, LDA, TAU,
$ A( 1, MA+1 ), LDA, WORK, INFO )
END IF
END IF
*
IF( ITEMP.LT.MN ) THEN
*
* Initialize partial column norms. The first n elements of
* work store the exact column norms.
*
DO 20 I = ITEMP + 1, N
WORK( I ) = DNRM2( M-ITEMP, A( ITEMP+1, I ), 1 )
WORK( N+I ) = WORK( I )
20 CONTINUE
*
* Compute factorization
*
DO 40 I = ITEMP + 1, MN
*
* Determine ith pivot column and swap if necessary
*
PVT = ( I-1 ) + IDAMAX( N-I+1, WORK( I ), 1 )
*
IF( PVT.NE.I ) THEN
CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
ITEMP = JPVT( PVT )
JPVT( PVT ) = JPVT( I )
JPVT( I ) = ITEMP
WORK( PVT ) = WORK( I )
WORK( N+PVT ) = WORK( N+I )
END IF
*
* Generate elementary reflector H(i)
*
IF( I.LT.M ) THEN
CALL DLARFG( M-I+1, A( I, I ), A( I+1, I ), 1,
$ TAU( I ) )
ELSE
CALL DLARFG( 1, A( M, M ), A( M, M ), 1, TAU( M ) )
END IF
*
IF( I.LT.N ) THEN
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( 'LEFT', M-I+1, N-I, A( I, I ), 1,
$ TAU( I ), A( I, I+1 ), LDA, WORK( 2*N+1 ) )
A( I, I ) = AII
END IF
*
* Update partial column norms
*
DO 30 J = I + 1, N
IF( WORK( J ).NE.ZERO ) THEN
*
* NOTE: The following 4 lines follow from the analysis in
* Lapack Working Note 176.
*
TEMP = ABS( A( I, J ) ) / WORK( J )
TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
TEMP2 = TEMP*( WORK( J ) / WORK( N+J ) )**2
IF( TEMP2 .LE. TOL3Z ) THEN
IF( M-I.GT.0 ) THEN
WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )
WORK( N+J ) = WORK( J )
ELSE
WORK( J ) = ZERO
WORK( N+J ) = ZERO
END IF
ELSE
WORK( J ) = WORK( J )*SQRT( TEMP )
END IF
END IF
30 CONTINUE
*
40 CONTINUE
END IF
RETURN
*
* End of DGEQPF
*
END
*> \brief \b DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQR2 computes a QR factorization of a real m-by-n matrix A:
*>
*> A = Q * ( R ),
*> ( 0 )
*>
*> where:
*>
*> Q is a m-by-m orthogonal matrix;
*> R is an upper-triangular n-by-n matrix;
*> 0 is a (m-n)-by-n zero matrix, if m > n.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(m,n) by n upper trapezoidal matrix R (R is
*> upper triangular if m >= n); the elements below the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqr2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQR2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = 1, K
*
* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAU( I ) )
IF( I.LT.N ) THEN
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK )
END IF
10 CONTINUE
RETURN
*
* End of DGEQR2
*
END
*> \brief \b DGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQR2P + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqr2p.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqr2p.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqr2p.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQR2P( M, N, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQR2P computes a QR factorization of a real m-by-n matrix A:
*>
*> A = Q * ( R ),
*> ( 0 )
*>
*> where:
*>
*> Q is a m-by-m orthogonal matrix;
*> R is an upper-triangular n-by-n matrix with nonnegative diagonal
*> entries;
*> 0 is a (m-n)-by-n zero matrix, if m > n.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(m,n) by n upper trapezoidal matrix R (R is
*> upper triangular if m >= n). The diagonal entries of R are
*> nonnegative; the elements below the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqr2p
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*> and tau in TAU(i).
*>
*> See Lapack Working Note 203 for details
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQR2P( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, DLARFGP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQR2P', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = 1, K
*
* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
CALL DLARFGP( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAU( I ) )
IF( I.LT.N ) THEN
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK )
END IF
10 CONTINUE
RETURN
*
* End of DGEQR2P
*
END
*> \brief \b DGEQRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQRF computes a QR factorization of a real M-by-N matrix A:
*>
*> A = Q * ( R ),
*> ( 0 )
*>
*> where:
*>
*> Q is a M-by-M orthogonal matrix;
*> R is an upper-triangular N-by-N matrix;
*> 0 is a (M-N)-by-N zero matrix, if M > N.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*> upper triangular if m >= n); the elements below the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of min(m,n) elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*> For optimum performance LWORK >= N*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DGEQR2, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
K = MIN( M, N )
INFO = 0
NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( .NOT.LQUERY ) THEN
IF( LWORK.LE.0 .OR. ( M.GT.0 .AND. LWORK.LT.MAX( 1, N ) ) )
$ INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
IF( K.EQ.0 ) THEN
LWKOPT = 1
ELSE
LWKOPT = N*NB
END IF
WORK( 1 ) = LWKOPT
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially
*
DO 10 I = 1, K - NX, NB
IB = MIN( K-I+1, NB )
*
* Compute the QR factorization of the current block
* A(i:m,i:i+ib-1)
*
CALL DGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
IF( I+IB.LE.N ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
$ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H**T to A(i:m,i+ib:n) from the left
*
CALL DLARFB( 'Left', 'Transpose', 'Forward',
$ 'Columnwise', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
$ LDA, WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
ELSE
I = 1
END IF
*
* Use unblocked code to factor the last or only block.
*
IF( I.LE.K )
$ CALL DGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGEQRF
*
END
*> \brief \b DGEQRFP
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRFP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrfp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrfp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrfp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQRFP( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQR2P computes a QR factorization of a real M-by-N matrix A:
*>
*> A = Q * ( R ),
*> ( 0 )
*>
*> where:
*>
*> Q is a M-by-M orthogonal matrix;
*> R is an upper-triangular N-by-N matrix with nonnegative diagonal
*> entries;
*> 0 is a (M-N)-by-N zero matrix, if M > N.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*> upper triangular if m >= n). The diagonal entries of R
*> are nonnegative; the elements below the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of min(m,n) elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*> For optimum performance LWORK >= N*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrfp
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*> and tau in TAU(i).
*>
*> See Lapack Working Note 203 for details
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQRFP( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKMIN, LWKOPT,
$ NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DGEQR2P, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
K = MIN( M, N )
IF( K.EQ.0 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
LWKMIN = N
LWKOPT = N*NB
END IF
WORK( 1 ) = LWKOPT
*
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRFP', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = LWKMIN
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially
*
DO 10 I = 1, K - NX, NB
IB = MIN( K-I+1, NB )
*
* Compute the QR factorization of the current block
* A(i:m,i:i+ib-1)
*
CALL DGEQR2P( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
IF( I+IB.LE.N ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
$ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H**T to A(i:m,i+ib:n) from the left
*
CALL DLARFB( 'Left', 'Transpose', 'Forward',
$ 'Columnwise', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
$ LDA, WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
ELSE
I = 1
END IF
*
* Use unblocked code to factor the last or only block.
*
IF( I.LE.K )
$ CALL DGEQR2P( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGEQRFP
*
END
*> \brief \b DGEQRT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQRT( M, N, NB, A, LDA, T, LDT, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQRT computes a blocked QR factorization of a real M-by-N matrix A
*> using the compact WY representation of Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*> upper triangular if M >= N); the elements below the diagonal
*> are the columns of V.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,MIN(M,N))
*> The upper triangular block reflectors stored in compact form
*> as a sequence of upper triangular blocks. See below
*> for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (NB*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrt
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix V stores the elementary reflectors H(i) in the i-th column
*> below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*> V = ( 1 )
*> ( v1 1 )
*> ( v1 v2 1 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> where the vi's represent the vectors which define H(i), which are returned
*> in the matrix A. The 1's along the diagonal of V are not stored in A.
*>
*> Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each
*> block is of order NB except for the last block, which is of order
*> IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block
*> reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
*> for the last block) T's are stored in the NB-by-K matrix T as
*>
*> T = (T1 T2 ... TB).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQRT( M, N, NB, A, LDA, T, LDT, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* ..
* .. Local Scalars ..
INTEGER I, IB, IINFO, K
LOGICAL USE_RECURSIVE_QR
PARAMETER( USE_RECURSIVE_QR=.TRUE. )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRT2, DGEQRT3, DLARFB, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NB.LT.1 .OR. ( NB.GT.MIN(M,N) .AND. MIN(M,N).GT.0 ) )THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDT.LT.NB ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRT', -INFO )
RETURN
END IF
*
* Quick return if possible
*
K = MIN( M, N )
IF( K.EQ.0 ) RETURN
*
* Blocked loop of length K
*
DO I = 1, K, NB
IB = MIN( K-I+1, NB )
*
* Compute the QR factorization of the current block A(I:M,I:I+IB-1)
*
IF( USE_RECURSIVE_QR ) THEN
CALL DGEQRT3( M-I+1, IB, A(I,I), LDA, T(1,I), LDT,
$ IINFO )
ELSE
CALL DGEQRT2( M-I+1, IB, A(I,I), LDA, T(1,I), LDT,
$ IINFO )
END IF
IF( I+IB.LE.N ) THEN
*
* Update by applying H**T to A(I:M,I+IB:N) from the left
*
CALL DLARFB( 'L', 'T', 'F', 'C', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, T( 1, I ), LDT,
$ A( I, I+IB ), LDA, WORK , N-I-IB+1 )
END IF
END DO
RETURN
*
* End of DGEQRT
*
END
*> \brief \b DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRT2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQRT2 computes a QR factorization of a real M-by-N matrix A,
*> using the compact WY representation of Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the real M-by-N matrix A. On exit, the elements on and
*> above the diagonal contain the N-by-N upper triangular matrix R; the
*> elements below the diagonal are the columns of V. See below for
*> further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The N-by-N upper triangular factor of the block reflector.
*> The elements on and above the diagonal contain the block
*> reflector T; the elements below the diagonal are not used.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrt2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix V stores the elementary reflectors H(i) in the i-th column
*> below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*> V = ( 1 )
*> ( v1 1 )
*> ( v1 v2 1 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> where the vi's represent the vectors which define H(i), which are returned
*> in the matrix A. The 1's along the diagonal of V are not stored in A. The
*> block reflector H is then given by
*>
*> H = I - V * T * V**T
*>
*> where V**T is the transpose of V.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDT, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER( ONE = 1.0D+00, ZERO = 0.0D+00 )
* ..
* .. Local Scalars ..
INTEGER I, K
DOUBLE PRECISION AII, ALPHA
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DGEMV, DGER, DTRMV, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( M.LT.N ) THEN
INFO = -1
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRT2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO I = 1, K
*
* Generate elem. refl. H(i) to annihilate A(i+1:m,i), tau(I) -> T(I,1)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ T( I, 1 ) )
IF( I.LT.N ) THEN
*
* Apply H(i) to A(I:M,I+1:N) from the left
*
AII = A( I, I )
A( I, I ) = ONE
*
* W(1:N-I) := A(I:M,I+1:N)^H * A(I:M,I) [W = T(:,N)]
*
CALL DGEMV( 'T',M-I+1, N-I, ONE, A( I, I+1 ), LDA,
$ A( I, I ), 1, ZERO, T( 1, N ), 1 )
*
* A(I:M,I+1:N) = A(I:m,I+1:N) + alpha*A(I:M,I)*W(1:N-1)^H
*
ALPHA = -(T( I, 1 ))
CALL DGER( M-I+1, N-I, ALPHA, A( I, I ), 1,
$ T( 1, N ), 1, A( I, I+1 ), LDA )
A( I, I ) = AII
END IF
END DO
*
DO I = 2, N
AII = A( I, I )
A( I, I ) = ONE
*
* T(1:I-1,I) := alpha * A(I:M,1:I-1)**T * A(I:M,I)
*
ALPHA = -T( I, 1 )
CALL DGEMV( 'T', M-I+1, I-1, ALPHA, A( I, 1 ), LDA,
$ A( I, I ), 1, ZERO, T( 1, I ), 1 )
A( I, I ) = AII
*
* T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
*
CALL DTRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 )
*
* T(I,I) = tau(I)
*
T( I, I ) = T( I, 1 )
T( I, 1) = ZERO
END DO
*
* End of DGEQRT2
*
END
*> \brief \b DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRT3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N, LDT
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQRT3 recursively computes a QR factorization of a real M-by-N
*> matrix A, using the compact WY representation of Q.
*>
*> Based on the algorithm of Elmroth and Gustavson,
*> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the real M-by-N matrix A. On exit, the elements on and
*> above the diagonal contain the N-by-N upper triangular matrix R; the
*> elements below the diagonal are the columns of V. See below for
*> further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The N-by-N upper triangular factor of the block reflector.
*> The elements on and above the diagonal contain the block
*> reflector T; the elements below the diagonal are not used.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrt3
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix V stores the elementary reflectors H(i) in the i-th column
*> below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*> V = ( 1 )
*> ( v1 1 )
*> ( v1 v2 1 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> where the vi's represent the vectors which define H(i), which are returned
*> in the matrix A. The 1's along the diagonal of V are not stored in A. The
*> block reflector H is then given by
*>
*> H = I - V * T * V**T
*>
*> where V**T is the transpose of V.
*>
*> For details of the algorithm, see Elmroth and Gustavson (cited above).
*> \endverbatim
*>
* =====================================================================
RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, LDT
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+00 )
* ..
* .. Local Scalars ..
INTEGER I, I1, J, J1, N1, N2, IINFO
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DTRMM, DGEMM, XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N .LT. 0 ) THEN
INFO = -2
ELSE IF( M .LT. N ) THEN
INFO = -1
ELSE IF( LDA .LT. MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LDT .LT. MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRT3', -INFO )
RETURN
END IF
*
IF( N.EQ.1 ) THEN
*
* Compute Householder transform when N=1
*
CALL DLARFG( M, A(1,1), A( MIN( 2, M ), 1 ), 1, T(1,1) )
*
ELSE
*
* Otherwise, split A into blocks...
*
N1 = N/2
N2 = N-N1
J1 = MIN( N1+1, N )
I1 = MIN( N+1, M )
*
* Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
*
CALL DGEQRT3( M, N1, A, LDA, T, LDT, IINFO )
*
* Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
*
DO J=1,N2
DO I=1,N1
T( I, J+N1 ) = A( I, J+N1 )
END DO
END DO
CALL DTRMM( 'L', 'L', 'T', 'U', N1, N2, ONE,
& A, LDA, T( 1, J1 ), LDT )
*
CALL DGEMM( 'T', 'N', N1, N2, M-N1, ONE, A( J1, 1 ), LDA,
& A( J1, J1 ), LDA, ONE, T( 1, J1 ), LDT)
*
CALL DTRMM( 'L', 'U', 'T', 'N', N1, N2, ONE,
& T, LDT, T( 1, J1 ), LDT )
*
CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( J1, 1 ), LDA,
& T( 1, J1 ), LDT, ONE, A( J1, J1 ), LDA )
*
CALL DTRMM( 'L', 'L', 'N', 'U', N1, N2, ONE,
& A, LDA, T( 1, J1 ), LDT )
*
DO J=1,N2
DO I=1,N1
A( I, J+N1 ) = A( I, J+N1 ) - T( I, J+N1 )
END DO
END DO
*
* Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
*
CALL DGEQRT3( M-N1, N2, A( J1, J1 ), LDA,
& T( J1, J1 ), LDT, IINFO )
*
* Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
*
DO I=1,N1
DO J=1,N2
T( I, J+N1 ) = (A( J+N1, I ))
END DO
END DO
*
CALL DTRMM( 'R', 'L', 'N', 'U', N1, N2, ONE,
& A( J1, J1 ), LDA, T( 1, J1 ), LDT )
*
CALL DGEMM( 'T', 'N', N1, N2, M-N, ONE, A( I1, 1 ), LDA,
& A( I1, J1 ), LDA, ONE, T( 1, J1 ), LDT )
*
CALL DTRMM( 'L', 'U', 'N', 'N', N1, N2, -ONE, T, LDT,
& T( 1, J1 ), LDT )
*
CALL DTRMM( 'R', 'U', 'N', 'N', N1, N2, ONE,
& T( J1, J1 ), LDT, T( 1, J1 ), LDT )
*
* Y = (Y1,Y2); R = [ R1 A(1:N1,J1:N) ]; T = [T1 T3]
* [ 0 R2 ] [ 0 T2]
*
END IF
*
RETURN
*
* End of DGEQRT3
*
END
*> \brief \b DGERFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGERFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
* X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGERFS improves the computed solution to a system of linear
*> equations and provides error bounds and backward error estimates for
*> the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The original N-by-N matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> The factors L and U from the factorization A = P*L*U
*> as computed by DGETRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices from DGETRF; for 1<=i<=N, row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DGETRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gerfs
*
* =====================================================================
SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
$ LDB,
$ X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
CHARACTER TRANST
INTEGER COUNT, I, J, K, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGETRS, DLACN2,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGERFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - op(A) * X,
* where op(A) = A, A**T, or A**H, depending on TRANS.
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
$ WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(op(A))*abs(X) + abs(B).
*
IF( NOTRAN ) THEN
DO 50 K = 1, N
XK = ABS( X( K, J ) )
DO 40 I = 1, N
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
40 CONTINUE
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
DO 60 I = 1, N
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
60 CONTINUE
WORK( K ) = WORK( K ) + S
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
$ INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(op(A)))*
* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(A)) is the inverse of op(A)
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(op(A)) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**T).
*
CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV,
$ WORK( N+1 ),
$ N, INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
110 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
120 CONTINUE
CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
$ N,
$ INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DGERFS
*
END
*> \brief \b DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGERQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGERQ2( M, N, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGERQ2 computes an RQ factorization of a real m by n matrix A:
*> A = R * Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, if m <= n, the upper triangle of the subarray
*> A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
*> if m >= n, the elements on and above the (m-n)-th subdiagonal
*> contain the m by n upper trapezoidal matrix R; the remaining
*> elements, with the array TAU, represent the orthogonal matrix
*> Q as a product of elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gerq2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
*> A(m-k+i,1:n-k+i-1), and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGERQ2( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
* ..
* .. External Subroutines ..
EXTERNAL DLARF1L, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGERQ2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = K, 1, -1
*
* Generate elementary reflector H(i) to annihilate
* A(m-k+i,1:n-k+i-1)
*
CALL DLARFG( N-K+I, A( M-K+I, N-K+I ), A( M-K+I, 1 ), LDA,
$ TAU( I ) )
*
* Apply H(i) to A(1:m-k+i-1,1:n-k+i) from the right
*
CALL DLARF1L( 'Right', M-K+I-1, N-K+I, A( M-K+I, 1 ), LDA,
$ TAU( I ), A, LDA, WORK )
10 CONTINUE
RETURN
*
* End of DGERQ2
*
END
*> \brief \b DGERQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGERQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGERQF computes an RQ factorization of a real M-by-N matrix A:
*> A = R * Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit,
*> if m <= n, the upper triangle of the subarray
*> A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
*> if m >= n, the elements on and above the (m-n)-th subdiagonal
*> contain the M-by-N upper trapezoidal matrix R;
*> the remaining elements, with the array TAU, represent the
*> orthogonal matrix Q as a product of min(m,n) elementary
*> reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
*> For optimum performance LWORK >= M*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gerqf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
*> A(m-k+i,1:n-k+i-1), and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
$ MU, NB, NBMIN, NU, NX
* ..
* .. External Subroutines ..
EXTERNAL DGERQ2, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
*
IF( INFO.EQ.0 ) THEN
K = MIN( M, N )
IF( K.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
LWKOPT = M*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( .NOT.LQUERY ) THEN
IF( LWORK.LE.0 .OR. ( N.GT.0 .AND. LWORK.LT.MAX( 1, M ) ) )
$ INFO = -7
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGERQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 ) THEN
RETURN
END IF
*
NBMIN = 2
NX = 1
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially.
* The last kk rows are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
IB = MIN( K-I+1, NB )
*
* Compute the RQ factorization of the current block
* A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1)
*
CALL DGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA,
$ TAU( I ),
$ WORK, IINFO )
IF( M-K+I.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
$ A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
CALL DLARFB( 'Right', 'No transpose', 'Backward',
$ 'Rowwise', M-K+I-1, N-K+I+IB-1, IB,
$ A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA,
$ WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
MU = M - K + I + NB - 1
NU = N - K + I + NB - 1
ELSE
MU = M
NU = N
END IF
*
* Use unblocked code to factor the last or only block
*
IF( MU.GT.0 .AND. NU.GT.0 )
$ CALL DGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGERQF
*
END
*> \brief \b DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESC2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesc2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesc2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesc2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESC2( N, A, LDA, RHS, IPIV, JPIV, SCALE )
*
* .. Scalar Arguments ..
* INTEGER LDA, N
* DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), JPIV( * )
* DOUBLE PRECISION A( LDA, * ), RHS( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGESC2 solves a system of linear equations
*>
*> A * X = scale* RHS
*>
*> with a general N-by-N matrix A using the LU factorization with
*> complete pivoting computed by DGETC2.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the LU part of the factorization of the n-by-n
*> matrix A computed by DGETC2: A = P * L * U * Q
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] RHS
*> \verbatim
*> RHS is DOUBLE PRECISION array, dimension (N).
*> On entry, the right hand side vector b.
*> On exit, the solution vector X.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N).
*> The pivot indices; for 1 <= i <= N, row i of the
*> matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] JPIV
*> \verbatim
*> JPIV is INTEGER array, dimension (N).
*> The pivot indices; for 1 <= j <= N, column j of the
*> matrix has been interchanged with column JPIV(j).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> On exit, SCALE contains the scale factor. SCALE is chosen
*> 0 <= SCALE <= 1 to prevent overflow in the solution.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesc2
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
SUBROUTINE DGESC2( N, A, LDA, RHS, IPIV, JPIV, SCALE )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LDA, N
DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
DOUBLE PRECISION A( LDA, * ), RHS( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, TWO
PARAMETER ( ONE = 1.0D+0, TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION BIGNUM, EPS, SMLNUM, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DLASWP, DSCAL
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL IDAMAX, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Set constant to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
BIGNUM = ONE / SMLNUM
*
* Apply permutations IPIV to RHS
*
CALL DLASWP( 1, RHS, LDA, 1, N-1, IPIV, 1 )
*
* Solve for L part
*
DO 20 I = 1, N - 1
DO 10 J = I + 1, N
RHS( J ) = RHS( J ) - A( J, I )*RHS( I )
10 CONTINUE
20 CONTINUE
*
* Solve for U part
*
SCALE = ONE
*
* Check for scaling
*
I = IDAMAX( N, RHS, 1 )
IF( TWO*SMLNUM*ABS( RHS( I ) ).GT.ABS( A( N, N ) ) ) THEN
TEMP = ( ONE / TWO ) / ABS( RHS( I ) )
CALL DSCAL( N, TEMP, RHS( 1 ), 1 )
SCALE = SCALE*TEMP
END IF
*
DO 40 I = N, 1, -1
TEMP = ONE / A( I, I )
RHS( I ) = RHS( I )*TEMP
DO 30 J = I + 1, N
RHS( I ) = RHS( I ) - RHS( J )*( A( I, J )*TEMP )
30 CONTINUE
40 CONTINUE
*
* Apply permutations JPIV to the solution (RHS)
*
CALL DLASWP( 1, RHS, LDA, 1, N-1, JPIV, -1 )
RETURN
*
* End of DGESC2
*
END
*> \brief \b DGESDD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESDD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesdd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesdd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesdd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
* WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
* INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGESDD computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, optionally computing the left and right singular
*> vectors. If singular vectors are desired, it uses a
*> divide-and-conquer algorithm.
*>
*> The SVD is written
*>
*> A = U * SIGMA * transpose(V)
*>
*> where SIGMA is an M-by-N matrix which is zero except for its
*> min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
*> V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA
*> are the singular values of A; they are real and non-negative, and
*> are returned in descending order. The first min(m,n) columns of
*> U and V are the left and right singular vectors of A.
*>
*> Note that the routine returns VT = V**T, not V.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> Specifies options for computing all or part of the matrix U:
*> = 'A': all M columns of U and all N rows of V**T are
*> returned in the arrays U and VT;
*> = 'S': the first min(M,N) columns of U and the first
*> min(M,N) rows of V**T are returned in the arrays U
*> and VT;
*> = 'O': If M >= N, the first N columns of U are overwritten
*> on the array A and all rows of V**T are returned in
*> the array VT;
*> otherwise, all columns of U are returned in the
*> array U and the first M rows of V**T are overwritten
*> in the array A;
*> = 'N': no columns of U or rows of V**T are computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit,
*> if JOBZ = 'O', A is overwritten with the first N columns
*> of U (the left singular vectors, stored
*> columnwise) if M >= N;
*> A is overwritten with the first M rows
*> of V**T (the right singular vectors, stored
*> rowwise) otherwise.
*> if JOBZ .ne. 'O', the contents of A are destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (min(M,N))
*> The singular values of A, sorted so that S(i) >= S(i+1).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,UCOL)
*> UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
*> UCOL = min(M,N) if JOBZ = 'S'.
*> If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
*> orthogonal matrix U;
*> if JOBZ = 'S', U contains the first min(M,N) columns of U
*> (the left singular vectors, stored columnwise);
*> if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= 1; if
*> JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
*> If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
*> N-by-N orthogonal matrix V**T;
*> if JOBZ = 'S', VT contains the first min(M,N) rows of
*> V**T (the right singular vectors, stored rowwise);
*> if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= 1;
*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*> if JOBZ = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
*> If LWORK = -1, a workspace query is assumed. The optimal
*> size for the WORK array is calculated and stored in WORK(1),
*> and no other work except argument checking is performed.
*>
*> Let mx = max(M,N) and mn = min(M,N).
*> If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
*> If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
*> If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
*> If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
*> These are not tight minimums in all cases; see comments inside code.
*> For good performance, LWORK should generally be larger;
*> a query is recommended.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (8*min(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = -4: if A had a NAN entry.
*> > 0: DBDSDC did not converge, updating process failed.
*> = 0: successful exit.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesdd
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
$ WORK, LWORK, IWORK, INFO )
implicit none
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IL,
$ IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR, NWORK, WRKBL
INTEGER LWORK_DGEBRD_MN, LWORK_DGEBRD_MM,
$ LWORK_DGEBRD_NN, LWORK_DGELQF_MN,
$ LWORK_DGEQRF_MN,
$ LWORK_DORGBR_P_MM, LWORK_DORGBR_Q_NN,
$ LWORK_DORGLQ_MN, LWORK_DORGLQ_NN,
$ LWORK_DORGQR_MM, LWORK_DORGQR_MN,
$ LWORK_DORMBR_PRT_MM, LWORK_DORMBR_QLN_MM,
$ LWORK_DORMBR_PRT_MN, LWORK_DORMBR_QLN_MN,
$ LWORK_DORMBR_PRT_NN, LWORK_DORMBR_QLN_NN
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSDC, DGEBRD, DGELQF, DGEMM, DGEQRF,
$ DLACPY,
$ DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DLAMCH, DLANGE, DROUNDUP_LWORK
EXTERNAL DLAMCH, DLANGE, LSAME, DISNAN,
$ DROUNDUP_LWORK
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
MINMN = MIN( M, N )
WNTQA = LSAME( JOBZ, 'A' )
WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
WNTQO = LSAME( JOBZ, 'O' )
WNTQN = LSAME( JOBZ, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDU.LT.1 .OR. ( WNTQAS .AND. LDU.LT.M ) .OR.
$ ( WNTQO .AND. M.LT.N .AND. LDU.LT.M ) ) THEN
INFO = -8
ELSE IF( LDVT.LT.1 .OR. ( WNTQA .AND. LDVT.LT.N ) .OR.
$ ( WNTQS .AND. LDVT.LT.MINMN ) .OR.
$ ( WNTQO .AND. M.GE.N .AND. LDVT.LT.N ) ) THEN
INFO = -10
END IF
*
* Compute workspace
* Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace allocated at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
*
IF( INFO.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
BDSPAC = 0
MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSDC
*
IF( WNTQN ) THEN
* dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
* keep 7*N for backwards compatibility.
BDSPAC = 7*N
ELSE
BDSPAC = 3*N*N + 4*N
END IF
*
* Compute space preferred for each routine
CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_MN = INT( DUM(1) )
*
CALL DGEBRD( N, N, DUM(1), N, DUM(1), DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_NN = INT( DUM(1) )
*
CALL DGEQRF( M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
LWORK_DGEQRF_MN = INT( DUM(1) )
*
CALL DORGBR( 'Q', N, N, N, DUM(1), N, DUM(1), DUM(1), -1,
$ IERR )
LWORK_DORGBR_Q_NN = INT( DUM(1) )
*
CALL DORGQR( M, M, N, DUM(1), M, DUM(1), DUM(1), -1,
$ IERR )
LWORK_DORGQR_MM = INT( DUM(1) )
*
CALL DORGQR( M, N, N, DUM(1), M, DUM(1), DUM(1), -1,
$ IERR )
LWORK_DORGQR_MN = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, DUM(1), N,
$ DUM(1), DUM(1), N, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, DUM(1), N,
$ DUM(1), DUM(1), N, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_NN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_MN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
IF( M.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
* Path 1 (M >> N, JOBZ='N')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
MAXWRK = MAX( WRKBL, BDSPAC + N )
MINWRK = BDSPAC + N
ELSE IF( WNTQO ) THEN
*
* Path 2 (M >> N, JOBZ='O')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + 2*N*N
MINWRK = BDSPAC + 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
* Path 3 (M >> N, JOBZ='S')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
MINWRK = BDSPAC + N*N + 3*N
ELSE IF( WNTQA ) THEN
*
* Path 4 (M >> N, JOBZ='A')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MM )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
MINWRK = N*N + MAX( 3*N + BDSPAC, N + M )
END IF
ELSE
*
* Path 5 (M >= N, but not much larger)
*
WRKBL = 3*N + LWORK_DGEBRD_MN
IF( WNTQN ) THEN
* Path 5n (M >= N, jobz='N')
MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQO ) THEN
* Path 5o (M >= N, jobz='O')
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + M*N
MINWRK = 3*N + MAX( M, N*N + BDSPAC )
ELSE IF( WNTQS ) THEN
* Path 5s (M >= N, jobz='S')
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQA ) THEN
* Path 5a (M >= N, jobz='A')
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
END IF
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSDC
*
IF( WNTQN ) THEN
* dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
* keep 7*N for backwards compatibility.
BDSPAC = 7*M
ELSE
BDSPAC = 3*M*M + 4*M
END IF
*
* Compute space preferred for each routine
CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_MN = INT( DUM(1) )
*
CALL DGEBRD( M, M, A, M, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_MM = INT( DUM(1) )
*
CALL DGELQF( M, N, A, M, DUM(1), DUM(1), -1, IERR )
LWORK_DGELQF_MN = INT( DUM(1) )
*
CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1,
$ IERR )
LWORK_DORGLQ_NN = INT( DUM(1) )
*
CALL DORGLQ( M, N, M, A, M, DUM(1), DUM(1), -1, IERR )
LWORK_DORGLQ_MN = INT( DUM(1) )
*
CALL DORGBR( 'P', M, M, M, A, N, DUM(1), DUM(1), -1,
$ IERR )
LWORK_DORGBR_P_MM = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', M, M, M, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_MM = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', M, N, M, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_MN = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', N, N, M, DUM(1), N,
$ DUM(1), DUM(1), N, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
IF( N.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
* Path 1t (N >> M, JOBZ='N')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
MAXWRK = MAX( WRKBL, BDSPAC + M )
MINWRK = BDSPAC + M
ELSE IF( WNTQO ) THEN
*
* Path 2t (N >> M, JOBZ='O')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_MN )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + 2*M*M
MINWRK = BDSPAC + 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
* Path 3t (N >> M, JOBZ='S')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_MN )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
MINWRK = BDSPAC + M*M + 3*M
ELSE IF( WNTQA ) THEN
*
* Path 4t (N >> M, JOBZ='A')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_NN )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
MINWRK = M*M + MAX( 3*M + BDSPAC, M + N )
END IF
ELSE
*
* Path 5t (N > M, but not much larger)
*
WRKBL = 3*M + LWORK_DGEBRD_MN
IF( WNTQN ) THEN
* Path 5tn (N > M, jobz='N')
MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQO ) THEN
* Path 5to (N > M, jobz='O')
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*N
MINWRK = 3*M + MAX( N, M*M + BDSPAC )
ELSE IF( WNTQS ) THEN
* Path 5ts (N > M, jobz='S')
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQA ) THEN
* Path 5ta (N > M, jobz='A')
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_NN )
MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
END IF
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
WORK( 1 ) = DROUNDUP_LWORK( MAXWRK )
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGESDD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
IF( DISNAN( ANRM ) ) THEN
INFO = -4
RETURN
END IF
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
ELSE IF( ANRM.GT.BIGNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
END IF
*
IF( M.GE.N ) THEN
*
* A has at least as many rows as columns. If A has sufficiently
* more rows than columns, first reduce using the QR
* decomposition (if sufficient workspace available)
*
IF( M.GE.MNTHR ) THEN
*
IF( WNTQN ) THEN
*
* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
* Workspace: need N [tau] + N [work]
* Workspace: prefer N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Zero out below R
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in A
* Workspace: need 3*N [e, tauq, taup] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
NWORK = IE + N
*
* Perform bidiagonal SVD, computing singular values only
* Workspace: need N [e] + BDSPAC
*
CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM,
$ 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
* Path 2 (M >> N, JOBZ = 'O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
IR = 1
*
* WORK(IR) is LDWRKR by N
*
IF( LWORK .GE. LDA*N + N*N + 3*N + BDSPAC ) THEN
LDWRKR = LDA
ELSE
LDWRKR = ( LWORK - N*N - 3*N - BDSPAC ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO,
$ WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* WORK(IU) is N by N
*
IU = NWORK
NWORK = IU + N*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
$ N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite WORK(IU) by left singular vectors of R
* and VT by right singular vectors of R
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ),
$ LDWRKR,
$ WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ),
$ LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U]
* Workspace: prefer M*N [R] + 3*N [e, tauq, taup] + N*N [U]
*
DO 10 I = 1, M, LDWRKR
CHUNK = MIN( M - I + 1, LDWRKR )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IU ), N, ZERO, WORK( IR ),
$ LDWRKR )
CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
$ A( I, 1 ), LDA )
10 CONTINUE
*
ELSE IF( WNTQS ) THEN
*
* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IR = 1
*
* WORK(IR) is N by N
*
LDWRKR = N
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO,
$ WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagoal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of R and VT
* by right singular vectors of R
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ),
$ LDWRKR,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ),
$ LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
* Workspace: need N*N [R]
*
CALL DLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
$ WORK( IR ),
$ LDWRKR, ZERO, U, LDU )
*
ELSE IF( WNTQA ) THEN
*
* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IU = 1
*
* WORK(IU) is N by N
*
LDWRKU = N
ITAU = IU + LDWRKU*N
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* Workspace: need N*N [U] + N [tau] + N [work]
* Workspace: prefer N*N [U] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* Workspace: need N*N [U] + N [tau] + M [work]
* Workspace: prefer N*N [U] + N [tau] + M*NB [work]
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce R in A, zeroing out other entries
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in A
* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
* Workspace: need N*N [U] + 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
$ N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite WORK(IU) by left singular vectors of R and VT
* by right singular vectors of R
* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
* Workspace: need N*N [U]
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
$ WORK( IU ),
$ LDWRKU, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
END IF
*
ELSE
*
* M .LT. MNTHR
*
* Path 5 (M >= N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize A
* Workspace: need 3*N [e, tauq, taup] + M [work]
* Workspace: prefer 3*N [e, tauq, taup] + (M+N)*NB [work]
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
* Path 5n (M >= N, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM,
$ 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
* Path 5o (M >= N, JOBZ='O')
IU = NWORK
IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
* WORK( IU ) is M by N
*
LDWRKU = M
NWORK = IU + LDWRKU*N
CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IU ),
$ LDWRKU )
* IR is unused; silence compile warnings
IR = -1
ELSE
*
* WORK( IU ) is N by N
*
LDWRKU = N
NWORK = IU + LDWRKU*N
*
* WORK(IR) is LDWRKR by N
*
IR = NWORK
LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
$ LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ),
$ IWORK, INFO )
*
* Overwrite VT by right singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
* Path 5o-fast
* Overwrite WORK(IU) by left singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + M*N [U] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + M*N [U] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy left singular vectors of A from WORK(IU) to A
*
CALL DLACPY( 'F', M, N, WORK( IU ), LDWRKU, A,
$ LDA )
ELSE
*
* Path 5o-slow
* Generate Q in A
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of
* bidiagonal matrix in WORK(IU), storing result in
* WORK(IR) and copying to A
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + NB*N [R]
* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + M*N [R]
*
DO 20 I = 1, M, LDWRKR
CHUNK = MIN( M - I + 1, LDWRKR )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I,
$ 1 ),
$ LDA, WORK( IU ), LDWRKU, ZERO,
$ WORK( IR ), LDWRKR )
CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
$ A( I, 1 ), LDA )
20 CONTINUE
END IF
*
ELSE IF( WNTQS ) THEN
*
* Path 5s (M >= N, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, N, ZERO, ZERO, U, LDU )
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
* Path 5a (M >= N, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, M, ZERO, ZERO, U, LDU )
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Set the right corner of U to identity matrix
*
IF( M.GT.N ) THEN
CALL DLASET( 'F', M - N, M - N, ZERO, ONE, U(N+1,
$ N+1),
$ LDU )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + M [work]
* Workspace: prefer 3*N [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
END IF
*
END IF
*
ELSE
*
* A has more columns than rows. If A has sufficiently more
* columns than rows, first reduce using the LQ decomposition (if
* sufficient workspace available)
*
IF( N.GE.MNTHR ) THEN
*
IF( WNTQN ) THEN
*
* Path 1t (N >> M, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + M
*
* Compute A=L*Q
* Workspace: need M [tau] + M [work]
* Workspace: prefer M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Zero out above L
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
$ LDA )
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in A
* Workspace: need 3*M [e, tauq, taup] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
NWORK = IE + M
*
* Perform bidiagonal SVD, computing singular values only
* Workspace: need M [e] + BDSPAC
*
CALL DBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM,
$ 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
* Path 2t (N >> M, JOBZ='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
IVT = 1
*
* WORK(IVT) is M by M
* WORK(IL) is M by M; it is later resized to M by chunk for gemm
*
IL = IVT + M*M
IF( LWORK .GE. M*N + M*M + 3*M + BDSPAC ) THEN
LDWRKL = M
CHUNK = N
ELSE
LDWRKL = M
CHUNK = ( LWORK - M*M ) / M
END IF
ITAU = IL + LDWRKL*M
NWORK = ITAU + M
*
* Compute A=L*Q
* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing about above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
$ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U, and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), M, DUM, IDUM, WORK( NWORK ),
$ IWORK, INFO )
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ),
$ LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ),
$ LDWRKL,
$ WORK( ITAUP ), WORK( IVT ), M,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by Q
* in A, storing result in WORK(IL) and copying to A
* Workspace: need M*M [VT] + M*M [L]
* Workspace: prefer M*M [VT] + M*N [L]
* At this point, L is resized as M by chunk.
*
DO 30 I = 1, N, CHUNK
BLK = MIN( N - I + 1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ),
$ M,
$ A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL )
CALL DLACPY( 'F', M, BLK, WORK( IL ), LDWRKL,
$ A( 1, I ), LDA )
30 CONTINUE
*
ELSE IF( WNTQS ) THEN
*
* Path 3t (N >> M, JOBZ='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IL = 1
*
* WORK(IL) is M by M
*
LDWRKL = M
ITAU = IL + LDWRKL*M
NWORK = ITAU + M
*
* Compute A=L*Q
* Workspace: need M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
$ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
* Workspace: need M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU).
* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of L and VT
* by right singular vectors of L
* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ),
$ LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ),
$ LDWRKL,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IL) by
* Q in A, storing result in VT
* Workspace: need M*M [L]
*
CALL DLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ),
$ LDWRKL,
$ A, LDA, ZERO, VT, LDVT )
*
ELSE IF( WNTQA ) THEN
*
* Path 4t (N >> M, JOBZ='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IVT = 1
*
* WORK(IVT) is M by M
*
LDWKVT = M
ITAU = IVT + LDWKVT*M
NWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* Workspace: need M*M [VT] + M [tau] + M [work]
* Workspace: prefer M*M [VT] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* Workspace: need M*M [VT] + M [tau] + N [work]
* Workspace: prefer M*M [VT] + M [tau] + N*NB [work]
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce L in A, zeroing out other entries
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
$ LDA )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in A
* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
$ WORK( NWORK ), IWORK, INFO )
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
* Workspace: need M*M [VT] + 3*M [e, tauq, taup]+ M [work]
* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup]+ M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by
* Q in VT, storing result in A
* Workspace: need M*M [VT]
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ),
$ LDWKVT,
$ VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
END IF
*
ELSE
*
* N .LT. MNTHR
*
* Path 5t (N > M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize A
* Workspace: need 3*M [e, tauq, taup] + N [work]
* Workspace: prefer 3*M [e, tauq, taup] + (M+N)*NB [work]
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
* Path 5tn (N > M, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM,
$ 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
* Path 5to (N > M, JOBZ='O')
LDWKVT = M
IVT = NWORK
IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
* WORK( IVT ) is M by N
*
CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ),
$ LDWKVT )
NWORK = IVT + LDWKVT*N
* IL is unused; silence compile warnings
IL = -1
ELSE
*
* WORK( IVT ) is M by M
*
NWORK = IVT + LDWKVT*M
IL = NWORK
*
* WORK(IL) is M by CHUNK
*
CHUNK = ( LWORK - M*M - 3*M ) / M
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + BDSPAC
*
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
$ WORK( NWORK ), IWORK, INFO )
*
* Overwrite U by left singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
* Path 5to-fast
* Overwrite WORK(IVT) by left singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + M*N [VT] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*N [VT] + M*NB [work]
*
CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy right singular vectors of A from WORK(IVT) to A
*
CALL DLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A,
$ LDA )
ELSE
*
* Path 5to-slow
* Generate P**T in A
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by right singular vectors of
* bidiagonal matrix in WORK(IVT), storing result in
* WORK(IL) and copying to A
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M*NB [L]
* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*N [L]
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N - I + 1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE,
$ WORK( IVT ),
$ LDWKVT, A( 1, I ), LDA, ZERO,
$ WORK( IL ), M )
CALL DLACPY( 'F', M, BLK, WORK( IL ), M, A( 1,
$ I ),
$ LDA )
40 CONTINUE
END IF
ELSE IF( WNTQS ) THEN
*
* Path 5ts (N > M, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, N, ZERO, ZERO, VT, LDVT )
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
* Path 5ta (N > M, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', N, N, ZERO, ZERO, VT, LDVT )
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Set the right corner of VT to identity matrix
*
IF( N.GT.M ) THEN
CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT(M+1,M+1),
$ LDVT )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + N [work]
* Workspace: prefer 3*M [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
END IF
*
END IF
*
END IF
*
* Undo scaling if necessary
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
IF( ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
END IF
*
* Return optimal workspace in WORK(1)
*
WORK( 1 ) = DROUNDUP_LWORK( MAXWRK )
*
RETURN
*
* End of DGESDD
*
END
*> \addtogroup gesv
*>
*> \brief <b> DGESV computes the solution to system of linear equations A * X = B for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGESV computes the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*>
*> The LU decomposition with partial pivoting and row interchanges is
*> used to factor A as
*> A = P * L * U,
*> where P is a permutation matrix, L is unit lower triangular, and U is
*> upper triangular. The factored form of A is then used to solve the
*> system of equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N coefficient matrix A.
*> On exit, the factors L and U from the factorization
*> A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices that define the permutation matrix P;
*> row i of the matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS matrix of right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, so the solution could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesv
*
* =====================================================================
SUBROUTINE DGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. External Subroutines ..
EXTERNAL DGETRF, DGETRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( NRHS.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGESV ', -INFO )
RETURN
END IF
*
* Compute the LU factorization of A.
*
CALL DGETRF( N, N, A, LDA, IPIV, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, B, LDB,
$ INFO )
END IF
RETURN
*
* End of DGESV
*
END
*> \brief <b> DGESVD computes the singular value decomposition (SVD) for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBU, JOBVT
* INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGESVD computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, optionally computing the left and/or right singular
*> vectors. The SVD is written
*>
*> A = U * SIGMA * transpose(V)
*>
*> where SIGMA is an M-by-N matrix which is zero except for its
*> min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
*> V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA
*> are the singular values of A; they are real and non-negative, and
*> are returned in descending order. The first min(m,n) columns of
*> U and V are the left and right singular vectors of A.
*>
*> Note that the routine returns V**T, not V.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> Specifies options for computing all or part of the matrix U:
*> = 'A': all M columns of U are returned in array U:
*> = 'S': the first min(m,n) columns of U (the left singular
*> vectors) are returned in the array U;
*> = 'O': the first min(m,n) columns of U (the left singular
*> vectors) are overwritten on the array A;
*> = 'N': no columns of U (no left singular vectors) are
*> computed.
*> \endverbatim
*>
*> \param[in] JOBVT
*> \verbatim
*> JOBVT is CHARACTER*1
*> Specifies options for computing all or part of the matrix
*> V**T:
*> = 'A': all N rows of V**T are returned in the array VT;
*> = 'S': the first min(m,n) rows of V**T (the right singular
*> vectors) are returned in the array VT;
*> = 'O': the first min(m,n) rows of V**T (the right singular
*> vectors) are overwritten on the array A;
*> = 'N': no rows of V**T (no right singular vectors) are
*> computed.
*>
*> JOBVT and JOBU cannot both be 'O'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit,
*> if JOBU = 'O', A is overwritten with the first min(m,n)
*> columns of U (the left singular vectors,
*> stored columnwise);
*> if JOBVT = 'O', A is overwritten with the first min(m,n)
*> rows of V**T (the right singular vectors,
*> stored rowwise);
*> if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
*> are destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (min(M,N))
*> The singular values of A, sorted so that S(i) >= S(i+1).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,UCOL)
*> (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
*> If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
*> if JOBU = 'S', U contains the first min(m,n) columns of U
*> (the left singular vectors, stored columnwise);
*> if JOBU = 'N' or 'O', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= 1; if
*> JOBU = 'S' or 'A', LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
*> If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
*> V**T;
*> if JOBVT = 'S', VT contains the first min(m,n) rows of
*> V**T (the right singular vectors, stored rowwise);
*> if JOBVT = 'N' or 'O', VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= 1; if
*> JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
*> if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged
*> superdiagonal elements of an upper bidiagonal matrix B
*> whose diagonal is in S (not necessarily sorted). B
*> satisfies A = U * B * VT, so it has the same singular values
*> as A, and singular vectors related by U and VT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
*> - PATH 1 (M much larger than N, JOBU='N')
*> - PATH 1t (N much larger than M, JOBVT='N')
*> LWORK >= MAX(1,3*MIN(M,N) + MAX(M,N),5*MIN(M,N)) for the other paths
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if DBDSQR did not converge, INFO specifies how many
*> superdiagonals of an intermediate bidiagonal form B
*> did not converge to zero. See the description of WORK
*> above for details.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesvd
*
* =====================================================================
SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU,
$ VT, LDVT, WORK, LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBU, JOBVT
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WNTUA, WNTUAS, WNTUN, WNTUO, WNTUS,
$ WNTVA, WNTVAS, WNTVN, WNTVO, WNTVS
INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IR, ISCL,
$ ITAU, ITAUP, ITAUQ, IU, IWORK, LDWRKR, LDWRKU,
$ MAXWRK, MINMN, MINWRK, MNTHR, NCU, NCVT, NRU,
$ NRVT, WRKBL
INTEGER LWORK_DGEQRF, LWORK_DORGQR_N, LWORK_DORGQR_M,
$ LWORK_DGEBRD, LWORK_DORGBR_P, LWORK_DORGBR_Q,
$ LWORK_DGELQF, LWORK_DORGLQ_N, LWORK_DORGLQ_M
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSQR, DGEBRD, DGELQF, DGEMM, DGEQRF,
$ DLACPY,
$ DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
MINMN = MIN( M, N )
WNTUA = LSAME( JOBU, 'A' )
WNTUS = LSAME( JOBU, 'S' )
WNTUAS = WNTUA .OR. WNTUS
WNTUO = LSAME( JOBU, 'O' )
WNTUN = LSAME( JOBU, 'N' )
WNTVA = LSAME( JOBVT, 'A' )
WNTVS = LSAME( JOBVT, 'S' )
WNTVAS = WNTVA .OR. WNTVS
WNTVO = LSAME( JOBVT, 'O' )
WNTVN = LSAME( JOBVT, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.( WNTUA .OR. WNTUS .OR. WNTUO .OR. WNTUN ) ) THEN
INFO = -1
ELSE IF( .NOT.( WNTVA .OR. WNTVS .OR. WNTVO .OR. WNTVN ) .OR.
$ ( WNTVO .AND. WNTUO ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDU.LT.1 .OR. ( WNTUAS .AND. LDU.LT.M ) ) THEN
INFO = -9
ELSE IF( LDVT.LT.1 .OR. ( WNTVA .AND. LDVT.LT.N ) .OR.
$ ( WNTVS .AND. LDVT.LT.MINMN ) ) THEN
INFO = -11
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.)
*
IF( INFO.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSQR
*
MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
BDSPAC = 5*N
* Compute space needed for DGEQRF
CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
LWORK_DGEQRF = INT( DUM(1) )
* Compute space needed for DORGQR
CALL DORGQR( M, N, N, A, LDA, DUM(1), DUM(1), -1, IERR )
LWORK_DORGQR_N = INT( DUM(1) )
CALL DORGQR( M, M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
LWORK_DORGQR_M = INT( DUM(1) )
* Compute space needed for DGEBRD
CALL DGEBRD( N, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD = INT( DUM(1) )
* Compute space needed for DORGBR P
CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_P = INT( DUM(1) )
* Compute space needed for DORGBR Q
CALL DORGBR( 'Q', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_Q = INT( DUM(1) )
*
IF( M.GE.MNTHR ) THEN
IF( WNTUN ) THEN
*
* Path 1 (M much larger than N, JOBU='N')
*
MAXWRK = N + LWORK_DGEQRF
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
IF( WNTVO .OR. WNTVAS )
$ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 4*N, BDSPAC )
ELSE IF( WNTUO .AND. WNTVN ) THEN
*
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or
* 'A')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVN ) THEN
*
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVO ) THEN
*
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*N*N + WRKBL
MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVAS ) THEN
*
* Path 6 (M much larger than N, JOBU='S', JOBVT='S' or
* 'A')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVN ) THEN
*
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVO ) THEN
*
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*N*N + WRKBL
MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVAS ) THEN
*
* Path 9 (M much larger than N, JOBU='A', JOBVT='S' or
* 'A')
*
WRKBL = N + LWORK_DGEQRF
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N + M, BDSPAC )
END IF
ELSE
*
* Path 10 (M at least N, but not much larger)
*
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD = INT( DUM(1) )
MAXWRK = 3*N + LWORK_DGEBRD
IF( WNTUS .OR. WNTUO ) THEN
CALL DORGBR( 'Q', M, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_Q = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
END IF
IF( WNTUA ) THEN
CALL DORGBR( 'Q', M, M, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_Q = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
END IF
IF( .NOT.WNTVN ) THEN
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
END IF
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 3*N + M, BDSPAC )
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSQR
*
MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
BDSPAC = 5*M
* Compute space needed for DGELQF
CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
LWORK_DGELQF = INT( DUM(1) )
* Compute space needed for DORGLQ
CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1,
$ IERR )
LWORK_DORGLQ_N = INT( DUM(1) )
CALL DORGLQ( M, N, M, A, LDA, DUM(1), DUM(1), -1, IERR )
LWORK_DORGLQ_M = INT( DUM(1) )
* Compute space needed for DGEBRD
CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD = INT( DUM(1) )
* Compute space needed for DORGBR P
CALL DORGBR( 'P', M, M, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_P = INT( DUM(1) )
* Compute space needed for DORGBR Q
CALL DORGBR( 'Q', M, M, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_Q = INT( DUM(1) )
IF( N.GE.MNTHR ) THEN
IF( WNTVN ) THEN
*
* Path 1t(N much larger than M, JOBVT='N')
*
MAXWRK = M + LWORK_DGELQF
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DGEBRD )
IF( WNTUO .OR. WNTUAS )
$ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 4*M, BDSPAC )
ELSE IF( WNTVO .AND. WNTUN ) THEN
*
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
* Path 3t(N much larger than M, JOBU='S' or 'A',
* JOBVT='O')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUN ) THEN
*
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUO ) THEN
*
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*M*M + WRKBL
MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUAS ) THEN
*
* Path 6t(N much larger than M, JOBU='S' or 'A',
* JOBVT='S')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUN ) THEN
*
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUO ) THEN
*
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*M*M + WRKBL
MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUAS ) THEN
*
* Path 9t(N much larger than M, JOBU='S' or 'A',
* JOBVT='A')
*
WRKBL = M + LWORK_DGELQF
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M + N, BDSPAC )
END IF
ELSE
*
* Path 10t(N greater than M, but not much larger)
*
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD = INT( DUM(1) )
MAXWRK = 3*M + LWORK_DGEBRD
IF( WNTVS .OR. WNTVO ) THEN
* Compute space needed for DORGBR P
CALL DORGBR( 'P', M, N, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_P = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
END IF
IF( WNTVA ) THEN
CALL DORGBR( 'P', N, N, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
LWORK_DORGBR_P = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
END IF
IF( .NOT.WNTUN ) THEN
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
END IF
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 3*M + N, BDSPAC )
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGESVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
ELSE IF( ANRM.GT.BIGNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
END IF
*
IF( M.GE.N ) THEN
*
* A has at least as many rows as columns. If A has sufficiently
* more rows than columns, first reduce using the QR
* decomposition (if sufficient workspace available)
*
IF( M.GE.MNTHR ) THEN
*
IF( WNTUN ) THEN
*
* Path 1 (M much larger than N, JOBU='N')
* No left singular vectors to be computed
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Zero out below R
*
IF( N .GT. 1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
END IF
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
NCVT = 0
IF( WNTVO .OR. WNTVAS ) THEN
*
* If right singular vectors desired, generate P'.
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
NCVT = N
END IF
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in A if desired
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, 0, 0, S, WORK( IE ), A,
$ LDA,
$ DUM, 1, DUM, 1, WORK( IWORK ), INFO )
*
* If right singular vectors desired in VT, copy them there
*
IF( WNTVAS )
$ CALL DLACPY( 'F', N, N, A, LDA, VT, LDVT )
*
ELSE IF( WNTUO .AND. WNTVN ) THEN
*
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
* N left singular vectors to be overwritten on A and
* no right singular vectors to be computed
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
*
* WORK(IU) is LDA by N, WORK(IR) is LDA by N
*
LDWRKU = LDA
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
*
* WORK(IU) is LDA by N, WORK(IR) is N by N
*
LDWRKU = LDA
LDWRKR = N
ELSE
*
* WORK(IU) is LDWRKU by N, WORK(IR) is N by N
*
LDWRKU = ( LWORK-N*N-N ) / N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IR) and zero out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IR+1 ),
$ LDWRKR )
*
* Generate Q in A
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
$ WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing R
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
$ 1,
$ WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + N
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in WORK(IU) and copying to A
* (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
*
DO 10 I = 1, M, LDWRKU
CHUNK = MIN( M-I+1, LDWRKU )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I,
$ 1 ),
$ LDA, WORK( IR ), LDWRKR, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
$ A( I, 1 ), LDA )
10 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize A
* (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing A
* (Workspace: need 4*N, prefer 3*N + N*NB)
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM,
$ 1,
$ A, LDA, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or 'A')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
LDWRKU = LDA
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
LDWRKU = LDA
LDWRKR = N
ELSE
*
* WORK(IU) is LDWRKU by N and WORK(IR) is N by N
*
LDWRKU = ( LWORK-N*N-N ) / N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
*
* Generate Q in A
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT, copying result to WORK(IR)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', N, N, VT, LDVT, WORK( IR ),
$ LDWRKR )
*
* Generate left vectors bidiagonalizing R in WORK(IR)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in VT
* (Workspace: need N*N + 4*N-1, prefer N*N + 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR) and computing right
* singular vectors of R in VT
* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
$ LDVT,
$ WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + N
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in WORK(IU) and copying to A
* (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
*
DO 20 I = 1, M, LDWRKU
CHUNK = MIN( M-I+1, LDWRKU )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I,
$ 1 ),
$ LDA, WORK( IR ), LDWRKR, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
$ A( I, 1 ), LDA )
20 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
*
* Generate Q in A
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in A by left vectors bidiagonalizing R
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), A, LDA, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in VT
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in A and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
$ LDVT,
$ A, LDA, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTUS ) THEN
*
IF( WNTVN ) THEN
*
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
* N left singular vectors to be computed in U and
* no right singular vectors to be computed
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IR) is LDA by N
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is N by N
*
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IR+1 ), LDWRKR )
*
* Generate Q in A
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing R in WORK(IR)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ),
$ DUM,
$ 1, WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
$ WORK( IR ), LDWRKR, ZERO, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
IF( N .GT. 1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ A( 2, 1 ), LDA )
END IF
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left vectors bidiagonalizing R
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ),
$ DUM,
$ 1, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVO ) THEN
*
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
* N left singular vectors to be computed in U and
* N right singular vectors to be overwritten on A
*
IF( LWORK.GE.2*N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = N
ELSE
*
* WORK(IU) is N by N and WORK(IR) is N by N
*
LDWRKU = N
IR = IU + LDWRKU*N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
*
* Generate Q in A
* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*N*N + 4*N,
* prefer 2*N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*N*N + 4*N-1,
* prefer 2*N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in WORK(IR)
* (Workspace: need 2*N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, WORK( IU ),
$ LDWRKU, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in U
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
$ WORK( IU ), LDWRKU, ZERO, U, LDU )
*
* Copy right singular vectors of R to A
* (Workspace: need N*N)
*
CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
IF( N .GT. 1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ A( 2, 1 ), LDA )
END IF
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left vectors bidiagonalizing R
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in A
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA,
$ WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
$ LDA, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVAS ) THEN
*
* Path 6 (M much larger than N, JOBU='S', JOBVT='S'
* or 'A')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IU) is LDA by N
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is N by N
*
LDWRKU = N
END IF
ITAU = IU + LDWRKU*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
*
* Generate Q in A
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to VT
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
$ LDVT )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need N*N + 4*N-1,
* prefer N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT,
$ WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in VT
* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in U
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
$ WORK( IU ), LDWRKU, ZERO, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in VT
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT,
$ WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
ELSE IF( WNTUA ) THEN
*
IF( WNTVN ) THEN
*
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
* M left singular vectors to be computed in U and
* no right singular vectors to be computed
*
IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IR) is LDA by N
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is N by N
*
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IR+1 ), LDWRKR )
*
* Generate Q in U
* (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ),
$ DUM,
$ 1, WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IR), storing result in A
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
$ WORK( IR ), LDWRKR, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N + M, prefer N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
IF( N .GT. 1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ A( 2, 1 ), LDA )
END IF
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in A
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ),
$ DUM,
$ 1, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVO ) THEN
*
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
* M left singular vectors to be computed in U and
* N right singular vectors to be overwritten on A
*
IF( LWORK.GE.2*N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = N
ELSE
*
* WORK(IU) is N by N and WORK(IR) is N by N
*
LDWRKU = N
IR = IU + LDWRKU*N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N*N + N + M, prefer 2*N*N + N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*N*N + 4*N,
* prefer 2*N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*N*N + 4*N-1,
* prefer 2*N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in WORK(IR)
* (Workspace: need 2*N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, WORK( IU ),
$ LDWRKU, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
$ WORK( IU ), LDWRKU, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
* Copy right singular vectors of R from WORK(IR) to A
*
CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N + M, prefer N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
IF( N .GT. 1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ A( 2, 1 ), LDA )
END IF
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in A
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in A
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA,
$ WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
$ LDA, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVAS ) THEN
*
* Path 9 (M much larger than N, JOBU='A', JOBVT='S'
* or 'A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IU) is LDA by N
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is N by N
*
LDWRKU = N
END IF
ITAU = IU + LDWRKU*N
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to VT
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
$ LDVT )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need N*N + 4*N-1,
* prefer N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT,
$ WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in VT
* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
$ WORK( IU ), LDWRKU, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N + M, prefer N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R from A to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in VT
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT,
$ WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
END IF
*
ELSE
*
* M .LT. MNTHR
*
* Path 10 (M at least N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize A
* (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
IF( WNTUAS ) THEN
*
* If left singular vectors desired in U, copy result to U
* and generate left bidiagonalizing vectors in U
* (Workspace: need 3*N + NCU, prefer 3*N + NCU*NB)
*
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
IF( WNTUS )
$ NCU = N
IF( WNTUA )
$ NCU = M
CALL DORGBR( 'Q', M, NCU, N, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVAS ) THEN
*
* If right singular vectors desired in VT, copy result to
* VT and generate right bidiagonalizing vectors in VT
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTUO ) THEN
*
* If left singular vectors desired in A, generate left
* bidiagonalizing vectors in A
* (Workspace: need 4*N, prefer 3*N + N*NB)
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVO ) THEN
*
* If right singular vectors desired in A, generate right
* bidiagonalizing vectors in A
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IWORK = IE + N
IF( WNTUAS .OR. WNTUO )
$ NRU = M
IF( WNTUN )
$ NRU = 0
IF( WNTVAS .OR. WNTVO )
$ NCVT = N
IF( WNTVN )
$ NCVT = 0
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), A,
$ LDA,
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in A and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
END IF
*
END IF
*
ELSE
*
* A has more columns than rows. If A has sufficiently more
* columns than rows, first reduce using the LQ decomposition (if
* sufficient workspace available)
*
IF( N.GE.MNTHR ) THEN
*
IF( WNTVN ) THEN
*
* Path 1t(N much larger than M, JOBVT='N')
* No right singular vectors to be computed
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Zero out above L
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
$ LDA )
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
IF( WNTUO .OR. WNTUAS ) THEN
*
* If left singular vectors desired, generate Q
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IWORK = IE + M
NRU = 0
IF( WNTUO .OR. WNTUAS )
$ NRU = M
*
* Perform bidiagonal QR iteration, computing left singular
* vectors of A in A if desired
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, 0, NRU, 0, S, WORK( IE ), DUM, 1,
$ A,
$ LDA, DUM, 1, WORK( IWORK ), INFO )
*
* If left singular vectors desired in U, copy them there
*
IF( WNTUAS )
$ CALL DLACPY( 'F', M, M, A, LDA, U, LDU )
*
ELSE IF( WNTVO .AND. WNTUN ) THEN
*
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
* M right singular vectors to be overwritten on A and
* no left singular vectors to be computed
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is M by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = M
ELSE
*
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
*
LDWRKU = M
CHUNK = ( LWORK-M*M-M ) / M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IR) and zero out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in A
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
$ WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing L
* (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + M
*
* Multiply right singular vectors of L in WORK(IR) by Q
* in A, storing result in WORK(IU) and copying to A
* (Workspace: need M*M + 2*M, prefer M*M + M*N + M)
*
DO 30 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE,
$ WORK( IR ),
$ LDWRKR, A( 1, I ), LDA, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
$ A( 1, I ), LDA )
30 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize A
* (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing A
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, N, 0, 0, S, WORK( IE ), A,
$ LDA,
$ DUM, 1, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
* Path 3t(N much larger than M, JOBU='S' or 'A', JOBVT='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is M by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = M
ELSE
*
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
*
LDWRKU = M
CHUNK = ( LWORK-M*M-M ) / M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing about above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
$ LDU )
*
* Generate Q in A
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U, copying result to WORK(IR)
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, M, U, LDU, WORK( IR ),
$ LDWRKR )
*
* Generate right vectors bidiagonalizing L in WORK(IR)
* (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing L in U
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U, and computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, U, LDU, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + M
*
* Multiply right singular vectors of L in WORK(IR) by Q
* in A, storing result in WORK(IU) and copying to A
* (Workspace: need M*M + 2*M, prefer M*M + M*N + M))
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE,
$ WORK( IR ),
$ LDWRKR, A( 1, I ), LDA, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
$ A( 1, I ), LDA )
40 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
$ LDU )
*
* Generate Q in A
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in A
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), A, LDA, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing L in U
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), A,
$ LDA,
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTVS ) THEN
*
IF( WNTUN ) THEN
*
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
* M right singular vectors to be computed in VT and
* no left singular vectors to be computed
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IR) is LDA by M
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is M by M
*
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IR), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in A
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing L in
* WORK(IR)
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IR) by
* Q in A, storing result in VT
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
$ LDWRKR, A, LDA, ZERO, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy result to VT
*
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
$ 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in VT
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
$ LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUO ) THEN
*
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be overwritten on A
*
IF( LWORK.GE.2*M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is M by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = M
ELSE
*
* WORK(IU) is M by M and WORK(IR) is M by M
*
LDWRKU = M
IR = IU + LDWRKU*M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out below it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
*
* Generate Q in A
* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*M*M + 4*M,
* prefer 2*M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*M*M + 4*M-1,
* prefer 2*M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in WORK(IR) and computing
* right singular vectors of L in WORK(IU)
* (Workspace: need 2*M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, WORK( IR ),
$ LDWRKR, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in A, storing result in VT
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, A, LDA, ZERO, VT, LDVT )
*
* Copy left singular vectors of L to A
* (Workspace: need M*M)
*
CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
$ 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in VT
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors of L in A
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA,
$ WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, compute left
* singular vectors of A in A and compute right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUAS ) THEN
*
* Path 6t(N much larger than M, JOBU='S' or 'A',
* JOBVT='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IU) is LDA by N
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is LDA by M
*
LDWRKU = M
END IF
ITAU = IU + LDWRKU*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
*
* Generate Q in A
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to U
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
$ LDU )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need M*M + 4*M-1,
* prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU,
$ WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U and computing right
* singular vectors of L in WORK(IU)
* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in A, storing result in VT
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, A, LDA, ZERO, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1,
$ 2 ),
$ LDU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in U by Q
* in VT
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU,
$ WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
ELSE IF( WNTVA ) THEN
*
IF( WNTUN ) THEN
*
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
* N right singular vectors to be computed in VT and
* no left singular vectors to be computed
*
IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IR) is LDA by M
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is M by M
*
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Copy L to WORK(IR), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in VT
* (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
* (Workspace: need M*M + 4*M-1,
* prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IR) by
* Q in VT, storing result in A
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
$ LDWRKR, VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M + N, prefer M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
$ 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in A by Q
* in VT
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
$ LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUO ) THEN
*
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be overwritten on A
*
IF( LWORK.GE.2*M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is M by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = M
ELSE
*
* WORK(IU) is M by M and WORK(IR) is M by M
*
LDWRKU = M
IR = IU + LDWRKU*M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M*M + M + N, prefer 2*M*M + M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*M*M + 4*M,
* prefer 2*M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*M*M + 4*M-1,
* prefer 2*M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in WORK(IR) and computing
* right singular vectors of L in WORK(IU)
* (Workspace: need 2*M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, WORK( IR ),
$ LDWRKR, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in VT, storing result in A
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
* Copy left singular vectors of A from WORK(IR) to A
*
CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M + N, prefer M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
$ 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in A by Q
* in VT
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in A
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA,
$ WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in A and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUAS ) THEN
*
* Path 9t(N much larger than M, JOBU='S' or 'A',
* JOBVT='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IU) is LDA by M
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is M by M
*
LDWRKU = M
END IF
ITAU = IU + LDWRKU*M
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to U
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
$ LDU )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU,
$ WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U and computing right
* singular vectors of L in WORK(IU)
* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in VT, storing result in A
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M + N, prefer M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1,
$ 2 ),
$ LDU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in U by Q
* in VT
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU,
$ WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
END IF
*
ELSE
*
* N .LT. MNTHR
*
* Path 10t(N greater than M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize A
* (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
IF( WNTUAS ) THEN
*
* If left singular vectors desired in U, copy result to U
* and generate left bidiagonalizing vectors in U
* (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DORGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVAS ) THEN
*
* If right singular vectors desired in VT, copy result to
* VT and generate right bidiagonalizing vectors in VT
* (Workspace: need 3*M + NRVT, prefer 3*M + NRVT*NB)
*
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
IF( WNTVA )
$ NRVT = N
IF( WNTVS )
$ NRVT = M
CALL DORGBR( 'P', NRVT, N, M, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTUO ) THEN
*
* If left singular vectors desired in A, generate left
* bidiagonalizing vectors in A
* (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
*
CALL DORGBR( 'Q', M, M, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVO ) THEN
*
* If right singular vectors desired in A, generate right
* bidiagonalizing vectors in A
* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IWORK = IE + M
IF( WNTUAS .OR. WNTUO )
$ NRU = M
IF( WNTUN )
$ NRU = 0
IF( WNTVAS .OR. WNTVO )
$ NCVT = N
IF( WNTVN )
$ NCVT = 0
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), A,
$ LDA,
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in A and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
END IF
*
END IF
*
END IF
*
* If DBDSQR failed to converge, copy unconverged superdiagonals
* to WORK( 2:MINMN )
*
IF( INFO.NE.0 ) THEN
IF( IE.GT.2 ) THEN
DO 50 I = 1, MINMN - 1
WORK( I+1 ) = WORK( I+IE-1 )
50 CONTINUE
END IF
IF( IE.LT.2 ) THEN
DO 60 I = MINMN - 1, 1, -1
WORK( I+1 ) = WORK( I+IE-1 )
60 CONTINUE
END IF
END IF
*
* Undo scaling if necessary
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
IF( INFO.NE.0 .AND. ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN-1, 1,
$ WORK( 2 ),
$ MINMN, IERR )
IF( ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
IF( INFO.NE.0 .AND. ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN-1, 1,
$ WORK( 2 ),
$ MINMN, IERR )
END IF
*
* Return optimal workspace in WORK(1)
*
WORK( 1 ) = MAXWRK
*
RETURN
*
* End of DGESVD
*
END
*> \brief \b DGESVJ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESVJ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvj.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvj.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvj.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
* LDV, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDV, LWORK, M, MV, N
* CHARACTER*1 JOBA, JOBU, JOBV
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), SVA( N ), V( LDV, * ),
* $ WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGESVJ computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, where M >= N. The SVD of A is written as
*> [++] [xx] [x0] [xx]
*> A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx]
*> [++] [xx]
*> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
*> matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
*> of SIGMA are the singular values of A. The columns of U and V are the
*> left and the right singular vectors of A, respectively.
*> DGESVJ can sometimes compute tiny singular values and their singular vectors much
*> more accurately than other SVD routines, see below under Further Details.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBA
*> \verbatim
*> JOBA is CHARACTER*1
*> Specifies the structure of A.
*> = 'L': The input matrix A is lower triangular;
*> = 'U': The input matrix A is upper triangular;
*> = 'G': The input matrix A is general M-by-N matrix, M >= N.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> Specifies whether to compute the left singular vectors
*> (columns of U):
*> = 'U': The left singular vectors corresponding to the nonzero
*> singular values are computed and returned in the leading
*> columns of A. See more details in the description of A.
*> The default numerical orthogonality threshold is set to
*> approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
*> = 'C': Analogous to JOBU='U', except that user can control the
*> level of numerical orthogonality of the computed left
*> singular vectors. TOL can be set to TOL = CTOL*EPS, where
*> CTOL is given on input in the array WORK.
*> No CTOL smaller than ONE is allowed. CTOL greater
*> than 1 / EPS is meaningless. The option 'C'
*> can be used if M*EPS is satisfactory orthogonality
*> of the computed left singular vectors, so CTOL=M could
*> save few sweeps of Jacobi rotations.
*> See the descriptions of A and WORK(1).
*> = 'N': The matrix U is not computed. However, see the
*> description of A.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether to compute the right singular vectors, that
*> is, the matrix V:
*> = 'V': the matrix V is computed and returned in the array V
*> = 'A': the Jacobi rotations are applied to the MV-by-N
*> array V. In other words, the right singular vector
*> matrix V is not computed explicitly, instead it is
*> applied to an MV-by-N matrix initially stored in the
*> first MV rows of V.
*> = 'N': the matrix V is not computed and the array V is not
*> referenced
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A.
*> M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit :
*> If JOBU = 'U' .OR. JOBU = 'C' :
*> If INFO = 0 :
*> RANKA orthonormal columns of U are returned in the
*> leading RANKA columns of the array A. Here RANKA <= N
*> is the number of computed singular values of A that are
*> above the underflow threshold DLAMCH('S'). The singular
*> vectors corresponding to underflowed or zero singular
*> values are not computed. The value of RANKA is returned
*> in the array WORK as RANKA=NINT(WORK(2)). Also see the
*> descriptions of SVA and WORK. The computed columns of U
*> are mutually numerically orthogonal up to approximately
*> TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
*> see the description of JOBU.
*> If INFO > 0 :
*> the procedure DGESVJ did not converge in the given number
*> of iterations (sweeps). In that case, the computed
*> columns of U may not be orthogonal up to TOL. The output
*> U (stored in A), SIGMA (given by the computed singular
*> values in SVA(1:N)) and V is still a decomposition of the
*> input matrix A in the sense that the residual
*> ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
*>
*> If JOBU = 'N' :
*> If INFO = 0 :
*> Note that the left singular vectors are 'for free' in the
*> one-sided Jacobi SVD algorithm. However, if only the
*> singular values are needed, the level of numerical
*> orthogonality of U is not an issue and iterations are
*> stopped when the columns of the iterated matrix are
*> numerically orthogonal up to approximately M*EPS. Thus,
*> on exit, A contains the columns of U scaled with the
*> corresponding singular values.
*> If INFO > 0 :
*> the procedure DGESVJ did not converge in the given number
*> of iterations (sweeps).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] SVA
*> \verbatim
*> SVA is DOUBLE PRECISION array, dimension (N)
*> On exit :
*> If INFO = 0 :
*> depending on the value SCALE = WORK(1), we have:
*> If SCALE = ONE :
*> SVA(1:N) contains the computed singular values of A.
*> During the computation SVA contains the Euclidean column
*> norms of the iterated matrices in the array A.
*> If SCALE .NE. ONE :
*> The singular values of A are SCALE*SVA(1:N), and this
*> factored representation is due to the fact that some of the
*> singular values of A might underflow or overflow.
*> If INFO > 0 :
*> the procedure DGESVJ did not converge in the given number of
*> iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*> MV is INTEGER
*> If JOBV = 'A', then the product of Jacobi rotations in DGESVJ
*> is applied to the first MV rows of V. See the description of JOBV.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,N)
*> If JOBV = 'V', then V contains on exit the N-by-N matrix of
*> the right singular vectors;
*> If JOBV = 'A', then V contains the product of the computed right
*> singular vector matrix and the initial matrix in
*> the array V.
*> If JOBV = 'N', then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V, LDV >= 1.
*> If JOBV = 'V', then LDV >= max(1,N).
*> If JOBV = 'A', then LDV >= max(1,MV) .
*> \endverbatim
*>
*> \param[in,out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On entry :
*> If JOBU = 'C' :
*> WORK(1) = CTOL, where CTOL defines the threshold for convergence.
*> The process stops if all columns of A are mutually
*> orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
*> It is required that CTOL >= ONE, i.e. it is not
*> allowed to force the routine to obtain orthogonality
*> below EPS.
*> On exit :
*> WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
*> are the computed singular values of A.
*> (See description of SVA().)
*> WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
*> singular values.
*> WORK(3) = NINT(WORK(3)) is the number of the computed singular
*> values that are larger than the underflow threshold.
*> WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
*> rotations needed for numerical convergence.
*> WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
*> This is useful information in cases when DGESVJ did
*> not converge, as it can be used to estimate whether
*> the output is still useful and for post festum analysis.
*> WORK(6) = the largest absolute value over all sines of the
*> Jacobi rotation angles in the last sweep. It can be
*> useful for a post festum analysis.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK.
*> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(6,M+N), otherwise.
*>
*> If on entry LWORK = -1, then a workspace query is assumed and
*> no computation is done; WORK(1) is set to the minial (and optimal)
*> length of WORK.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, then the i-th argument had an illegal value
*> > 0: DGESVJ did not converge in the maximal allowed number (30)
*> of sweeps. The output may still be useful. See the
*> description of WORK.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesvj
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
*> rotations. The rotations are implemented as fast scaled rotations of
*> Anda and Park [1]. In the case of underflow of the Jacobi angle, a
*> modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
*> column interchanges of de Rijk [2]. The relative accuracy of the computed
*> singular values and the accuracy of the computed singular vectors (in
*> angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
*> The condition number that determines the accuracy in the full rank case
*> is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
*> spectral condition number. The best performance of this Jacobi SVD
*> procedure is achieved if used in an accelerated version of Drmac and
*> Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
*> Some tuning parameters (marked with [TP]) are available for the
*> implementer.
*> The computational range for the nonzero singular values is the machine
*> number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
*> denormalized singular values can be computed with the corresponding
*> gradual loss of accurate digits.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> ============
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*> \endverbatim
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
*> SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
*> [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
*> singular value decomposition on a vector computer.
*> SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
*> [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
*> [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
*> value computation in floating point arithmetic.
*> SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
*> [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
*> LAPACK Working note 169.
*> [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
*> LAPACK Working note 170.
*> [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*> QSVD, (H,K)-SVD computations.
*> Department of Mathematics, University of Zagreb, 2008.
*> \endverbatim
*
*> \par Bugs, examples and comments:
* =================================
*>
*> \verbatim
*> ===========================
*> Please report all bugs and send interesting test examples and comments to
*> drmac@math.hr. Thank you.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
$ LDV, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDV, LWORK, M, MV, N
CHARACTER*1 JOBA, JOBU, JOBV
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SVA( N ), V( LDV, * ),
$ WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Local Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
INTEGER NSWEEP
PARAMETER ( NSWEEP = 30 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
$ BIGTHETA, CS, CTOL, EPSLN, LARGE, MXAAPQ,
$ MXSINJ, ROOTBIG, ROOTEPS, ROOTSFMIN, ROOTTOL,
$ SKL, SFMIN, SMALL, SN, T, TEMP1, THETA,
$ THSIGN, TOL
INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
$ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, N2, N34,
$ N4, NBL, NOTROT, p, PSKIPPED, q, ROWSKIP,
$ SWBAND, MINMN, LWMIN
LOGICAL APPLV, GOSCALE, LOWER, LQUERY, LSVEC, NOSCALE,
$ ROTOK, RSVEC, UCTOL, UPPER
* ..
* .. Local Arrays ..
DOUBLE PRECISION FASTR( 5 )
* ..
* .. Intrinsic Functions ..
INTRINSIC DABS, MAX, MIN, DBLE, DSIGN, DSQRT
* ..
* .. External Functions ..
* ..
* from BLAS
DOUBLE PRECISION DDOT, DNRM2
EXTERNAL DDOT, DNRM2
INTEGER IDAMAX
EXTERNAL IDAMAX
* from LAPACK
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
* ..
* from BLAS
EXTERNAL DAXPY, DCOPY, DROTM, DSCAL, DSWAP
* from LAPACK
EXTERNAL DLASCL, DLASET, DLASSQ, XERBLA
*
EXTERNAL DGSVJ0, DGSVJ1
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
LSVEC = LSAME( JOBU, 'U' )
UCTOL = LSAME( JOBU, 'C' )
RSVEC = LSAME( JOBV, 'V' )
APPLV = LSAME( JOBV, 'A' )
UPPER = LSAME( JOBA, 'U' )
LOWER = LSAME( JOBA, 'L' )
*
MINMN = MIN( M, N )
IF( MINMN.EQ.0 ) THEN
LWMIN = 1
ELSE
LWMIN = MAX( 6, M+N )
END IF
*
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.( UPPER .OR. LOWER .OR. LSAME( JOBA, 'G' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LSVEC .OR.
$ UCTOL .OR.
$ LSAME( JOBU, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( RSVEC .OR.
$ APPLV .OR.
$ LSAME( JOBV, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
INFO = -5
ELSE IF( LDA.LT.M ) THEN
INFO = -7
ELSE IF( MV.LT.0 ) THEN
INFO = -9
ELSE IF( ( RSVEC .AND. ( LDV.LT.N ) ) .OR.
$ ( APPLV .AND. ( LDV.LT.MV ) ) ) THEN
INFO = -11
ELSE IF( UCTOL .AND. ( WORK( 1 ).LE.ONE ) ) THEN
INFO = -12
ELSE IF( LWORK.LT.LWMIN .AND. ( .NOT.LQUERY ) ) THEN
INFO = -13
ELSE
INFO = 0
END IF
*
* #:(
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGESVJ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
WORK( 1 ) = LWMIN
RETURN
END IF
*
* #:) Quick return for void matrix
*
IF( MINMN.EQ.0 ) RETURN
*
* Set numerical parameters
* The stopping criterion for Jacobi rotations is
*
* max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS
*
* where EPS is the round-off and CTOL is defined as follows:
*
IF( UCTOL ) THEN
* ... user controlled
CTOL = WORK( 1 )
ELSE
* ... default
IF( LSVEC .OR. RSVEC .OR. APPLV ) THEN
CTOL = DSQRT( DBLE( M ) )
ELSE
CTOL = DBLE( M )
END IF
END IF
* ... and the machine dependent parameters are
*[!] (Make sure that DLAMCH() works properly on the target machine.)
*
EPSLN = DLAMCH( 'Epsilon' )
ROOTEPS = DSQRT( EPSLN )
SFMIN = DLAMCH( 'SafeMinimum' )
ROOTSFMIN = DSQRT( SFMIN )
SMALL = SFMIN / EPSLN
BIG = DLAMCH( 'Overflow' )
* BIG = ONE / SFMIN
ROOTBIG = ONE / ROOTSFMIN
LARGE = BIG / DSQRT( DBLE( M*N ) )
BIGTHETA = ONE / ROOTEPS
*
TOL = CTOL*EPSLN
ROOTTOL = DSQRT( TOL )
*
IF( DBLE( M )*EPSLN.GE.ONE ) THEN
INFO = -4
CALL XERBLA( 'DGESVJ', -INFO )
RETURN
END IF
*
* Initialize the right singular vector matrix.
*
IF( RSVEC ) THEN
MVL = N
CALL DLASET( 'A', MVL, N, ZERO, ONE, V, LDV )
ELSE IF( APPLV ) THEN
MVL = MV
END IF
RSVEC = RSVEC .OR. APPLV
*
* Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N )
*(!) If necessary, scale A to protect the largest singular value
* from overflow. It is possible that saving the largest singular
* value destroys the information about the small ones.
* This initial scaling is almost minimal in the sense that the
* goal is to make sure that no column norm overflows, and that
* DSQRT(N)*max_i SVA(i) does not overflow. If INFinite entries
* in A are detected, the procedure returns with INFO=-6.
*
SKL= ONE / DSQRT( DBLE( M )*DBLE( N ) )
NOSCALE = .TRUE.
GOSCALE = .TRUE.
*
IF( LOWER ) THEN
* the input matrix is M-by-N lower triangular (trapezoidal)
DO 1874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL DLASSQ( M-p+1, A( p, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'DGESVJ', -INFO )
RETURN
END IF
AAQQ = DSQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL)
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 1873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
1873 CONTINUE
END IF
END IF
1874 CONTINUE
ELSE IF( UPPER ) THEN
* the input matrix is M-by-N upper triangular (trapezoidal)
DO 2874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL DLASSQ( p, A( 1, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'DGESVJ', -INFO )
RETURN
END IF
AAQQ = DSQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL)
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 2873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
2873 CONTINUE
END IF
END IF
2874 CONTINUE
ELSE
* the input matrix is M-by-N general dense
DO 3874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL DLASSQ( M, A( 1, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'DGESVJ', -INFO )
RETURN
END IF
AAQQ = DSQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL)
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 3873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
3873 CONTINUE
END IF
END IF
3874 CONTINUE
END IF
*
IF( NOSCALE )SKL= ONE
*
* Move the smaller part of the spectrum from the underflow threshold
*(!) Start by determining the position of the nonzero entries of the
* array SVA() relative to ( SFMIN, BIG ).
*
AAPP = ZERO
AAQQ = BIG
DO 4781 p = 1, N
IF( SVA( p ).NE.ZERO )AAQQ = MIN( AAQQ, SVA( p ) )
AAPP = MAX( AAPP, SVA( p ) )
4781 CONTINUE
*
* #:) Quick return for zero matrix
*
IF( AAPP.EQ.ZERO ) THEN
IF( LSVEC )CALL DLASET( 'G', M, N, ZERO, ONE, A, LDA )
WORK( 1 ) = ONE
WORK( 2 ) = ZERO
WORK( 3 ) = ZERO
WORK( 4 ) = ZERO
WORK( 5 ) = ZERO
WORK( 6 ) = ZERO
RETURN
END IF
*
* #:) Quick return for one-column matrix
*
IF( N.EQ.1 ) THEN
IF( LSVEC )CALL DLASCL( 'G', 0, 0, SVA( 1 ), SKL, M, 1,
$ A( 1, 1 ), LDA, IERR )
WORK( 1 ) = ONE / SKL
IF( SVA( 1 ).GE.SFMIN ) THEN
WORK( 2 ) = ONE
ELSE
WORK( 2 ) = ZERO
END IF
WORK( 3 ) = ZERO
WORK( 4 ) = ZERO
WORK( 5 ) = ZERO
WORK( 6 ) = ZERO
RETURN
END IF
*
* Protect small singular values from underflow, and try to
* avoid underflows/overflows in computing Jacobi rotations.
*
SN = DSQRT( SFMIN / EPSLN )
TEMP1 = DSQRT( BIG / DBLE( N ) )
IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR.
$ ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN
TEMP1 = MIN( BIG, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN
TEMP1 = MIN( SN / AAQQ, BIG / ( AAPP*DSQRT( DBLE( N ) ) ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
TEMP1 = MAX( SN / AAQQ, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
TEMP1 = MIN( SN / AAQQ, BIG / ( DSQRT( DBLE( N ) )*AAPP ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE
TEMP1 = ONE
END IF
*
* Scale, if necessary
*
IF( TEMP1.NE.ONE ) THEN
CALL DLASCL( 'G', 0, 0, ONE, TEMP1, N, 1, SVA, N, IERR )
END IF
SKL= TEMP1*SKL
IF( SKL.NE.ONE ) THEN
CALL DLASCL( JOBA, 0, 0, ONE, SKL, M, N, A, LDA, IERR )
SKL= ONE / SKL
END IF
*
* Row-cyclic Jacobi SVD algorithm with column pivoting
*
EMPTSW = ( N*( N-1 ) ) / 2
NOTROT = 0
FASTR( 1 ) = ZERO
*
* A is represented in factored form A = A * diag(WORK), where diag(WORK)
* is initialized to identity. WORK is updated during fast scaled
* rotations.
*
DO 1868 q = 1, N
WORK( q ) = ONE
1868 CONTINUE
*
*
SWBAND = 3
*[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective
* if DGESVJ is used as a computational routine in the preconditioned
* Jacobi SVD algorithm DGESVJ. For sweeps i=1:SWBAND the procedure
* works on pivots inside a band-like region around the diagonal.
* The boundaries are determined dynamically, based on the number of
* pivots above a threshold.
*
KBL = MIN( 8, N )
*[TP] KBL is a tuning parameter that defines the tile size in the
* tiling of the p-q loops of pivot pairs. In general, an optimal
* value of KBL depends on the matrix dimensions and on the
* parameters of the computer's memory.
*
NBL = N / KBL
IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
*
BLSKIP = KBL**2
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
*
ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
*
LKAHEAD = 1
*[TP] LKAHEAD is a tuning parameter.
*
* Quasi block transformations, using the lower (upper) triangular
* structure of the input matrix. The quasi-block-cycling usually
* invokes cubic convergence. Big part of this cycle is done inside
* canonical subspaces of dimensions less than M.
*
IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX( 64, 4*KBL ) ) ) THEN
*[TP] The number of partition levels and the actual partition are
* tuning parameters.
N4 = N / 4
N2 = N / 2
N34 = 3*N4
IF( APPLV ) THEN
q = 0
ELSE
q = 1
END IF
*
IF( LOWER ) THEN
*
* This works very well on lower triangular matrices, in particular
* in the framework of the preconditioned Jacobi SVD (xGEJSV).
* The idea is simple:
* [+ 0 0 0] Note that Jacobi transformations of [0 0]
* [+ + 0 0] [0 0]
* [+ + x 0] actually work on [x 0] [x 0]
* [+ + x x] [x x]. [x x]
*
CALL DGSVJ0( JOBV, M-N34, N-N34, A( N34+1, N34+1 ), LDA,
$ WORK( N34+1 ), SVA( N34+1 ), MVL,
$ V( N34*q+1, N34+1 ), LDV, EPSLN, SFMIN, TOL,
$ 2, WORK( N+1 ), LWORK-N, IERR )
*
CALL DGSVJ0( JOBV, M-N2, N34-N2, A( N2+1, N2+1 ), LDA,
$ WORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 2,
$ WORK( N+1 ), LWORK-N, IERR )
*
CALL DGSVJ1( JOBV, M-N2, N-N2, N4, A( N2+1, N2+1 ), LDA,
$ WORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ WORK( N+1 ), LWORK-N, IERR )
*
CALL DGSVJ0( JOBV, M-N4, N2-N4, A( N4+1, N4+1 ), LDA,
$ WORK( N4+1 ), SVA( N4+1 ), MVL,
$ V( N4*q+1, N4+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ WORK( N+1 ), LWORK-N, IERR )
*
CALL DGSVJ0( JOBV, M, N4, A, LDA, WORK, SVA, MVL, V, LDV,
$ EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N,
$ IERR )
*
CALL DGSVJ1( JOBV, M, N2, N4, A, LDA, WORK, SVA, MVL, V,
$ LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ),
$ LWORK-N, IERR )
*
*
ELSE IF( UPPER ) THEN
*
*
CALL DGSVJ0( JOBV, N4, N4, A, LDA, WORK, SVA, MVL, V,
$ LDV,
$ EPSLN, SFMIN, TOL, 2, WORK( N+1 ), LWORK-N,
$ IERR )
*
CALL DGSVJ0( JOBV, N2, N4, A( 1, N4+1 ), LDA,
$ WORK( N4+1 ),
$ SVA( N4+1 ), MVL, V( N4*q+1, N4+1 ), LDV,
$ EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N,
$ IERR )
*
CALL DGSVJ1( JOBV, N2, N2, N4, A, LDA, WORK, SVA, MVL, V,
$ LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ),
$ LWORK-N, IERR )
*
CALL DGSVJ0( JOBV, N2+N4, N4, A( 1, N2+1 ), LDA,
$ WORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ WORK( N+1 ), LWORK-N, IERR )
END IF
*
END IF
*
* .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
DO 1993 i = 1, NSWEEP
*
* .. go go go ...
*
MXAAPQ = ZERO
MXSINJ = ZERO
ISWROT = 0
*
NOTROT = 0
PSKIPPED = 0
*
* Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs
* 1 <= p < q <= N. This is the first step toward a blocked implementation
* of the rotations. New implementation, based on block transformations,
* is under development.
*
DO 2000 ibr = 1, NBL
*
igl = ( ibr-1 )*KBL + 1
*
DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr )
*
igl = igl + ir1*KBL
*
DO 2001 p = igl, MIN( igl+KBL-1, N-1 )
*
* .. de Rijk's pivoting
*
q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
IF( p.NE.q ) THEN
CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
$ V( 1, q ), 1 )
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = WORK( p )
WORK( p ) = WORK( q )
WORK( q ) = TEMP1
END IF
*
IF( ir1.EQ.0 ) THEN
*
* Column norms are periodically updated by explicit
* norm computation.
* Caveat:
* Unfortunately, some BLAS implementations compute DNRM2(M,A(1,p),1)
* as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to
* overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and to
* underflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
* Hence, DNRM2 cannot be trusted, not even in the case when
* the true norm is far from the under(over)flow boundaries.
* If properly implemented DNRM2 is available, the IF-THEN-ELSE
* below should read "AAPP = DNRM2( M, A(1,p), 1 ) * WORK(p)".
*
IF( ( SVA( p ).LT.ROOTBIG ) .AND.
$ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
SVA( p ) = DNRM2( M, A( 1, p ), 1 )*WORK( p )
ELSE
TEMP1 = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
SVA( p ) = TEMP1*DSQRT( AAPP )*WORK( p )
END IF
AAPP = SVA( p )
ELSE
AAPP = SVA( p )
END IF
*
IF( AAPP.GT.ZERO ) THEN
*
PSKIPPED = 0
*
DO 2002 q = p + 1, MIN( igl+KBL-1, N )
*
AAQQ = SVA( q )
*
IF( AAQQ.GT.ZERO ) THEN
*
AAPP0 = AAPP
IF( AAQQ.GE.ONE ) THEN
ROTOK = ( SMALL*AAPP ).LE.AAQQ
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL DCOPY( M, A( 1, p ), 1,
$ WORK( N+1 ), 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ WORK( p ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = DDOT( M, WORK( N+1 ), 1,
$ A( 1, q ), 1 )*WORK( q ) / AAQQ
END IF
ELSE
ROTOK = AAPP.LE.( AAQQ / SMALL )
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL DCOPY( M, A( 1, q ), 1,
$ WORK( N+1 ), 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ WORK( q ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = DDOT( M, WORK( N+1 ), 1,
$ A( 1, p ), 1 )*WORK( p ) / AAPP
END IF
END IF
*
MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( DABS( AAPQ ).GT.TOL ) THEN
*
* .. rotate
*[RTD] ROTATED = ROTATED + ONE
*
IF( ir1.EQ.0 ) THEN
NOTROT = 0
PSKIPPED = 0
ISWROT = ISWROT + 1
END IF
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*DABS(AQOAP-APOAQ)/AAPQ
*
IF( DABS( THETA ).GT.BIGTHETA ) THEN
*
T = HALF / THETA
FASTR( 3 ) = T*WORK( p ) / WORK( q )
FASTR( 4 ) = -T*WORK( q ) /
$ WORK( p )
CALL DROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = MAX( MXSINJ, DABS( T ) )
*
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -DSIGN( ONE, AAPQ )
T = ONE / ( THETA+THSIGN*
$ DSQRT( ONE+THETA*THETA ) )
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
*
MXSINJ = MAX( MXSINJ, DABS( SN ) )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = WORK( p ) / WORK( q )
AQOAP = WORK( q ) / WORK( p )
IF( WORK( p ).GE.ONE ) THEN
IF( WORK( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( q )*CS
CALL DROTM( M, A( 1, p ),
$ 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( q ) / CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
END IF
ELSE
IF( WORK( q ).GE.ONE ) THEN
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( q )*CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
ELSE
IF( WORK( p ).GE.WORK( q ) )
$ THEN
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M,
$ CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( q ) / CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( q )*CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
*
ELSE
* .. have to use modified Gram-Schmidt like transformation
CALL DCOPY( M, A( 1, p ), 1,
$ WORK( N+1 ), 1 )
CALL DLASCL( 'G', 0, 0, AAPP, ONE,
$ M,
$ 1, WORK( N+1 ), LDA,
$ IERR )
CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
$ M,
$ 1, A( 1, q ), LDA, IERR )
TEMP1 = -AAPQ*WORK( p ) / WORK( q )
CALL DAXPY( M, TEMP1, WORK( N+1 ),
$ 1,
$ A( 1, q ), 1 )
CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
$ M,
$ 1, A( 1, q ), LDA, IERR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q), SVA(p)
* recompute SVA(q), SVA(p).
*
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
$ THEN
IF( ( AAQQ.LT.ROOTBIG ) .AND.
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
SVA( q ) = DNRM2( M, A( 1, q ),
$ 1 )*
$ WORK( q )
ELSE
T = ZERO
AAQQ = ONE
CALL DLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*DSQRT( AAQQ )*WORK( q )
END IF
END IF
IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = DNRM2( M, A( 1, p ), 1 )*
$ WORK( p )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*DSQRT( AAPP )*WORK( p )
END IF
SVA( p ) = AAPP
END IF
*
ELSE
* A(:,p) and A(:,q) already numerically orthogonal
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
*[RTD] SKIPPED = SKIPPED + 1
PSKIPPED = PSKIPPED + 1
END IF
ELSE
* A(:,q) is zero column
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
END IF
*
IF( ( i.LE.SWBAND ) .AND.
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
IF( ir1.EQ.0 )AAPP = -AAPP
NOTROT = 0
GO TO 2103
END IF
*
2002 CONTINUE
* END q-LOOP
*
2103 CONTINUE
* bailed out of q-loop
*
SVA( p ) = AAPP
*
ELSE
SVA( p ) = AAPP
IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
$ NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p
END IF
*
2001 CONTINUE
* end of the p-loop
* end of doing the block ( ibr, ibr )
1002 CONTINUE
* end of ir1-loop
*
* ... go to the off diagonal blocks
*
igl = ( ibr-1 )*KBL + 1
*
DO 2010 jbc = ibr + 1, NBL
*
jgl = ( jbc-1 )*KBL + 1
*
* doing the block at ( ibr, jbc )
*
IJBLSK = 0
DO 2100 p = igl, MIN( igl+KBL-1, N )
*
AAPP = SVA( p )
IF( AAPP.GT.ZERO ) THEN
*
PSKIPPED = 0
*
DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
AAQQ = SVA( q )
IF( AAQQ.GT.ZERO ) THEN
AAPP0 = AAPP
*
* .. M x 2 Jacobi SVD ..
*
* Safe Gram matrix computation
*
IF( AAQQ.GE.ONE ) THEN
IF( AAPP.GE.AAQQ ) THEN
ROTOK = ( SMALL*AAPP ).LE.AAQQ
ELSE
ROTOK = ( SMALL*AAQQ ).LE.AAPP
END IF
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL DCOPY( M, A( 1, p ), 1,
$ WORK( N+1 ), 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ WORK( p ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = DDOT( M, WORK( N+1 ), 1,
$ A( 1, q ), 1 )*WORK( q ) / AAQQ
END IF
ELSE
IF( AAPP.GE.AAQQ ) THEN
ROTOK = AAPP.LE.( AAQQ / SMALL )
ELSE
ROTOK = AAQQ.LE.( AAPP / SMALL )
END IF
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL DCOPY( M, A( 1, q ), 1,
$ WORK( N+1 ), 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ WORK( q ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = DDOT( M, WORK( N+1 ), 1,
$ A( 1, p ), 1 )*WORK( p ) / AAPP
END IF
END IF
*
MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( DABS( AAPQ ).GT.TOL ) THEN
NOTROT = 0
*[RTD] ROTATED = ROTATED + 1
PSKIPPED = 0
ISWROT = ISWROT + 1
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*DABS(AQOAP-APOAQ)/AAPQ
IF( AAQQ.GT.AAPP0 )THETA = -THETA
*
IF( DABS( THETA ).GT.BIGTHETA ) THEN
T = HALF / THETA
FASTR( 3 ) = T*WORK( p ) / WORK( q )
FASTR( 4 ) = -T*WORK( q ) /
$ WORK( p )
CALL DROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = MAX( MXSINJ, DABS( T ) )
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -DSIGN( ONE, AAPQ )
IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
T = ONE / ( THETA+THSIGN*
$ DSQRT( ONE+THETA*THETA ) )
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
MXSINJ = MAX( MXSINJ, DABS( SN ) )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = WORK( p ) / WORK( q )
AQOAP = WORK( q ) / WORK( p )
IF( WORK( p ).GE.ONE ) THEN
*
IF( WORK( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( q )*CS
CALL DROTM( M, A( 1, p ),
$ 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( q ) / CS
END IF
ELSE
IF( WORK( q ).GE.ONE ) THEN
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( q )*CS
ELSE
IF( WORK( p ).GE.WORK( q ) )
$ THEN
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M,
$ CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( q ) / CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( q )*CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
*
ELSE
IF( AAPP.GT.AAQQ ) THEN
CALL DCOPY( M, A( 1, p ), 1,
$ WORK( N+1 ), 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ ONE,
$ M, 1, WORK( N+1 ), LDA,
$ IERR )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ ONE,
$ M, 1, A( 1, q ), LDA,
$ IERR )
TEMP1 = -AAPQ*WORK( p ) / WORK( q )
CALL DAXPY( M, TEMP1,
$ WORK( N+1 ),
$ 1, A( 1, q ), 1 )
CALL DLASCL( 'G', 0, 0, ONE,
$ AAQQ,
$ M, 1, A( 1, q ), LDA,
$ IERR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
ELSE
CALL DCOPY( M, A( 1, q ), 1,
$ WORK( N+1 ), 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ ONE,
$ M, 1, WORK( N+1 ), LDA,
$ IERR )
CALL DLASCL( 'G', 0, 0, AAPP,
$ ONE,
$ M, 1, A( 1, p ), LDA,
$ IERR )
TEMP1 = -AAPQ*WORK( q ) / WORK( p )
CALL DAXPY( M, TEMP1,
$ WORK( N+1 ),
$ 1, A( 1, p ), 1 )
CALL DLASCL( 'G', 0, 0, ONE,
$ AAPP,
$ M, 1, A( 1, p ), LDA,
$ IERR )
SVA( p ) = AAPP*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
END IF
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q)
* .. recompute SVA(q)
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
$ THEN
IF( ( AAQQ.LT.ROOTBIG ) .AND.
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
SVA( q ) = DNRM2( M, A( 1, q ),
$ 1 )*
$ WORK( q )
ELSE
T = ZERO
AAQQ = ONE
CALL DLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*DSQRT( AAQQ )*WORK( q )
END IF
END IF
IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = DNRM2( M, A( 1, p ), 1 )*
$ WORK( p )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*DSQRT( AAPP )*WORK( p )
END IF
SVA( p ) = AAPP
END IF
* end of OK rotation
ELSE
NOTROT = NOTROT + 1
*[RTD] SKIPPED = SKIPPED + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
ELSE
NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
*
IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
$ THEN
SVA( p ) = AAPP
NOTROT = 0
GO TO 2011
END IF
IF( ( i.LE.SWBAND ) .AND.
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
AAPP = -AAPP
NOTROT = 0
GO TO 2203
END IF
*
2200 CONTINUE
* end of the q-loop
2203 CONTINUE
*
SVA( p ) = AAPP
*
ELSE
*
IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
$ MIN( jgl+KBL-1, N ) - jgl + 1
IF( AAPP.LT.ZERO )NOTROT = 0
*
END IF
*
2100 CONTINUE
* end of the p-loop
2010 CONTINUE
* end of the jbc-loop
2011 CONTINUE
*2011 bailed out of the jbc-loop
DO 2012 p = igl, MIN( igl+KBL-1, N )
SVA( p ) = DABS( SVA( p ) )
2012 CONTINUE
***
2000 CONTINUE
*2000 :: end of the ibr-loop
*
* .. update SVA(N)
IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
$ THEN
SVA( N ) = DNRM2( M, A( 1, N ), 1 )*WORK( N )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
SVA( N ) = T*DSQRT( AAPP )*WORK( N )
END IF
*
* Additional steering devices
*
IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
$ ( ISWROT.LE.N ) ) )SWBAND = i
*
IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DSQRT( DBLE( N ) )*
$ TOL ) .AND. ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
GO TO 1994
END IF
*
IF( NOTROT.GE.EMPTSW )GO TO 1994
*
1993 CONTINUE
* end i=1:NSWEEP loop
*
* #:( Reaching this point means that the procedure has not converged.
INFO = NSWEEP - 1
GO TO 1995
*
1994 CONTINUE
* #:) Reaching this point means numerical convergence after the i-th
* sweep.
*
INFO = 0
* #:) INFO = 0 confirms successful iterations.
1995 CONTINUE
*
* Sort the singular values and find how many are above
* the underflow threshold.
*
N2 = 0
N4 = 0
DO 5991 p = 1, N - 1
q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
IF( p.NE.q ) THEN
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = WORK( p )
WORK( p ) = WORK( q )
WORK( q ) = TEMP1
CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
END IF
IF( SVA( p ).NE.ZERO ) THEN
N4 = N4 + 1
IF( SVA( p )*SKL.GT.SFMIN )N2 = N2 + 1
END IF
5991 CONTINUE
IF( SVA( N ).NE.ZERO ) THEN
N4 = N4 + 1
IF( SVA( N )*SKL.GT.SFMIN )N2 = N2 + 1
END IF
*
* Normalize the left singular vectors.
*
IF( LSVEC .OR. UCTOL ) THEN
DO 1998 p = 1, N2
CALL DSCAL( M, WORK( p ) / SVA( p ), A( 1, p ), 1 )
1998 CONTINUE
END IF
*
* Scale the product of Jacobi rotations (assemble the fast rotations).
*
IF( RSVEC ) THEN
IF( APPLV ) THEN
DO 2398 p = 1, N
CALL DSCAL( MVL, WORK( p ), V( 1, p ), 1 )
2398 CONTINUE
ELSE
DO 2399 p = 1, N
TEMP1 = ONE / DNRM2( MVL, V( 1, p ), 1 )
CALL DSCAL( MVL, TEMP1, V( 1, p ), 1 )
2399 CONTINUE
END IF
END IF
*
* Undo scaling, if necessary (and possible).
IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG / SKL) ) )
$ .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( MAX( N2, 1 ) ) .GT.
$ ( SFMIN / SKL) ) ) ) THEN
DO 2400 p = 1, N
SVA( P ) = SKL*SVA( P )
2400 CONTINUE
SKL= ONE
END IF
*
WORK( 1 ) = SKL
* The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE
* then some of the singular values may overflow or underflow and
* the spectrum is given in this factored representation.
*
WORK( 2 ) = DBLE( N4 )
* N4 is the number of computed nonzero singular values of A.
*
WORK( 3 ) = DBLE( N2 )
* N2 is the number of singular values of A greater than SFMIN.
* If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers
* that may carry some information.
*
WORK( 4 ) = DBLE( i )
* i is the index of the last sweep before declaring convergence.
*
WORK( 5 ) = MXAAPQ
* MXAAPQ is the largest absolute value of scaled pivots in the
* last sweep
*
WORK( 6 ) = MXSINJ
* MXSINJ is the largest absolute value of the sines of Jacobi angles
* in the last sweep
*
RETURN
* ..
* .. END OF DGESVJ
* ..
END
*> \brief <b> DGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
* EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, TRANS
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ BERR( * ), C( * ), FERR( * ), R( * ),
* $ WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGESVX uses the LU factorization to compute the solution to a real
*> system of linear equations
*> A * X = B,
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*> Whether or not the system will be equilibrated depends on the
*> scaling of the matrix A, but if equilibration is used, A is
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
*> or diag(C)*B (if TRANS = 'T' or 'C').
*>
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
*> matrix A (after equilibration if FACT = 'E') as
*> A = P * L * U,
*> where P is a permutation matrix, L is a unit lower triangular
*> matrix, and U is upper triangular.
*>
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
*> returns with INFO = i. Otherwise, the factored form of A is used
*> to estimate the condition number of the matrix A. If the
*> reciprocal of the condition number is less than machine precision,
*> INFO = N+1 is returned as a warning, but the routine still goes on
*> to solve for X and compute error bounds as described below.
*>
*> 4. The system of equations is solved for X using the factored form
*> of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
*> that it solves the original system before equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of the matrix A is
*> supplied on entry, and if not, whether the matrix A should be
*> equilibrated before it is factored.
*> = 'F': On entry, AF and IPIV contain the factored form of A.
*> If EQUED is not 'N', the matrix A has been
*> equilibrated with scaling factors given by R and C.
*> A, AF, and IPIV are not modified.
*> = 'N': The matrix A will be copied to AF and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> copied to AF and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
*> not 'N', then A must have been equilibrated by the scaling
*> factors in R and/or C. A is not modified if FACT = 'F' or
*> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*>
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
*> EQUED = 'R': A := diag(R) * A
*> EQUED = 'C': A := A * diag(C)
*> EQUED = 'B': A := diag(R) * A * diag(C).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> If FACT = 'F', then AF is an input argument and on entry
*> contains the factors L and U from the factorization
*> A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
*> AF is the factored form of the equilibrated matrix A.
*>
*> If FACT = 'N', then AF is an output argument and on exit
*> returns the factors L and U from the factorization A = P*L*U
*> of the original matrix A.
*>
*> If FACT = 'E', then AF is an output argument and on exit
*> returns the factors L and U from the factorization A = P*L*U
*> of the equilibrated matrix A (see the description of A for
*> the form of the equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains the pivot indices from the factorization A = P*L*U
*> as computed by DGETRF; row i of the matrix was interchanged
*> with row IPIV(i).
*>
*> If FACT = 'N', then IPIV is an output argument and on exit
*> contains the pivot indices from the factorization A = P*L*U
*> of the original matrix A.
*>
*> If FACT = 'E', then IPIV is an output argument and on exit
*> contains the pivot indices from the factorization A = P*L*U
*> of the equilibrated matrix A.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration (always true if FACT = 'N').
*> = 'R': Row equilibration, i.e., A has been premultiplied by
*> diag(R).
*> = 'C': Column equilibration, i.e., A has been postmultiplied
*> by diag(C).
*> = 'B': Both row and column equilibration, i.e., A has been
*> replaced by diag(R) * A * diag(C).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (N)
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
*> is not accessed. R is an input argument if FACT = 'F';
*> otherwise, R is an output argument. If FACT = 'F' and
*> EQUED = 'R' or 'B', each element of R must be positive.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
*> is not accessed. C is an input argument if FACT = 'F';
*> otherwise, C is an output argument. If FACT = 'F' and
*> EQUED = 'C' or 'B', each element of C must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit,
*> if EQUED = 'N', B is not modified;
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
*> diag(R)*B;
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
*> overwritten by diag(C)*B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
*> to the original system of equations. Note that A and B are
*> modified on exit if EQUED .ne. 'N', and the solution to the
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
*> and EQUED = 'R' or 'B'.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A after equilibration (if done). If RCOND is less than the
*> machine precision (in particular, if RCOND = 0), the matrix
*> is singular to working precision. This condition is
*> indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,4*N))
*> On exit, WORK(1) contains the reciprocal pivot growth
*> factor norm(A)/norm(U). The "max absolute element" norm is
*> used. If WORK(1) is much less than 1, then the stability
*> of the LU factorization of the (equilibrated) matrix A
*> could be poor. This also means that the solution X, condition
*> estimator RCOND, and forward error bound FERR could be
*> unreliable. If factorization fails with 0<INFO<=N, then
*> WORK(1) contains the reciprocal pivot growth factor for the
*> leading INFO columns of A.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: U(i,i) is exactly zero. The factorization has
*> been completed, but the factor U is exactly
*> singular, so the solution and error bounds
*> could not be computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesvx
*
* =====================================================================
SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF,
$ IPIV,
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
$ WORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), C( * ), FERR( * ), R( * ),
$ WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
CHARACTER NORM
INTEGER I, INFEQU, J
DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
$ ROWCND, RPVGRW, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE, DLANTR
EXTERNAL LSAME, DLAMCH, DLANGE, DLANTR
* ..
* .. External Subroutines ..
EXTERNAL DGECON, DGEEQU, DGERFS, DGETRF, DGETRS,
$ DLACPY,
$ DLAQGE, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
NOTRAN = LSAME( TRANS, 'N' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
ROWEQU = .FALSE.
COLEQU = .FALSE.
ELSE
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND.
$ .NOT.EQUIL .AND.
$ .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -10
ELSE
IF( ROWEQU ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 10 J = 1, N
RCMIN = MIN( RCMIN, R( J ) )
RCMAX = MAX( RCMAX, R( J ) )
10 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -11
ELSE IF( N.GT.0 ) THEN
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
ROWCND = ONE
END IF
END IF
IF( COLEQU .AND. INFO.EQ.0 ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 20 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
20 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -12
ELSE IF( N.GT.0 ) THEN
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
COLCND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -16
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGESVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
$ INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
$ EQUED )
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
END IF
END IF
*
* Scale the right hand side.
*
IF( NOTRAN ) THEN
IF( ROWEQU ) THEN
DO 40 J = 1, NRHS
DO 30 I = 1, N
B( I, J ) = R( I )*B( I, J )
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( COLEQU ) THEN
DO 60 J = 1, NRHS
DO 50 I = 1, N
B( I, J ) = C( I )*B( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the LU factorization of A.
*
CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF )
CALL DGETRF( N, N, AF, LDAF, IPIV, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 ) THEN
*
* Compute the reciprocal pivot growth factor of the
* leading rank-deficient INFO columns of A.
*
RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
$ WORK )
IF( RPVGRW.EQ.ZERO ) THEN
RPVGRW = ONE
ELSE
RPVGRW = DLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW
END IF
WORK( 1 ) = RPVGRW
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A and the
* reciprocal pivot growth factor RPVGRW.
*
IF( NOTRAN ) THEN
NORM = '1'
ELSE
NORM = 'I'
END IF
ANORM = DLANGE( NORM, N, N, A, LDA, WORK )
RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK )
IF( RPVGRW.EQ.ZERO ) THEN
RPVGRW = ONE
ELSE
RPVGRW = DLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW
END IF
*
* Compute the reciprocal of the condition number of A.
*
CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
$ LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( NOTRAN ) THEN
IF( COLEQU ) THEN
DO 80 J = 1, NRHS
DO 70 I = 1, N
X( I, J ) = C( I )*X( I, J )
70 CONTINUE
80 CONTINUE
DO 90 J = 1, NRHS
FERR( J ) = FERR( J ) / COLCND
90 CONTINUE
END IF
ELSE IF( ROWEQU ) THEN
DO 110 J = 1, NRHS
DO 100 I = 1, N
X( I, J ) = R( I )*X( I, J )
100 CONTINUE
110 CONTINUE
DO 120 J = 1, NRHS
FERR( J ) = FERR( J ) / ROWCND
120 CONTINUE
END IF
*
WORK( 1 ) = RPVGRW
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
RETURN
*
* End of DGESVX
*
END
*> \brief \b DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETC2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetc2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetc2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetc2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGETC2( N, A, LDA, IPIV, JPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), JPIV( * )
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGETC2 computes an LU factorization with complete pivoting of the
*> n-by-n matrix A. The factorization has the form A = P * L * U * Q,
*> where P and Q are permutation matrices, L is lower triangular with
*> unit diagonal elements and U is upper triangular.
*>
*> This is the Level 2 BLAS algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the n-by-n matrix A to be factored.
*> On exit, the factors L and U from the factorization
*> A = P*L*U*Q; the unit diagonal elements of L are not stored.
*> If U(k, k) appears to be less than SMIN, U(k, k) is given the
*> value of SMIN, i.e., giving a nonsingular perturbed system.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension(N).
*> The pivot indices; for 1 <= i <= N, row i of the
*> matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] JPIV
*> \verbatim
*> JPIV is INTEGER array, dimension(N).
*> The pivot indices; for 1 <= j <= N, column j of the
*> matrix has been interchanged with column JPIV(j).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = k, U(k, k) is likely to produce overflow if
*> we try to solve for x in Ax = b. So U is perturbed to
*> avoid the overflow.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getc2
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
SUBROUTINE DGETC2( N, A, LDA, IPIV, JPIV, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IP, IPV, J, JP, JPV
DOUBLE PRECISION BIGNUM, EPS, SMIN, SMLNUM, XMAX
* ..
* .. External Subroutines ..
EXTERNAL DGER, DSWAP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
BIGNUM = ONE / SMLNUM
*
* Handle the case N=1 by itself
*
IF( N.EQ.1 ) THEN
IPIV( 1 ) = 1
JPIV( 1 ) = 1
IF( ABS( A( 1, 1 ) ).LT.SMLNUM ) THEN
INFO = 1
A( 1, 1 ) = SMLNUM
END IF
RETURN
END IF
*
* Factorize A using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
DO 40 I = 1, N - 1
*
* Find max element in matrix A
*
XMAX = ZERO
DO 20 JP = I, N
DO 10 IP = I, N
IF( ABS( A( IP, JP ) ).GE.XMAX ) THEN
XMAX = ABS( A( IP, JP ) )
IPV = IP
JPV = JP
END IF
10 CONTINUE
20 CONTINUE
IF( I.EQ.1 )
$ SMIN = MAX( EPS*XMAX, SMLNUM )
*
* Swap rows
*
IF( IPV.NE.I )
$ CALL DSWAP( N, A( IPV, 1 ), LDA, A( I, 1 ), LDA )
IPIV( I ) = IPV
*
* Swap columns
*
IF( JPV.NE.I )
$ CALL DSWAP( N, A( 1, JPV ), 1, A( 1, I ), 1 )
JPIV( I ) = JPV
*
* Check for singularity
*
IF( ABS( A( I, I ) ).LT.SMIN ) THEN
INFO = I
A( I, I ) = SMIN
END IF
DO 30 J = I + 1, N
A( J, I ) = A( J, I ) / A( I, I )
30 CONTINUE
CALL DGER( N-I, N-I, -ONE, A( I+1, I ), 1, A( I, I+1 ), LDA,
$ A( I+1, I+1 ), LDA )
40 CONTINUE
*
IF( ABS( A( N, N ) ).LT.SMIN ) THEN
INFO = N
A( N, N ) = SMIN
END IF
*
* Set last pivots to N
*
IPIV( N ) = N
JPIV( N ) = N
*
RETURN
*
* End of DGETC2
*
END
*> \brief \b DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGETF2( M, N, A, LDA, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGETF2 computes an LU factorization of a general m-by-n matrix A
*> using partial pivoting with row interchanges.
*>
*> The factorization has the form
*> A = P * L * U
*> where P is a permutation matrix, L is lower triangular with unit
*> diagonal elements (lower trapezoidal if m > n), and U is upper
*> triangular (upper trapezoidal if m < n).
*>
*> This is the right-looking Level 2 BLAS version of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix to be factored.
*> On exit, the factors L and U from the factorization
*> A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (min(M,N))
*> The pivot indices; for 1 <= i <= min(M,N), row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, U(k,k) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and division by zero will occur if it is used
*> to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getf2
*
* =====================================================================
SUBROUTINE DGETF2( M, N, A, LDA, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION SFMIN
INTEGER I, J, JP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
INTEGER IDAMAX
EXTERNAL DLAMCH, IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DGER, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Compute machine safe minimum
*
SFMIN = DLAMCH('S')
*
DO 10 J = 1, MIN( M, N )
*
* Find pivot and test for singularity.
*
JP = J - 1 + IDAMAX( M-J+1, A( J, J ), 1 )
IPIV( J ) = JP
IF( A( JP, J ).NE.ZERO ) THEN
*
* Apply the interchange to columns 1:N.
*
IF( JP.NE.J )
$ CALL DSWAP( N, A( J, 1 ), LDA, A( JP, 1 ), LDA )
*
* Compute elements J+1:M of J-th column.
*
IF( J.LT.M ) THEN
IF( ABS(A( J, J )) .GE. SFMIN ) THEN
CALL DSCAL( M-J, ONE / A( J, J ), A( J+1, J ), 1 )
ELSE
DO 20 I = 1, M-J
A( J+I, J ) = A( J+I, J ) / A( J, J )
20 CONTINUE
END IF
END IF
*
ELSE IF( INFO.EQ.0 ) THEN
*
INFO = J
END IF
*
IF( J.LT.MIN( M, N ) ) THEN
*
* Update trailing submatrix.
*
CALL DGER( M-J, N-J, -ONE, A( J+1, J ), 1, A( J, J+1 ),
$ LDA,
$ A( J+1, J+1 ), LDA )
END IF
10 CONTINUE
RETURN
*
* End of DGETF2
*
END
*> \brief \b DGETRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGETRF computes an LU factorization of a general M-by-N matrix A
*> using partial pivoting with row interchanges.
*>
*> The factorization has the form
*> A = P * L * U
*> where P is a permutation matrix, L is lower triangular with unit
*> diagonal elements (lower trapezoidal if m > n), and U is upper
*> triangular (upper trapezoidal if m < n).
*>
*> This is the right-looking Level 3 BLAS version of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix to be factored.
*> On exit, the factors L and U from the factorization
*> A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (min(M,N))
*> The pivot indices; for 1 <= i <= min(M,N), row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and division by zero will occur if it is used
*> to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getrf
*
* =====================================================================
SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IINFO, J, JB, NB
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DGETRF2, DLASWP, DTRSM,
$ XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DGETRF', ' ', M, N, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.MIN( M, N ) ) THEN
*
* Use unblocked code.
*
CALL DGETRF2( M, N, A, LDA, IPIV, INFO )
ELSE
*
* Use blocked code.
*
DO 20 J = 1, MIN( M, N ), NB
JB = MIN( MIN( M, N )-J+1, NB )
*
* Factor diagonal and subdiagonal blocks and test for exact
* singularity.
*
CALL DGETRF2( M-J+1, JB, A( J, J ), LDA, IPIV( J ),
$ IINFO )
*
* Adjust INFO and the pivot indices.
*
IF( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO + J - 1
DO 10 I = J, MIN( M, J+JB-1 )
IPIV( I ) = J - 1 + IPIV( I )
10 CONTINUE
*
* Apply interchanges to columns 1:J-1.
*
CALL DLASWP( J-1, A, LDA, J, J+JB-1, IPIV, 1 )
*
IF( J+JB.LE.N ) THEN
*
* Apply interchanges to columns J+JB:N.
*
CALL DLASWP( N-J-JB+1, A( 1, J+JB ), LDA, J, J+JB-1,
$ IPIV, 1 )
*
* Compute block row of U.
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
$ JB,
$ N-J-JB+1, ONE, A( J, J ), LDA, A( J, J+JB ),
$ LDA )
IF( J+JB.LE.M ) THEN
*
* Update trailing submatrix.
*
CALL DGEMM( 'No transpose', 'No transpose',
$ M-J-JB+1,
$ N-J-JB+1, JB, -ONE, A( J+JB, J ), LDA,
$ A( J, J+JB ), LDA, ONE, A( J+JB, J+JB ),
$ LDA )
END IF
END IF
20 CONTINUE
END IF
RETURN
*
* End of DGETRF
*
END
*> \brief \b DGETRF2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DGETRF2( M, N, A, LDA, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGETRF2 computes an LU factorization of a general M-by-N matrix A
*> using partial pivoting with row interchanges.
*>
*> The factorization has the form
*> A = P * L * U
*> where P is a permutation matrix, L is lower triangular with unit
*> diagonal elements (lower trapezoidal if m > n), and U is upper
*> triangular (upper trapezoidal if m < n).
*>
*> This is the recursive version of the algorithm. It divides
*> the matrix into four submatrices:
*>
*> [ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2
*> A = [ -----|----- ] with n1 = min(m,n)/2
*> [ A21 | A22 ] n2 = n-n1
*>
*> [ A11 ]
*> The subroutine calls itself to factor [ --- ],
*> [ A12 ]
*> [ A12 ]
*> do the swaps on [ --- ], solve A12, update A22,
*> [ A22 ]
*>
*> then calls itself to factor A22 and do the swaps on A21.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix to be factored.
*> On exit, the factors L and U from the factorization
*> A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (min(M,N))
*> The pivot indices; for 1 <= i <= min(M,N), row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and division by zero will occur if it is used
*> to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getrf2
*
* =====================================================================
RECURSIVE SUBROUTINE DGETRF2( M, N, A, LDA, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION SFMIN, TEMP
INTEGER I, IINFO, N1, N2
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
INTEGER IDAMAX
EXTERNAL DLAMCH, IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DSCAL, DLASWP, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETRF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
IF ( M.EQ.1 ) THEN
*
* Use unblocked code for one row case
* Just need to handle IPIV and INFO
*
IPIV( 1 ) = 1
IF ( A(1,1).EQ.ZERO )
$ INFO = 1
*
ELSE IF( N.EQ.1 ) THEN
*
* Use unblocked code for one column case
*
*
* Compute machine safe minimum
*
SFMIN = DLAMCH('S')
*
* Find pivot and test for singularity
*
I = IDAMAX( M, A( 1, 1 ), 1 )
IPIV( 1 ) = I
IF( A( I, 1 ).NE.ZERO ) THEN
*
* Apply the interchange
*
IF( I.NE.1 ) THEN
TEMP = A( 1, 1 )
A( 1, 1 ) = A( I, 1 )
A( I, 1 ) = TEMP
END IF
*
* Compute elements 2:M of the column
*
IF( ABS(A( 1, 1 )) .GE. SFMIN ) THEN
CALL DSCAL( M-1, ONE / A( 1, 1 ), A( 2, 1 ), 1 )
ELSE
DO 10 I = 1, M-1
A( 1+I, 1 ) = A( 1+I, 1 ) / A( 1, 1 )
10 CONTINUE
END IF
*
ELSE
INFO = 1
END IF
*
ELSE
*
* Use recursive code
*
N1 = MIN( M, N ) / 2
N2 = N-N1
*
* [ A11 ]
* Factor [ --- ]
* [ A21 ]
*
CALL DGETRF2( M, N1, A, LDA, IPIV, IINFO )
IF ( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO
*
* [ A12 ]
* Apply interchanges to [ --- ]
* [ A22 ]
*
CALL DLASWP( N2, A( 1, N1+1 ), LDA, 1, N1, IPIV, 1 )
*
* Solve A12
*
CALL DTRSM( 'L', 'L', 'N', 'U', N1, N2, ONE, A, LDA,
$ A( 1, N1+1 ), LDA )
*
* Update A22
*
CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( N1+1, 1 ), LDA,
$ A( 1, N1+1 ), LDA, ONE, A( N1+1, N1+1 ), LDA )
*
* Factor A22
*
CALL DGETRF2( M-N1, N2, A( N1+1, N1+1 ), LDA, IPIV( N1+1 ),
$ IINFO )
*
* Adjust INFO and the pivot indices
*
IF ( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO + N1
DO 20 I = N1+1, MIN( M, N )
IPIV( I ) = IPIV( I ) + N1
20 CONTINUE
*
* Apply interchanges to A21
*
CALL DLASWP( N1, A( 1, 1 ), LDA, N1+1, MIN( M, N), IPIV, 1 )
*
END IF
RETURN
*
* End of DGETRF2
*
END
*> \brief \b DGETRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGETRI computes the inverse of a matrix using the LU factorization
*> computed by DGETRF.
*>
*> This method inverts U and then computes inv(A) by solving the system
*> inv(A)*L = inv(U) for inv(A).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the factors L and U from the factorization
*> A = P*L*U as computed by DGETRF.
*> On exit, if INFO = 0, the inverse of the original matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices from DGETRF; for 1<=i<=N, row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO=0, then WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> For optimal performance LWORK >= N*NB, where NB is
*> the optimal blocksize returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, U(i,i) is exactly zero; the matrix is
*> singular and its inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getri
*
* =====================================================================
SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IWS, J, JB, JJ, JP, LDWORK, LWKOPT, NB,
$ NBMIN, NN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DGEMV, DSWAP, DTRSM, DTRTRI,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NB = ILAENV( 1, 'DGETRI', ' ', N, -1, -1, -1 )
LWKOPT = MAX( 1, N*NB )
WORK( 1 ) = LWKOPT
*
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -3
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETRI', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form inv(U). If INFO > 0 from DTRTRI, then U is singular,
* and the inverse is not computed.
*
CALL DTRTRI( 'Upper', 'Non-unit', N, A, LDA, INFO )
IF( INFO.GT.0 )
$ RETURN
*
NBMIN = 2
LDWORK = N
IF( NB.GT.1 .AND. NB.LT.N ) THEN
IWS = MAX( LDWORK*NB, 1 )
IF( LWORK.LT.IWS ) THEN
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGETRI', ' ', N, -1, -1,
$ -1 ) )
END IF
ELSE
IWS = N
END IF
*
* Solve the equation inv(A)*L = inv(U) for inv(A).
*
IF( NB.LT.NBMIN .OR. NB.GE.N ) THEN
*
* Use unblocked code.
*
DO 20 J = N, 1, -1
*
* Copy current column of L to WORK and replace with zeros.
*
DO 10 I = J + 1, N
WORK( I ) = A( I, J )
A( I, J ) = ZERO
10 CONTINUE
*
* Compute current column of inv(A).
*
IF( J.LT.N )
$ CALL DGEMV( 'No transpose', N, N-J, -ONE, A( 1, J+1 ),
$ LDA, WORK( J+1 ), 1, ONE, A( 1, J ), 1 )
20 CONTINUE
ELSE
*
* Use blocked code.
*
NN = ( ( N-1 ) / NB )*NB + 1
DO 50 J = NN, 1, -NB
JB = MIN( NB, N-J+1 )
*
* Copy current block column of L to WORK and replace with
* zeros.
*
DO 40 JJ = J, J + JB - 1
DO 30 I = JJ + 1, N
WORK( I+( JJ-J )*LDWORK ) = A( I, JJ )
A( I, JJ ) = ZERO
30 CONTINUE
40 CONTINUE
*
* Compute current block column of inv(A).
*
IF( J+JB.LE.N )
$ CALL DGEMM( 'No transpose', 'No transpose', N, JB,
$ N-J-JB+1, -ONE, A( 1, J+JB ), LDA,
$ WORK( J+JB ), LDWORK, ONE, A( 1, J ), LDA )
CALL DTRSM( 'Right', 'Lower', 'No transpose', 'Unit', N,
$ JB,
$ ONE, WORK( J ), LDWORK, A( 1, J ), LDA )
50 CONTINUE
END IF
*
* Apply column interchanges.
*
DO 60 J = N - 1, 1, -1
JP = IPIV( J )
IF( JP.NE.J )
$ CALL DSWAP( N, A( 1, J ), 1, A( 1, JP ), 1 )
60 CONTINUE
*
WORK( 1 ) = IWS
RETURN
*
* End of DGETRI
*
END
*> \brief \b DGETRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGETRS solves a system of linear equations
*> A * X = B or A**T * X = B
*> with a general N-by-N matrix A using the LU factorization computed
*> by DGETRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T* X = B (Transpose)
*> = 'C': A**T* X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The factors L and U from the factorization A = P*L*U
*> as computed by DGETRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices from DGETRF; for 1<=i<=N, row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getrs
*
* =====================================================================
SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLASWP, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( NOTRAN ) THEN
*
* Solve A * X = B.
*
* Apply row interchanges to the right hand sides.
*
CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, 1 )
*
* Solve L*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', N,
$ NRHS,
$ ONE, A, LDA, B, LDB )
*
* Solve U*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
$ NRHS, ONE, A, LDA, B, LDB )
ELSE
*
* Solve A**T * X = B.
*
* Solve U**T *X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N,
$ NRHS,
$ ONE, A, LDA, B, LDB )
*
* Solve L**T *X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Unit', N, NRHS,
$ ONE,
$ A, LDA, B, LDB )
*
* Apply row interchanges to the solution vectors.
*
CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, -1 )
END IF
*
RETURN
*
* End of DGETRS
*
END
*> \brief \b DGGBAK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGBAK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggbak.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggbak.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggbak.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGBAK( JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V,
* LDV, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB, SIDE
* INTEGER IHI, ILO, INFO, LDV, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION LSCALE( * ), RSCALE( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGBAK forms the right or left eigenvectors of a real generalized
*> eigenvalue problem A*x = lambda*B*x, by backward transformation on
*> the computed eigenvectors of the balanced pair of matrices output by
*> DGGBAL.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies the type of backward transformation required:
*> = 'N': do nothing, return immediately;
*> = 'P': do backward transformation for permutation only;
*> = 'S': do backward transformation for scaling only;
*> = 'B': do backward transformations for both permutation and
*> scaling.
*> JOB must be the same as the argument JOB supplied to DGGBAL.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': V contains right eigenvectors;
*> = 'L': V contains left eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrix V. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> The integers ILO and IHI determined by DGGBAL.
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in] LSCALE
*> \verbatim
*> LSCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and/or scaling factors applied
*> to the left side of A and B, as returned by DGGBAL.
*> \endverbatim
*>
*> \param[in] RSCALE
*> \verbatim
*> RSCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and/or scaling factors applied
*> to the right side of A and B, as returned by DGGBAL.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of columns of the matrix V. M >= 0.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,M)
*> On entry, the matrix of right or left eigenvectors to be
*> transformed, as returned by DTGEVC.
*> On exit, V is overwritten by the transformed eigenvectors.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the matrix V. LDV >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggbak
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> See R.C. Ward, Balancing the generalized eigenvalue problem,
*> SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGBAK( JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M,
$ V,
$ LDV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOB, SIDE
INTEGER IHI, ILO, INFO, LDV, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION LSCALE( * ), RSCALE( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LEFTV, RIGHTV
INTEGER I, K
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, INT
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
RIGHTV = LSAME( SIDE, 'R' )
LEFTV = LSAME( SIDE, 'L' )
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND.
$ .NOT.LSAME( JOB, 'P' ) .AND.
$ .NOT.LSAME( JOB, 'S' ) .AND.
$ .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 ) THEN
INFO = -4
ELSE IF( N.EQ.0 .AND. IHI.EQ.0 .AND. ILO.NE.1 ) THEN
INFO = -4
ELSE IF( N.GT.0 .AND. ( IHI.LT.ILO .OR. IHI.GT.MAX( 1, N ) ) )
$ THEN
INFO = -5
ELSE IF( N.EQ.0 .AND. ILO.EQ.1 .AND. IHI.NE.0 ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -8
ELSE IF( LDV.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGBAK', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( M.EQ.0 )
$ RETURN
IF( LSAME( JOB, 'N' ) )
$ RETURN
*
IF( ILO.EQ.IHI )
$ GO TO 30
*
* Backward balance
*
IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
* Backward transformation on right eigenvectors
*
IF( RIGHTV ) THEN
DO 10 I = ILO, IHI
CALL DSCAL( M, RSCALE( I ), V( I, 1 ), LDV )
10 CONTINUE
END IF
*
* Backward transformation on left eigenvectors
*
IF( LEFTV ) THEN
DO 20 I = ILO, IHI
CALL DSCAL( M, LSCALE( I ), V( I, 1 ), LDV )
20 CONTINUE
END IF
END IF
*
* Backward permutation
*
30 CONTINUE
IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
* Backward permutation on right eigenvectors
*
IF( RIGHTV ) THEN
IF( ILO.EQ.1 )
$ GO TO 50
*
DO 40 I = ILO - 1, 1, -1
K = INT(RSCALE( I ))
IF( K.EQ.I )
$ GO TO 40
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
40 CONTINUE
*
50 CONTINUE
IF( IHI.EQ.N )
$ GO TO 70
DO 60 I = IHI + 1, N
K = INT(RSCALE( I ))
IF( K.EQ.I )
$ GO TO 60
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
60 CONTINUE
END IF
*
* Backward permutation on left eigenvectors
*
70 CONTINUE
IF( LEFTV ) THEN
IF( ILO.EQ.1 )
$ GO TO 90
DO 80 I = ILO - 1, 1, -1
K = INT(LSCALE( I ))
IF( K.EQ.I )
$ GO TO 80
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
80 CONTINUE
*
90 CONTINUE
IF( IHI.EQ.N )
$ GO TO 110
DO 100 I = IHI + 1, N
K = INT(LSCALE( I ))
IF( K.EQ.I )
$ GO TO 100
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
100 CONTINUE
END IF
END IF
*
110 CONTINUE
*
RETURN
*
* End of DGGBAK
*
END
*> \brief \b DGGBAL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGBAL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggbal.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggbal.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggbal.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
* RSCALE, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB
* INTEGER IHI, ILO, INFO, LDA, LDB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), LSCALE( * ),
* $ RSCALE( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGBAL balances a pair of general real matrices (A,B). This
*> involves, first, permuting A and B by similarity transformations to
*> isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
*> elements on the diagonal; and second, applying a diagonal similarity
*> transformation to rows and columns ILO to IHI to make the rows
*> and columns as close in norm as possible. Both steps are optional.
*>
*> Balancing may reduce the 1-norm of the matrices, and improve the
*> accuracy of the computed eigenvalues and/or eigenvectors in the
*> generalized eigenvalue problem A*x = lambda*B*x.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies the operations to be performed on A and B:
*> = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
*> and RSCALE(I) = 1.0 for i = 1,...,N.
*> = 'P': permute only;
*> = 'S': scale only;
*> = 'B': both permute and scale.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the input matrix A.
*> On exit, A is overwritten by the balanced matrix.
*> If JOB = 'N', A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the input matrix B.
*> On exit, B is overwritten by the balanced matrix.
*> If JOB = 'N', B is not referenced.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI are set to integers such that on exit
*> A(i,j) = 0 and B(i,j) = 0 if i > j and
*> j = 1,...,ILO-1 or i = IHI+1,...,N.
*> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] LSCALE
*> \verbatim
*> LSCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and scaling factors applied
*> to the left side of A and B. If P(j) is the index of the
*> row interchanged with row j, and D(j)
*> is the scaling factor applied to row j, then
*> LSCALE(j) = P(j) for J = 1,...,ILO-1
*> = D(j) for J = ILO,...,IHI
*> = P(j) for J = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] RSCALE
*> \verbatim
*> RSCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and scaling factors applied
*> to the right side of A and B. If P(j) is the index of the
*> column interchanged with column j, and D(j)
*> is the scaling factor applied to column j, then
*> LSCALE(j) = P(j) for J = 1,...,ILO-1
*> = D(j) for J = ILO,...,IHI
*> = P(j) for J = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (lwork)
*> lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
*> at least 1 when JOB = 'N' or 'P'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggbal
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> See R.C. WARD, Balancing the generalized eigenvalue problem,
*> SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
$ RSCALE, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, LDB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), LSCALE( * ),
$ RSCALE( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
DOUBLE PRECISION THREE, SCLFAC
PARAMETER ( THREE = 3.0D+0, SCLFAC = 1.0D+1 )
* ..
* .. Local Scalars ..
INTEGER I, ICAB, IFLOW, IP1, IR, IRAB, IT, J, JC, JP1,
$ K, KOUNT, L, LCAB, LM1, LRAB, LSFMAX, LSFMIN,
$ M, NR, NRP2
DOUBLE PRECISION ALPHA, BASL, BETA, CAB, CMAX, COEF, COEF2,
$ COEF5, COR, EW, EWC, GAMMA, PGAMMA, RAB, SFMAX,
$ SFMIN, SUM, T, TA, TB, TC
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG10, MAX, MIN, SIGN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND.
$ .NOT.LSAME( JOB, 'P' ) .AND.
$ .NOT.LSAME( JOB, 'S' ) .AND.
$ .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGBAL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
ILO = 1
IHI = N
RETURN
END IF
*
IF( N.EQ.1 ) THEN
ILO = 1
IHI = N
LSCALE( 1 ) = ONE
RSCALE( 1 ) = ONE
RETURN
END IF
*
IF( LSAME( JOB, 'N' ) ) THEN
ILO = 1
IHI = N
DO 10 I = 1, N
LSCALE( I ) = ONE
RSCALE( I ) = ONE
10 CONTINUE
RETURN
END IF
*
K = 1
L = N
IF( LSAME( JOB, 'S' ) )
$ GO TO 190
*
GO TO 30
*
* Permute the matrices A and B to isolate the eigenvalues.
*
* Find row with one nonzero in columns 1 through L
*
20 CONTINUE
L = LM1
IF( L.NE.1 )
$ GO TO 30
*
RSCALE( 1 ) = ONE
LSCALE( 1 ) = ONE
GO TO 190
*
30 CONTINUE
LM1 = L - 1
DO 80 I = L, 1, -1
DO 40 J = 1, LM1
JP1 = J + 1
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 50
40 CONTINUE
J = L
GO TO 70
*
50 CONTINUE
DO 60 J = JP1, L
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 80
60 CONTINUE
J = JP1 - 1
*
70 CONTINUE
M = L
IFLOW = 1
GO TO 160
80 CONTINUE
GO TO 100
*
* Find column with one nonzero in rows K through N
*
90 CONTINUE
K = K + 1
*
100 CONTINUE
DO 150 J = K, L
DO 110 I = K, LM1
IP1 = I + 1
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 120
110 CONTINUE
I = L
GO TO 140
120 CONTINUE
DO 130 I = IP1, L
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 150
130 CONTINUE
I = IP1 - 1
140 CONTINUE
M = K
IFLOW = 2
GO TO 160
150 CONTINUE
GO TO 190
*
* Permute rows M and I
*
160 CONTINUE
LSCALE( M ) = I
IF( I.EQ.M )
$ GO TO 170
CALL DSWAP( N-K+1, A( I, K ), LDA, A( M, K ), LDA )
CALL DSWAP( N-K+1, B( I, K ), LDB, B( M, K ), LDB )
*
* Permute columns M and J
*
170 CONTINUE
RSCALE( M ) = J
IF( J.EQ.M )
$ GO TO 180
CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
CALL DSWAP( L, B( 1, J ), 1, B( 1, M ), 1 )
*
180 CONTINUE
GO TO ( 20, 90 )IFLOW
*
190 CONTINUE
ILO = K
IHI = L
*
IF( LSAME( JOB, 'P' ) ) THEN
DO 195 I = ILO, IHI
LSCALE( I ) = ONE
RSCALE( I ) = ONE
195 CONTINUE
RETURN
END IF
*
IF( ILO.EQ.IHI )
$ RETURN
*
* Balance the submatrix in rows ILO to IHI.
*
NR = IHI - ILO + 1
DO 200 I = ILO, IHI
RSCALE( I ) = ZERO
LSCALE( I ) = ZERO
*
WORK( I ) = ZERO
WORK( I+N ) = ZERO
WORK( I+2*N ) = ZERO
WORK( I+3*N ) = ZERO
WORK( I+4*N ) = ZERO
WORK( I+5*N ) = ZERO
200 CONTINUE
*
* Compute right side vector in resulting linear equations
*
BASL = LOG10( SCLFAC )
DO 240 I = ILO, IHI
DO 230 J = ILO, IHI
TB = B( I, J )
TA = A( I, J )
IF( TA.EQ.ZERO )
$ GO TO 210
TA = LOG10( ABS( TA ) ) / BASL
210 CONTINUE
IF( TB.EQ.ZERO )
$ GO TO 220
TB = LOG10( ABS( TB ) ) / BASL
220 CONTINUE
WORK( I+4*N ) = WORK( I+4*N ) - TA - TB
WORK( J+5*N ) = WORK( J+5*N ) - TA - TB
230 CONTINUE
240 CONTINUE
*
COEF = ONE / DBLE( 2*NR )
COEF2 = COEF*COEF
COEF5 = HALF*COEF2
NRP2 = NR + 2
BETA = ZERO
IT = 1
*
* Start generalized conjugate gradient iteration
*
250 CONTINUE
*
GAMMA = DDOT( NR, WORK( ILO+4*N ), 1, WORK( ILO+4*N ), 1 ) +
$ DDOT( NR, WORK( ILO+5*N ), 1, WORK( ILO+5*N ), 1 )
*
EW = ZERO
EWC = ZERO
DO 260 I = ILO, IHI
EW = EW + WORK( I+4*N )
EWC = EWC + WORK( I+5*N )
260 CONTINUE
*
GAMMA = COEF*GAMMA - COEF2*( EW**2+EWC**2 ) - COEF5*( EW-EWC )**2
IF( GAMMA.EQ.ZERO )
$ GO TO 350
IF( IT.NE.1 )
$ BETA = GAMMA / PGAMMA
T = COEF5*( EWC-THREE*EW )
TC = COEF5*( EW-THREE*EWC )
*
CALL DSCAL( NR, BETA, WORK( ILO ), 1 )
CALL DSCAL( NR, BETA, WORK( ILO+N ), 1 )
*
CALL DAXPY( NR, COEF, WORK( ILO+4*N ), 1, WORK( ILO+N ), 1 )
CALL DAXPY( NR, COEF, WORK( ILO+5*N ), 1, WORK( ILO ), 1 )
*
DO 270 I = ILO, IHI
WORK( I ) = WORK( I ) + TC
WORK( I+N ) = WORK( I+N ) + T
270 CONTINUE
*
* Apply matrix to vector
*
DO 300 I = ILO, IHI
KOUNT = 0
SUM = ZERO
DO 290 J = ILO, IHI
IF( A( I, J ).EQ.ZERO )
$ GO TO 280
KOUNT = KOUNT + 1
SUM = SUM + WORK( J )
280 CONTINUE
IF( B( I, J ).EQ.ZERO )
$ GO TO 290
KOUNT = KOUNT + 1
SUM = SUM + WORK( J )
290 CONTINUE
WORK( I+2*N ) = DBLE( KOUNT )*WORK( I+N ) + SUM
300 CONTINUE
*
DO 330 J = ILO, IHI
KOUNT = 0
SUM = ZERO
DO 320 I = ILO, IHI
IF( A( I, J ).EQ.ZERO )
$ GO TO 310
KOUNT = KOUNT + 1
SUM = SUM + WORK( I+N )
310 CONTINUE
IF( B( I, J ).EQ.ZERO )
$ GO TO 320
KOUNT = KOUNT + 1
SUM = SUM + WORK( I+N )
320 CONTINUE
WORK( J+3*N ) = DBLE( KOUNT )*WORK( J ) + SUM
330 CONTINUE
*
SUM = DDOT( NR, WORK( ILO+N ), 1, WORK( ILO+2*N ), 1 ) +
$ DDOT( NR, WORK( ILO ), 1, WORK( ILO+3*N ), 1 )
ALPHA = GAMMA / SUM
*
* Determine correction to current iteration
*
CMAX = ZERO
DO 340 I = ILO, IHI
COR = ALPHA*WORK( I+N )
IF( ABS( COR ).GT.CMAX )
$ CMAX = ABS( COR )
LSCALE( I ) = LSCALE( I ) + COR
COR = ALPHA*WORK( I )
IF( ABS( COR ).GT.CMAX )
$ CMAX = ABS( COR )
RSCALE( I ) = RSCALE( I ) + COR
340 CONTINUE
IF( CMAX.LT.HALF )
$ GO TO 350
*
CALL DAXPY( NR, -ALPHA, WORK( ILO+2*N ), 1, WORK( ILO+4*N ),
$ 1 )
CALL DAXPY( NR, -ALPHA, WORK( ILO+3*N ), 1, WORK( ILO+5*N ),
$ 1 )
*
PGAMMA = GAMMA
IT = IT + 1
IF( IT.LE.NRP2 )
$ GO TO 250
*
* End generalized conjugate gradient iteration
*
350 CONTINUE
SFMIN = DLAMCH( 'S' )
SFMAX = ONE / SFMIN
LSFMIN = INT( LOG10( SFMIN ) / BASL+ONE )
LSFMAX = INT( LOG10( SFMAX ) / BASL )
DO 360 I = ILO, IHI
IRAB = IDAMAX( N-ILO+1, A( I, ILO ), LDA )
RAB = ABS( A( I, IRAB+ILO-1 ) )
IRAB = IDAMAX( N-ILO+1, B( I, ILO ), LDB )
RAB = MAX( RAB, ABS( B( I, IRAB+ILO-1 ) ) )
LRAB = INT( LOG10( RAB+SFMIN ) / BASL+ONE )
IR = INT(LSCALE( I ) + SIGN( HALF, LSCALE( I ) ))
IR = MIN( MAX( IR, LSFMIN ), LSFMAX, LSFMAX-LRAB )
LSCALE( I ) = SCLFAC**IR
ICAB = IDAMAX( IHI, A( 1, I ), 1 )
CAB = ABS( A( ICAB, I ) )
ICAB = IDAMAX( IHI, B( 1, I ), 1 )
CAB = MAX( CAB, ABS( B( ICAB, I ) ) )
LCAB = INT( LOG10( CAB+SFMIN ) / BASL+ONE )
JC = INT(RSCALE( I ) + SIGN( HALF, RSCALE( I ) ))
JC = MIN( MAX( JC, LSFMIN ), LSFMAX, LSFMAX-LCAB )
RSCALE( I ) = SCLFAC**JC
360 CONTINUE
*
* Row scaling of matrices A and B
*
DO 370 I = ILO, IHI
CALL DSCAL( N-ILO+1, LSCALE( I ), A( I, ILO ), LDA )
CALL DSCAL( N-ILO+1, LSCALE( I ), B( I, ILO ), LDB )
370 CONTINUE
*
* Column scaling of matrices A and B
*
DO 380 J = ILO, IHI
CALL DSCAL( IHI, RSCALE( J ), A( 1, J ), 1 )
CALL DSCAL( IHI, RSCALE( J ), B( 1, J ), 1 )
380 CONTINUE
*
RETURN
*
* End of DGGBAL
*
END
*> \brief <b> DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGES + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgges.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgges.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgges.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
* SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
* LDVSR, WORK, LWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR, SORT
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
* $ VSR( LDVSR, * ), WORK( * )
* ..
* .. Function Arguments ..
* LOGICAL SELCTG
* EXTERNAL SELCTG
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
*> the generalized eigenvalues, the generalized real Schur form (S,T),
*> optionally, the left and/or right matrices of Schur vectors (VSL and
*> VSR). This gives the generalized Schur factorization
*>
*> (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
*>
*> Optionally, it also orders the eigenvalues so that a selected cluster
*> of eigenvalues appears in the leading diagonal blocks of the upper
*> quasi-triangular matrix S and the upper triangular matrix T.The
*> leading columns of VSL and VSR then form an orthonormal basis for the
*> corresponding left and right eigenspaces (deflating subspaces).
*>
*> (If only the generalized eigenvalues are needed, use the driver
*> DGGEV instead, which is faster.)
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
*> usually represented as the pair (alpha,beta), as there is a
*> reasonable interpretation for beta=0 or both being zero.
*>
*> A pair of matrices (S,T) is in generalized real Schur form if T is
*> upper triangular with non-negative diagonal and S is block upper
*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
*> "standardized" by making the corresponding elements of T have the
*> form:
*> [ a 0 ]
*> [ 0 b ]
*>
*> and the pair of corresponding 2-by-2 blocks in S and T will have a
*> complex conjugate pair of generalized eigenvalues.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors.
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*> SORT is CHARACTER*1
*> Specifies whether or not to order the eigenvalues on the
*> diagonal of the generalized Schur form.
*> = 'N': Eigenvalues are not ordered;
*> = 'S': Eigenvalues are ordered (see SELCTG);
*> \endverbatim
*>
*> \param[in] SELCTG
*> \verbatim
*> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
*> SELCTG must be declared EXTERNAL in the calling subroutine.
*> If SORT = 'N', SELCTG is not referenced.
*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
*> to the top left of the Schur form.
*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
*> one of a complex conjugate pair of eigenvalues is selected,
*> then both complex eigenvalues are selected.
*>
*> Note that in the ill-conditioned case, a selected complex
*> eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
*> BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
*> in this case.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the first of the pair of matrices.
*> On exit, A has been overwritten by its generalized Schur
*> form S.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the second of the pair of matrices.
*> On exit, B has been overwritten by its generalized Schur
*> form T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*> SDIM is INTEGER
*> If SORT = 'N', SDIM = 0.
*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*> for which SELCTG is true. (Complex conjugate pairs for which
*> SELCTG is true for either eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
*> and BETA(j),j=1,...,N are the diagonals of the complex Schur
*> form (S,T) that would result if the 2-by-2 diagonal blocks of
*> the real Schur form of (A,B) were further reduced to
*> triangular form using 2-by-2 complex unitary transformations.
*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*> positive, then the j-th and (j+1)-st eigenvalues are a
*> complex conjugate pair, with ALPHAI(j+1) negative.
*>
*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*> may easily over- or underflow, and BETA(j) may even be zero.
*> Thus, the user should avoid naively computing the ratio.
*> However, ALPHAR and ALPHAI will be always less than and
*> usually comparable with norm(A) in magnitude, and BETA always
*> less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >=1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N = 0, LWORK >= 1, else LWORK >= MAX(8*N,6*N+16).
*> For good performance, LWORK must generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. (A,B) are not in Schur
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*> be correct for j=INFO+1,...,N.
*> > N: =N+1: other than QZ iteration failed in DHGEQZ.
*> =N+2: after reordering, roundoff changed values of
*> some complex eigenvalues so that leading
*> eigenvalues in the Generalized Schur form no
*> longer satisfy SELCTG=.TRUE. This could also
*> be caused due to scaling.
*> =N+3: reordering failed in DTGSEN.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gges
*
* =====================================================================
SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
$ LDB,
$ SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
$ LDVSR, WORK, LWORK, BWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
* .. Function Arguments ..
LOGICAL SELCTG
EXTERNAL SELCTG
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
$ LQUERY, LST2SL, WANTST
INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
$ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
$ MINWRK
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
$ PVSR, SAFMAX, SAFMIN, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DIF( 2 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
$ DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, DTGSEN, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
WANTST = LSAME( SORT, 'S' )
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( ( .NOT.WANTST ) .AND.
$ ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -15
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -17
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.)
*
IF( INFO.EQ.0 ) THEN
IF( N.GT.0 )THEN
MINWRK = MAX( 8*N, 6*N + 16 )
MAXWRK = MINWRK - N +
$ N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 )
MAXWRK = MAX( MAXWRK, MINWRK - N +
$ N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, -1 ) )
IF( ILVSL ) THEN
MAXWRK = MAX( MAXWRK, MINWRK - N +
$ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N,
$ -1 ) )
END IF
ELSE
MINWRK = 1
MAXWRK = 1
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
$ INFO = -19
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGES ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SDIM = 0
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SMLNUM = SQRT( SAFMIN ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
IF( ILASCL )
$ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrix to make it more nearly triangular
* (Workspace: need 6*N + 2*N space for storing balancing factors)
*
ILEFT = 1
IRIGHT = N + 1
IWRK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWRK ), IERR )
*
* Reduce B to triangular form (QR decomposition of B)
* (Workspace: need N, prefer N*NB)
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWRK
IWRK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to matrix A
* (Workspace: need N, prefer N*NB)
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VSL
* (Workspace: need N, prefer N*NB)
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
IF( IROWS.GT.1 ) THEN
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
END IF
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VSR
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IERR )
*
* Perform QZ algorithm, computing Schur vectors if desired
* (Workspace: need N)
*
IWRK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
INFO = IERR
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
INFO = IERR - N
ELSE
INFO = N + 1
END IF
GO TO 50
END IF
*
* Sort eigenvalues ALPHA/BETA if desired
* (Workspace: need 4*N+16 )
*
SDIM = 0
IF( WANTST ) THEN
*
* Undo scaling on eigenvalues before SELCTGing
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N,
$ IERR )
*
* Select eigenvalues
*
DO 10 I = 1, N
BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ),
$ BETA( I ) )
10 CONTINUE
*
CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB,
$ ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL,
$ PVSR, DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
$ IERR )
IF( IERR.EQ.1 )
$ INFO = N + 3
*
END IF
*
* Apply back-permutation to VSL and VSR
* (Workspace: none needed)
*
IF( ILVSL )
$ CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IERR )
*
IF( ILVSR )
$ CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
* Check if unscaling would cause over/underflow, if so, rescale
* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
*
IF( ILASCL ) THEN
DO 20 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
$ ( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
$ THEN
WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
20 CONTINUE
END IF
*
IF( ILBSCL ) THEN
DO 30 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
$ ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
30 CONTINUE
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
IF( WANTST ) THEN
*
* Check if reordering is correct
*
LASTSL = .TRUE.
LST2SL = .TRUE.
SDIM = 0
IP = 0
DO 40 I = 1, N
CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
IF( ALPHAI( I ).EQ.ZERO ) THEN
IF( CURSL )
$ SDIM = SDIM + 1
IP = 0
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
ELSE
IF( IP.EQ.1 ) THEN
*
* Last eigenvalue of conjugate pair
*
CURSL = CURSL .OR. LASTSL
LASTSL = CURSL
IF( CURSL )
$ SDIM = SDIM + 2
IP = -1
IF( CURSL .AND. .NOT.LST2SL )
$ INFO = N + 2
ELSE
*
* First eigenvalue of conjugate pair
*
IP = 1
END IF
END IF
LST2SL = LASTSL
LASTSL = CURSL
40 CONTINUE
*
END IF
*
50 CONTINUE
*
WORK( 1 ) = MAXWRK
*
RETURN
*
* End of DGGES
*
END
*> \brief <b> DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGESX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggesx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggesx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggesx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
* B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
* VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
* LIWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR, SENSE, SORT
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
* $ SDIM
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), RCONDE( 2 ),
* $ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
* $ WORK( * )
* ..
* .. Function Arguments ..
* LOGICAL SELCTG
* EXTERNAL SELCTG
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGESX computes for a pair of N-by-N real nonsymmetric matrices
*> (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
*> optionally, the left and/or right matrices of Schur vectors (VSL and
*> VSR). This gives the generalized Schur factorization
*>
*> (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
*>
*> Optionally, it also orders the eigenvalues so that a selected cluster
*> of eigenvalues appears in the leading diagonal blocks of the upper
*> quasi-triangular matrix S and the upper triangular matrix T; computes
*> a reciprocal condition number for the average of the selected
*> eigenvalues (RCONDE); and computes a reciprocal condition number for
*> the right and left deflating subspaces corresponding to the selected
*> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
*> an orthonormal basis for the corresponding left and right eigenspaces
*> (deflating subspaces).
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
*> usually represented as the pair (alpha,beta), as there is a
*> reasonable interpretation for beta=0 or for both being zero.
*>
*> A pair of matrices (S,T) is in generalized real Schur form if T is
*> upper triangular with non-negative diagonal and S is block upper
*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
*> "standardized" by making the corresponding elements of T have the
*> form:
*> [ a 0 ]
*> [ 0 b ]
*>
*> and the pair of corresponding 2-by-2 blocks in S and T will have a
*> complex conjugate pair of generalized eigenvalues.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors.
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*> SORT is CHARACTER*1
*> Specifies whether or not to order the eigenvalues on the
*> diagonal of the generalized Schur form.
*> = 'N': Eigenvalues are not ordered;
*> = 'S': Eigenvalues are ordered (see SELCTG).
*> \endverbatim
*>
*> \param[in] SELCTG
*> \verbatim
*> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
*> SELCTG must be declared EXTERNAL in the calling subroutine.
*> If SORT = 'N', SELCTG is not referenced.
*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
*> to the top left of the Schur form.
*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
*> one of a complex conjugate pair of eigenvalues is selected,
*> then both complex eigenvalues are selected.
*> Note that a selected complex eigenvalue may no longer satisfy
*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
*> since ordering may change the value of complex eigenvalues
*> (especially if the eigenvalue is ill-conditioned), in this
*> case INFO is set to N+3.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*> SENSE is CHARACTER*1
*> Determines which reciprocal condition numbers are computed.
*> = 'N': None are computed;
*> = 'E': Computed for average of selected eigenvalues only;
*> = 'V': Computed for selected deflating subspaces only;
*> = 'B': Computed for both.
*> If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the first of the pair of matrices.
*> On exit, A has been overwritten by its generalized Schur
*> form S.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the second of the pair of matrices.
*> On exit, B has been overwritten by its generalized Schur
*> form T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*> SDIM is INTEGER
*> If SORT = 'N', SDIM = 0.
*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*> for which SELCTG is true. (Complex conjugate pairs for which
*> SELCTG is true for either eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
*> and BETA(j),j=1,...,N are the diagonals of the complex Schur
*> form (S,T) that would result if the 2-by-2 diagonal blocks of
*> the real Schur form of (A,B) were further reduced to
*> triangular form using 2-by-2 complex unitary transformations.
*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*> positive, then the j-th and (j+1)-st eigenvalues are a
*> complex conjugate pair, with ALPHAI(j+1) negative.
*>
*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*> may easily over- or underflow, and BETA(j) may even be zero.
*> Thus, the user should avoid naively computing the ratio.
*> However, ALPHAR and ALPHAI will be always less than and
*> usually comparable with norm(A) in magnitude, and BETA always
*> less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >=1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*> RCONDE is DOUBLE PRECISION array, dimension ( 2 )
*> If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
*> reciprocal condition numbers for the average of the selected
*> eigenvalues.
*> Not referenced if SENSE = 'N' or 'V'.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*> RCONDV is DOUBLE PRECISION array, dimension ( 2 )
*> If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
*> reciprocal condition numbers for the selected deflating
*> subspaces.
*> Not referenced if SENSE = 'N' or 'E'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
*> LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
*> LWORK >= max( 8*N, 6*N+16 ).
*> Note that 2*SDIM*(N-SDIM) <= N*N/2.
*> Note also that an error is only returned if
*> LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
*> this may not be large enough.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the bound on the optimal size of the WORK
*> array and the minimum size of the IWORK array, returns these
*> values as the first entries of the WORK and IWORK arrays, and
*> no error message related to LWORK or LIWORK is issued by
*> XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
*> LIWORK >= N+6.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the bound on the optimal size of the
*> WORK array and the minimum size of the IWORK array, returns
*> these values as the first entries of the WORK and IWORK
*> arrays, and no error message related to LWORK or LIWORK is
*> issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. (A,B) are not in Schur
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*> be correct for j=INFO+1,...,N.
*> > N: =N+1: other than QZ iteration failed in DHGEQZ
*> =N+2: after reordering, roundoff changed values of
*> some complex eigenvalues so that leading
*> eigenvalues in the Generalized Schur form no
*> longer satisfy SELCTG=.TRUE. This could also
*> be caused due to scaling.
*> =N+3: reordering failed in DTGSEN.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggesx
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> An approximate (asymptotic) bound on the average absolute error of
*> the selected eigenvalues is
*>
*> EPS * norm((A, B)) / RCONDE( 1 ).
*>
*> An approximate (asymptotic) bound on the maximum angular error in
*> the computed deflating subspaces is
*>
*> EPS * norm((A, B)) / RCONDV( 2 ).
*>
*> See LAPACK User's Guide, section 4.11 for more information.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A,
$ LDA,
$ B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
$ VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
$ LIWORK, BWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR, SENSE, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
$ SDIM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), RCONDE( 2 ),
$ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
$ WORK( * )
* ..
* .. Function Arguments ..
LOGICAL SELCTG
EXTERNAL SELCTG
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
$ LQUERY, LST2SL, WANTSB, WANTSE, WANTSN, WANTST,
$ WANTSV
INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
$ ILEFT, ILO, IP, IRIGHT, IROWS, ITAU, IWRK,
$ LIWMIN, LWRK, MAXWRK, MINWRK
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
$ PR, SAFMAX, SAFMIN, SMLNUM
* ..
* .. Local Arrays ..
DOUBLE PRECISION DIF( 2 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
$ DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, DTGSEN, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
WANTST = LSAME( SORT, 'S' )
WANTSN = LSAME( SENSE, 'N' )
WANTSE = LSAME( SENSE, 'E' )
WANTSV = LSAME( SENSE, 'V' )
WANTSB = LSAME( SENSE, 'B' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
IF( WANTSN ) THEN
IJOB = 0
ELSE IF( WANTSE ) THEN
IJOB = 1
ELSE IF( WANTSV ) THEN
IJOB = 2
ELSE IF( WANTSB ) THEN
IJOB = 4
END IF
*
* Test the input arguments
*
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( ( .NOT.WANTST ) .AND.
$ ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
INFO = -3
ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
$ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
INFO = -5
ELSE IF( N.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -16
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -18
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.)
*
IF( INFO.EQ.0 ) THEN
IF( N.GT.0) THEN
MINWRK = MAX( 8*N, 6*N + 16 )
MAXWRK = MINWRK - N +
$ N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 )
MAXWRK = MAX( MAXWRK, MINWRK - N +
$ N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, -1 ) )
IF( ILVSL ) THEN
MAXWRK = MAX( MAXWRK, MINWRK - N +
$ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) )
END IF
LWRK = MAXWRK
IF( IJOB.GE.1 )
$ LWRK = MAX( LWRK, N*N/2 )
ELSE
MINWRK = 1
MAXWRK = 1
LWRK = 1
END IF
WORK( 1 ) = LWRK
IF( WANTSN .OR. N.EQ.0 ) THEN
LIWMIN = 1
ELSE
LIWMIN = N + 6
END IF
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -22
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -24
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGESX', -INFO )
RETURN
ELSE IF (LQUERY) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SDIM = 0
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SMLNUM = SQRT( SAFMIN ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
IF( ILASCL )
$ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrix to make it more nearly triangular
* (Workspace: need 6*N + 2*N for permutation parameters)
*
ILEFT = 1
IRIGHT = N + 1
IWRK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWRK ), IERR )
*
* Reduce B to triangular form (QR decomposition of B)
* (Workspace: need N, prefer N*NB)
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWRK
IWRK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to matrix A
* (Workspace: need N, prefer N*NB)
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VSL
* (Workspace: need N, prefer N*NB)
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
IF( IROWS.GT.1 ) THEN
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
END IF
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VSR
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IERR )
*
SDIM = 0
*
* Perform QZ algorithm, computing Schur vectors if desired
* (Workspace: need N)
*
IWRK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
INFO = IERR
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
INFO = IERR - N
ELSE
INFO = N + 1
END IF
GO TO 60
END IF
*
* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
* condition number(s)
* (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
* otherwise, need 8*(N+1) )
*
IF( WANTST ) THEN
*
* Undo scaling on eigenvalues before SELCTGing
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N,
$ IERR )
*
* Select eigenvalues
*
DO 10 I = 1, N
BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ),
$ BETA( I ) )
10 CONTINUE
*
* Reorder eigenvalues, transform Generalized Schur vectors, and
* compute reciprocal condition numbers
*
CALL DTGSEN( IJOB, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ SDIM, PL, PR, DIF, WORK( IWRK ), LWORK-IWRK+1,
$ IWORK, LIWORK, IERR )
*
IF( IJOB.GE.1 )
$ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
IF( IERR.EQ.-22 ) THEN
*
* not enough real workspace
*
INFO = -22
ELSE
IF( IJOB.EQ.1 .OR. IJOB.EQ.4 ) THEN
RCONDE( 1 ) = PL
RCONDE( 2 ) = PR
END IF
IF( IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
RCONDV( 1 ) = DIF( 1 )
RCONDV( 2 ) = DIF( 2 )
END IF
IF( IERR.EQ.1 )
$ INFO = N + 3
END IF
*
END IF
*
* Apply permutation to VSL and VSR
* (Workspace: none needed)
*
IF( ILVSL )
$ CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IERR )
*
IF( ILVSR )
$ CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
* Check if unscaling would cause over/underflow, if so, rescale
* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
*
IF( ILASCL ) THEN
DO 20 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
$ ( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
$ THEN
WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
20 CONTINUE
END IF
*
IF( ILBSCL ) THEN
DO 30 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
$ ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
30 CONTINUE
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
IF( WANTST ) THEN
*
* Check if reordering is correct
*
LASTSL = .TRUE.
LST2SL = .TRUE.
SDIM = 0
IP = 0
DO 50 I = 1, N
CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
IF( ALPHAI( I ).EQ.ZERO ) THEN
IF( CURSL )
$ SDIM = SDIM + 1
IP = 0
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
ELSE
IF( IP.EQ.1 ) THEN
*
* Last eigenvalue of conjugate pair
*
CURSL = CURSL .OR. LASTSL
LASTSL = CURSL
IF( CURSL )
$ SDIM = SDIM + 2
IP = -1
IF( CURSL .AND. .NOT.LST2SL )
$ INFO = N + 2
ELSE
*
* First eigenvalue of conjugate pair
*
IP = 1
END IF
END IF
LST2SL = LASTSL
LASTSL = CURSL
50 CONTINUE
*
END IF
*
60 CONTINUE
*
WORK( 1 ) = MAXWRK
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DGGESX
*
END
*> \brief <b> DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggev.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
*> the generalized eigenvalues, and optionally, the left and/or right
*> generalized eigenvectors.
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
*> singular. It is usually represented as the pair (alpha,beta), as
*> there is a reasonable interpretation for beta=0, and even for both
*> being zero.
*>
*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*> A * v(j) = lambda(j) * B * v(j).
*>
*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*> u(j)**H * A = lambda(j) * u(j)**H * B .
*>
*> where u(j)**H is the conjugate-transpose of u(j).
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the matrix A in the pair (A,B).
*> On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the matrix B in the pair (A,B).
*> On exit, B has been overwritten.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*> be the generalized eigenvalues. If ALPHAI(j) is zero, then
*> the j-th eigenvalue is real; if positive, then the j-th and
*> (j+1)-st eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) negative.
*>
*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*> may easily over- or underflow, and BETA(j) may even be zero.
*> Thus, the user should avoid naively computing the ratio
*> alpha/beta. However, ALPHAR and ALPHAI will be always less
*> than and usually comparable with norm(A) in magnitude, and
*> BETA always less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
*> after another in the columns of VL, in the same order as
*> their eigenvalues. If the j-th eigenvalue is real, then
*> u(j) = VL(:,j), the j-th column of VL. If the j-th and
*> (j+1)-th eigenvalues form a complex conjugate pair, then
*> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
*> Each eigenvector is scaled so the largest component has
*> abs(real part)+abs(imag. part)=1.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
*> after another in the columns of VR, in the same order as
*> their eigenvalues. If the j-th eigenvalue is real, then
*> v(j) = VR(:,j), the j-th column of VR. If the j-th and
*> (j+1)-th eigenvalues form a complex conjugate pair, then
*> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
*> Each eigenvector is scaled so the largest component has
*> abs(real part)+abs(imag. part)=1.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,8*N).
*> For good performance, LWORK must generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*> should be correct for j=INFO+1,...,N.
*> > N: =N+1: other than QZ iteration failed in DHGEQZ.
*> =N+2: error return from DTGEVC.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggev
*
* =====================================================================
SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI,
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK,
$ MINWRK
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SMLNUM, TEMP
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
$ DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -14
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV. The workspace is
* computed assuming ILO = 1 and IHI = N, the worst case.)
*
IF( INFO.EQ.0 ) THEN
MINWRK = MAX( 1, 8*N )
MAXWRK = MAX( 1, N*( 7 +
$ ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) ) )
MAXWRK = MAX( MAXWRK, N*( 7 +
$ ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) ) )
IF( ILVL ) THEN
MAXWRK = MAX( MAXWRK, N*( 7 +
$ ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) ) )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
$ INFO = -16
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
IF( ILASCL )
$ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrices A, B to isolate eigenvalues if possible
* (Workspace: need 6*N)
*
ILEFT = 1
IRIGHT = N + 1
IWRK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWRK ), IERR )
*
* Reduce B to triangular form (QR decomposition of B)
* (Workspace: need N, prefer N*NB)
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = IWRK
IWRK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to matrix A
* (Workspace: need N, prefer N*NB)
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VL
* (Workspace: need N, prefer N*NB)
*
IF( ILVL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
IF( IROWS.GT.1 ) THEN
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
END IF
CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VR
*
IF( ILVR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IERR )
ELSE
CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
END IF
*
* Perform QZ algorithm (Compute eigenvalues, and optionally, the
* Schur forms and Schur vectors)
* (Workspace: need N)
*
IWRK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
INFO = IERR
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
INFO = IERR - N
ELSE
INFO = N + 1
END IF
GO TO 110
END IF
*
* Compute Eigenvectors
* (Workspace: need 6*N)
*
IF( ILV ) THEN
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
$ LDVL,
$ VR, LDVR, N, IN, WORK( IWRK ), IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 110
END IF
*
* Undo balancing on VL and VR and normalization
* (Workspace: none needed)
*
IF( ILVL ) THEN
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VL, LDVL, IERR )
DO 50 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 50
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
10 CONTINUE
ELSE
DO 20 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
$ ABS( VL( JR, JC+1 ) ) )
20 CONTINUE
END IF
IF( TEMP.LT.SMLNUM )
$ GO TO 50
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 30 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
30 CONTINUE
ELSE
DO 40 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
40 CONTINUE
END IF
50 CONTINUE
END IF
IF( ILVR ) THEN
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VR, LDVR, IERR )
DO 100 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 100
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 60 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
60 CONTINUE
ELSE
DO 70 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
$ ABS( VR( JR, JC+1 ) ) )
70 CONTINUE
END IF
IF( TEMP.LT.SMLNUM )
$ GO TO 100
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 80 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
80 CONTINUE
ELSE
DO 90 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
90 CONTINUE
END IF
100 CONTINUE
END IF
*
* End of eigenvector calculation
*
END IF
*
* Undo scaling if necessary
*
110 CONTINUE
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGGEV
*
END
*> \brief <b> DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
* ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
* IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
* RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER BALANC, JOBVL, JOBVR, SENSE
* INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* DOUBLE PRECISION ABNRM, BBNRM
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), LSCALE( * ),
* $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
* $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
*> the generalized eigenvalues, and optionally, the left and/or right
*> generalized eigenvectors.
*>
*> Optionally also, it computes a balancing transformation to improve
*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
*> the eigenvalues (RCONDE), and reciprocal condition numbers for the
*> right eigenvectors (RCONDV).
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
*> singular. It is usually represented as the pair (alpha,beta), as
*> there is a reasonable interpretation for beta=0, and even for both
*> being zero.
*>
*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*> A * v(j) = lambda(j) * B * v(j) .
*>
*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*> u(j)**H * A = lambda(j) * u(j)**H * B.
*>
*> where u(j)**H is the conjugate-transpose of u(j).
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] BALANC
*> \verbatim
*> BALANC is CHARACTER*1
*> Specifies the balance option to be performed.
*> = 'N': do not diagonally scale or permute;
*> = 'P': permute only;
*> = 'S': scale only;
*> = 'B': both permute and scale.
*> Computed reciprocal condition numbers will be for the
*> matrices after permuting and/or balancing. Permuting does
*> not change condition numbers (in exact arithmetic), but
*> balancing does.
*> \endverbatim
*>
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*> SENSE is CHARACTER*1
*> Determines which reciprocal condition numbers are computed.
*> = 'N': none are computed;
*> = 'E': computed for eigenvalues only;
*> = 'V': computed for eigenvectors only;
*> = 'B': computed for eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the matrix A in the pair (A,B).
*> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
*> or both, then A contains the first part of the real Schur
*> form of the "balanced" versions of the input A and B.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the matrix B in the pair (A,B).
*> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
*> or both, then B contains the second part of the real Schur
*> form of the "balanced" versions of the input A and B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*> be the generalized eigenvalues. If ALPHAI(j) is zero, then
*> the j-th eigenvalue is real; if positive, then the j-th and
*> (j+1)-st eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) negative.
*>
*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*> may easily over- or underflow, and BETA(j) may even be zero.
*> Thus, the user should avoid naively computing the ratio
*> ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
*> than and usually comparable with norm(A) in magnitude, and
*> BETA always less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
*> after another in the columns of VL, in the same order as
*> their eigenvalues. If the j-th eigenvalue is real, then
*> u(j) = VL(:,j), the j-th column of VL. If the j-th and
*> (j+1)-th eigenvalues form a complex conjugate pair, then
*> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
*> Each eigenvector will be scaled so the largest component have
*> abs(real part) + abs(imag. part) = 1.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
*> after another in the columns of VR, in the same order as
*> their eigenvalues. If the j-th eigenvalue is real, then
*> v(j) = VR(:,j), the j-th column of VR. If the j-th and
*> (j+1)-th eigenvalues form a complex conjugate pair, then
*> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
*> Each eigenvector will be scaled so the largest component have
*> abs(real part) + abs(imag. part) = 1.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI are integer values such that on exit
*> A(i,j) = 0 and B(i,j) = 0 if i > j and
*> j = 1,...,ILO-1 or i = IHI+1,...,N.
*> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] LSCALE
*> \verbatim
*> LSCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and scaling factors applied
*> to the left side of A and B. If PL(j) is the index of the
*> row interchanged with row j, and DL(j) is the scaling
*> factor applied to row j, then
*> LSCALE(j) = PL(j) for j = 1,...,ILO-1
*> = DL(j) for j = ILO,...,IHI
*> = PL(j) for j = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] RSCALE
*> \verbatim
*> RSCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and scaling factors applied
*> to the right side of A and B. If PR(j) is the index of the
*> column interchanged with column j, and DR(j) is the scaling
*> factor applied to column j, then
*> RSCALE(j) = PR(j) for j = 1,...,ILO-1
*> = DR(j) for j = ILO,...,IHI
*> = PR(j) for j = IHI+1,...,N
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] ABNRM
*> \verbatim
*> ABNRM is DOUBLE PRECISION
*> The one-norm of the balanced matrix A.
*> \endverbatim
*>
*> \param[out] BBNRM
*> \verbatim
*> BBNRM is DOUBLE PRECISION
*> The one-norm of the balanced matrix B.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*> RCONDE is DOUBLE PRECISION array, dimension (N)
*> If SENSE = 'E' or 'B', the reciprocal condition numbers of
*> the eigenvalues, stored in consecutive elements of the array.
*> For a complex conjugate pair of eigenvalues two consecutive
*> elements of RCONDE are set to the same value. Thus RCONDE(j),
*> RCONDV(j), and the j-th columns of VL and VR all correspond
*> to the j-th eigenpair.
*> If SENSE = 'N or 'V', RCONDE is not referenced.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*> RCONDV is DOUBLE PRECISION array, dimension (N)
*> If SENSE = 'V' or 'B', the estimated reciprocal condition
*> numbers of the eigenvectors, stored in consecutive elements
*> of the array. For a complex eigenvector two consecutive
*> elements of RCONDV are set to the same value. If the
*> eigenvalues cannot be reordered to compute RCONDV(j),
*> RCONDV(j) is set to 0; this can only occur when the true
*> value would be very small anyway.
*> If SENSE = 'N' or 'E', RCONDV is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,2*N).
*> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
*> LWORK >= max(1,6*N).
*> If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
*> If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N+6)
*> If SENSE = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> If SENSE = 'N', BWORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*> should be correct for j=INFO+1,...,N.
*> > N: =N+1: other than QZ iteration failed in DHGEQZ.
*> =N+2: error return from DTGEVC.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggevx
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Balancing a matrix pair (A,B) includes, first, permuting rows and
*> columns to isolate eigenvalues, second, applying diagonal similarity
*> transformation to the rows and columns to make the rows and columns
*> as close in norm as possible. The computed reciprocal condition
*> numbers correspond to the balanced matrix. Permuting rows and columns
*> will not change the condition numbers (in exact arithmetic) but
*> diagonal scaling will. For further explanation of balancing, see
*> section 4.11.1.2 of LAPACK Users' Guide.
*>
*> An approximate error bound on the chordal distance between the i-th
*> computed generalized eigenvalue w and the corresponding exact
*> eigenvalue lambda is
*>
*> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
*>
*> An approximate error bound for the angle between the i-th computed
*> eigenvector VL(i) or VR(i) is given by
*>
*> EPS * norm(ABNRM, BBNRM) / DIF(i).
*>
*> For further explanation of the reciprocal condition numbers RCONDE
*> and RCONDV, see section 4.11 of LAPACK User's Guide.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B,
$ LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
$ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
$ RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION ABNRM, BBNRM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), LSCALE( * ),
$ RCONDE( * ), RCONDV( * ), RSCALE( * ),
$ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
$ PAIR, WANTSB, WANTSE, WANTSN, WANTSV
CHARACTER CHTEMP
INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
$ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
$ MINWRK, MM
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SMLNUM, TEMP
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
$ DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, DTGSNA,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
WANTSN = LSAME( SENSE, 'N' )
WANTSE = LSAME( SENSE, 'E' )
WANTSV = LSAME( SENSE, 'V' )
WANTSB = LSAME( SENSE, 'B' )
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
$ 'S' ) .OR.
$ LSAME( BALANC, 'P' ) .OR.
$ LSAME( BALANC, 'B' ) ) )
$ THEN
INFO = -1
ELSE IF( IJOBVL.LE.0 ) THEN
INFO = -2
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -3
ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
$ THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -14
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -16
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV. The workspace is
* computed assuming ILO = 1 and IHI = N, the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
IF( NOSCL .AND. .NOT.ILV ) THEN
MINWRK = 2*N
ELSE
MINWRK = 6*N
END IF
IF( WANTSE .OR. WANTSB ) THEN
MINWRK = 10*N
END IF
IF( WANTSV .OR. WANTSB ) THEN
MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
END IF
MAXWRK = MINWRK
MAXWRK = MAX( MAXWRK,
$ N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N,
$ 0 ) )
MAXWRK = MAX( MAXWRK,
$ N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N,
$ 0 ) )
IF( ILVL ) THEN
MAXWRK = MAX( MAXWRK, N +
$ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N,
$ 0 ) )
END IF
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -26
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGEVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
IF( ILASCL )
$ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute and/or balance the matrix pair (A,B)
* (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
*
CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
$ RSCALE,
$ WORK, IERR )
*
* Compute ABNRM and BBNRM
*
ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
IF( ILASCL ) THEN
WORK( 1 ) = ABNRM
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
$ IERR )
ABNRM = WORK( 1 )
END IF
*
BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
IF( ILBSCL ) THEN
WORK( 1 ) = BBNRM
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
$ IERR )
BBNRM = WORK( 1 )
END IF
*
* Reduce B to triangular form (QR decomposition of B)
* (Workspace: need N, prefer N*NB )
*
IROWS = IHI + 1 - ILO
IF( ILV .OR. .NOT.WANTSN ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = 1
IWRK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to A
* (Workspace: need N, prefer N*NB)
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VL and/or VR
* (Workspace: need N, prefer N*NB)
*
IF( ILVL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
IF( IROWS.GT.1 ) THEN
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
END IF
CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
IF( ILVR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
IF( ILV .OR. .NOT.WANTSN ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IERR )
ELSE
CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
END IF
*
* Perform QZ algorithm (Compute eigenvalues, and optionally, the
* Schur forms and Schur vectors)
* (Workspace: need N)
*
IF( ILV .OR. .NOT.WANTSN ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
*
CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
$ LWORK, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
INFO = IERR
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
INFO = IERR - N
ELSE
INFO = N + 1
END IF
GO TO 130
END IF
*
* Compute Eigenvectors and estimate condition numbers if desired
* (Workspace: DTGEVC: need 6*N
* DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
* need N otherwise )
*
IF( ILV .OR. .NOT.WANTSN ) THEN
IF( ILV ) THEN
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, N, IN, WORK, IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 130
END IF
END IF
*
IF( .NOT.WANTSN ) THEN
*
* compute eigenvectors (DTGEVC) and estimate condition
* numbers (DTGSNA). Note that the definition of the condition
* number is not invariant under transformation (u,v) to
* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
* Schur form (S,T), Q and Z are orthogonal matrices. In order
* to avoid using extra 2*N*N workspace, we have to recalculate
* eigenvectors and estimate one condition numbers at a time.
*
PAIR = .FALSE.
DO 20 I = 1, N
*
IF( PAIR ) THEN
PAIR = .FALSE.
GO TO 20
END IF
MM = 1
IF( I.LT.N ) THEN
IF( A( I+1, I ).NE.ZERO ) THEN
PAIR = .TRUE.
MM = 2
END IF
END IF
*
DO 10 J = 1, N
BWORK( J ) = .FALSE.
10 CONTINUE
IF( MM.EQ.1 ) THEN
BWORK( I ) = .TRUE.
ELSE IF( MM.EQ.2 ) THEN
BWORK( I ) = .TRUE.
BWORK( I+1 ) = .TRUE.
END IF
*
IWRK = MM*N + 1
IWRK1 = IWRK + MM*N
*
* Compute a pair of left and right eigenvectors.
* (compute workspace: need up to 4*N + 6*N)
*
IF( WANTSE .OR. WANTSB ) THEN
CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
$ WORK( 1 ), N, WORK( IWRK ), N, MM, M,
$ WORK( IWRK1 ), IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 130
END IF
END IF
*
CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
$ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
$ RCONDV( I ), MM, M, WORK( IWRK1 ),
$ LWORK-IWRK1+1, IWORK, IERR )
*
20 CONTINUE
END IF
END IF
*
* Undo balancing on VL and VR and normalization
* (Workspace: none needed)
*
IF( ILVL ) THEN
CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N,
$ VL,
$ LDVL, IERR )
*
DO 70 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 70
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 30 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
30 CONTINUE
ELSE
DO 40 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
$ ABS( VL( JR, JC+1 ) ) )
40 CONTINUE
END IF
IF( TEMP.LT.SMLNUM )
$ GO TO 70
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 50 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
50 CONTINUE
ELSE
DO 60 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
60 CONTINUE
END IF
70 CONTINUE
END IF
IF( ILVR ) THEN
CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N,
$ VR,
$ LDVR, IERR )
DO 120 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 120
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 80 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
80 CONTINUE
ELSE
DO 90 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
$ ABS( VR( JR, JC+1 ) ) )
90 CONTINUE
END IF
IF( TEMP.LT.SMLNUM )
$ GO TO 120
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 100 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
100 CONTINUE
ELSE
DO 110 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
110 CONTINUE
END IF
120 CONTINUE
END IF
*
* Undo scaling if necessary
*
130 CONTINUE
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGGEVX
*
END
*> \brief \b DGGGLM
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGGLM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggglm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggglm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggglm.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
* $ X( * ), Y( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGGLM solves a general Gauss-Markov linear model (GLM) problem:
*>
*> minimize || y ||_2 subject to d = A*x + B*y
*> x
*>
*> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
*> given N-vector. It is assumed that M <= N <= M+P, and
*>
*> rank(A) = M and rank( A B ) = N.
*>
*> Under these assumptions, the constrained equation is always
*> consistent, and there is a unique solution x and a minimal 2-norm
*> solution y, which is obtained using a generalized QR factorization
*> of the matrices (A, B) given by
*>
*> A = Q*(R), B = Q*T*Z.
*> (0)
*>
*> In particular, if matrix B is square nonsingular, then the problem
*> GLM is equivalent to the following weighted linear least squares
*> problem
*>
*> minimize || inv(B)*(d-A*x) ||_2
*> x
*>
*> where inv(B) denotes the inverse of B.
*>
*> Callers of this subroutine should note that the singularity/rank-deficiency checks
*> implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this
*> subroutine only signals a failure due to singularity if the problem is exactly singular.
*>
*> It is conceivable for one (or more) of the factors involved in the generalized QR
*> factorization of the pair (A, B) to be subnormally close to singularity without this
*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
*> problems may be less accurate due to a loss of numerical precision.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of columns of the matrix A. 0 <= M <= N.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of columns of the matrix B. P >= N-M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,M)
*> On entry, the N-by-M matrix A.
*> On exit, the upper triangular part of the array A contains
*> the M-by-M upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,P)
*> On entry, the N-by-P matrix B.
*> On exit, if N <= P, the upper triangle of the subarray
*> B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
*> if N > P, the elements on and above the (N-P)th subdiagonal
*> contain the N-by-P upper trapezoidal matrix T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D is the left hand side of the GLM equation.
*> On exit, D is destroyed.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (P)
*>
*> On exit, X and Y are the solutions of the GLM problem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N+M+P).
*> For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
*> where NB is an upper bound for the optimal blocksizes for
*> DGEQRF, SGERQF, DORMQR and SORMRQ.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1: the upper triangular factor R associated with A in the
*> generalized QR factorization of the pair (A, B) is exactly
*> singular, so that rank(A) < M; the least squares
*> solution could not be computed.
*> = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal
*> factor T associated with B in the generalized QR
*> factorization of the pair (A, B) is exactly singular, so that
*> rank( A B ) < N; the least squares solution could not
*> be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggglm
*
* =====================================================================
SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK,
$ LWORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
$ X( * ), Y( * )
* ..
*
* ===================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
$ NB4, NP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DGGQRF, DORMQR, DORMRQ,
$ DTRTRS,
$ XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NP = MIN( N, P )
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
INFO = -2
ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
*
* Calculate workspace
*
IF( INFO.EQ.0) THEN
IF( N.EQ.0 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 )
NB2 = ILAENV( 1, 'DGERQF', ' ', N, M, -1, -1 )
NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 )
NB4 = ILAENV( 1, 'DORMRQ', ' ', N, M, P, -1 )
NB = MAX( NB1, NB2, NB3, NB4 )
LWKMIN = M + N + P
LWKOPT = M + NP + MAX( N, P )*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGGLM', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
DO I = 1, M
X(I) = ZERO
END DO
DO I = 1, P
Y(I) = ZERO
END DO
RETURN
END IF
*
* Compute the GQR factorization of matrices A and B:
*
* Q**T*A = ( R11 ) M, Q**T*B*Z**T = ( T11 T12 ) M
* ( 0 ) N-M ( 0 T22 ) N-M
* M M+P-N N-M
*
* where R11 and T22 are upper triangular, and Q and Z are
* orthogonal.
*
CALL DGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
$ WORK( M+NP+1 ), LWORK-M-NP, INFO )
LOPT = INT( WORK( M+NP+1 ) )
*
* Update left-hand-side vector d = Q**T*d = ( d1 ) M
* ( d2 ) N-M
*
CALL DORMQR( 'Left', 'Transpose', N, 1, M, A, LDA, WORK, D,
$ MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) )
*
* Solve T22*y2 = d2 for y2
*
IF( N.GT.M ) THEN
CALL DTRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1,
$ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 1
RETURN
END IF
*
CALL DCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 )
END IF
*
* Set y1 = 0
*
DO 10 I = 1, M + P - N
Y( I ) = ZERO
10 CONTINUE
*
* Update d1 = d1 - T12*y2
*
CALL DGEMV( 'No transpose', M, N-M, -ONE, B( 1, M+P-N+1 ), LDB,
$ Y( M+P-N+1 ), 1, ONE, D, 1 )
*
* Solve triangular system: R11*x = d1
*
IF( M.GT.0 ) THEN
CALL DTRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A,
$ LDA,
$ D, M, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 2
RETURN
END IF
*
* Copy D to X
*
CALL DCOPY( M, D, 1, X, 1 )
END IF
*
* Backward transformation y = Z**T *y
*
CALL DORMRQ( 'Left', 'Transpose', P, 1, NP,
$ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
$ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) )
*
RETURN
*
* End of DGGGLM
*
END
*> \brief \b DGGHRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGHRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgghrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgghrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgghrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
* LDQ, Z, LDZ, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPQ, COMPZ
* INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGHRD reduces a pair of real matrices (A,B) to generalized upper
*> Hessenberg form using orthogonal transformations, where A is a
*> general matrix and B is upper triangular. The form of the
*> generalized eigenvalue problem is
*> A*x = lambda*B*x,
*> and B is typically made upper triangular by computing its QR
*> factorization and moving the orthogonal matrix Q to the left side
*> of the equation.
*>
*> This subroutine simultaneously reduces A to a Hessenberg matrix H:
*> Q**T*A*Z = H
*> and transforms B to another upper triangular matrix T:
*> Q**T*B*Z = T
*> in order to reduce the problem to its standard form
*> H*y = lambda*T*y
*> where y = Z**T*x.
*>
*> The orthogonal matrices Q and Z are determined as products of Givens
*> rotations. They may either be formed explicitly, or they may be
*> postmultiplied into input matrices Q1 and Z1, so that
*>
*> Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
*>
*> Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
*>
*> If Q1 is the orthogonal matrix from the QR factorization of B in the
*> original equation A*x = lambda*B*x, then DGGHRD reduces the original
*> problem to generalized Hessenberg form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> = 'N': do not compute Q;
*> = 'I': Q is initialized to the unit matrix, and the
*> orthogonal matrix Q is returned;
*> = 'V': Q must contain an orthogonal matrix Q1 on entry,
*> and the product Q1*Q is returned.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': do not compute Z;
*> = 'I': Z is initialized to the unit matrix, and the
*> orthogonal matrix Z is returned;
*> = 'V': Z must contain an orthogonal matrix Z1 on entry,
*> and the product Z1*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> ILO and IHI mark the rows and columns of A which are to be
*> reduced. It is assumed that A is already upper triangular
*> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
*> normally set by a previous call to DGGBAL; otherwise they
*> should be set to 1 and N respectively.
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the N-by-N general matrix to be reduced.
*> On exit, the upper triangle and the first subdiagonal of A
*> are overwritten with the upper Hessenberg matrix H, and the
*> rest is set to zero.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the N-by-N upper triangular matrix B.
*> On exit, the upper triangular matrix T = Q**T B Z. The
*> elements below the diagonal are set to zero.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, if COMPQ = 'V', the orthogonal matrix Q1,
*> typically from the QR factorization of B.
*> On exit, if COMPQ='I', the orthogonal matrix Q, and if
*> COMPQ = 'V', the product Q1*Q.
*> Not referenced if COMPQ='N'.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', the orthogonal matrix Z1.
*> On exit, if COMPZ='I', the orthogonal matrix Z, and if
*> COMPZ = 'V', the product Z1*Z.
*> Not referenced if COMPZ='N'.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z.
*> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gghrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> This routine reduces A to Hessenberg and B to triangular form by
*> an unblocked reduction, as described in _Matrix_Computations_,
*> by Golub and Van Loan (Johns Hopkins Press.)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
$ Q,
$ LDQ, Z, LDZ, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ILQ, ILZ
INTEGER ICOMPQ, ICOMPZ, JCOL, JROW
DOUBLE PRECISION C, S, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARTG, DLASET, DROT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode COMPQ
*
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
*
* Decode COMPZ
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
*
* Test the input parameters.
*
INFO = 0
IF( ICOMPQ.LE.0 ) THEN
INFO = -1
ELSE IF( ICOMPZ.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 ) THEN
INFO = -4
ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN
INFO = -11
ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGHRD', -INFO )
RETURN
END IF
*
* Initialize Q and Z if desired.
*
IF( ICOMPQ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
* Zero out lower triangle of B
*
DO 20 JCOL = 1, N - 1
DO 10 JROW = JCOL + 1, N
B( JROW, JCOL ) = ZERO
10 CONTINUE
20 CONTINUE
*
* Reduce A and B
*
DO 40 JCOL = ILO, IHI - 2
*
DO 30 JROW = IHI, JCOL + 2, -1
*
* Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
*
TEMP = A( JROW-1, JCOL )
CALL DLARTG( TEMP, A( JROW, JCOL ), C, S,
$ A( JROW-1, JCOL ) )
A( JROW, JCOL ) = ZERO
CALL DROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA,
$ A( JROW, JCOL+1 ), LDA, C, S )
CALL DROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB,
$ B( JROW, JROW-1 ), LDB, C, S )
IF( ILQ )
$ CALL DROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C,
$ S )
*
* Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
*
TEMP = B( JROW, JROW )
CALL DLARTG( TEMP, B( JROW, JROW-1 ), C, S,
$ B( JROW, JROW ) )
B( JROW, JROW-1 ) = ZERO
CALL DROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C,
$ S )
CALL DROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C,
$ S )
IF( ILZ )
$ CALL DROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C,
$ S )
30 CONTINUE
40 CONTINUE
*
RETURN
*
* End of DGGHRD
*
END
*> \brief <b> DGGLSE solves overdetermined or underdetermined systems for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGLSE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgglse.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgglse.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgglse.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ),
* $ WORK( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGLSE solves the linear equality-constrained least squares (LSE)
*> problem:
*>
*> minimize || c - A*x ||_2 subject to B*x = d
*>
*> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
*> M-vector, and d is a given P-vector. It is assumed that
*> P <= N <= M+P, and
*>
*> rank(B) = P and rank( (A) ) = N.
*> ( (B) )
*>
*> These conditions ensure that the LSE problem has a unique solution,
*> which is obtained using a generalized RQ factorization of the
*> matrices (B, A) given by
*>
*> B = (0 R)*Q, A = Z*T*Q.
*>
*> Callers of this subroutine should note that the singularity/rank-deficiency checks
*> implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this
*> subroutine only signals a failure due to singularity if the problem is exactly singular.
*>
*> It is conceivable for one (or more) of the factors involved in the generalized RQ
*> factorization of the pair (B, A) to be subnormally close to singularity without this
*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
*> problems may be less accurate due to a loss of numerical precision.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. 0 <= P <= N <= M+P.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(M,N)-by-N upper trapezoidal matrix T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
*> contains the P-by-P upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (M)
*> On entry, C contains the right hand side vector for the
*> least squares part of the LSE problem.
*> On exit, the residual sum of squares for the solution
*> is given by the sum of squares of elements N-P+1 to M of
*> vector C.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (P)
*> On entry, D contains the right hand side vector for the
*> constrained equation.
*> On exit, D is destroyed.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> On exit, X is the solution of the LSE problem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M+N+P).
*> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
*> where NB is an upper bound for the optimal blocksizes for
*> DGEQRF, SGERQF, DORMQR and SORMRQ.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1: the upper triangular factor R associated with B in the
*> generalized RQ factorization of the pair (B, A) is exactly
*> singular, so that rank(B) < P; the least squares
*> solution could not be computed.
*> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor
*> T associated with A in the generalized RQ factorization
*> of the pair (B, A) is exactly singular, so that
*> rank( (A) ) < N; the least squares solution could not
*> ( (B) )
*> be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gglse
*
* =====================================================================
SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK,
$ LWORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ),
$ WORK( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3,
$ NB4, NR
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGGRQF, DORMQR,
$ DORMRQ,
$ DTRMV, DTRTRS, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
MN = MIN( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 .OR. P.GT.N .OR. P.LT.N-M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -7
END IF
*
* Calculate workspace
*
IF( INFO.EQ.0) THEN
IF( N.EQ.0 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, P, -1 )
NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 )
NB = MAX( NB1, NB2, NB3, NB4 )
LWKMIN = M + N + P
LWKOPT = P + MN + MAX( M, N )*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGLSE', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Compute the GRQ factorization of matrices B and A:
*
* B*Q**T = ( 0 T12 ) P Z**T*A*Q**T = ( R11 R12 ) N-P
* N-P P ( 0 R22 ) M+P-N
* N-P P
*
* where T12 and R11 are upper triangular, and Q and Z are
* orthogonal.
*
CALL DGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ),
$ WORK( P+MN+1 ), LWORK-P-MN, INFO )
LOPT = INT( WORK( P+MN+1 ) )
*
* Update c = Z**T *c = ( c1 ) N-P
* ( c2 ) M+P-N
*
CALL DORMQR( 'Left', 'Transpose', M, 1, MN, A, LDA,
$ WORK( P+1 ),
$ C, MAX( 1, M ), WORK( P+MN+1 ), LWORK-P-MN, INFO )
LOPT = MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
* Solve T12*x2 = d for x2
*
IF( P.GT.0 ) THEN
CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', P, 1,
$ B( 1, N-P+1 ), LDB, D, P, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 1
RETURN
END IF
*
* Put the solution in X
*
CALL DCOPY( P, D, 1, X( N-P+1 ), 1 )
*
* Update c1
*
CALL DGEMV( 'No transpose', N-P, P, -ONE, A( 1, N-P+1 ),
$ LDA,
$ D, 1, ONE, C, 1 )
END IF
*
* Solve R11*x1 = c1 for x1
*
IF( N.GT.P ) THEN
CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N-P, 1,
$ A, LDA, C, N-P, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 2
RETURN
END IF
*
* Put the solutions in X
*
CALL DCOPY( N-P, C, 1, X, 1 )
END IF
*
* Compute the residual vector:
*
IF( M.LT.N ) THEN
NR = M + P - N
IF( NR.GT.0 )
$ CALL DGEMV( 'No transpose', NR, N-M, -ONE, A( N-P+1,
$ M+1 ),
$ LDA, D( NR+1 ), 1, ONE, C( N-P+1 ), 1 )
ELSE
NR = P
END IF
IF( NR.GT.0 ) THEN
CALL DTRMV( 'Upper', 'No transpose', 'Non unit', NR,
$ A( N-P+1, N-P+1 ), LDA, D, 1 )
CALL DAXPY( NR, -ONE, D, 1, C( N-P+1 ), 1 )
END IF
*
* Backward transformation x = Q**T*x
*
CALL DORMRQ( 'Left', 'Transpose', N, 1, P, B, LDB, WORK( 1 ),
$ X,
$ N, WORK( P+MN+1 ), LWORK-P-MN, INFO )
WORK( 1 ) = P + MN + MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
RETURN
*
* End of DGGLSE
*
END
*> \brief \b DGGQRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGQRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggqrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggqrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggqrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGQRF computes a generalized QR factorization of an N-by-M matrix A
*> and an N-by-P matrix B:
*>
*> A = Q*R, B = Q*T*Z,
*>
*> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
*> matrix, and R and T assume one of the forms:
*>
*> if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
*> ( 0 ) N-M N M-N
*> M
*>
*> where R11 is upper triangular, and
*>
*> if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
*> P-N N ( T21 ) P
*> P
*>
*> where T12 or T21 is upper triangular.
*>
*> In particular, if B is square and nonsingular, the GQR factorization
*> of A and B implicitly gives the QR factorization of inv(B)*A:
*>
*> inv(B)*A = Z**T*(inv(T)*R)
*>
*> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
*> transpose of the matrix Z.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of columns of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of columns of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,M)
*> On entry, the N-by-M matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(N,M)-by-M upper trapezoidal matrix R (R is
*> upper triangular if N >= M); the elements below the diagonal,
*> with the array TAUA, represent the orthogonal matrix Q as a
*> product of min(N,M) elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAUA
*> \verbatim
*> TAUA is DOUBLE PRECISION array, dimension (min(N,M))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q (see Further Details).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,P)
*> On entry, the N-by-P matrix B.
*> On exit, if N <= P, the upper triangle of the subarray
*> B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
*> if N > P, the elements on and above the (N-P)-th subdiagonal
*> contain the N-by-P upper trapezoidal matrix T; the remaining
*> elements, with the array TAUB, represent the orthogonal
*> matrix Z as a product of elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAUB
*> \verbatim
*> TAUB is DOUBLE PRECISION array, dimension (min(N,P))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Z (see Further Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N,M,P).
*> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
*> where NB1 is the optimal blocksize for the QR factorization
*> of an N-by-M matrix, NB2 is the optimal blocksize for the
*> RQ factorization of an N-by-P matrix, and NB3 is the optimal
*> blocksize for a call of DORMQR.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggqrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(n,m).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taua * v * v**T
*>
*> where taua is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*> and taua in TAUA(i).
*> To form Q explicitly, use LAPACK subroutine DORGQR.
*> To use Q to update another matrix, use LAPACK subroutine DORMQR.
*>
*> The matrix Z is represented as a product of elementary reflectors
*>
*> Z = H(1) H(2) . . . H(k), where k = min(n,p).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taub * v * v**T
*>
*> where taub is a real scalar, and v is a real vector with
*> v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
*> B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
*> To form Z explicitly, use LAPACK subroutine DORGRQ.
*> To use Z to update another matrix, use LAPACK subroutine DORMRQ.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
$ LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGERQF, DORMQR, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 )
NB2 = ILAENV( 1, 'DGERQF', ' ', N, P, -1, -1 )
NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 )
NB = MAX( NB1, NB2, NB3 )
LWKOPT = MAX( 1, MAX( N, M, P )*NB )
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGQRF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* QR factorization of N-by-M matrix A: A = Q*R
*
CALL DGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
LOPT = INT( WORK( 1 ) )
*
* Update B := Q**T*B.
*
CALL DORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, LDA,
$ TAUA,
$ B, LDB, WORK, LWORK, INFO )
LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
*
* RQ factorization of N-by-P matrix B: B = T*Z.
*
CALL DGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
*
WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
*
RETURN
*
* End of DGGQRF
*
END
*> \brief \b DGGRQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGRQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggrqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggrqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggrqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
*> and a P-by-N matrix B:
*>
*> A = R*Q, B = Z*T*Q,
*>
*> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
*> matrix, and R and T assume one of the forms:
*>
*> if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
*> N-M M ( R21 ) N
*> N
*>
*> where R12 or R21 is upper triangular, and
*>
*> if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
*> ( 0 ) P-N P N-P
*> N
*>
*> where T11 is upper triangular.
*>
*> In particular, if B is square and nonsingular, the GRQ factorization
*> of A and B implicitly gives the RQ factorization of A*inv(B):
*>
*> A*inv(B) = (R*inv(T))*Z**T
*>
*> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
*> transpose of the matrix Z.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, if M <= N, the upper triangle of the subarray
*> A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
*> if M > N, the elements on and above the (M-N)-th subdiagonal
*> contain the M-by-N upper trapezoidal matrix R; the remaining
*> elements, with the array TAUA, represent the orthogonal
*> matrix Q as a product of elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAUA
*> \verbatim
*> TAUA is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q (see Further Details).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(P,N)-by-N upper trapezoidal matrix T (T is
*> upper triangular if P >= N); the elements below the diagonal,
*> with the array TAUB, represent the orthogonal matrix Z as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] TAUB
*> \verbatim
*> TAUB is DOUBLE PRECISION array, dimension (min(P,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Z (see Further Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N,M,P).
*> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
*> where NB1 is the optimal blocksize for the RQ factorization
*> of an M-by-N matrix, NB2 is the optimal blocksize for the
*> QR factorization of a P-by-N matrix, and NB3 is the optimal
*> blocksize for a call of DORMRQ.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INF0= -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggrqf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taua * v * v**T
*>
*> where taua is a real scalar, and v is a real vector with
*> v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
*> A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
*> To form Q explicitly, use LAPACK subroutine DORGRQ.
*> To use Q to update another matrix, use LAPACK subroutine DORMRQ.
*>
*> The matrix Z is represented as a product of elementary reflectors
*>
*> Z = H(1) H(2) . . . H(k), where k = min(p,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taub * v * v**T
*>
*> where taub is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
*> and taub in TAUB(i).
*> To form Z explicitly, use LAPACK subroutine DORGQR.
*> To use Z to update another matrix, use LAPACK subroutine DORMQR.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
$ LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGERQF, DORMRQ, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB1 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'DGEQRF', ' ', P, N, -1, -1 )
NB3 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 )
NB = MAX( NB1, NB2, NB3 )
LWKOPT = MAX( 1, MAX( N, M, P )*NB )
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( P.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGRQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* RQ factorization of M-by-N matrix A: A = R*Q
*
CALL DGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
LOPT = INT( WORK( 1 ) )
*
* Update B := B*Q**T
*
CALL DORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ),
$ A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK,
$ LWORK, INFO )
LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
*
* QR factorization of P-by-N matrix B: B = Z*T
*
CALL DGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO )
WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
*
RETURN
*
* End of DGGRQF
*
END
*> \brief <b> DGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGSVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
* LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBQ, JOBU, JOBV
* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
* $ V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGSVD3.
*>
*> DGGSVD computes the generalized singular value decomposition (GSVD)
*> of an M-by-N real matrix A and P-by-N real matrix B:
*>
*> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
*>
*> where U, V and Q are orthogonal matrices.
*> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
*> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
*> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
*> following structures, respectively:
*>
*> If M-K-L >= 0,
*>
*> K L
*> D1 = K ( I 0 )
*> L ( 0 C )
*> M-K-L ( 0 0 )
*>
*> K L
*> D2 = L ( 0 S )
*> P-L ( 0 0 )
*>
*> N-K-L K L
*> ( 0 R ) = K ( 0 R11 R12 )
*> L ( 0 0 R22 )
*>
*> where
*>
*> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
*> S = diag( BETA(K+1), ... , BETA(K+L) ),
*> C**2 + S**2 = I.
*>
*> R is stored in A(1:K+L,N-K-L+1:N) on exit.
*>
*> If M-K-L < 0,
*>
*> K M-K K+L-M
*> D1 = K ( I 0 0 )
*> M-K ( 0 C 0 )
*>
*> K M-K K+L-M
*> D2 = M-K ( 0 S 0 )
*> K+L-M ( 0 0 I )
*> P-L ( 0 0 0 )
*>
*> N-K-L K M-K K+L-M
*> ( 0 R ) = K ( 0 R11 R12 R13 )
*> M-K ( 0 0 R22 R23 )
*> K+L-M ( 0 0 0 R33 )
*>
*> where
*>
*> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
*> S = diag( BETA(K+1), ... , BETA(M) ),
*> C**2 + S**2 = I.
*>
*> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
*> ( 0 R22 R23 )
*> in B(M-K+1:L,N+M-K-L+1:N) on exit.
*>
*> The routine computes C, S, R, and optionally the orthogonal
*> transformation matrices U, V and Q.
*>
*> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
*> A and B implicitly gives the SVD of A*inv(B):
*> A*inv(B) = U*(D1*inv(D2))*V**T.
*> If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is
*> also equal to the CS decomposition of A and B. Furthermore, the GSVD
*> can be used to derive the solution of the eigenvalue problem:
*> A**T*A x = lambda* B**T*B x.
*> In some literature, the GSVD of A and B is presented in the form
*> U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
*> where U and V are orthogonal and X is nonsingular, D1 and D2 are
*> ``diagonal''. The former GSVD form can be converted to the latter
*> form by taking the nonsingular matrix X as
*>
*> X = Q*( I 0 )
*> ( 0 inv(R) ).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> = 'U': Orthogonal matrix U is computed;
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> = 'V': Orthogonal matrix V is computed;
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*> JOBQ is CHARACTER*1
*> = 'Q': Orthogonal matrix Q is computed;
*> = 'N': Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> \endverbatim
*>
*> \param[out] L
*> \verbatim
*> L is INTEGER
*>
*> On exit, K and L specify the dimension of the subblocks
*> described in Purpose.
*> K + L = effective numerical rank of (A**T,B**T)**T.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, A contains the triangular matrix R, or part of R.
*> See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, B contains the triangular matrix R if M-K-L < 0.
*> See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*>
*> On exit, ALPHA and BETA contain the generalized singular
*> value pairs of A and B;
*> ALPHA(1:K) = 1,
*> BETA(1:K) = 0,
*> and if M-K-L >= 0,
*> ALPHA(K+1:K+L) = C,
*> BETA(K+1:K+L) = S,
*> or if M-K-L < 0,
*> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
*> BETA(K+1:M) =S, BETA(M+1:K+L) =1
*> and
*> ALPHA(K+L+1:N) = 0
*> BETA(K+L+1:N) = 0
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,M)
*> If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
*> If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,M) if
*> JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,P)
*> If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
*> If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,P) if
*> JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
*> If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N) if
*> JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> dimension (max(3*N,M,P)+N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> On exit, IWORK stores the sorting information. More
*> precisely, the following loop will sort ALPHA
*> for I = K+1, min(M,K+L)
*> swap ALPHA(I) and ALPHA(IWORK(I))
*> endfor
*> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, the Jacobi-type procedure failed to
*> converge. For further details, see subroutine DTGSJA.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> TOLA DOUBLE PRECISION
*> TOLB DOUBLE PRECISION
*> TOLA and TOLB are the thresholds to determine the effective
*> rank of (A',B')**T. Generally, they are set to
*> TOLA = MAX(M,N)*norm(A)*MAZHEPS,
*> TOLB = MAX(P,N)*norm(B)*MAZHEPS.
*> The size of TOLA and TOLB may affect the size of backward
*> errors of the decomposition.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERsing
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
$ IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), Q( LDQ, * ), U( LDU, * ),
$ V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL WANTQ, WANTU, WANTV
INTEGER I, IBND, ISUB, J, NCYCLE
DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTU = LSAME( JOBU, 'U' )
WANTV = LSAME( JOBV, 'V' )
WANTQ = LSAME( JOBQ, 'Q' )
*
INFO = 0
IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( P.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -12
ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
INFO = -16
ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
INFO = -18
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -20
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGSVD', -INFO )
RETURN
END IF
*
* Compute the Frobenius norm of matrices A and B
*
ANORM = DLANGE( '1', M, N, A, LDA, WORK )
BNORM = DLANGE( '1', P, N, B, LDB, WORK )
*
* Get machine precision and set up threshold for determining
* the effective numerical rank of the matrices A and B.
*
ULP = DLAMCH( 'Precision' )
UNFL = DLAMCH( 'Safe Minimum' )
TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
*
* Preprocessing
*
CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
$ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
$ WORK( N+1 ), INFO )
*
* Compute the GSVD of two upper "triangular" matrices
*
CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
$ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
$ WORK, NCYCLE, INFO )
*
* Sort the singular values and store the pivot indices in IWORK
* Copy ALPHA to WORK, then sort ALPHA in WORK
*
CALL DCOPY( N, ALPHA, 1, WORK, 1 )
IBND = MIN( L, M-K )
DO 20 I = 1, IBND
*
* Scan for largest ALPHA(K+I)
*
ISUB = I
SMAX = WORK( K+I )
DO 10 J = I + 1, IBND
TEMP = WORK( K+J )
IF( TEMP.GT.SMAX ) THEN
ISUB = J
SMAX = TEMP
END IF
10 CONTINUE
IF( ISUB.NE.I ) THEN
WORK( K+ISUB ) = WORK( K+I )
WORK( K+I ) = SMAX
IWORK( K+I ) = K + ISUB
ELSE
IWORK( K+I ) = K + I
END IF
20 CONTINUE
*
RETURN
*
* End of DGGSVD
*
END
*> \brief \b DGGSVP
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGSVP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
* TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
* IWORK, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBQ, JOBU, JOBV
* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
* DOUBLE PRECISION TOLA, TOLB
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGSVP3.
*>
*> DGGSVP computes orthogonal matrices U, V and Q such that
*>
*> N-K-L K L
*> U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
*> L ( 0 0 A23 )
*> M-K-L ( 0 0 0 )
*>
*> N-K-L K L
*> = K ( 0 A12 A13 ) if M-K-L < 0;
*> M-K ( 0 0 A23 )
*>
*> N-K-L K L
*> V**T*B*Q = L ( 0 0 B13 )
*> P-L ( 0 0 0 )
*>
*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
*> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
*> numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
*>
*> This decomposition is the preprocessing step for computing the
*> Generalized Singular Value Decomposition (GSVD), see subroutine
*> DGGSVD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> = 'U': Orthogonal matrix U is computed;
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> = 'V': Orthogonal matrix V is computed;
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*> JOBQ is CHARACTER*1
*> = 'Q': Orthogonal matrix Q is computed;
*> = 'N': Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, A contains the triangular (or trapezoidal) matrix
*> described in the Purpose section.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, B contains the triangular matrix described in
*> the Purpose section.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in] TOLA
*> \verbatim
*> TOLA is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] TOLB
*> \verbatim
*> TOLB is DOUBLE PRECISION
*>
*> TOLA and TOLB are the thresholds to determine the effective
*> numerical rank of matrix B and a subblock of A. Generally,
*> they are set to
*> TOLA = MAX(M,N)*norm(A)*MACHEPS,
*> TOLB = MAX(P,N)*norm(B)*MACHEPS.
*> The size of TOLA and TOLB may affect the size of backward
*> errors of the decomposition.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> \endverbatim
*>
*> \param[out] L
*> \verbatim
*> L is INTEGER
*>
*> On exit, K and L specify the dimension of the subblocks
*> described in Purpose section.
*> K + L = effective numerical rank of (A**T,B**T)**T.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,M)
*> If JOBU = 'U', U contains the orthogonal matrix U.
*> If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,M) if
*> JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,P)
*> If JOBV = 'V', V contains the orthogonal matrix V.
*> If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,P) if
*> JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> If JOBQ = 'Q', Q contains the orthogonal matrix Q.
*> If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N) if
*> JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(3*N,M,P))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
*> with column pivoting to detect the effective numerical rank of the
*> a matrix. It may be replaced by a better rank determination strategy.
*>
* =====================================================================
SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
$ IWORK, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
DOUBLE PRECISION TOLA, TOLB
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL FORWRD, WANTQ, WANTU, WANTV
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
$ DORG2R, DORM2R, DORMR2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTU = LSAME( JOBU, 'U' )
WANTV = LSAME( JOBV, 'V' )
WANTQ = LSAME( JOBQ, 'Q' )
FORWRD = .TRUE.
*
INFO = 0
IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 ) THEN
INFO = -5
ELSE IF( N.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -10
ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
INFO = -16
ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
INFO = -18
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -20
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGSVP', -INFO )
RETURN
END IF
*
* QR with column pivoting of B: B*P = V*( S11 S12 )
* ( 0 0 )
*
DO 10 I = 1, N
IWORK( I ) = 0
10 CONTINUE
CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
*
* Update A := A*P
*
CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
*
* Determine the effective rank of matrix B.
*
L = 0
DO 20 I = 1, MIN( P, N )
IF( ABS( B( I, I ) ).GT.TOLB )
$ L = L + 1
20 CONTINUE
*
IF( WANTV ) THEN
*
* Copy the details of V, and form V.
*
CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
IF( P.GT.1 )
$ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
$ LDV )
CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
END IF
*
* Clean up B
*
DO 40 J = 1, L - 1
DO 30 I = J + 1, L
B( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
IF( P.GT.L )
$ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
*
IF( WANTQ ) THEN
*
* Set Q = I and Update Q := Q*P
*
CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
END IF
*
IF( P.GE.L .AND. N.NE.L ) THEN
*
* RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
*
CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
*
* Update A := A*Z**T
*
CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
$ LDA, WORK, INFO )
*
IF( WANTQ ) THEN
*
* Update Q := Q*Z**T
*
CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU,
$ Q, LDQ, WORK, INFO )
END IF
*
* Clean up B
*
CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
DO 60 J = N - L + 1, N
DO 50 I = J - N + L + 1, L
B( I, J ) = ZERO
50 CONTINUE
60 CONTINUE
*
END IF
*
* Let N-L L
* A = ( A11 A12 ) M,
*
* then the following does the complete QR decomposition of A11:
*
* A11 = U*( 0 T12 )*P1**T
* ( 0 0 )
*
DO 70 I = 1, N - L
IWORK( I ) = 0
70 CONTINUE
CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
*
* Determine the effective rank of A11
*
K = 0
DO 80 I = 1, MIN( M, N-L )
IF( ABS( A( I, I ) ).GT.TOLA )
$ K = K + 1
80 CONTINUE
*
* Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
*
CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
$ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
*
IF( WANTU ) THEN
*
* Copy the details of U, and form U
*
CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
IF( M.GT.1 )
$ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA,
$ U( 2, 1 ), LDU )
CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK,
$ INFO )
END IF
*
IF( WANTQ ) THEN
*
* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
*
CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
END IF
*
* Clean up A: set the strictly lower triangular part of
* A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
*
DO 100 J = 1, K - 1
DO 90 I = J + 1, K
A( I, J ) = ZERO
90 CONTINUE
100 CONTINUE
IF( M.GT.K )
$ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ),
$ LDA )
*
IF( N-L.GT.K ) THEN
*
* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
*
CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
*
IF( WANTQ ) THEN
*
* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
*
CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA,
$ TAU, Q, LDQ, WORK, INFO )
END IF
*
* Clean up A
*
CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
DO 120 J = N - L - K + 1, N - L
DO 110 I = J - N + L + K + 1, K
A( I, J ) = ZERO
110 CONTINUE
120 CONTINUE
*
END IF
*
IF( M.GT.K ) THEN
*
* QR factorization of A( K+1:M,N-L+1:N )
*
CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
*
IF( WANTU ) THEN
*
* Update U(:,K+1:M) := U(:,K+1:M)*U1
*
CALL DORM2R( 'Right', 'No transpose', M, M-K,
$ MIN( M-K, L ), A( K+1, N-L+1 ), LDA, TAU,
$ U( 1, K+1 ), LDU, WORK, INFO )
END IF
*
* Clean up
*
DO 140 J = N - L + 1, N
DO 130 I = J - N + K + L + 1, M
A( I, J ) = ZERO
130 CONTINUE
140 CONTINUE
*
END IF
*
RETURN
*
* End of DGGSVP
*
END
*> \brief \b DGSVJ0 pre-processor for the routine dgesvj.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGSVJ0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgsvj0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgsvj0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgsvj0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
* SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
* DOUBLE PRECISION EPS, SFMIN, TOL
* CHARACTER*1 JOBV
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
* $ WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
*> purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
*> it does not check convergence (stopping criterion). Few tuning
*> parameters (marked by [TP]) are available for the implementer.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether the output from this procedure is used
*> to compute the matrix V:
*> = 'V': the product of the Jacobi rotations is accumulated
*> by postmultiplying the N-by-N array V.
*> (See the description of V.)
*> = 'A': the product of the Jacobi rotations is accumulated
*> by postmultiplying the MV-by-N array V.
*> (See the descriptions of MV and V.)
*> = 'N': the Jacobi rotations are not accumulated.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A.
*> M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, M-by-N matrix A, such that A*diag(D) represents
*> the input matrix.
*> On exit,
*> A_onexit * D_onexit represents the input matrix A*diag(D)
*> post-multiplied by a sequence of Jacobi rotations, where the
*> rotation threshold and the total number of sweeps are given in
*> TOL and NSWEEP, respectively.
*> (See the descriptions of D, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The array D accumulates the scaling factors from the fast scaled
*> Jacobi rotations.
*> On entry, A*diag(D) represents the input matrix.
*> On exit, A_onexit*diag(D_onexit) represents the input matrix
*> post-multiplied by a sequence of Jacobi rotations, where the
*> rotation threshold and the total number of sweeps are given in
*> TOL and NSWEEP, respectively.
*> (See the descriptions of A, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in,out] SVA
*> \verbatim
*> SVA is DOUBLE PRECISION array, dimension (N)
*> On entry, SVA contains the Euclidean norms of the columns of
*> the matrix A*diag(D).
*> On exit, SVA contains the Euclidean norms of the columns of
*> the matrix onexit*diag(D_onexit).
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*> MV is INTEGER
*> If JOBV = 'A', then MV rows of V are post-multiplied by a
*> sequence of Jacobi rotations.
*> If JOBV = 'N', then MV is not referenced.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,N)
*> If JOBV = 'V' then N rows of V are post-multiplied by a
*> sequence of Jacobi rotations.
*> If JOBV = 'A' then MV rows of V are post-multiplied by a
*> sequence of Jacobi rotations.
*> If JOBV = 'N', then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V, LDV >= 1.
*> If JOBV = 'V', LDV >= N.
*> If JOBV = 'A', LDV >= MV.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*> EPS is DOUBLE PRECISION
*> EPS = DLAMCH('Epsilon')
*> \endverbatim
*>
*> \param[in] SFMIN
*> \verbatim
*> SFMIN is DOUBLE PRECISION
*> SFMIN = DLAMCH('Safe Minimum')
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> TOL is the threshold for Jacobi rotations. For a pair
*> A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
*> applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.
*> \endverbatim
*>
*> \param[in] NSWEEP
*> \verbatim
*> NSWEEP is INTEGER
*> NSWEEP is the number of sweeps of Jacobi rotations to be
*> performed.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> LWORK is the dimension of WORK. LWORK >= M.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, then the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gsvj0
*
*> \par Further Details:
* =====================
*>
*> DGSVJ0 is used just to enable DGESVJ to call a simplified version of
*> itself to work on a submatrix of the original matrix.
*>
*> \par Contributors:
* ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*>
*> \par Bugs, Examples and Comments:
* =================================
*>
*> Please report all bugs and send interesting test examples and comments to
*> drmac@math.hr. Thank you.
*
* =====================================================================
SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
$ SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
DOUBLE PRECISION EPS, SFMIN, TOL
CHARACTER*1 JOBV
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
$ WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Local Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
$ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
$ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
$ THSIGN
INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
$ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
$ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
LOGICAL APPLV, ROTOK, RSVEC
* ..
* .. Local Arrays ..
DOUBLE PRECISION FASTR( 5 )
* ..
* .. Intrinsic Functions ..
INTRINSIC DABS, MAX, DBLE, MIN, DSIGN, DSQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DDOT, DNRM2
INTEGER IDAMAX
LOGICAL LSAME
EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM,
$ DSWAP,
$ XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
APPLV = LSAME( JOBV, 'A' )
RSVEC = LSAME( JOBV, 'V' )
IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
INFO = -3
ELSE IF( LDA.LT.M ) THEN
INFO = -5
ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
INFO = -8
ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
$ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
INFO = -10
ELSE IF( TOL.LE.EPS ) THEN
INFO = -13
ELSE IF( NSWEEP.LT.0 ) THEN
INFO = -14
ELSE IF( LWORK.LT.M ) THEN
INFO = -16
ELSE
INFO = 0
END IF
*
* #:(
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGSVJ0', -INFO )
RETURN
END IF
*
IF( RSVEC ) THEN
MVL = N
ELSE IF( APPLV ) THEN
MVL = MV
END IF
RSVEC = RSVEC .OR. APPLV
ROOTEPS = DSQRT( EPS )
ROOTSFMIN = DSQRT( SFMIN )
SMALL = SFMIN / EPS
BIG = ONE / SFMIN
ROOTBIG = ONE / ROOTSFMIN
BIGTHETA = ONE / ROOTEPS
ROOTTOL = DSQRT( TOL )
*
* -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#-
*
EMPTSW = ( N*( N-1 ) ) / 2
NOTROT = 0
FASTR( 1 ) = ZERO
*
* -#- Row-cyclic pivot strategy with de Rijk's pivoting -#-
*
SWBAND = 0
*[TP] SWBAND is a tuning parameter. It is meaningful and effective
* if SGESVJ is used as a computational routine in the preconditioned
* Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
* ......
KBL = MIN( 8, N )
*[TP] KBL is a tuning parameter that defines the tile size in the
* tiling of the p-q loops of pivot pairs. In general, an optimal
* value of KBL depends on the matrix dimensions and on the
* parameters of the computer's memory.
*
NBL = N / KBL
IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
BLSKIP = ( KBL**2 ) + 1
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
LKAHEAD = 1
*[TP] LKAHEAD is a tuning parameter.
SWBAND = 0
PSKIPPED = 0
*
DO 1993 i = 1, NSWEEP
* .. go go go ...
*
MXAAPQ = ZERO
MXSINJ = ZERO
ISWROT = 0
*
NOTROT = 0
PSKIPPED = 0
*
DO 2000 ibr = 1, NBL
igl = ( ibr-1 )*KBL + 1
*
DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr )
*
igl = igl + ir1*KBL
*
DO 2001 p = igl, MIN( igl+KBL-1, N-1 )
* .. de Rijk's pivoting
q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
IF( p.NE.q ) THEN
CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
$ V( 1, q ), 1 )
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = D( p )
D( p ) = D( q )
D( q ) = TEMP1
END IF
*
IF( ir1.EQ.0 ) THEN
*
* Column norms are periodically updated by explicit
* norm computation.
* Caveat:
* Some BLAS implementations compute DNRM2(M,A(1,p),1)
* as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in
* overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and
* underflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
* Hence, DNRM2 cannot be trusted, not even in the case when
* the true norm is far from the under(over)flow boundaries.
* If properly implemented DNRM2 is available, the IF-THEN-ELSE
* below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)".
*
IF( ( SVA( p ).LT.ROOTBIG ) .AND.
$ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p )
ELSE
TEMP1 = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
SVA( p ) = TEMP1*DSQRT( AAPP )*D( p )
END IF
AAPP = SVA( p )
ELSE
AAPP = SVA( p )
END IF
*
IF( AAPP.GT.ZERO ) THEN
*
PSKIPPED = 0
*
DO 2002 q = p + 1, MIN( igl+KBL-1, N )
*
AAQQ = SVA( q )
IF( AAQQ.GT.ZERO ) THEN
*
AAPP0 = AAPP
IF( AAQQ.GE.ONE ) THEN
ROTOK = ( SMALL*AAPP ).LE.AAQQ
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL DCOPY( M, A( 1, p ), 1, WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ D( p ),
$ M, 1, WORK, LDA, IERR )
AAPQ = DDOT( M, WORK, 1, A( 1, q ),
$ 1 )*D( q ) / AAQQ
END IF
ELSE
ROTOK = AAPP.LE.( AAQQ / SMALL )
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL DCOPY( M, A( 1, q ), 1, WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ D( q ),
$ M, 1, WORK, LDA, IERR )
AAPQ = DDOT( M, WORK, 1, A( 1, p ),
$ 1 )*D( p ) / AAPP
END IF
END IF
*
MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( DABS( AAPQ ).GT.TOL ) THEN
*
* .. rotate
* ROTATED = ROTATED + ONE
*
IF( ir1.EQ.0 ) THEN
NOTROT = 0
PSKIPPED = 0
ISWROT = ISWROT + 1
END IF
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
*
IF( DABS( THETA ).GT.BIGTHETA ) THEN
*
T = HALF / THETA
FASTR( 3 ) = T*D( p ) / D( q )
FASTR( 4 ) = -T*D( q ) / D( p )
CALL DROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = MAX( MXSINJ, DABS( T ) )
*
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -DSIGN( ONE, AAPQ )
T = ONE / ( THETA+THSIGN*
$ DSQRT( ONE+THETA*THETA ) )
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
*
MXSINJ = MAX( MXSINJ, DABS( SN ) )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = D( p ) / D( q )
AQOAP = D( q ) / D( p )
IF( D( p ).GE.ONE ) THEN
IF( D( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
D( p ) = D( p )*CS
D( q ) = D( q )*CS
CALL DROTM( M, A( 1, p ),
$ 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
END IF
ELSE
IF( D( q ).GE.ONE ) THEN
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
ELSE
IF( D( p ).GE.D( q ) ) THEN
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M,
$ CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
*
ELSE
* .. have to use modified Gram-Schmidt like transformation
CALL DCOPY( M, A( 1, p ), 1, WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAPP, ONE,
$ M,
$ 1, WORK, LDA, IERR )
CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
$ M,
$ 1, A( 1, q ), LDA, IERR )
TEMP1 = -AAPQ*D( p ) / D( q )
CALL DAXPY( M, TEMP1, WORK, 1,
$ A( 1, q ), 1 )
CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
$ M,
$ 1, A( 1, q ), LDA, IERR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q), SVA(p)
* recompute SVA(q), SVA(p).
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
$ THEN
IF( ( AAQQ.LT.ROOTBIG ) .AND.
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
SVA( q ) = DNRM2( M, A( 1, q ),
$ 1 )*
$ D( q )
ELSE
T = ZERO
AAQQ = ONE
CALL DLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*DSQRT( AAQQ )*D( q )
END IF
END IF
IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = DNRM2( M, A( 1, p ), 1 )*
$ D( p )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*DSQRT( AAPP )*D( p )
END IF
SVA( p ) = AAPP
END IF
*
ELSE
* A(:,p) and A(:,q) already numerically orthogonal
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
END IF
ELSE
* A(:,q) is zero column
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
END IF
*
IF( ( i.LE.SWBAND ) .AND.
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
IF( ir1.EQ.0 )AAPP = -AAPP
NOTROT = 0
GO TO 2103
END IF
*
2002 CONTINUE
* END q-LOOP
*
2103 CONTINUE
* bailed out of q-loop
SVA( p ) = AAPP
ELSE
SVA( p ) = AAPP
IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
$ NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p
END IF
*
2001 CONTINUE
* end of the p-loop
* end of doing the block ( ibr, ibr )
1002 CONTINUE
* end of ir1-loop
*
*........................................................
* ... go to the off diagonal blocks
*
igl = ( ibr-1 )*KBL + 1
*
DO 2010 jbc = ibr + 1, NBL
*
jgl = ( jbc-1 )*KBL + 1
*
* doing the block at ( ibr, jbc )
*
IJBLSK = 0
DO 2100 p = igl, MIN( igl+KBL-1, N )
*
AAPP = SVA( p )
*
IF( AAPP.GT.ZERO ) THEN
*
PSKIPPED = 0
*
DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
AAQQ = SVA( q )
*
IF( AAQQ.GT.ZERO ) THEN
AAPP0 = AAPP
*
* -#- M x 2 Jacobi SVD -#-
*
* -#- Safe Gram matrix computation -#-
*
IF( AAQQ.GE.ONE ) THEN
IF( AAPP.GE.AAQQ ) THEN
ROTOK = ( SMALL*AAPP ).LE.AAQQ
ELSE
ROTOK = ( SMALL*AAQQ ).LE.AAPP
END IF
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL DCOPY( M, A( 1, p ), 1, WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ D( p ),
$ M, 1, WORK, LDA, IERR )
AAPQ = DDOT( M, WORK, 1, A( 1, q ),
$ 1 )*D( q ) / AAQQ
END IF
ELSE
IF( AAPP.GE.AAQQ ) THEN
ROTOK = AAPP.LE.( AAQQ / SMALL )
ELSE
ROTOK = AAQQ.LE.( AAPP / SMALL )
END IF
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL DCOPY( M, A( 1, q ), 1, WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ D( q ),
$ M, 1, WORK, LDA, IERR )
AAPQ = DDOT( M, WORK, 1, A( 1, p ),
$ 1 )*D( p ) / AAPP
END IF
END IF
*
MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( DABS( AAPQ ).GT.TOL ) THEN
NOTROT = 0
* ROTATED = ROTATED + 1
PSKIPPED = 0
ISWROT = ISWROT + 1
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
IF( AAQQ.GT.AAPP0 )THETA = -THETA
*
IF( DABS( THETA ).GT.BIGTHETA ) THEN
T = HALF / THETA
FASTR( 3 ) = T*D( p ) / D( q )
FASTR( 4 ) = -T*D( q ) / D( p )
CALL DROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = MAX( MXSINJ, DABS( T ) )
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -DSIGN( ONE, AAPQ )
IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
T = ONE / ( THETA+THSIGN*
$ DSQRT( ONE+THETA*THETA ) )
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
MXSINJ = MAX( MXSINJ, DABS( SN ) )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = D( p ) / D( q )
AQOAP = D( q ) / D( p )
IF( D( p ).GE.ONE ) THEN
*
IF( D( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
D( p ) = D( p )*CS
D( q ) = D( q )*CS
CALL DROTM( M, A( 1, p ),
$ 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
END IF
ELSE
IF( D( q ).GE.ONE ) THEN
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
ELSE
IF( D( p ).GE.D( q ) ) THEN
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M,
$ CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
*
ELSE
IF( AAPP.GT.AAQQ ) THEN
CALL DCOPY( M, A( 1, p ), 1,
$ WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ ONE,
$ M, 1, WORK, LDA, IERR )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ ONE,
$ M, 1, A( 1, q ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( p ) / D( q )
CALL DAXPY( M, TEMP1, WORK, 1,
$ A( 1, q ), 1 )
CALL DLASCL( 'G', 0, 0, ONE,
$ AAQQ,
$ M, 1, A( 1, q ), LDA,
$ IERR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
ELSE
CALL DCOPY( M, A( 1, q ), 1,
$ WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ ONE,
$ M, 1, WORK, LDA, IERR )
CALL DLASCL( 'G', 0, 0, AAPP,
$ ONE,
$ M, 1, A( 1, p ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( q ) / D( p )
CALL DAXPY( M, TEMP1, WORK, 1,
$ A( 1, p ), 1 )
CALL DLASCL( 'G', 0, 0, ONE,
$ AAPP,
$ M, 1, A( 1, p ), LDA,
$ IERR )
SVA( p ) = AAPP*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
END IF
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q)
* .. recompute SVA(q)
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
$ THEN
IF( ( AAQQ.LT.ROOTBIG ) .AND.
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
SVA( q ) = DNRM2( M, A( 1, q ),
$ 1 )*
$ D( q )
ELSE
T = ZERO
AAQQ = ONE
CALL DLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*DSQRT( AAQQ )*D( q )
END IF
END IF
IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = DNRM2( M, A( 1, p ), 1 )*
$ D( p )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*DSQRT( AAPP )*D( p )
END IF
SVA( p ) = AAPP
END IF
* end of OK rotation
ELSE
NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
ELSE
NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
*
IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
$ THEN
SVA( p ) = AAPP
NOTROT = 0
GO TO 2011
END IF
IF( ( i.LE.SWBAND ) .AND.
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
AAPP = -AAPP
NOTROT = 0
GO TO 2203
END IF
*
2200 CONTINUE
* end of the q-loop
2203 CONTINUE
*
SVA( p ) = AAPP
*
ELSE
IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
$ MIN( jgl+KBL-1, N ) - jgl + 1
IF( AAPP.LT.ZERO )NOTROT = 0
END IF
2100 CONTINUE
* end of the p-loop
2010 CONTINUE
* end of the jbc-loop
2011 CONTINUE
*2011 bailed out of the jbc-loop
DO 2012 p = igl, MIN( igl+KBL-1, N )
SVA( p ) = DABS( SVA( p ) )
2012 CONTINUE
*
2000 CONTINUE
*2000 :: end of the ibr-loop
*
* .. update SVA(N)
IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
$ THEN
SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
SVA( N ) = T*DSQRT( AAPP )*D( N )
END IF
*
* Additional steering devices
*
IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
$ ( ISWROT.LE.N ) ) )SWBAND = i
*
IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
$ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
GO TO 1994
END IF
*
IF( NOTROT.GE.EMPTSW )GO TO 1994
1993 CONTINUE
* end i=1:NSWEEP loop
* #:) Reaching this point means that the procedure has completed the given
* number of iterations.
INFO = NSWEEP - 1
GO TO 1995
1994 CONTINUE
* #:) Reaching this point means that during the i-th sweep all pivots were
* below the given tolerance, causing early exit.
*
INFO = 0
* #:) INFO = 0 confirms successful iterations.
1995 CONTINUE
*
* Sort the vector D.
DO 5991 p = 1, N - 1
q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
IF( p.NE.q ) THEN
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = D( p )
D( p ) = D( q )
D( q ) = TEMP1
CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
END IF
5991 CONTINUE
*
RETURN
* ..
* .. END OF DGSVJ0
* ..
END
*> \brief \b DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGSVJ1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgsvj1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgsvj1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgsvj1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
* EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION EPS, SFMIN, TOL
* INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
* CHARACTER*1 JOBV
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
* $ WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGSVJ1 is called from DGESVJ as a pre-processor and that is its main
*> purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
*> it targets only particular pivots and it does not check convergence
*> (stopping criterion). Few tuning parameters (marked by [TP]) are
*> available for the implementer.
*>
*> Further Details
*> ~~~~~~~~~~~~~~~
*> DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
*> the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
*> off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
*> block-entries (tiles) of the (1,2) off-diagonal block are marked by the
*> [x]'s in the following scheme:
*>
*> | * * * [x] [x] [x]|
*> | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
*> | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
*> |[x] [x] [x] * * * |
*> |[x] [x] [x] * * * |
*> |[x] [x] [x] * * * |
*>
*> In terms of the columns of A, the first N1 columns are rotated 'against'
*> the remaining N-N1 columns, trying to increase the angle between the
*> corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
*> tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
*> The number of sweeps is given in NSWEEP and the orthogonality threshold
*> is given in TOL.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether the output from this procedure is used
*> to compute the matrix V:
*> = 'V': the product of the Jacobi rotations is accumulated
*> by postmultiplying the N-by-N array V.
*> (See the description of V.)
*> = 'A': the product of the Jacobi rotations is accumulated
*> by postmultiplying the MV-by-N array V.
*> (See the descriptions of MV and V.)
*> = 'N': the Jacobi rotations are not accumulated.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A.
*> M >= N >= 0.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> N1 specifies the 2 x 2 block partition, the first N1 columns are
*> rotated 'against' the remaining N-N1 columns of A.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, M-by-N matrix A, such that A*diag(D) represents
*> the input matrix.
*> On exit,
*> A_onexit * D_onexit represents the input matrix A*diag(D)
*> post-multiplied by a sequence of Jacobi rotations, where the
*> rotation threshold and the total number of sweeps are given in
*> TOL and NSWEEP, respectively.
*> (See the descriptions of N1, D, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The array D accumulates the scaling factors from the fast scaled
*> Jacobi rotations.
*> On entry, A*diag(D) represents the input matrix.
*> On exit, A_onexit*diag(D_onexit) represents the input matrix
*> post-multiplied by a sequence of Jacobi rotations, where the
*> rotation threshold and the total number of sweeps are given in
*> TOL and NSWEEP, respectively.
*> (See the descriptions of N1, A, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in,out] SVA
*> \verbatim
*> SVA is DOUBLE PRECISION array, dimension (N)
*> On entry, SVA contains the Euclidean norms of the columns of
*> the matrix A*diag(D).
*> On exit, SVA contains the Euclidean norms of the columns of
*> the matrix onexit*diag(D_onexit).
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*> MV is INTEGER
*> If JOBV = 'A', then MV rows of V are post-multiplied by a
*> sequence of Jacobi rotations.
*> If JOBV = 'N', then MV is not referenced.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,N)
*> If JOBV = 'V', then N rows of V are post-multiplied by a
*> sequence of Jacobi rotations.
*> If JOBV = 'A', then MV rows of V are post-multiplied by a
*> sequence of Jacobi rotations.
*> If JOBV = 'N', then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V, LDV >= 1.
*> If JOBV = 'V', LDV >= N.
*> If JOBV = 'A', LDV >= MV.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*> EPS is DOUBLE PRECISION
*> EPS = DLAMCH('Epsilon')
*> \endverbatim
*>
*> \param[in] SFMIN
*> \verbatim
*> SFMIN is DOUBLE PRECISION
*> SFMIN = DLAMCH('Safe Minimum')
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> TOL is the threshold for Jacobi rotations. For a pair
*> A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
*> applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.
*> \endverbatim
*>
*> \param[in] NSWEEP
*> \verbatim
*> NSWEEP is INTEGER
*> NSWEEP is the number of sweeps of Jacobi rotations to be
*> performed.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> LWORK is the dimension of WORK. LWORK >= M.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, then the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gsvj1
*
*> \par Contributors:
* ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
* =====================================================================
SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
$ EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION EPS, SFMIN, TOL
INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
CHARACTER*1 JOBV
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
$ WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Local Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
$ BIGTHETA, CS, LARGE, MXAAPQ, MXSINJ, ROOTBIG,
$ ROOTEPS, ROOTSFMIN, ROOTTOL, SMALL, SN, T,
$ TEMP1, THETA, THSIGN
INTEGER BLSKIP, EMPTSW, i, ibr, igl, IERR, IJBLSK,
$ ISWROT, jbc, jgl, KBL, MVL, NOTROT, nblc, nblr,
$ p, PSKIPPED, q, ROWSKIP, SWBAND
LOGICAL APPLV, ROTOK, RSVEC
* ..
* .. Local Arrays ..
DOUBLE PRECISION FASTR( 5 )
* ..
* .. Intrinsic Functions ..
INTRINSIC DABS, MAX, DBLE, MIN, DSIGN, DSQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DDOT, DNRM2
INTEGER IDAMAX
LOGICAL LSAME
EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM,
$ DSWAP,
$ XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
APPLV = LSAME( JOBV, 'A' )
RSVEC = LSAME( JOBV, 'V' )
IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
INFO = -3
ELSE IF( N1.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.M ) THEN
INFO = -6
ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
INFO = -9
ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
$ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
INFO = -11
ELSE IF( TOL.LE.EPS ) THEN
INFO = -14
ELSE IF( NSWEEP.LT.0 ) THEN
INFO = -15
ELSE IF( LWORK.LT.M ) THEN
INFO = -17
ELSE
INFO = 0
END IF
*
* #:(
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGSVJ1', -INFO )
RETURN
END IF
*
IF( RSVEC ) THEN
MVL = N
ELSE IF( APPLV ) THEN
MVL = MV
END IF
RSVEC = RSVEC .OR. APPLV
ROOTEPS = DSQRT( EPS )
ROOTSFMIN = DSQRT( SFMIN )
SMALL = SFMIN / EPS
BIG = ONE / SFMIN
ROOTBIG = ONE / ROOTSFMIN
LARGE = BIG / DSQRT( DBLE( M*N ) )
BIGTHETA = ONE / ROOTEPS
ROOTTOL = DSQRT( TOL )
*
* .. Initialize the right singular vector matrix ..
*
* RSVEC = LSAME( JOBV, 'Y' )
*
EMPTSW = N1*( N-N1 )
NOTROT = 0
FASTR( 1 ) = ZERO
*
* .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
KBL = MIN( 8, N )
NBLR = N1 / KBL
IF( ( NBLR*KBL ).NE.N1 )NBLR = NBLR + 1
* .. the tiling is nblr-by-nblc [tiles]
NBLC = ( N-N1 ) / KBL
IF( ( NBLC*KBL ).NE.( N-N1 ) )NBLC = NBLC + 1
BLSKIP = ( KBL**2 ) + 1
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
SWBAND = 0
*[TP] SWBAND is a tuning parameter. It is meaningful and effective
* if SGESVJ is used as a computational routine in the preconditioned
* Jacobi SVD algorithm SGESVJ.
*
*
* | * * * [x] [x] [x]|
* | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
* | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
* |[x] [x] [x] * * * |
* |[x] [x] [x] * * * |
* |[x] [x] [x] * * * |
*
*
DO 1993 i = 1, NSWEEP
* .. go go go ...
*
MXAAPQ = ZERO
MXSINJ = ZERO
ISWROT = 0
*
NOTROT = 0
PSKIPPED = 0
*
DO 2000 ibr = 1, NBLR
igl = ( ibr-1 )*KBL + 1
*
*
*........................................................
* ... go to the off diagonal blocks
igl = ( ibr-1 )*KBL + 1
DO 2010 jbc = 1, NBLC
jgl = N1 + ( jbc-1 )*KBL + 1
* doing the block at ( ibr, jbc )
IJBLSK = 0
DO 2100 p = igl, MIN( igl+KBL-1, N1 )
AAPP = SVA( p )
IF( AAPP.GT.ZERO ) THEN
PSKIPPED = 0
DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
AAQQ = SVA( q )
IF( AAQQ.GT.ZERO ) THEN
AAPP0 = AAPP
*
* .. M x 2 Jacobi SVD ..
*
* .. Safe Gram matrix computation ..
*
IF( AAQQ.GE.ONE ) THEN
IF( AAPP.GE.AAQQ ) THEN
ROTOK = ( SMALL*AAPP ).LE.AAQQ
ELSE
ROTOK = ( SMALL*AAQQ ).LE.AAPP
END IF
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL DCOPY( M, A( 1, p ), 1, WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ D( p ),
$ M, 1, WORK, LDA, IERR )
AAPQ = DDOT( M, WORK, 1, A( 1, q ),
$ 1 )*D( q ) / AAQQ
END IF
ELSE
IF( AAPP.GE.AAQQ ) THEN
ROTOK = AAPP.LE.( AAQQ / SMALL )
ELSE
ROTOK = AAQQ.LE.( AAPP / SMALL )
END IF
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( DDOT( M, A( 1, p ), 1,
$ A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL DCOPY( M, A( 1, q ), 1, WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ D( q ),
$ M, 1, WORK, LDA, IERR )
AAPQ = DDOT( M, WORK, 1, A( 1, p ),
$ 1 )*D( p ) / AAPP
END IF
END IF
MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( DABS( AAPQ ).GT.TOL ) THEN
NOTROT = 0
* ROTATED = ROTATED + 1
PSKIPPED = 0
ISWROT = ISWROT + 1
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*DABS(AQOAP-APOAQ) / AAPQ
IF( AAQQ.GT.AAPP0 )THETA = -THETA
IF( DABS( THETA ).GT.BIGTHETA ) THEN
T = HALF / THETA
FASTR( 3 ) = T*D( p ) / D( q )
FASTR( 4 ) = -T*D( q ) / D( p )
CALL DROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = MAX( MXSINJ, DABS( T ) )
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -DSIGN( ONE, AAPQ )
IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
T = ONE / ( THETA+THSIGN*
$ DSQRT( ONE+THETA*THETA ) )
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
MXSINJ = MAX( MXSINJ, DABS( SN ) )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
APOAQ = D( p ) / D( q )
AQOAP = D( q ) / D( p )
IF( D( p ).GE.ONE ) THEN
*
IF( D( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
D( p ) = D( p )*CS
D( q ) = D( q )*CS
CALL DROTM( M, A( 1, p ),
$ 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL DROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
END IF
ELSE
IF( D( q ).GE.ONE ) THEN
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
ELSE
IF( D( p ).GE.D( q ) ) THEN
CALL DAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL DAXPY( M,
$ CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL DAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL DAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL DAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL DAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL DAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
ELSE
IF( AAPP.GT.AAQQ ) THEN
CALL DCOPY( M, A( 1, p ), 1,
$ WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAPP,
$ ONE,
$ M, 1, WORK, LDA, IERR )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ ONE,
$ M, 1, A( 1, q ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( p ) / D( q )
CALL DAXPY( M, TEMP1, WORK, 1,
$ A( 1, q ), 1 )
CALL DLASCL( 'G', 0, 0, ONE,
$ AAQQ,
$ M, 1, A( 1, q ), LDA,
$ IERR )
SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
ELSE
CALL DCOPY( M, A( 1, q ), 1,
$ WORK,
$ 1 )
CALL DLASCL( 'G', 0, 0, AAQQ,
$ ONE,
$ M, 1, WORK, LDA, IERR )
CALL DLASCL( 'G', 0, 0, AAPP,
$ ONE,
$ M, 1, A( 1, p ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( q ) / D( p )
CALL DAXPY( M, TEMP1, WORK, 1,
$ A( 1, p ), 1 )
CALL DLASCL( 'G', 0, 0, ONE,
$ AAPP,
$ M, 1, A( 1, p ), LDA,
$ IERR )
SVA( p ) = AAPP*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
END IF
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q)
* .. recompute SVA(q)
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
$ THEN
IF( ( AAQQ.LT.ROOTBIG ) .AND.
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
SVA( q ) = DNRM2( M, A( 1, q ),
$ 1 )*
$ D( q )
ELSE
T = ZERO
AAQQ = ONE
CALL DLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*DSQRT( AAQQ )*D( q )
END IF
END IF
IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = DNRM2( M, A( 1, p ), 1 )*
$ D( p )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*DSQRT( AAPP )*D( p )
END IF
SVA( p ) = AAPP
END IF
* end of OK rotation
ELSE
NOTROT = NOTROT + 1
* SKIPPED = SKIPPED + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
ELSE
NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
* IF ( NOTROT .GE. EMPTSW ) GO TO 2011
IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
$ THEN
SVA( p ) = AAPP
NOTROT = 0
GO TO 2011
END IF
IF( ( i.LE.SWBAND ) .AND.
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
AAPP = -AAPP
NOTROT = 0
GO TO 2203
END IF
*
2200 CONTINUE
* end of the q-loop
2203 CONTINUE
SVA( p ) = AAPP
*
ELSE
IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
$ MIN( jgl+KBL-1, N ) - jgl + 1
IF( AAPP.LT.ZERO )NOTROT = 0
*** IF ( NOTROT .GE. EMPTSW ) GO TO 2011
END IF
2100 CONTINUE
* end of the p-loop
2010 CONTINUE
* end of the jbc-loop
2011 CONTINUE
*2011 bailed out of the jbc-loop
DO 2012 p = igl, MIN( igl+KBL-1, N )
SVA( p ) = DABS( SVA( p ) )
2012 CONTINUE
*** IF ( NOTROT .GE. EMPTSW ) GO TO 1994
2000 CONTINUE
*2000 :: end of the ibr-loop
*
* .. update SVA(N)
IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
$ THEN
SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
ELSE
T = ZERO
AAPP = ONE
CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
SVA( N ) = T*DSQRT( AAPP )*D( N )
END IF
*
* Additional steering devices
*
IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
$ ( ISWROT.LE.N ) ) )SWBAND = i
IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
$ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
GO TO 1994
END IF
*
IF( NOTROT.GE.EMPTSW )GO TO 1994
1993 CONTINUE
* end i=1:NSWEEP loop
* #:) Reaching this point means that the procedure has completed the given
* number of sweeps.
INFO = NSWEEP - 1
GO TO 1995
1994 CONTINUE
* #:) Reaching this point means that during the i-th sweep all pivots were
* below the given threshold, causing early exit.
INFO = 0
* #:) INFO = 0 confirms successful iterations.
1995 CONTINUE
*
* Sort the vector D
*
DO 5991 p = 1, N - 1
q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
IF( p.NE.q ) THEN
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = D( p )
D( p ) = D( q )
D( q ) = TEMP1
CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
END IF
5991 CONTINUE
*
RETURN
* ..
* .. END OF DGSVJ1
* ..
END
*> \brief \b DGTCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGTCON( NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER INFO, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION D( * ), DL( * ), DU( * ), DU2( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTCON estimates the reciprocal of the condition number of a real
*> tridiagonal matrix A using the LU factorization as computed by
*> DGTTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies whether the 1-norm condition number or the
*> infinity-norm condition number is required:
*> = '1' or 'O': 1-norm;
*> = 'I': Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) multipliers that define the matrix L from the
*> LU factorization of A as computed by DGTTRF.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the upper triangular matrix U from
*> the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*> DU2 is DOUBLE PRECISION array, dimension (N-2)
*> The (n-2) elements of the second superdiagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= n, row i of the matrix was
*> interchanged with row IPIV(i). IPIV(i) will always be either
*> i or i+1; IPIV(i) = i indicates a row interchange was not
*> required.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> If NORM = '1' or 'O', the 1-norm of the original matrix A.
*> If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*> estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtcon
*
* =====================================================================
SUBROUTINE DGTCON( NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND,
$ WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER INFO, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION D( * ), DL( * ), DU( * ), DU2( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ONENRM
INTEGER I, KASE, KASE1
DOUBLE PRECISION AINVNM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGTTRS, DLACN2, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGTCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
* Check that D(1:N) is non-zero.
*
DO 10 I = 1, N
IF( D( I ).EQ.ZERO )
$ RETURN
10 CONTINUE
*
AINVNM = ZERO
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KASE = 0
20 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(U)*inv(L).
*
CALL DGTTRS( 'No transpose', N, 1, DL, D, DU, DU2, IPIV,
$ WORK, N, INFO )
ELSE
*
* Multiply by inv(L**T)*inv(U**T).
*
CALL DGTTRS( 'Transpose', N, 1, DL, D, DU, DU2, IPIV,
$ WORK,
$ N, INFO )
END IF
GO TO 20
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
RETURN
*
* End of DGTCON
*
END
*> \brief \b DGTRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
* IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
* $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is tridiagonal, and provides
*> error bounds and backward error estimates for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) superdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] DLF
*> \verbatim
*> DLF is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) multipliers that define the matrix L from the
*> LU factorization of A as computed by DGTTRF.
*> \endverbatim
*>
*> \param[in] DF
*> \verbatim
*> DF is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the upper triangular matrix U from
*> the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DUF
*> \verbatim
*> DUF is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*> DU2 is DOUBLE PRECISION array, dimension (N-2)
*> The (n-2) elements of the second superdiagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= n, row i of the matrix was
*> interchanged with row IPIV(i). IPIV(i) will always be either
*> i or i+1; IPIV(i) = i indicates a row interchange was not
*> required.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DGTTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtrfs
*
* =====================================================================
SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
$ DU2,
$ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
CHARACTER TRANSN, TRANST
INTEGER COUNT, I, J, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGTTRS, DLACN2, DLAGTM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -15
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGTRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANSN = 'N'
TRANST = 'T'
ELSE
TRANSN = 'T'
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = 4
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 110 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - op(A) * X,
* where op(A) = A, A**T, or A**H, depending on TRANS.
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX,
$ ONE,
$ WORK( N+1 ), N )
*
* Compute abs(op(A))*abs(x) + abs(b) for use in the backward
* error bound.
*
IF( NOTRAN ) THEN
IF( N.EQ.1 ) THEN
WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) )
ELSE
WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) +
$ ABS( DU( 1 )*X( 2, J ) )
DO 30 I = 2, N - 1
WORK( I ) = ABS( B( I, J ) ) +
$ ABS( DL( I-1 )*X( I-1, J ) ) +
$ ABS( D( I )*X( I, J ) ) +
$ ABS( DU( I )*X( I+1, J ) )
30 CONTINUE
WORK( N ) = ABS( B( N, J ) ) +
$ ABS( DL( N-1 )*X( N-1, J ) ) +
$ ABS( D( N )*X( N, J ) )
END IF
ELSE
IF( N.EQ.1 ) THEN
WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) )
ELSE
WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) +
$ ABS( DL( 1 )*X( 2, J ) )
DO 40 I = 2, N - 1
WORK( I ) = ABS( B( I, J ) ) +
$ ABS( DU( I-1 )*X( I-1, J ) ) +
$ ABS( D( I )*X( I, J ) ) +
$ ABS( DL( I )*X( I+1, J ) )
40 CONTINUE
WORK( N ) = ABS( B( N, J ) ) +
$ ABS( DU( N-1 )*X( N-1, J ) ) +
$ ABS( D( N )*X( N, J ) )
END IF
END IF
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
S = ZERO
DO 50 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
50 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV,
$ WORK( N+1 ), N, INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(op(A)))*
* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(A)) is the inverse of op(A)
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(op(A)) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 60 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
60 CONTINUE
*
KASE = 0
70 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**T).
*
CALL DGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV,
$ WORK( N+1 ), N, INFO )
DO 80 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
80 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 90 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
90 CONTINUE
CALL DGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV,
$ WORK( N+1 ), N, INFO )
END IF
GO TO 70
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 100 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
100 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
110 CONTINUE
*
RETURN
*
* End of DGTRFS
*
END
*> \brief <b> DGTSV computes the solution to system of linear equations A * X = B for GT matrices </b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGTSV( N, NRHS, DL, D, DU, B, LDB, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTSV solves the equation
*>
*> A*X = B,
*>
*> where A is an n by n tridiagonal matrix, by Gaussian elimination with
*> partial pivoting.
*>
*> Note that the equation A**T*X = B may be solved by interchanging the
*> order of the arguments DU and DL.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> On entry, DL must contain the (n-1) sub-diagonal elements of
*> A.
*>
*> On exit, DL is overwritten by the (n-2) elements of the
*> second super-diagonal of the upper triangular matrix U from
*> the LU factorization of A, in DL(1), ..., DL(n-2).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D must contain the diagonal elements of A.
*>
*> On exit, D is overwritten by the n diagonal elements of U.
*> \endverbatim
*>
*> \param[in,out] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> On entry, DU must contain the (n-1) super-diagonal elements
*> of A.
*>
*> On exit, DU is overwritten by the (n-1) elements of the first
*> super-diagonal of U.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N by NRHS matrix of right hand side matrix B.
*> On exit, if INFO = 0, the N by NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, U(i,i) is exactly zero, and the solution
*> has not been computed. The factorization has not been
*> completed unless i = N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtsv
*
* =====================================================================
SUBROUTINE DGTSV( N, NRHS, DL, D, DU, B, LDB, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION FACT, TEMP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( NRHS.LT.0 ) THEN
INFO = -2
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGTSV ', -INFO )
RETURN
END IF
*
IF( N.EQ.0 )
$ RETURN
*
IF( NRHS.EQ.1 ) THEN
DO 10 I = 1, N - 2
IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
*
* No row interchange required
*
IF( D( I ).NE.ZERO ) THEN
FACT = DL( I ) / D( I )
D( I+1 ) = D( I+1 ) - FACT*DU( I )
B( I+1, 1 ) = B( I+1, 1 ) - FACT*B( I, 1 )
ELSE
INFO = I
RETURN
END IF
DL( I ) = ZERO
ELSE
*
* Interchange rows I and I+1
*
FACT = D( I ) / DL( I )
D( I ) = DL( I )
TEMP = D( I+1 )
D( I+1 ) = DU( I ) - FACT*TEMP
DL( I ) = DU( I+1 )
DU( I+1 ) = -FACT*DL( I )
DU( I ) = TEMP
TEMP = B( I, 1 )
B( I, 1 ) = B( I+1, 1 )
B( I+1, 1 ) = TEMP - FACT*B( I+1, 1 )
END IF
10 CONTINUE
IF( N.GT.1 ) THEN
I = N - 1
IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
IF( D( I ).NE.ZERO ) THEN
FACT = DL( I ) / D( I )
D( I+1 ) = D( I+1 ) - FACT*DU( I )
B( I+1, 1 ) = B( I+1, 1 ) - FACT*B( I, 1 )
ELSE
INFO = I
RETURN
END IF
ELSE
FACT = D( I ) / DL( I )
D( I ) = DL( I )
TEMP = D( I+1 )
D( I+1 ) = DU( I ) - FACT*TEMP
DU( I ) = TEMP
TEMP = B( I, 1 )
B( I, 1 ) = B( I+1, 1 )
B( I+1, 1 ) = TEMP - FACT*B( I+1, 1 )
END IF
END IF
IF( D( N ).EQ.ZERO ) THEN
INFO = N
RETURN
END IF
ELSE
DO 40 I = 1, N - 2
IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
*
* No row interchange required
*
IF( D( I ).NE.ZERO ) THEN
FACT = DL( I ) / D( I )
D( I+1 ) = D( I+1 ) - FACT*DU( I )
DO 20 J = 1, NRHS
B( I+1, J ) = B( I+1, J ) - FACT*B( I, J )
20 CONTINUE
ELSE
INFO = I
RETURN
END IF
DL( I ) = ZERO
ELSE
*
* Interchange rows I and I+1
*
FACT = D( I ) / DL( I )
D( I ) = DL( I )
TEMP = D( I+1 )
D( I+1 ) = DU( I ) - FACT*TEMP
DL( I ) = DU( I+1 )
DU( I+1 ) = -FACT*DL( I )
DU( I ) = TEMP
DO 30 J = 1, NRHS
TEMP = B( I, J )
B( I, J ) = B( I+1, J )
B( I+1, J ) = TEMP - FACT*B( I+1, J )
30 CONTINUE
END IF
40 CONTINUE
IF( N.GT.1 ) THEN
I = N - 1
IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
IF( D( I ).NE.ZERO ) THEN
FACT = DL( I ) / D( I )
D( I+1 ) = D( I+1 ) - FACT*DU( I )
DO 50 J = 1, NRHS
B( I+1, J ) = B( I+1, J ) - FACT*B( I, J )
50 CONTINUE
ELSE
INFO = I
RETURN
END IF
ELSE
FACT = D( I ) / DL( I )
D( I ) = DL( I )
TEMP = D( I+1 )
D( I+1 ) = DU( I ) - FACT*TEMP
DU( I ) = TEMP
DO 60 J = 1, NRHS
TEMP = B( I, J )
B( I, J ) = B( I+1, J )
B( I+1, J ) = TEMP - FACT*B( I+1, J )
60 CONTINUE
END IF
END IF
IF( D( N ).EQ.ZERO ) THEN
INFO = N
RETURN
END IF
END IF
*
* Back solve with the matrix U from the factorization.
*
IF( NRHS.LE.2 ) THEN
J = 1
70 CONTINUE
B( N, J ) = B( N, J ) / D( N )
IF( N.GT.1 )
$ B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) / D( N-1 )
DO 80 I = N - 2, 1, -1
B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DL( I )*
$ B( I+2, J ) ) / D( I )
80 CONTINUE
IF( J.LT.NRHS ) THEN
J = J + 1
GO TO 70
END IF
ELSE
DO 100 J = 1, NRHS
B( N, J ) = B( N, J ) / D( N )
IF( N.GT.1 )
$ B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
$ D( N-1 )
DO 90 I = N - 2, 1, -1
B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DL( I )*
$ B( I+2, J ) ) / D( I )
90 CONTINUE
100 CONTINUE
END IF
*
RETURN
*
* End of DGTSV
*
END
*> \brief <b> DGTSVX computes the solution to system of linear equations A * X = B for GT matrices </b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
* DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER FACT, TRANS
* INTEGER INFO, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
* $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTSVX uses the LU factorization to compute the solution to a real
*> system of linear equations A * X = B or A**T * X = B,
*> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
*> matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
*> as A = L * U, where L is a product of permutation and unit lower
*> bidiagonal matrices and U is upper triangular with nonzeros in
*> only the main diagonal and first two superdiagonals.
*>
*> 2. If some U(i,i)=0, so that U is exactly singular, then the routine
*> returns with INFO = i. Otherwise, the factored form of A is used
*> to estimate the condition number of the matrix A. If the
*> reciprocal of the condition number is less than machine precision,
*> INFO = N+1 is returned as a warning, but the routine still goes on
*> to solve for X and compute error bounds as described below.
*>
*> 3. The system of equations is solved for X using the factored form
*> of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of A has been
*> supplied on entry.
*> = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored
*> form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
*> will not be modified.
*> = 'N': The matrix will be copied to DLF, DF, and DUF
*> and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) superdiagonal elements of A.
*> \endverbatim
*>
*> \param[in,out] DLF
*> \verbatim
*> DLF is DOUBLE PRECISION array, dimension (N-1)
*> If FACT = 'F', then DLF is an input argument and on entry
*> contains the (n-1) multipliers that define the matrix L from
*> the LU factorization of A as computed by DGTTRF.
*>
*> If FACT = 'N', then DLF is an output argument and on exit
*> contains the (n-1) multipliers that define the matrix L from
*> the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] DF
*> \verbatim
*> DF is DOUBLE PRECISION array, dimension (N)
*> If FACT = 'F', then DF is an input argument and on entry
*> contains the n diagonal elements of the upper triangular
*> matrix U from the LU factorization of A.
*>
*> If FACT = 'N', then DF is an output argument and on exit
*> contains the n diagonal elements of the upper triangular
*> matrix U from the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] DUF
*> \verbatim
*> DUF is DOUBLE PRECISION array, dimension (N-1)
*> If FACT = 'F', then DUF is an input argument and on entry
*> contains the (n-1) elements of the first superdiagonal of U.
*>
*> If FACT = 'N', then DUF is an output argument and on exit
*> contains the (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in,out] DU2
*> \verbatim
*> DU2 is DOUBLE PRECISION array, dimension (N-2)
*> If FACT = 'F', then DU2 is an input argument and on entry
*> contains the (n-2) elements of the second superdiagonal of
*> U.
*>
*> If FACT = 'N', then DU2 is an output argument and on exit
*> contains the (n-2) elements of the second superdiagonal of
*> U.
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains the pivot indices from the LU factorization of A as
*> computed by DGTTRF.
*>
*> If FACT = 'N', then IPIV is an output argument and on exit
*> contains the pivot indices from the LU factorization of A;
*> row i of the matrix was interchanged with row IPIV(i).
*> IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
*> a row interchange was not required.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A. If RCOND is less than the machine precision (in
*> particular, if RCOND = 0), the matrix is singular to working
*> precision. This condition is indicated by a return code of
*> INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: U(i,i) is exactly zero. The factorization
*> has not been completed unless i = N, but the
*> factor U is exactly singular, so the solution
*> and error bounds could not be computed.
*> RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtsvx
*
* =====================================================================
SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF,
$ DUF,
$ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
$ WORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER FACT, TRANS
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOFACT, NOTRAN
CHARACTER NORM
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGT
EXTERNAL LSAME, DLAMCH, DLANGT
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGTCON, DGTRFS, DGTTRF, DGTTRS,
$ DLACPY,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGTSVX', -INFO )
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the LU factorization of A.
*
CALL DCOPY( N, D, 1, DF, 1 )
IF( N.GT.1 ) THEN
CALL DCOPY( N-1, DL, 1, DLF, 1 )
CALL DCOPY( N-1, DU, 1, DUF, 1 )
END IF
CALL DGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
IF( NOTRAN ) THEN
NORM = '1'
ELSE
NORM = 'I'
END IF
ANORM = DLANGT( NORM, N, DL, D, DU )
*
* Compute the reciprocal of the condition number of A.
*
CALL DGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND,
$ WORK,
$ IWORK, INFO )
*
* Compute the solution vectors X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
$ INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
$ IPIV,
$ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DGTSVX
*
END
*> \brief \b DGTTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgttrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgttrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgttrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGTTRF( N, DL, D, DU, DU2, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION D( * ), DL( * ), DU( * ), DU2( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTTRF computes an LU factorization of a real tridiagonal matrix A
*> using elimination with partial pivoting and row interchanges.
*>
*> The factorization has the form
*> A = L * U
*> where L is a product of permutation and unit lower bidiagonal
*> matrices and U is upper triangular with nonzeros in only the main
*> diagonal and first two superdiagonals.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in,out] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> On entry, DL must contain the (n-1) sub-diagonal elements of
*> A.
*>
*> On exit, DL is overwritten by the (n-1) multipliers that
*> define the matrix L from the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D must contain the diagonal elements of A.
*>
*> On exit, D is overwritten by the n diagonal elements of the
*> upper triangular matrix U from the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> On entry, DU must contain the (n-1) super-diagonal elements
*> of A.
*>
*> On exit, DU is overwritten by the (n-1) elements of the first
*> super-diagonal of U.
*> \endverbatim
*>
*> \param[out] DU2
*> \verbatim
*> DU2 is DOUBLE PRECISION array, dimension (N-2)
*> On exit, DU2 is overwritten by the (n-2) elements of the
*> second super-diagonal of U.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= n, row i of the matrix was
*> interchanged with row IPIV(i). IPIV(i) will always be either
*> i or i+1; IPIV(i) = i indicates a row interchange was not
*> required.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, U(k,k) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and division by zero will occur if it is used
*> to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gttrf
*
* =====================================================================
SUBROUTINE DGTTRF( N, DL, D, DU, DU2, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION D( * ), DL( * ), DU( * ), DU2( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION FACT, TEMP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DGTTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Initialize IPIV(i) = i and DU2(I) = 0
*
DO 10 I = 1, N
IPIV( I ) = I
10 CONTINUE
DO 20 I = 1, N - 2
DU2( I ) = ZERO
20 CONTINUE
*
DO 30 I = 1, N - 2
IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
*
* No row interchange required, eliminate DL(I)
*
IF( D( I ).NE.ZERO ) THEN
FACT = DL( I ) / D( I )
DL( I ) = FACT
D( I+1 ) = D( I+1 ) - FACT*DU( I )
END IF
ELSE
*
* Interchange rows I and I+1, eliminate DL(I)
*
FACT = D( I ) / DL( I )
D( I ) = DL( I )
DL( I ) = FACT
TEMP = DU( I )
DU( I ) = D( I+1 )
D( I+1 ) = TEMP - FACT*D( I+1 )
DU2( I ) = DU( I+1 )
DU( I+1 ) = -FACT*DU( I+1 )
IPIV( I ) = I + 1
END IF
30 CONTINUE
IF( N.GT.1 ) THEN
I = N - 1
IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
IF( D( I ).NE.ZERO ) THEN
FACT = DL( I ) / D( I )
DL( I ) = FACT
D( I+1 ) = D( I+1 ) - FACT*DU( I )
END IF
ELSE
FACT = D( I ) / DL( I )
D( I ) = DL( I )
DL( I ) = FACT
TEMP = DU( I )
DU( I ) = D( I+1 )
D( I+1 ) = TEMP - FACT*D( I+1 )
IPIV( I ) = I + 1
END IF
END IF
*
* Check for a zero on the diagonal of U.
*
DO 40 I = 1, N
IF( D( I ).EQ.ZERO ) THEN
INFO = I
GO TO 50
END IF
40 CONTINUE
50 CONTINUE
*
RETURN
*
* End of DGTTRF
*
END
*> \brief \b DGTTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgttrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgttrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgttrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGTTRS( TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTTRS solves one of the systems of equations
*> A*X = B or A**T*X = B,
*> with a tridiagonal matrix A using the LU factorization computed
*> by DGTTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations.
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T* X = B (Transpose)
*> = 'C': A**T* X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) multipliers that define the matrix L from the
*> LU factorization of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the upper triangular matrix U from
*> the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) elements of the first super-diagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*> DU2 is DOUBLE PRECISION array, dimension (N-2)
*> The (n-2) elements of the second super-diagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= n, row i of the matrix was
*> interchanged with row IPIV(i). IPIV(i) will always be either
*> i or i+1; IPIV(i) = i indicates a row interchange was not
*> required.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the matrix of right hand side vectors B.
*> On exit, B is overwritten by the solution vectors X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gttrs
*
* =====================================================================
SUBROUTINE DGTTRS( TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B,
$ LDB,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER ITRANS, J, JB, NB
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGTTS2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOTRAN = ( TRANS.EQ.'N' .OR. TRANS.EQ.'n' )
IF( .NOT.NOTRAN .AND. .NOT.( TRANS.EQ.'T' .OR. TRANS.EQ.
$ 't' ) .AND. .NOT.( TRANS.EQ.'C' .OR. TRANS.EQ.'c' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( N, 1 ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGTTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
* Decode TRANS
*
IF( NOTRAN ) THEN
ITRANS = 0
ELSE
ITRANS = 1
END IF
*
* Determine the number of right-hand sides to solve at a time.
*
IF( NRHS.EQ.1 ) THEN
NB = 1
ELSE
NB = MAX( 1, ILAENV( 1, 'DGTTRS', TRANS, N, NRHS, -1, -1 ) )
END IF
*
IF( NB.GE.NRHS ) THEN
CALL DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB )
ELSE
DO 10 J = 1, NRHS, NB
JB = MIN( NRHS-J+1, NB )
CALL DGTTS2( ITRANS, N, JB, DL, D, DU, DU2, IPIV, B( 1,
$ J ),
$ LDB )
10 CONTINUE
END IF
*
* End of DGTTRS
*
END
*> \brief \b DGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTTS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtts2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtts2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtts2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB )
*
* .. Scalar Arguments ..
* INTEGER ITRANS, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTTS2 solves one of the systems of equations
*> A*X = B or A**T*X = B,
*> with a tridiagonal matrix A using the LU factorization computed
*> by DGTTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITRANS
*> \verbatim
*> ITRANS is INTEGER
*> Specifies the form of the system of equations.
*> = 0: A * X = B (No transpose)
*> = 1: A**T* X = B (Transpose)
*> = 2: A**T* X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) multipliers that define the matrix L from the
*> LU factorization of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the upper triangular matrix U from
*> the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) elements of the first super-diagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*> DU2 is DOUBLE PRECISION array, dimension (N-2)
*> The (n-2) elements of the second super-diagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= n, row i of the matrix was
*> interchanged with row IPIV(i). IPIV(i) will always be either
*> i or i+1; IPIV(i) = i indicates a row interchange was not
*> required.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the matrix of right hand side vectors B.
*> On exit, B is overwritten by the solution vectors X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtts2
*
* =====================================================================
SUBROUTINE DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B,
$ LDB )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER ITRANS, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IP, J
DOUBLE PRECISION TEMP
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( ITRANS.EQ.0 ) THEN
*
* Solve A*X = B using the LU factorization of A,
* overwriting each right hand side vector with its solution.
*
IF( NRHS.LE.1 ) THEN
J = 1
10 CONTINUE
*
* Solve L*x = b.
*
DO 20 I = 1, N - 1
IP = IPIV( I )
TEMP = B( I+1-IP+I, J ) - DL( I )*B( IP, J )
B( I, J ) = B( IP, J )
B( I+1, J ) = TEMP
20 CONTINUE
*
* Solve U*x = b.
*
B( N, J ) = B( N, J ) / D( N )
IF( N.GT.1 )
$ B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
$ D( N-1 )
DO 30 I = N - 2, 1, -1
B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )*
$ B( I+2, J ) ) / D( I )
30 CONTINUE
IF( J.LT.NRHS ) THEN
J = J + 1
GO TO 10
END IF
ELSE
DO 60 J = 1, NRHS
*
* Solve L*x = b.
*
DO 40 I = 1, N - 1
IF( IPIV( I ).EQ.I ) THEN
B( I+1, J ) = B( I+1, J ) - DL( I )*B( I, J )
ELSE
TEMP = B( I, J )
B( I, J ) = B( I+1, J )
B( I+1, J ) = TEMP - DL( I )*B( I, J )
END IF
40 CONTINUE
*
* Solve U*x = b.
*
B( N, J ) = B( N, J ) / D( N )
IF( N.GT.1 )
$ B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
$ D( N-1 )
DO 50 I = N - 2, 1, -1
B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )*
$ B( I+2, J ) ) / D( I )
50 CONTINUE
60 CONTINUE
END IF
ELSE
*
* Solve A**T * X = B.
*
IF( NRHS.LE.1 ) THEN
*
* Solve U**T*x = b.
*
J = 1
70 CONTINUE
B( 1, J ) = B( 1, J ) / D( 1 )
IF( N.GT.1 )
$ B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 )
DO 80 I = 3, N
B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )-DU2( I-2 )*
$ B( I-2, J ) ) / D( I )
80 CONTINUE
*
* Solve L**T*x = b.
*
DO 90 I = N - 1, 1, -1
IP = IPIV( I )
TEMP = B( I, J ) - DL( I )*B( I+1, J )
B( I, J ) = B( IP, J )
B( IP, J ) = TEMP
90 CONTINUE
IF( J.LT.NRHS ) THEN
J = J + 1
GO TO 70
END IF
*
ELSE
DO 120 J = 1, NRHS
*
* Solve U**T*x = b.
*
B( 1, J ) = B( 1, J ) / D( 1 )
IF( N.GT.1 )
$ B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 )
DO 100 I = 3, N
B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )-
$ DU2( I-2 )*B( I-2, J ) ) / D( I )
100 CONTINUE
DO 110 I = N - 1, 1, -1
IF( IPIV( I ).EQ.I ) THEN
B( I, J ) = B( I, J ) - DL( I )*B( I+1, J )
ELSE
TEMP = B( I+1, J )
B( I+1, J ) = B( I, J ) - DL( I )*TEMP
B( I, J ) = TEMP
END IF
110 CONTINUE
120 CONTINUE
END IF
END IF
*
* End of DGTTS2
*
END
*> \brief \b DHGEQZ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DHGEQZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhgeqz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhgeqz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhgeqz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPQ, COMPZ, JOB
* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
* $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
* $ WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
*> where H is an upper Hessenberg matrix and T is upper triangular,
*> using the double-shift QZ method.
*> Matrix pairs of this type are produced by the reduction to
*> generalized upper Hessenberg form of a real matrix pair (A,B):
*>
*> A = Q1*H*Z1**T, B = Q1*T*Z1**T,
*>
*> as computed by DGGHRD.
*>
*> If JOB='S', then the Hessenberg-triangular pair (H,T) is
*> also reduced to generalized Schur form,
*>
*> H = Q*S*Z**T, T = Q*P*Z**T,
*>
*> where Q and Z are orthogonal matrices, P is an upper triangular
*> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
*> diagonal blocks.
*>
*> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
*> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
*> eigenvalues.
*>
*> Additionally, the 2-by-2 upper triangular diagonal blocks of P
*> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
*> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
*> P(j,j) > 0, and P(j+1,j+1) > 0.
*>
*> Optionally, the orthogonal matrix Q from the generalized Schur
*> factorization may be postmultiplied into an input matrix Q1, and the
*> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
*> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
*> the matrix pair (A,B) to generalized upper Hessenberg form, then the
*> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
*> generalized Schur factorization of (A,B):
*>
*> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
*>
*> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
*> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
*> complex and beta real.
*> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
*> generalized nonsymmetric eigenvalue problem (GNEP)
*> A*x = lambda*B*x
*> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
*> alternate form of the GNEP
*> mu*A*y = B*y.
*> Real eigenvalues can be read directly from the generalized Schur
*> form:
*> alpha = S(i,i), beta = P(i,i).
*>
*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
*> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
*> pp. 241--256.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> = 'E': Compute eigenvalues only;
*> = 'S': Compute eigenvalues and the Schur form.
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> = 'N': Left Schur vectors (Q) are not computed;
*> = 'I': Q is initialized to the unit matrix and the matrix Q
*> of left Schur vectors of (H,T) is returned;
*> = 'V': Q must contain an orthogonal matrix Q1 on entry and
*> the product Q1*Q is returned.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Right Schur vectors (Z) are not computed;
*> = 'I': Z is initialized to the unit matrix and the matrix Z
*> of right Schur vectors of (H,T) is returned;
*> = 'V': Z must contain an orthogonal matrix Z1 on entry and
*> the product Z1*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices H, T, Q, and Z. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI mark the rows and columns of H which are in
*> Hessenberg form. It is assumed that A is already upper
*> triangular in rows and columns 1:ILO-1 and IHI+1:N.
*> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH, N)
*> On entry, the N-by-N upper Hessenberg matrix H.
*> On exit, if JOB = 'S', H contains the upper quasi-triangular
*> matrix S from the generalized Schur factorization.
*> If JOB = 'E', the diagonal blocks of H match those of S, but
*> the rest of H is unspecified.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT, N)
*> On entry, the N-by-N upper triangular matrix T.
*> On exit, if JOB = 'S', T contains the upper triangular
*> matrix P from the generalized Schur factorization;
*> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
*> are reduced to positive diagonal form, i.e., if H(j+1,j) is
*> non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
*> T(j+1,j+1) > 0.
*> If JOB = 'E', the diagonal blocks of T match those of P, but
*> the rest of T is unspecified.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue
*> of GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> The imaginary parts of each scalar alpha defining an
*> eigenvalue of GNEP.
*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*> positive, then the j-th and (j+1)-st eigenvalues are a
*> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> The scalars beta that define the eigenvalues of GNEP.
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
*> pair (A,B), in one of the forms lambda = alpha/beta or
*> mu = beta/alpha. Since either lambda or mu may overflow,
*> they should not, in general, be computed.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
*> the reduction of (A,B) to generalized Hessenberg form.
*> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
*> vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
*> of left Schur vectors of (A,B).
*> Not referenced if COMPQ = 'N'.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= 1.
*> If COMPQ='V' or 'I', then LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
*> the reduction of (A,B) to generalized Hessenberg form.
*> On exit, if COMPZ = 'I', the orthogonal matrix of
*> right Schur vectors of (H,T), and if COMPZ = 'V', the
*> orthogonal matrix of right Schur vectors of (A,B).
*> Not referenced if COMPZ = 'N'.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1.
*> If COMPZ='V' or 'I', then LDZ >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> = 1,...,N: the QZ iteration did not converge. (H,T) is not
*> in Schur form, but ALPHAR(i), ALPHAI(i), and
*> BETA(i), i=INFO+1,...,N should be correct.
*> = N+1,...,2*N: the shift calculation failed. (H,T) is not
*> in Schur form, but ALPHAR(i), ALPHAI(i), and
*> BETA(i), i=INFO-N+1,...,N should be correct.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hgeqz
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Iteration counters:
*>
*> JITER -- counts iterations.
*> IITER -- counts iterations run since ILAST was last
*> changed. This is therefore reset only when a 1-by-1 or
*> 2-by-2 block deflates off the bottom.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T,
$ LDT,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
$ LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
$ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
* $ SAFETY = 1.0E+0 )
DOUBLE PRECISION HALF, ZERO, ONE, SAFETY
PARAMETER ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0,
$ SAFETY = 1.0D+2 )
* ..
* .. Local Scalars ..
LOGICAL ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
$ LQUERY
INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
$ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
$ JR, MAXIT
DOUBLE PRECISION A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
$ AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
$ AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
$ B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
$ BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
$ CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
$ SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1,
$ T2, T3, TAU, TEMP, TEMP2, TEMPI, TEMPR, U1,
$ U12, U12L, U2, ULP, VS, W11, W12, W21, W22,
$ WABS, WI, WR, WR2
* ..
* .. Local Arrays ..
DOUBLE PRECISION V( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2, DLAPY3
EXTERNAL LSAME, DLAMCH, DLANHS, DLAPY2,
$ DLAPY3
* ..
* .. External Subroutines ..
EXTERNAL DLAG2, DLARFG, DLARTG, DLASET, DLASV2,
$ DROT,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Decode JOB, COMPQ, COMPZ
*
IF( LSAME( JOB, 'E' ) ) THEN
ILSCHR = .FALSE.
ISCHUR = 1
ELSE IF( LSAME( JOB, 'S' ) ) THEN
ILSCHR = .TRUE.
ISCHUR = 2
ELSE
ISCHUR = 0
END IF
*
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
*
* Check Argument Values
*
INFO = 0
WORK( 1 ) = MAX( 1, N )
LQUERY = ( LWORK.EQ.-1 )
IF( ISCHUR.EQ.0 ) THEN
INFO = -1
ELSE IF( ICOMPQ.EQ.0 ) THEN
INFO = -2
ELSE IF( ICOMPZ.EQ.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( ILO.LT.1 ) THEN
INFO = -5
ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
INFO = -6
ELSE IF( LDH.LT.N ) THEN
INFO = -8
ELSE IF( LDT.LT.N ) THEN
INFO = -10
ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
INFO = -15
ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
INFO = -17
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DHGEQZ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = DBLE( 1 )
RETURN
END IF
*
* Initialize Q and Z
*
IF( ICOMPQ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
* Machine Constants
*
IN = IHI + 1 - ILO
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
ANORM = DLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
BNORM = DLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK )
ATOL = MAX( SAFMIN, ULP*ANORM )
BTOL = MAX( SAFMIN, ULP*BNORM )
ASCALE = ONE / MAX( SAFMIN, ANORM )
BSCALE = ONE / MAX( SAFMIN, BNORM )
*
* Set Eigenvalues IHI+1:N
*
DO 30 J = IHI + 1, N
IF( T( J, J ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 10 JR = 1, J
H( JR, J ) = -H( JR, J )
T( JR, J ) = -T( JR, J )
10 CONTINUE
ELSE
H( J, J ) = -H( J, J )
T( J, J ) = -T( J, J )
END IF
IF( ILZ ) THEN
DO 20 JR = 1, N
Z( JR, J ) = -Z( JR, J )
20 CONTINUE
END IF
END IF
ALPHAR( J ) = H( J, J )
ALPHAI( J ) = ZERO
BETA( J ) = T( J, J )
30 CONTINUE
*
* If IHI < ILO, skip QZ steps
*
IF( IHI.LT.ILO )
$ GO TO 380
*
* MAIN QZ ITERATION LOOP
*
* Initialize dynamic indices
*
* Eigenvalues ILAST+1:N have been found.
* Column operations modify rows IFRSTM:whatever.
* Row operations modify columns whatever:ILASTM.
*
* If only eigenvalues are being computed, then
* IFRSTM is the row of the last splitting row above row ILAST;
* this is always at least ILO.
* IITER counts iterations since the last eigenvalue was found,
* to tell when to use an extraordinary shift.
* MAXIT is the maximum number of QZ sweeps allowed.
*
ILAST = IHI
IF( ILSCHR ) THEN
IFRSTM = 1
ILASTM = N
ELSE
IFRSTM = ILO
ILASTM = IHI
END IF
IITER = 0
ESHIFT = ZERO
MAXIT = 30*( IHI-ILO+1 )
*
DO 360 JITER = 1, MAXIT
*
* Split the matrix if possible.
*
* Two tests:
* 1: H(j,j-1)=0 or j=ILO
* 2: T(j,j)=0
*
IF( ILAST.EQ.ILO ) THEN
*
* Special case: j=ILAST
*
GO TO 80
ELSE
IF( ABS( H( ILAST, ILAST-1 ) ).LE.MAX( SAFMIN, ULP*(
$ ABS( H( ILAST, ILAST ) ) + ABS( H( ILAST-1, ILAST-1 ) )
$ ) ) ) THEN
H( ILAST, ILAST-1 ) = ZERO
GO TO 80
END IF
END IF
*
IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
T( ILAST, ILAST ) = ZERO
GO TO 70
END IF
*
* General case: j<ILAST
*
DO 60 J = ILAST - 1, ILO, -1
*
* Test 1: for H(j,j-1)=0 or j=ILO
*
IF( J.EQ.ILO ) THEN
ILAZRO = .TRUE.
ELSE
IF( ABS( H( J, J-1 ) ).LE.MAX( SAFMIN, ULP*(
$ ABS( H( J, J ) ) + ABS( H( J-1, J-1 ) )
$ ) ) ) THEN
H( J, J-1 ) = ZERO
ILAZRO = .TRUE.
ELSE
ILAZRO = .FALSE.
END IF
END IF
*
* Test 2: for T(j,j)=0
*
IF( ABS( T( J, J ) ).LT.BTOL ) THEN
T( J, J ) = ZERO
*
* Test 1a: Check for 2 consecutive small subdiagonals in A
*
ILAZR2 = .FALSE.
IF( .NOT.ILAZRO ) THEN
TEMP = ABS( H( J, J-1 ) )
TEMP2 = ABS( H( J, J ) )
TEMPR = MAX( TEMP, TEMP2 )
IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
TEMP = TEMP / TEMPR
TEMP2 = TEMP2 / TEMPR
END IF
IF( TEMP*( ASCALE*ABS( H( J+1, J ) ) ).LE.TEMP2*
$ ( ASCALE*ATOL ) )ILAZR2 = .TRUE.
END IF
*
* If both tests pass (1 & 2), i.e., the leading diagonal
* element of B in the block is zero, split a 1x1 block off
* at the top. (I.e., at the J-th row/column) The leading
* diagonal element of the remainder can also be zero, so
* this may have to be done repeatedly.
*
IF( ILAZRO .OR. ILAZR2 ) THEN
DO 40 JCH = J, ILAST - 1
TEMP = H( JCH, JCH )
CALL DLARTG( TEMP, H( JCH+1, JCH ), C, S,
$ H( JCH, JCH ) )
H( JCH+1, JCH ) = ZERO
CALL DROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
$ H( JCH+1, JCH+1 ), LDH, C, S )
CALL DROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
$ T( JCH+1, JCH+1 ), LDT, C, S )
IF( ILQ )
$ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ),
$ 1,
$ C, S )
IF( ILAZR2 )
$ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
ILAZR2 = .FALSE.
IF( ABS( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
IF( JCH+1.GE.ILAST ) THEN
GO TO 80
ELSE
IFIRST = JCH + 1
GO TO 110
END IF
END IF
T( JCH+1, JCH+1 ) = ZERO
40 CONTINUE
GO TO 70
ELSE
*
* Only test 2 passed -- chase the zero to T(ILAST,ILAST)
* Then process as in the case T(ILAST,ILAST)=0
*
DO 50 JCH = J, ILAST - 1
TEMP = T( JCH, JCH+1 )
CALL DLARTG( TEMP, T( JCH+1, JCH+1 ), C, S,
$ T( JCH, JCH+1 ) )
T( JCH+1, JCH+1 ) = ZERO
IF( JCH.LT.ILASTM-1 )
$ CALL DROT( ILASTM-JCH-1, T( JCH, JCH+2 ),
$ LDT,
$ T( JCH+1, JCH+2 ), LDT, C, S )
CALL DROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
$ H( JCH+1, JCH-1 ), LDH, C, S )
IF( ILQ )
$ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ),
$ 1,
$ C, S )
TEMP = H( JCH+1, JCH )
CALL DLARTG( TEMP, H( JCH+1, JCH-1 ), C, S,
$ H( JCH+1, JCH ) )
H( JCH+1, JCH-1 ) = ZERO
CALL DROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
$ H( IFRSTM, JCH-1 ), 1, C, S )
CALL DROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
$ T( IFRSTM, JCH-1 ), 1, C, S )
IF( ILZ )
$ CALL DROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ),
$ 1,
$ C, S )
50 CONTINUE
GO TO 70
END IF
ELSE IF( ILAZRO ) THEN
*
* Only test 1 passed -- work on J:ILAST
*
IFIRST = J
GO TO 110
END IF
*
* Neither test passed -- try next J
*
60 CONTINUE
*
* (Drop-through is "impossible")
*
INFO = N + 1
GO TO 420
*
* T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
* 1x1 block.
*
70 CONTINUE
TEMP = H( ILAST, ILAST )
CALL DLARTG( TEMP, H( ILAST, ILAST-1 ), C, S,
$ H( ILAST, ILAST ) )
H( ILAST, ILAST-1 ) = ZERO
CALL DROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
$ H( IFRSTM, ILAST-1 ), 1, C, S )
CALL DROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
$ T( IFRSTM, ILAST-1 ), 1, C, S )
IF( ILZ )
$ CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C,
$ S )
*
* H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI,
* and BETA
*
80 CONTINUE
IF( T( ILAST, ILAST ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 90 J = IFRSTM, ILAST
H( J, ILAST ) = -H( J, ILAST )
T( J, ILAST ) = -T( J, ILAST )
90 CONTINUE
ELSE
H( ILAST, ILAST ) = -H( ILAST, ILAST )
T( ILAST, ILAST ) = -T( ILAST, ILAST )
END IF
IF( ILZ ) THEN
DO 100 J = 1, N
Z( J, ILAST ) = -Z( J, ILAST )
100 CONTINUE
END IF
END IF
ALPHAR( ILAST ) = H( ILAST, ILAST )
ALPHAI( ILAST ) = ZERO
BETA( ILAST ) = T( ILAST, ILAST )
*
* Go to next block -- exit if finished.
*
ILAST = ILAST - 1
IF( ILAST.LT.ILO )
$ GO TO 380
*
* Reset counters
*
IITER = 0
ESHIFT = ZERO
IF( .NOT.ILSCHR ) THEN
ILASTM = ILAST
IF( IFRSTM.GT.ILAST )
$ IFRSTM = ILO
END IF
GO TO 350
*
* QZ step
*
* This iteration only involves rows/columns IFIRST:ILAST. We
* assume IFIRST < ILAST, and that the diagonal of B is non-zero.
*
110 CONTINUE
IITER = IITER + 1
IF( .NOT.ILSCHR ) THEN
IFRSTM = IFIRST
END IF
*
* Compute single shifts.
*
* At this point, IFIRST < ILAST, and the diagonal elements of
* T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
* magnitude)
*
IF( ( IITER / 10 )*10.EQ.IITER ) THEN
*
* Exceptional shift. Chosen for no particularly good reason.
* (Single shift only.)
*
IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST, ILAST-1 ) ).LT.
$ ABS( T( ILAST-1, ILAST-1 ) ) ) THEN
ESHIFT = H( ILAST, ILAST-1 ) /
$ T( ILAST-1, ILAST-1 )
ELSE
ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) )
END IF
S1 = ONE
WR = ESHIFT
*
ELSE
*
* Shifts based on the generalized eigenvalues of the
* bottom-right 2x2 block of A and B. The first eigenvalue
* returned by DLAG2 is the Wilkinson shift (AEP p.512),
*
CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
$ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
$ S2, WR, WR2, WI )
*
IF ( ABS( (WR/S1)*T( ILAST, ILAST ) - H( ILAST, ILAST ) )
$ .GT. ABS( (WR2/S2)*T( ILAST, ILAST )
$ - H( ILAST, ILAST ) ) ) THEN
TEMP = WR
WR = WR2
WR2 = TEMP
TEMP = S1
S1 = S2
S2 = TEMP
END IF
TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) )
IF( WI.NE.ZERO )
$ GO TO 200
END IF
*
* Fiddle with shift to avoid overflow
*
TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX )
IF( S1.GT.TEMP ) THEN
SCALE = TEMP / S1
ELSE
SCALE = ONE
END IF
*
TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX )
IF( ABS( WR ).GT.TEMP )
$ SCALE = MIN( SCALE, TEMP / ABS( WR ) )
S1 = SCALE*S1
WR = SCALE*WR
*
* Now check for two consecutive small subdiagonals.
*
DO 120 J = ILAST - 1, IFIRST + 1, -1
ISTART = J
TEMP = ABS( S1*H( J, J-1 ) )
TEMP2 = ABS( S1*H( J, J )-WR*T( J, J ) )
TEMPR = MAX( TEMP, TEMP2 )
IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
TEMP = TEMP / TEMPR
TEMP2 = TEMP2 / TEMPR
END IF
IF( ABS( ( ASCALE*H( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )*
$ TEMP2 )GO TO 130
120 CONTINUE
*
ISTART = IFIRST
130 CONTINUE
*
* Do an implicit single-shift QZ sweep.
*
* Initial Q
*
TEMP = S1*H( ISTART, ISTART ) - WR*T( ISTART, ISTART )
TEMP2 = S1*H( ISTART+1, ISTART )
CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
*
* Sweep
*
DO 190 J = ISTART, ILAST - 1
IF( J.GT.ISTART ) THEN
TEMP = H( J, J-1 )
CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
H( J+1, J-1 ) = ZERO
END IF
*
DO 140 JC = J, ILASTM
TEMP = C*H( J, JC ) + S*H( J+1, JC )
H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
H( J, JC ) = TEMP
TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
T( J, JC ) = TEMP2
140 CONTINUE
IF( ILQ ) THEN
DO 150 JR = 1, N
TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
Q( JR, J ) = TEMP
150 CONTINUE
END IF
*
TEMP = T( J+1, J+1 )
CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
T( J+1, J ) = ZERO
*
DO 160 JR = IFRSTM, MIN( J+2, ILAST )
TEMP = C*H( JR, J+1 ) + S*H( JR, J )
H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
H( JR, J+1 ) = TEMP
160 CONTINUE
DO 170 JR = IFRSTM, J
TEMP = C*T( JR, J+1 ) + S*T( JR, J )
T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
T( JR, J+1 ) = TEMP
170 CONTINUE
IF( ILZ ) THEN
DO 180 JR = 1, N
TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
Z( JR, J+1 ) = TEMP
180 CONTINUE
END IF
190 CONTINUE
*
GO TO 350
*
* Use Francis double-shift
*
* Note: the Francis double-shift should work with real shifts,
* but only if the block is at least 3x3.
* This code may break if this point is reached with
* a 2x2 block with real eigenvalues.
*
200 CONTINUE
IF( IFIRST+1.EQ.ILAST ) THEN
*
* Special case -- 2x2 block with complex eigenvectors
*
* Step 1: Standardize, that is, rotate so that
*
* ( B11 0 )
* B = ( ) with B11 non-negative.
* ( 0 B22 )
*
CALL DLASV2( T( ILAST-1, ILAST-1 ), T( ILAST-1, ILAST ),
$ T( ILAST, ILAST ), B22, B11, SR, CR, SL, CL )
*
IF( B11.LT.ZERO ) THEN
CR = -CR
SR = -SR
B11 = -B11
B22 = -B22
END IF
*
CALL DROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH,
$ H( ILAST, ILAST-1 ), LDH, CL, SL )
CALL DROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1,
$ H( IFRSTM, ILAST ), 1, CR, SR )
*
IF( ILAST.LT.ILASTM )
$ CALL DROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT,
$ T( ILAST, ILAST+1 ), LDT, CL, SL )
IF( IFRSTM.LT.ILAST-1 )
$ CALL DROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1,
$ T( IFRSTM, ILAST ), 1, CR, SR )
*
IF( ILQ )
$ CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1,
$ CL,
$ SL )
IF( ILZ )
$ CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1,
$ CR,
$ SR )
*
T( ILAST-1, ILAST-1 ) = B11
T( ILAST-1, ILAST ) = ZERO
T( ILAST, ILAST-1 ) = ZERO
T( ILAST, ILAST ) = B22
*
* If B22 is negative, negate column ILAST
*
IF( B22.LT.ZERO ) THEN
DO 210 J = IFRSTM, ILAST
H( J, ILAST ) = -H( J, ILAST )
T( J, ILAST ) = -T( J, ILAST )
210 CONTINUE
*
IF( ILZ ) THEN
DO 220 J = 1, N
Z( J, ILAST ) = -Z( J, ILAST )
220 CONTINUE
END IF
B22 = -B22
END IF
*
* Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.)
*
* Recompute shift
*
CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
$ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
$ TEMP, WR, TEMP2, WI )
*
* If standardization has perturbed the shift onto real line,
* do another (real single-shift) QR step.
*
IF( WI.EQ.ZERO )
$ GO TO 350
S1INV = ONE / S1
*
* Do EISPACK (QZVAL) computation of alpha and beta
*
A11 = H( ILAST-1, ILAST-1 )
A21 = H( ILAST, ILAST-1 )
A12 = H( ILAST-1, ILAST )
A22 = H( ILAST, ILAST )
*
* Compute complex Givens rotation on right
* (Assume some element of C = (sA - wB) > unfl )
* __
* (sA - wB) ( CZ -SZ )
* ( SZ CZ )
*
C11R = S1*A11 - WR*B11
C11I = -WI*B11
C12 = S1*A12
C21 = S1*A21
C22R = S1*A22 - WR*B22
C22I = -WI*B22
*
IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+
$ ABS( C22R )+ABS( C22I ) ) THEN
T1 = DLAPY3( C12, C11R, C11I )
CZ = C12 / T1
SZR = -C11R / T1
SZI = -C11I / T1
ELSE
CZ = DLAPY2( C22R, C22I )
IF( CZ.LE.SAFMIN ) THEN
CZ = ZERO
SZR = ONE
SZI = ZERO
ELSE
TEMPR = C22R / CZ
TEMPI = C22I / CZ
T1 = DLAPY2( CZ, C21 )
CZ = CZ / T1
SZR = -C21*TEMPR / T1
SZI = C21*TEMPI / T1
END IF
END IF
*
* Compute Givens rotation on left
*
* ( CQ SQ )
* ( __ ) A or B
* ( -SQ CQ )
*
AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 )
BN = ABS( B11 ) + ABS( B22 )
WABS = ABS( WR ) + ABS( WI )
IF( S1*AN.GT.WABS*BN ) THEN
CQ = CZ*B11
SQR = SZR*B22
SQI = -SZI*B22
ELSE
A1R = CZ*A11 + SZR*A12
A1I = SZI*A12
A2R = CZ*A21 + SZR*A22
A2I = SZI*A22
CQ = DLAPY2( A1R, A1I )
IF( CQ.LE.SAFMIN ) THEN
CQ = ZERO
SQR = ONE
SQI = ZERO
ELSE
TEMPR = A1R / CQ
TEMPI = A1I / CQ
SQR = TEMPR*A2R + TEMPI*A2I
SQI = TEMPI*A2R - TEMPR*A2I
END IF
END IF
T1 = DLAPY3( CQ, SQR, SQI )
CQ = CQ / T1
SQR = SQR / T1
SQI = SQI / T1
*
* Compute diagonal elements of QBZ
*
TEMPR = SQR*SZR - SQI*SZI
TEMPI = SQR*SZI + SQI*SZR
B1R = CQ*CZ*B11 + TEMPR*B22
B1I = TEMPI*B22
B1A = DLAPY2( B1R, B1I )
B2R = CQ*CZ*B22 + TEMPR*B11
B2I = -TEMPI*B11
B2A = DLAPY2( B2R, B2I )
*
* Normalize so beta > 0, and Im( alpha1 ) > 0
*
BETA( ILAST-1 ) = B1A
BETA( ILAST ) = B2A
ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV
ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV
ALPHAR( ILAST ) = ( WR*B2A )*S1INV
ALPHAI( ILAST ) = -( WI*B2A )*S1INV
*
* Step 3: Go to next block -- exit if finished.
*
ILAST = IFIRST - 1
IF( ILAST.LT.ILO )
$ GO TO 380
*
* Reset counters
*
IITER = 0
ESHIFT = ZERO
IF( .NOT.ILSCHR ) THEN
ILASTM = ILAST
IF( IFRSTM.GT.ILAST )
$ IFRSTM = ILO
END IF
GO TO 350
ELSE
*
* Usual case: 3x3 or larger block, using Francis implicit
* double-shift
*
* 2
* Eigenvalue equation is w - c w + d = 0,
*
* -1 2 -1
* so compute 1st column of (A B ) - c A B + d
* using the formula in QZIT (from EISPACK)
*
* We assume that the block is at least 3x3
*
AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
$ ( BSCALE*T( ILAST-1, ILAST-1 ) )
AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
$ ( BSCALE*T( ILAST-1, ILAST-1 ) )
AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
$ ( BSCALE*T( ILAST, ILAST ) )
AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
$ ( BSCALE*T( ILAST, ILAST ) )
U12 = T( ILAST-1, ILAST ) / T( ILAST, ILAST )
AD11L = ( ASCALE*H( IFIRST, IFIRST ) ) /
$ ( BSCALE*T( IFIRST, IFIRST ) )
AD21L = ( ASCALE*H( IFIRST+1, IFIRST ) ) /
$ ( BSCALE*T( IFIRST, IFIRST ) )
AD12L = ( ASCALE*H( IFIRST, IFIRST+1 ) ) /
$ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
AD22L = ( ASCALE*H( IFIRST+1, IFIRST+1 ) ) /
$ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
AD32L = ( ASCALE*H( IFIRST+2, IFIRST+1 ) ) /
$ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
U12L = T( IFIRST, IFIRST+1 ) / T( IFIRST+1, IFIRST+1 )
*
V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 +
$ AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L
V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )-
$ ( AD22-AD11L )+AD21*U12 )*AD21L
V( 3 ) = AD32L*AD21L
*
ISTART = IFIRST
*
CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU )
V( 1 ) = ONE
*
* Sweep
*
DO 290 J = ISTART, ILAST - 2
*
* All but last elements: use 3x3 Householder transforms.
*
* Zero (j-1)st column of A
*
IF( J.GT.ISTART ) THEN
V( 1 ) = H( J, J-1 )
V( 2 ) = H( J+1, J-1 )
V( 3 ) = H( J+2, J-1 )
*
CALL DLARFG( 3, H( J, J-1 ), V( 2 ), 1, TAU )
V( 1 ) = ONE
H( J+1, J-1 ) = ZERO
H( J+2, J-1 ) = ZERO
END IF
*
T2 = TAU*V( 2 )
T3 = TAU*V( 3 )
DO 230 JC = J, ILASTM
TEMP = H( J, JC )+V( 2 )*H( J+1, JC )+V( 3 )*
$ H( J+2, JC )
H( J, JC ) = H( J, JC ) - TEMP*TAU
H( J+1, JC ) = H( J+1, JC ) - TEMP*T2
H( J+2, JC ) = H( J+2, JC ) - TEMP*T3
TEMP2 = T( J, JC )+V( 2 )*T( J+1, JC )+V( 3 )*
$ T( J+2, JC )
T( J, JC ) = T( J, JC ) - TEMP2*TAU
T( J+1, JC ) = T( J+1, JC ) - TEMP2*T2
T( J+2, JC ) = T( J+2, JC ) - TEMP2*T3
230 CONTINUE
IF( ILQ ) THEN
DO 240 JR = 1, N
TEMP = Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )*
$ Q( JR, J+2 )
Q( JR, J ) = Q( JR, J ) - TEMP*TAU
Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*T2
Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*T3
240 CONTINUE
END IF
*
* Zero j-th column of B (see DLAGBC for details)
*
* Swap rows to pivot
*
ILPIVT = .FALSE.
TEMP = MAX( ABS( T( J+1, J+1 ) ), ABS( T( J+1, J+2 ) ) )
TEMP2 = MAX( ABS( T( J+2, J+1 ) ), ABS( T( J+2, J+2 ) ) )
IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN
SCALE = ZERO
U1 = ONE
U2 = ZERO
GO TO 250
ELSE IF( TEMP.GE.TEMP2 ) THEN
W11 = T( J+1, J+1 )
W21 = T( J+2, J+1 )
W12 = T( J+1, J+2 )
W22 = T( J+2, J+2 )
U1 = T( J+1, J )
U2 = T( J+2, J )
ELSE
W21 = T( J+1, J+1 )
W11 = T( J+2, J+1 )
W22 = T( J+1, J+2 )
W12 = T( J+2, J+2 )
U2 = T( J+1, J )
U1 = T( J+2, J )
END IF
*
* Swap columns if nec.
*
IF( ABS( W12 ).GT.ABS( W11 ) ) THEN
ILPIVT = .TRUE.
TEMP = W12
TEMP2 = W22
W12 = W11
W22 = W21
W11 = TEMP
W21 = TEMP2
END IF
*
* LU-factor
*
TEMP = W21 / W11
U2 = U2 - TEMP*U1
W22 = W22 - TEMP*W12
W21 = ZERO
*
* Compute SCALE
*
SCALE = ONE
IF( ABS( W22 ).LT.SAFMIN ) THEN
SCALE = ZERO
U2 = ONE
U1 = -W12 / W11
GO TO 250
END IF
IF( ABS( W22 ).LT.ABS( U2 ) )
$ SCALE = ABS( W22 / U2 )
IF( ABS( W11 ).LT.ABS( U1 ) )
$ SCALE = MIN( SCALE, ABS( W11 / U1 ) )
*
* Solve
*
U2 = ( SCALE*U2 ) / W22
U1 = ( SCALE*U1-W12*U2 ) / W11
*
250 CONTINUE
IF( ILPIVT ) THEN
TEMP = U2
U2 = U1
U1 = TEMP
END IF
*
* Compute Householder Vector
*
T1 = SQRT( SCALE**2+U1**2+U2**2 )
TAU = ONE + SCALE / T1
VS = -ONE / ( SCALE+T1 )
V( 1 ) = ONE
V( 2 ) = VS*U1
V( 3 ) = VS*U2
*
* Apply transformations from the right.
*
T2 = TAU*V(2)
T3 = TAU*V(3)
DO 260 JR = IFRSTM, MIN( J+3, ILAST )
TEMP = H( JR, J )+V( 2 )*H( JR, J+1 )+V( 3 )*
$ H( JR, J+2 )
H( JR, J ) = H( JR, J ) - TEMP*TAU
H( JR, J+1 ) = H( JR, J+1 ) - TEMP*T2
H( JR, J+2 ) = H( JR, J+2 ) - TEMP*T3
260 CONTINUE
DO 270 JR = IFRSTM, J + 2
TEMP = T( JR, J )+V( 2 )*T( JR, J+1 )+V( 3 )*
$ T( JR, J+2 )
T( JR, J ) = T( JR, J ) - TEMP*TAU
T( JR, J+1 ) = T( JR, J+1 ) - TEMP*T2
T( JR, J+2 ) = T( JR, J+2 ) - TEMP*T3
270 CONTINUE
IF( ILZ ) THEN
DO 280 JR = 1, N
TEMP = Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )*
$ Z( JR, J+2 )
Z( JR, J ) = Z( JR, J ) - TEMP*TAU
Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*T2
Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*T3
280 CONTINUE
END IF
T( J+1, J ) = ZERO
T( J+2, J ) = ZERO
290 CONTINUE
*
* Last elements: Use Givens rotations
*
* Rotations from the left
*
J = ILAST - 1
TEMP = H( J, J-1 )
CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
H( J+1, J-1 ) = ZERO
*
DO 300 JC = J, ILASTM
TEMP = C*H( J, JC ) + S*H( J+1, JC )
H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
H( J, JC ) = TEMP
TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
T( J, JC ) = TEMP2
300 CONTINUE
IF( ILQ ) THEN
DO 310 JR = 1, N
TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
Q( JR, J ) = TEMP
310 CONTINUE
END IF
*
* Rotations from the right.
*
TEMP = T( J+1, J+1 )
CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
T( J+1, J ) = ZERO
*
DO 320 JR = IFRSTM, ILAST
TEMP = C*H( JR, J+1 ) + S*H( JR, J )
H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
H( JR, J+1 ) = TEMP
320 CONTINUE
DO 330 JR = IFRSTM, ILAST - 1
TEMP = C*T( JR, J+1 ) + S*T( JR, J )
T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
T( JR, J+1 ) = TEMP
330 CONTINUE
IF( ILZ ) THEN
DO 340 JR = 1, N
TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
Z( JR, J+1 ) = TEMP
340 CONTINUE
END IF
*
* End of Double-Shift code
*
END IF
*
GO TO 350
*
* End of iteration loop
*
350 CONTINUE
360 CONTINUE
*
* Drop-through = non-convergence
*
INFO = ILAST
GO TO 420
*
* Successful completion of all QZ steps
*
380 CONTINUE
*
* Set Eigenvalues 1:ILO-1
*
DO 410 J = 1, ILO - 1
IF( T( J, J ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 390 JR = 1, J
H( JR, J ) = -H( JR, J )
T( JR, J ) = -T( JR, J )
390 CONTINUE
ELSE
H( J, J ) = -H( J, J )
T( J, J ) = -T( J, J )
END IF
IF( ILZ ) THEN
DO 400 JR = 1, N
Z( JR, J ) = -Z( JR, J )
400 CONTINUE
END IF
END IF
ALPHAR( J ) = H( J, J )
ALPHAI( J ) = ZERO
BETA( J ) = T( J, J )
410 CONTINUE
*
* Normal Termination
*
INFO = 0
*
* Exit (other than argument error) -- return optimal workspace size
*
420 CONTINUE
WORK( 1 ) = DBLE( N )
RETURN
*
* End of DHGEQZ
*
END
*> \brief \b DHSEIN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DHSEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhsein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhsein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhsein.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI,
* VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL,
* IFAILR, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EIGSRC, INITV, SIDE
* INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* INTEGER IFAILL( * ), IFAILR( * )
* DOUBLE PRECISION H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WI( * ), WORK( * ), WR( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DHSEIN uses inverse iteration to find specified right and/or left
*> eigenvectors of a real upper Hessenberg matrix H.
*>
*> The right eigenvector x and the left eigenvector y of the matrix H
*> corresponding to an eigenvalue w are defined by:
*>
*> H * x = w * x, y**h * H = w * y**h
*>
*> where y**h denotes the conjugate transpose of the vector y.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': compute right eigenvectors only;
*> = 'L': compute left eigenvectors only;
*> = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] EIGSRC
*> \verbatim
*> EIGSRC is CHARACTER*1
*> Specifies the source of eigenvalues supplied in (WR,WI):
*> = 'Q': the eigenvalues were found using DHSEQR; thus, if
*> H has zero subdiagonal elements, and so is
*> block-triangular, then the j-th eigenvalue can be
*> assumed to be an eigenvalue of the block containing
*> the j-th row/column. This property allows DHSEIN to
*> perform inverse iteration on just one diagonal block.
*> = 'N': no assumptions are made on the correspondence
*> between eigenvalues and diagonal blocks. In this
*> case, DHSEIN must always perform inverse iteration
*> using the whole matrix H.
*> \endverbatim
*>
*> \param[in] INITV
*> \verbatim
*> INITV is CHARACTER*1
*> = 'N': no initial vectors are supplied;
*> = 'U': user-supplied initial vectors are stored in the arrays
*> VL and/or VR.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> Specifies the eigenvectors to be computed. To select the
*> real eigenvector corresponding to a real eigenvalue WR(j),
*> SELECT(j) must be set to .TRUE.. To select the complex
*> eigenvector corresponding to a complex eigenvalue
*> (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
*> either SELECT(j) or SELECT(j+1) or both must be set to
*> .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
*> .FALSE..
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> The upper Hessenberg matrix H.
*> If a NaN is detected in H, the routine will return with INFO=-6.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*>
*> On entry, the real and imaginary parts of the eigenvalues of
*> H; a complex conjugate pair of eigenvalues must be stored in
*> consecutive elements of WR and WI.
*> On exit, WR may have been altered since close eigenvalues
*> are perturbed slightly in searching for independent
*> eigenvectors.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*> On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
*> contain starting vectors for the inverse iteration for the
*> left eigenvectors; the starting vector for each eigenvector
*> must be in the same column(s) in which the eigenvector will
*> be stored.
*> On exit, if SIDE = 'L' or 'B', the left eigenvectors
*> specified by SELECT will be stored consecutively in the
*> columns of VL, in the same order as their eigenvalues. A
*> complex eigenvector corresponding to a complex eigenvalue is
*> stored in two consecutive columns, the first holding the real
*> part and the second the imaginary part.
*> If SIDE = 'R', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL.
*> LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*> On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
*> contain starting vectors for the inverse iteration for the
*> right eigenvectors; the starting vector for each eigenvector
*> must be in the same column(s) in which the eigenvector will
*> be stored.
*> On exit, if SIDE = 'R' or 'B', the right eigenvectors
*> specified by SELECT will be stored consecutively in the
*> columns of VR, in the same order as their eigenvalues. A
*> complex eigenvector corresponding to a complex eigenvalue is
*> stored in two consecutive columns, the first holding the real
*> part and the second the imaginary part.
*> If SIDE = 'L', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR.
*> LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of columns in the arrays VL and/or VR required to
*> store the eigenvectors; each selected real eigenvector
*> occupies one column and each selected complex eigenvector
*> occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension ((N+2)*N)
*> \endverbatim
*>
*> \param[out] IFAILL
*> \verbatim
*> IFAILL is INTEGER array, dimension (MM)
*> If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
*> eigenvector in the i-th column of VL (corresponding to the
*> eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
*> eigenvector converged satisfactorily. If the i-th and (i+1)th
*> columns of VL hold a complex eigenvector, then IFAILL(i) and
*> IFAILL(i+1) are set to the same value.
*> If SIDE = 'R', IFAILL is not referenced.
*> \endverbatim
*>
*> \param[out] IFAILR
*> \verbatim
*> IFAILR is INTEGER array, dimension (MM)
*> If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
*> eigenvector in the i-th column of VR (corresponding to the
*> eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
*> eigenvector converged satisfactorily. If the i-th and (i+1)th
*> columns of VR hold a complex eigenvector, then IFAILR(i) and
*> IFAILR(i+1) are set to the same value.
*> If SIDE = 'L', IFAILR is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, i is the number of eigenvectors which
*> failed to converge; see IFAILL and IFAILR for further
*> details.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hsein
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Each eigenvector is normalized so that the element of largest
*> magnitude has magnitude 1; here the magnitude of a complex number
*> (x,y) is taken to be |x|+|y|.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR,
$ WI,
$ VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL,
$ IFAILR, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EIGSRC, INITV, SIDE
INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IFAILL( * ), IFAILR( * )
DOUBLE PRECISION H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WI( * ), WORK( * ), WR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, PAIR, RIGHTV
INTEGER I, IINFO, K, KL, KLN, KR, KSI, KSR, LDWORK
DOUBLE PRECISION BIGNUM, EPS3, HNORM, SMLNUM, ULP, UNFL, WKI,
$ WKR
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DLAMCH, DLANHS
EXTERNAL LSAME, DLAMCH, DLANHS, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLAEIN, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters.
*
BOTHV = LSAME( SIDE, 'B' )
RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
FROMQR = LSAME( EIGSRC, 'Q' )
*
NOINIT = LSAME( INITV, 'N' )
*
* Set M to the number of columns required to store the selected
* eigenvectors, and standardize the array SELECT.
*
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
SELECT( K ) = .FALSE.
ELSE
IF( WI( K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) ) THEN
SELECT( K ) = .TRUE.
M = M + 2
END IF
END IF
END IF
10 CONTINUE
*
INFO = 0
IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -1
ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
INFO = -13
ELSE IF( MM.LT.M ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DHSEIN', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* Set machine-dependent constants.
*
UNFL = DLAMCH( 'Safe minimum' )
ULP = DLAMCH( 'Precision' )
SMLNUM = UNFL*( N / ULP )
BIGNUM = ( ONE-ULP ) / SMLNUM
*
LDWORK = N + 1
*
KL = 1
KLN = 0
IF( FROMQR ) THEN
KR = 0
ELSE
KR = N
END IF
KSR = 1
*
DO 120 K = 1, N
IF( SELECT( K ) ) THEN
*
* Compute eigenvector(s) corresponding to W(K).
*
IF( FROMQR ) THEN
*
* If affiliation of eigenvalues is known, check whether
* the matrix splits.
*
* Determine KL and KR such that 1 <= KL <= K <= KR <= N
* and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or
* KR = N).
*
* Then inverse iteration can be performed with the
* submatrix H(KL:N,KL:N) for a left eigenvector, and with
* the submatrix H(1:KR,1:KR) for a right eigenvector.
*
DO 20 I = K, KL + 1, -1
IF( H( I, I-1 ).EQ.ZERO )
$ GO TO 30
20 CONTINUE
30 CONTINUE
KL = I
IF( K.GT.KR ) THEN
DO 40 I = K, N - 1
IF( H( I+1, I ).EQ.ZERO )
$ GO TO 50
40 CONTINUE
50 CONTINUE
KR = I
END IF
END IF
*
IF( KL.NE.KLN ) THEN
KLN = KL
*
* Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it
* has not ben computed before.
*
HNORM = DLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, WORK )
IF( DISNAN( HNORM ) ) THEN
INFO = -6
RETURN
ELSE IF( HNORM.GT.ZERO ) THEN
EPS3 = HNORM*ULP
ELSE
EPS3 = SMLNUM
END IF
END IF
*
* Perturb eigenvalue if it is close to any previous
* selected eigenvalues affiliated to the submatrix
* H(KL:KR,KL:KR). Close roots are modified by EPS3.
*
WKR = WR( K )
WKI = WI( K )
60 CONTINUE
DO 70 I = K - 1, KL, -1
IF( SELECT( I ) .AND. ABS( WR( I )-WKR )+
$ ABS( WI( I )-WKI ).LT.EPS3 ) THEN
WKR = WKR + EPS3
GO TO 60
END IF
70 CONTINUE
WR( K ) = WKR
*
PAIR = WKI.NE.ZERO
IF( PAIR ) THEN
KSI = KSR + 1
ELSE
KSI = KSR
END IF
IF( LEFTV ) THEN
*
* Compute left eigenvector.
*
CALL DLAEIN( .FALSE., NOINIT, N-KL+1, H( KL, KL ),
$ LDH,
$ WKR, WKI, VL( KL, KSR ), VL( KL, KSI ),
$ WORK, LDWORK, WORK( N*N+N+1 ), EPS3, SMLNUM,
$ BIGNUM, IINFO )
IF( IINFO.GT.0 ) THEN
IF( PAIR ) THEN
INFO = INFO + 2
ELSE
INFO = INFO + 1
END IF
IFAILL( KSR ) = K
IFAILL( KSI ) = K
ELSE
IFAILL( KSR ) = 0
IFAILL( KSI ) = 0
END IF
DO 80 I = 1, KL - 1
VL( I, KSR ) = ZERO
80 CONTINUE
IF( PAIR ) THEN
DO 90 I = 1, KL - 1
VL( I, KSI ) = ZERO
90 CONTINUE
END IF
END IF
IF( RIGHTV ) THEN
*
* Compute right eigenvector.
*
CALL DLAEIN( .TRUE., NOINIT, KR, H, LDH, WKR, WKI,
$ VR( 1, KSR ), VR( 1, KSI ), WORK, LDWORK,
$ WORK( N*N+N+1 ), EPS3, SMLNUM, BIGNUM,
$ IINFO )
IF( IINFO.GT.0 ) THEN
IF( PAIR ) THEN
INFO = INFO + 2
ELSE
INFO = INFO + 1
END IF
IFAILR( KSR ) = K
IFAILR( KSI ) = K
ELSE
IFAILR( KSR ) = 0
IFAILR( KSI ) = 0
END IF
DO 100 I = KR + 1, N
VR( I, KSR ) = ZERO
100 CONTINUE
IF( PAIR ) THEN
DO 110 I = KR + 1, N
VR( I, KSI ) = ZERO
110 CONTINUE
END IF
END IF
*
IF( PAIR ) THEN
KSR = KSR + 2
ELSE
KSR = KSR + 1
END IF
END IF
120 CONTINUE
*
RETURN
*
* End of DHSEIN
*
END
*> \brief \b DHSEQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DHSEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhseqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhseqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
* LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
* CHARACTER COMPZ, JOB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DHSEQR computes the eigenvalues of a Hessenberg matrix H
*> and, optionally, the matrices T and Z from the Schur decomposition
*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*> Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*> Optionally Z may be postmultiplied into an input orthogonal
*> matrix Q so that this routine can give the Schur factorization
*> of a matrix A which has been reduced to the Hessenberg form H
*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> = 'E': compute eigenvalues only;
*> = 'S': compute eigenvalues and the Schur form T.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': no Schur vectors are computed;
*> = 'I': Z is initialized to the unit matrix and the matrix Z
*> of Schur vectors of H is returned;
*> = 'V': Z must contain an orthogonal matrix Q on entry, and
*> the product Q*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*> set by a previous call to DGEBAL, and then passed to ZGEHRD
*> when the matrix output by DGEBAL is reduced to Hessenberg
*> form. Otherwise ILO and IHI should be set to 1 and N
*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO = 0 and JOB = 'S', then H contains the
*> upper quasi-triangular matrix T from the Schur decomposition
*> (the Schur form); 2-by-2 diagonal blocks (corresponding to
*> complex conjugate pairs of eigenvalues) are returned in
*> standard form, with H(i,i) = H(i+1,i+1) and
*> H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
*> contents of H are unspecified on exit. (The output value of
*> H when INFO > 0 is given under the description of INFO
*> below.)
*>
*> Unlike earlier versions of DHSEQR, this subroutine may
*> explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
*> or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*>
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues. If two eigenvalues are computed as a complex
*> conjugate pair, they are stored in consecutive elements of
*> WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and
*> WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in
*> the same order as on the diagonal of the Schur form returned
*> in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
*> diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*> WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> If COMPZ = 'N', Z is not referenced.
*> If COMPZ = 'I', on entry Z need not be set and on exit,
*> if INFO = 0, Z contains the orthogonal matrix Z of the Schur
*> vectors of H. If COMPZ = 'V', on entry Z must contain an
*> N-by-N matrix Q, which is assumed to be equal to the unit
*> matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
*> if INFO = 0, Z contains Q*Z.
*> Normally Q is the orthogonal matrix generated by DORGHR
*> after the call to DGEHRD which formed the Hessenberg matrix
*> H. (The output value of Z when INFO > 0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if COMPZ = 'I' or
*> COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns an estimate of
*> the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N)
*> is sufficient and delivers very good and sometimes
*> optimal performance. However, LWORK as large as 11*N
*> may be required for optimal performance. A workspace
*> query is recommended to determine the optimal workspace
*> size.
*>
*> If LWORK = -1, then DHSEQR does a workspace query.
*> In this case, DHSEQR checks the input parameters and
*> estimates the optimal workspace size for the given
*> values of N, ILO and IHI. The estimate is returned
*> in WORK(1). No error message related to LWORK is
*> issued by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal
*> value
*> > 0: if INFO = i, DHSEQR failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO > 0 and JOB = 'E', then on exit, the
*> remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO > 0 and JOB = 'S', then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
*> where U is an orthogonal matrix. The final
*> value of H is upper Hessenberg and quasi-triangular
*> in rows and columns INFO+1 through IHI.
*>
*> If INFO > 0 and COMPZ = 'V', then on exit
*>
*> (final value of Z) = (initial value of Z)*U
*>
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of JOB.)
*>
*> If INFO > 0 and COMPZ = 'I', then on exit
*> (final value of Z) = U
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of JOB.)
*>
*> If INFO > 0 and COMPZ = 'N', then Z is not
*> accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hseqr
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Default values supplied by
*> ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
*> It is suggested that these defaults be adjusted in order
*> to attain best performance in each particular
*> computational environment.
*>
*> ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
*> Default: 75. (Must be at least 11.)
*>
*> ISPEC=13: Recommended deflation window size.
*> This depends on ILO, IHI and NS. NS is the
*> number of simultaneous shifts returned
*> by ILAENV(ISPEC=15). (See ISPEC=15 below.)
*> The default for (IHI-ILO+1) <= 500 is NS.
*> The default for (IHI-ILO+1) > 500 is 3*NS/2.
*>
*> ISPEC=14: Nibble crossover point. (See IPARMQ for
*> details.) Default: 14% of deflation window
*> size.
*>
*> ISPEC=15: Number of simultaneous shifts in a multishift
*> QR iteration.
*>
*> If IHI-ILO+1 is ...
*>
*> greater than ...but less ... the
*> or equal to ... than default is
*>
*> 1 30 NS = 2(+)
*> 30 60 NS = 4(+)
*> 60 150 NS = 10(+)
*> 150 590 NS = **
*> 590 3000 NS = 64
*> 3000 6000 NS = 128
*> 6000 infinity NS = 256
*>
*> (+) By default some or all matrices of this order
*> are passed to the implicit double shift routine
*> DLAHQR and this parameter is ignored. See
*> ISPEC=12 above and comments in IPARMQ for
*> details.
*>
*> (**) The asterisks (**) indicate an ad-hoc
*> function of N increasing from 10 to 64.
*>
*> ISPEC=16: Select structured matrix multiply.
*> If the number of simultaneous shifts (specified
*> by ISPEC=15) is less than 14, then the default
*> for ISPEC=16 is 0. Otherwise the default for
*> ISPEC=16 is 2.
*> \endverbatim
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*> \n
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*> of Matrix Analysis, volume 23, pages 948--973, 2002.
*
* =====================================================================
SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
$ LDZ, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
CHARACTER COMPZ, JOB
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 15 )
*
* ==== NL allocates some local workspace to help small matrices
* . through a rare DLAHQR failure. NL > NTINY = 15 is
* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
* . mended. (The default value of NMIN is 75.) Using NL = 49
* . allows up to six simultaneous shifts and a 16-by-16
* . deflation window. ====
INTEGER NL
PARAMETER ( NL = 49 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Arrays ..
DOUBLE PRECISION HL( NL, NL ), WORKL( NL )
* ..
* .. Local Scalars ..
INTEGER I, KBOT, NMIN
LOGICAL INITZ, LQUERY, WANTT, WANTZ
* ..
* .. External Functions ..
INTEGER ILAENV
LOGICAL LSAME
EXTERNAL ILAENV, LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAHQR, DLAQR0, DLASET,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* ==== Decode and check the input parameters. ====
*
WANTT = LSAME( JOB, 'S' )
INITZ = LSAME( COMPZ, 'I' )
WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
WORK( 1 ) = DBLE( MAX( 1, N ) )
LQUERY = LWORK.EQ.-1
*
INFO = 0
IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -5
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.NE.0 ) THEN
*
* ==== Quick return in case of invalid argument. ====
*
CALL XERBLA( 'DHSEQR', -INFO )
RETURN
*
ELSE IF( N.EQ.0 ) THEN
*
* ==== Quick return in case N = 0; nothing to do. ====
*
RETURN
*
ELSE IF( LQUERY ) THEN
*
* ==== Quick return in case of a workspace query ====
*
CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
$ IHI, Z, LDZ, WORK, LWORK, INFO )
* ==== Ensure reported workspace size is backward-compatible with
* . previous LAPACK versions. ====
WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
RETURN
*
ELSE
*
* ==== copy eigenvalues isolated by DGEBAL ====
*
DO 10 I = 1, ILO - 1
WR( I ) = H( I, I )
WI( I ) = ZERO
10 CONTINUE
DO 20 I = IHI + 1, N
WR( I ) = H( I, I )
WI( I ) = ZERO
20 CONTINUE
*
* ==== Initialize Z, if requested ====
*
IF( INITZ )
$ CALL DLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
*
* ==== Quick return if possible ====
*
IF( ILO.EQ.IHI ) THEN
WR( ILO ) = H( ILO, ILO )
WI( ILO ) = ZERO
RETURN
END IF
*
* ==== DLAHQR/DLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'DHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
$ ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== DLAQR0 for big matrices; DLAHQR for small ones ====
*
IF( N.GT.NMIN ) THEN
CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILO,
$ IHI, Z, LDZ, WORK, LWORK, INFO )
ELSE
*
* ==== Small matrix ====
*
CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILO,
$ IHI, Z, LDZ, INFO )
*
IF( INFO.GT.0 ) THEN
*
* ==== A rare DLAHQR failure! DLAQR0 sometimes succeeds
* . when DLAHQR fails. ====
*
KBOT = INFO
*
IF( N.GE.NL ) THEN
*
* ==== Larger matrices have enough subdiagonal scratch
* . space to call DLAQR0 directly. ====
*
CALL DLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH,
$ WR,
$ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
*
ELSE
*
* ==== Tiny matrices don't have enough subdiagonal
* . scratch space to benefit from DLAQR0. Hence,
* . tiny matrices must be copied into a larger
* . array before calling DLAQR0. ====
*
CALL DLACPY( 'A', N, N, H, LDH, HL, NL )
HL( N+1, N ) = ZERO
CALL DLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1,
$ N+1 ),
$ NL )
CALL DLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL,
$ WR,
$ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
IF( WANTT .OR. INFO.NE.0 )
$ CALL DLACPY( 'A', N, N, HL, NL, H, LDH )
END IF
END IF
END IF
*
* ==== Clear out the trash, if necessary. ====
*
IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
$ CALL DLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
*
* ==== Ensure reported workspace size is backward-compatible with
* . previous LAPACK versions. ====
*
WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
END IF
*
* ==== End of DHSEQR ====
*
END
*> \brief \b DISNAN tests input for NaN.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DISNAN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/disnan.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/disnan.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/disnan.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* LOGICAL FUNCTION DISNAN( DIN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION, INTENT(IN) :: DIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DISNAN returns .TRUE. if its argument is NaN, and .FALSE.
*> otherwise. To be replaced by the Fortran 2003 intrinsic in the
*> future.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIN
*> \verbatim
*> DIN is DOUBLE PRECISION
*> Input to test for NaN.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup isnan
*
* =====================================================================
LOGICAL FUNCTION DISNAN( DIN )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION, INTENT(IN) :: DIN
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL DLAISNAN
EXTERNAL DLAISNAN
* ..
* .. Executable Statements ..
DISNAN = DLAISNAN(DIN,DIN)
RETURN
END
*> \brief \b DLABAD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLABAD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabad.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabad.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabad.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLABAD( SMALL, LARGE )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION LARGE, SMALL
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLABAD is a no-op and kept for compatibility reasons. It used
*> to correct the overflow/underflow behavior of machines that
*> are not IEEE-754 compliant.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] SMALL
*> \verbatim
*> SMALL is DOUBLE PRECISION
*> On entry, the underflow threshold as computed by DLAMCH.
*> On exit, the unchanged value SMALL.
*> \endverbatim
*>
*> \param[in,out] LARGE
*> \verbatim
*> LARGE is DOUBLE PRECISION
*> On entry, the overflow threshold as computed by DLAMCH.
*> On exit, the unchanged value LARGE.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup labad
*
* =====================================================================
SUBROUTINE DLABAD( SMALL, LARGE )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION LARGE, SMALL
* ..
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC LOG10, SQRT
* ..
* .. Executable Statements ..
*
* If it looks like we're on a Cray, take the square root of
* SMALL and LARGE to avoid overflow and underflow problems.
*
* IF( LOG10( LARGE ).GT.2000.D0 ) THEN
* SMALL = SQRT( SMALL )
* LARGE = SQRT( LARGE )
* END IF
*
RETURN
*
* End of DLABAD
*
END
*> \brief \b DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLABRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
* LDY )
*
* .. Scalar Arguments ..
* INTEGER LDA, LDX, LDY, M, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLABRD reduces the first NB rows and columns of a real general
*> m by n matrix A to upper or lower bidiagonal form by an orthogonal
*> transformation Q**T * A * P, and returns the matrices X and Y which
*> are needed to apply the transformation to the unreduced part of A.
*>
*> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
*> bidiagonal form.
*>
*> This is an auxiliary routine called by DGEBRD
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of leading rows and columns of A to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n general matrix to be reduced.
*> On exit, the first NB rows and columns of the matrix are
*> overwritten; the rest of the array is unchanged.
*> If m >= n, elements on and below the diagonal in the first NB
*> columns, with the array TAUQ, represent the orthogonal
*> matrix Q as a product of elementary reflectors; and
*> elements above the diagonal in the first NB rows, with the
*> array TAUP, represent the orthogonal matrix P as a product
*> of elementary reflectors.
*> If m < n, elements below the diagonal in the first NB
*> columns, with the array TAUQ, represent the orthogonal
*> matrix Q as a product of elementary reflectors, and
*> elements on and above the diagonal in the first NB rows,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (NB)
*> The diagonal elements of the first NB rows and columns of
*> the reduced matrix. D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (NB)
*> The off-diagonal elements of the first NB rows and columns of
*> the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is DOUBLE PRECISION array, dimension (NB)
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is DOUBLE PRECISION array, dimension (NB)
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NB)
*> The m-by-nb matrix X required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,M).
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (LDY,NB)
*> The n-by-nb matrix Y required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*> LDY is INTEGER
*> The leading dimension of the array Y. LDY >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup labrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors.
*>
*> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
*> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
*> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The elements of the vectors v and u together form the m-by-nb matrix
*> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
*> the transformation to the unreduced part of the matrix, using a block
*> update of the form: A := A - V*Y**T - X*U**T.
*>
*> The contents of A on exit are illustrated by the following examples
*> with nb = 2:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
*> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
*> ( v1 v2 a a a ) ( v1 1 a a a a )
*> ( v1 v2 a a a ) ( v1 v2 a a a a )
*> ( v1 v2 a a a ) ( v1 v2 a a a a )
*> ( v1 v2 a a a )
*>
*> where a denotes an element of the original matrix which is unchanged,
*> vi denotes an element of the vector defining H(i), and ui an element
*> of the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX,
$ Y,
$ LDY )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LDA, LDX, LDY, M, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), X( LDX, * ), Y( LDY, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DLARFG, DSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( M.GE.N ) THEN
*
* Reduce to upper bidiagonal form
*
DO 10 I = 1, NB
*
* Update A(i:m,i)
*
CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+1:m,i)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAUQ( I ) )
D( I ) = A( I, I )
IF( I.LT.N ) THEN
A( I, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
$ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ),
$ LDA,
$ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1,
$ 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ),
$ LDX,
$ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
$ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
*
* Update A(i,i+1:n)
*
CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
$ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
*
* Generate reflection P(i) to annihilate A(i,i+2:n)
*
CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
$ LDA, TAUP( I ) )
E( I ) = A( I, I+1 )
A( I, I+1 ) = ONE
*
* Compute X(i+1:m,i)
*
CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1,
$ I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ),
$ LDY,
$ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1,
$ I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1,
$ 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
END IF
10 CONTINUE
ELSE
*
* Reduce to lower bidiagonal form
*
DO 20 I = 1, NB
*
* Update A(i,i:n)
*
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ),
$ LDA,
$ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
*
* Generate reflection P(i) to annihilate A(i,i+1:n)
*
CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ),
$ LDA,
$ TAUP( I ) )
D( I ) = A( I, I )
IF( I.LT.M ) THEN
A( I, I ) = ONE
*
* Compute X(i+1:m,i)
*
CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1,
$ I ),
$ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ),
$ LDY,
$ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1,
$ 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1,
$ I ),
$ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1,
$ 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
*
* Update A(i+1:m,i)
*
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1,
$ 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+2:m,i)
*
CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ),
$ 1,
$ TAUQ( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
$ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ),
$ LDA,
$ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1,
$ 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ),
$ LDX,
$ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ),
$ LDA,
$ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
END IF
20 CONTINUE
END IF
RETURN
*
* End of DLABRD
*
END
*> \brief \b DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACN2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlacn2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlacn2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlacn2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLACN2( N, V, X, ISGN, EST, KASE, ISAVE )
*
* .. Scalar Arguments ..
* INTEGER KASE, N
* DOUBLE PRECISION EST
* ..
* .. Array Arguments ..
* INTEGER ISGN( * ), ISAVE( 3 )
* DOUBLE PRECISION V( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLACN2 estimates the 1-norm of a square, real matrix A.
*> Reverse communication is used for evaluating matrix-vector products.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 1.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (N)
*> On the final return, V = A*W, where EST = norm(V)/norm(W)
*> (W is not returned).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> On an intermediate return, X should be overwritten by
*> A * X, if KASE=1,
*> A**T * X, if KASE=2,
*> and DLACN2 must be re-called with all the other parameters
*> unchanged.
*> \endverbatim
*>
*> \param[out] ISGN
*> \verbatim
*> ISGN is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[in,out] EST
*> \verbatim
*> EST is DOUBLE PRECISION
*> On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
*> unchanged from the previous call to DLACN2.
*> On exit, EST is an estimate (a lower bound) for norm(A).
*> \endverbatim
*>
*> \param[in,out] KASE
*> \verbatim
*> KASE is INTEGER
*> On the initial call to DLACN2, KASE should be 0.
*> On an intermediate return, KASE will be 1 or 2, indicating
*> whether X should be overwritten by A * X or A**T * X.
*> On the final return from DLACN2, KASE will again be 0.
*> \endverbatim
*>
*> \param[in,out] ISAVE
*> \verbatim
*> ISAVE is INTEGER array, dimension (3)
*> ISAVE is used to save variables between calls to DLACN2
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lacn2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Originally named SONEST, dated March 16, 1988.
*>
*> This is a thread safe version of DLACON, which uses the array ISAVE
*> in place of a SAVE statement, as follows:
*>
*> DLACON DLACN2
*> JUMP ISAVE(1)
*> J ISAVE(2)
*> ITER ISAVE(3)
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Nick Higham, University of Manchester
*
*> \par References:
* ================
*>
*> N.J. Higham, "FORTRAN codes for estimating the one-norm of
*> a real or complex matrix, with applications to condition estimation",
*> ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
*>
* =====================================================================
SUBROUTINE DLACN2( N, V, X, ISGN, EST, KASE, ISAVE )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER KASE, N
DOUBLE PRECISION EST
* ..
* .. Array Arguments ..
INTEGER ISGN( * ), ISAVE( 3 )
DOUBLE PRECISION V( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, JLAST
DOUBLE PRECISION ALTSGN, ESTOLD, TEMP, XS
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DASUM
EXTERNAL IDAMAX, DASUM
* ..
* .. External Subroutines ..
EXTERNAL DCOPY
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, NINT
* ..
* .. Executable Statements ..
*
IF( KASE.EQ.0 ) THEN
DO 10 I = 1, N
X( I ) = ONE / DBLE( N )
10 CONTINUE
KASE = 1
ISAVE( 1 ) = 1
RETURN
END IF
*
GO TO ( 20, 40, 70, 110, 140 )ISAVE( 1 )
*
* ................ ENTRY (ISAVE( 1 ) = 1)
* FIRST ITERATION. X HAS BEEN OVERWRITTEN BY A*X.
*
20 CONTINUE
IF( N.EQ.1 ) THEN
V( 1 ) = X( 1 )
EST = ABS( V( 1 ) )
* ... QUIT
GO TO 150
END IF
EST = DASUM( N, X, 1 )
*
DO 30 I = 1, N
IF( X(I).GE.ZERO ) THEN
X(I) = ONE
ELSE
X(I) = -ONE
END IF
ISGN( I ) = NINT( X( I ) )
30 CONTINUE
KASE = 2
ISAVE( 1 ) = 2
RETURN
*
* ................ ENTRY (ISAVE( 1 ) = 2)
* FIRST ITERATION. X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
40 CONTINUE
ISAVE( 2 ) = IDAMAX( N, X, 1 )
ISAVE( 3 ) = 2
*
* MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
*
50 CONTINUE
DO 60 I = 1, N
X( I ) = ZERO
60 CONTINUE
X( ISAVE( 2 ) ) = ONE
KASE = 1
ISAVE( 1 ) = 3
RETURN
*
* ................ ENTRY (ISAVE( 1 ) = 3)
* X HAS BEEN OVERWRITTEN BY A*X.
*
70 CONTINUE
CALL DCOPY( N, X, 1, V, 1 )
ESTOLD = EST
EST = DASUM( N, V, 1 )
DO 80 I = 1, N
IF( X(I).GE.ZERO ) THEN
XS = ONE
ELSE
XS = -ONE
END IF
IF( NINT( XS ).NE.ISGN( I ) )
$ GO TO 90
80 CONTINUE
* REPEATED SIGN VECTOR DETECTED, HENCE ALGORITHM HAS CONVERGED.
GO TO 120
*
90 CONTINUE
* TEST FOR CYCLING.
IF( EST.LE.ESTOLD )
$ GO TO 120
*
DO 100 I = 1, N
IF( X(I).GE.ZERO ) THEN
X(I) = ONE
ELSE
X(I) = -ONE
END IF
ISGN( I ) = NINT( X( I ) )
100 CONTINUE
KASE = 2
ISAVE( 1 ) = 4
RETURN
*
* ................ ENTRY (ISAVE( 1 ) = 4)
* X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
110 CONTINUE
JLAST = ISAVE( 2 )
ISAVE( 2 ) = IDAMAX( N, X, 1 )
IF( ( X( JLAST ).NE.ABS( X( ISAVE( 2 ) ) ) ) .AND.
$ ( ISAVE( 3 ).LT.ITMAX ) ) THEN
ISAVE( 3 ) = ISAVE( 3 ) + 1
GO TO 50
END IF
*
* ITERATION COMPLETE. FINAL STAGE.
*
120 CONTINUE
ALTSGN = ONE
DO 130 I = 1, N
X( I ) = ALTSGN*( ONE+DBLE( I-1 ) / DBLE( N-1 ) )
ALTSGN = -ALTSGN
130 CONTINUE
KASE = 1
ISAVE( 1 ) = 5
RETURN
*
* ................ ENTRY (ISAVE( 1 ) = 5)
* X HAS BEEN OVERWRITTEN BY A*X.
*
140 CONTINUE
TEMP = TWO*( DASUM( N, X, 1 ) / DBLE( 3*N ) )
IF( TEMP.GT.EST ) THEN
CALL DCOPY( N, X, 1, V, 1 )
EST = TEMP
END IF
*
150 CONTINUE
KASE = 0
RETURN
*
* End of DLACN2
*
END
*> \brief \b DLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlacon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlacon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlacon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLACON( N, V, X, ISGN, EST, KASE )
*
* .. Scalar Arguments ..
* INTEGER KASE, N
* DOUBLE PRECISION EST
* ..
* .. Array Arguments ..
* INTEGER ISGN( * )
* DOUBLE PRECISION V( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLACON estimates the 1-norm of a square, real matrix A.
*> Reverse communication is used for evaluating matrix-vector products.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 1.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (N)
*> On the final return, V = A*W, where EST = norm(V)/norm(W)
*> (W is not returned).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> On an intermediate return, X should be overwritten by
*> A * X, if KASE=1,
*> A**T * X, if KASE=2,
*> and DLACON must be re-called with all the other parameters
*> unchanged.
*> \endverbatim
*>
*> \param[out] ISGN
*> \verbatim
*> ISGN is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[in,out] EST
*> \verbatim
*> EST is DOUBLE PRECISION
*> On entry with KASE = 1 or 2 and JUMP = 3, EST should be
*> unchanged from the previous call to DLACON.
*> On exit, EST is an estimate (a lower bound) for norm(A).
*> \endverbatim
*>
*> \param[in,out] KASE
*> \verbatim
*> KASE is INTEGER
*> On the initial call to DLACON, KASE should be 0.
*> On an intermediate return, KASE will be 1 or 2, indicating
*> whether X should be overwritten by A * X or A**T * X.
*> On the final return from DLACON, KASE will again be 0.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lacon
*
*> \par Contributors:
* ==================
*>
*> Nick Higham, University of Manchester. \n
*> Originally named SONEST, dated March 16, 1988.
*
*> \par References:
* ================
*>
*> N.J. Higham, "FORTRAN codes for estimating the one-norm of
*> a real or complex matrix, with applications to condition estimation",
*> ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
*>
* =====================================================================
SUBROUTINE DLACON( N, V, X, ISGN, EST, KASE )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER KASE, N
DOUBLE PRECISION EST
* ..
* .. Array Arguments ..
INTEGER ISGN( * )
DOUBLE PRECISION V( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, ITER, J, JLAST, JUMP
DOUBLE PRECISION ALTSGN, ESTOLD, TEMP
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DASUM
EXTERNAL IDAMAX, DASUM
* ..
* .. External Subroutines ..
EXTERNAL DCOPY
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, NINT, SIGN
* ..
* .. Save statement ..
SAVE
* ..
* .. Executable Statements ..
*
IF( KASE.EQ.0 ) THEN
DO 10 I = 1, N
X( I ) = ONE / DBLE( N )
10 CONTINUE
KASE = 1
JUMP = 1
RETURN
END IF
*
GO TO ( 20, 40, 70, 110, 140 )JUMP
*
* ................ ENTRY (JUMP = 1)
* FIRST ITERATION. X HAS BEEN OVERWRITTEN BY A*X.
*
20 CONTINUE
IF( N.EQ.1 ) THEN
V( 1 ) = X( 1 )
EST = ABS( V( 1 ) )
* ... QUIT
GO TO 150
END IF
EST = DASUM( N, X, 1 )
*
DO 30 I = 1, N
X( I ) = SIGN( ONE, X( I ) )
ISGN( I ) = NINT( X( I ) )
30 CONTINUE
KASE = 2
JUMP = 2
RETURN
*
* ................ ENTRY (JUMP = 2)
* FIRST ITERATION. X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
40 CONTINUE
J = IDAMAX( N, X, 1 )
ITER = 2
*
* MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
*
50 CONTINUE
DO 60 I = 1, N
X( I ) = ZERO
60 CONTINUE
X( J ) = ONE
KASE = 1
JUMP = 3
RETURN
*
* ................ ENTRY (JUMP = 3)
* X HAS BEEN OVERWRITTEN BY A*X.
*
70 CONTINUE
CALL DCOPY( N, X, 1, V, 1 )
ESTOLD = EST
EST = DASUM( N, V, 1 )
DO 80 I = 1, N
IF( NINT( SIGN( ONE, X( I ) ) ).NE.ISGN( I ) )
$ GO TO 90
80 CONTINUE
* REPEATED SIGN VECTOR DETECTED, HENCE ALGORITHM HAS CONVERGED.
GO TO 120
*
90 CONTINUE
* TEST FOR CYCLING.
IF( EST.LE.ESTOLD )
$ GO TO 120
*
DO 100 I = 1, N
X( I ) = SIGN( ONE, X( I ) )
ISGN( I ) = NINT( X( I ) )
100 CONTINUE
KASE = 2
JUMP = 4
RETURN
*
* ................ ENTRY (JUMP = 4)
* X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
110 CONTINUE
JLAST = J
J = IDAMAX( N, X, 1 )
IF( ( X( JLAST ).NE.ABS( X( J ) ) ) .AND. ( ITER.LT.ITMAX ) ) THEN
ITER = ITER + 1
GO TO 50
END IF
*
* ITERATION COMPLETE. FINAL STAGE.
*
120 CONTINUE
ALTSGN = ONE
DO 130 I = 1, N
X( I ) = ALTSGN*( ONE+DBLE( I-1 ) / DBLE( N-1 ) )
ALTSGN = -ALTSGN
130 CONTINUE
KASE = 1
JUMP = 5
RETURN
*
* ................ ENTRY (JUMP = 5)
* X HAS BEEN OVERWRITTEN BY A*X.
*
140 CONTINUE
TEMP = TWO*( DASUM( N, X, 1 ) / DBLE( 3*N ) )
IF( TEMP.GT.EST ) THEN
CALL DCOPY( N, X, 1, V, 1 )
EST = TEMP
END IF
*
150 CONTINUE
KASE = 0
RETURN
*
* End of DLACON
*
END
*> \brief \b DLACPY copies all or part of one two-dimensional array to another.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACPY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlacpy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlacpy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlacpy.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDB, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLACPY copies all or part of a two-dimensional matrix A to another
*> matrix B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies the part of the matrix A to be copied to B.
*> = 'U': Upper triangular part
*> = 'L': Lower triangular part
*> Otherwise: All of the matrix A
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A. If UPLO = 'U', only the upper triangle
*> or trapezoid is accessed; if UPLO = 'L', only the lower
*> triangle or trapezoid is accessed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On exit, B = A in the locations specified by UPLO.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lacpy
*
* =====================================================================
SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, MIN( J, M )
B( I, J ) = A( I, J )
10 CONTINUE
20 CONTINUE
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
DO 40 J = 1, N
DO 30 I = J, M
B( I, J ) = A( I, J )
30 CONTINUE
40 CONTINUE
ELSE
DO 60 J = 1, N
DO 50 I = 1, M
B( I, J ) = A( I, J )
50 CONTINUE
60 CONTINUE
END IF
RETURN
*
* End of DLACPY
*
END
*> \brief \b DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLADIV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dladiv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dladiv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dladiv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLADIV( A, B, C, D, P, Q )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, D, P, Q
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLADIV performs complex division in real arithmetic
*>
*> a + i*b
*> p + i*q = ---------
*> c + i*d
*>
*> The algorithm is due to Michael Baudin and Robert L. Smith
*> and can be found in the paper
*> "A Robust Complex Division in Scilab"
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION
*> The scalars a, b, c, and d in the above expression.
*> \endverbatim
*>
*> \param[out] P
*> \verbatim
*> P is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION
*> The scalars p and q in the above expression.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ladiv
*
* =====================================================================
SUBROUTINE DLADIV( A, B, C, D, P, Q )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, D, P, Q
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION BS
PARAMETER ( BS = 2.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
*
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, CC, DD, AB, CD, S, OV, UN, BE, EPS
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLADIV1
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
AA = A
BB = B
CC = C
DD = D
AB = MAX( ABS(A), ABS(B) )
CD = MAX( ABS(C), ABS(D) )
S = 1.0D0
OV = DLAMCH( 'Overflow threshold' )
UN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Epsilon' )
BE = BS / (EPS*EPS)
IF( AB >= HALF*OV ) THEN
AA = HALF * AA
BB = HALF * BB
S = TWO * S
END IF
IF( CD >= HALF*OV ) THEN
CC = HALF * CC
DD = HALF * DD
S = HALF * S
END IF
IF( AB <= UN*BS/EPS ) THEN
AA = AA * BE
BB = BB * BE
S = S / BE
END IF
IF( CD <= UN*BS/EPS ) THEN
CC = CC * BE
DD = DD * BE
S = S * BE
END IF
IF( ABS( D ).LE.ABS( C ) ) THEN
CALL DLADIV1(AA, BB, CC, DD, P, Q)
ELSE
CALL DLADIV1(BB, AA, DD, CC, P, Q)
Q = -Q
END IF
P = P * S
Q = Q * S
*
RETURN
*
* End of DLADIV
*
END
*> \ingroup ladiv
SUBROUTINE DLADIV1( A, B, C, D, P, Q )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, D, P, Q
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
*
* .. Local Scalars ..
DOUBLE PRECISION R, T
* ..
* .. External Functions ..
DOUBLE PRECISION DLADIV2
EXTERNAL DLADIV2
* ..
* .. Executable Statements ..
*
R = D / C
T = ONE / (C + D * R)
P = DLADIV2(A, B, C, D, R, T)
A = -A
Q = DLADIV2(B, A, C, D, R, T)
*
RETURN
*
* End of DLADIV1
*
END
*> \ingroup ladiv
DOUBLE PRECISION FUNCTION DLADIV2( A, B, C, D, R, T )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, D, R, T
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
*
* .. Local Scalars ..
DOUBLE PRECISION BR
* ..
* .. Executable Statements ..
*
IF( R.NE.ZERO ) THEN
BR = B * R
IF( BR.NE.ZERO ) THEN
DLADIV2 = (A + BR) * T
ELSE
DLADIV2 = A * T + (B * T) * R
END IF
ELSE
DLADIV2 = (A + D * (B / C)) * T
END IF
*
RETURN
*
* End of DLADIV2
*
END
*> \brief \b DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAE2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlae2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlae2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlae2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, RT1, RT2
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix
*> [ A B ]
*> [ B C ].
*> On return, RT1 is the eigenvalue of larger absolute value, and RT2
*> is the eigenvalue of smaller absolute value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION
*> The (1,2) and (2,1) elements of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*> RT1 is DOUBLE PRECISION
*> The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*> RT2 is DOUBLE PRECISION
*> The eigenvalue of smaller absolute value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lae2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> RT1 is accurate to a few ulps barring over/underflow.
*>
*> RT2 may be inaccurate if there is massive cancellation in the
*> determinant A*C-B*B; higher precision or correctly rounded or
*> correctly truncated arithmetic would be needed to compute RT2
*> accurately in all cases.
*>
*> Overflow is possible only if RT1 is within a factor of 5 of overflow.
*> Underflow is harmless if the input data is 0 or exceeds
*> underflow_threshold / macheps.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, RT1, RT2
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AB, ACMN, ACMX, ADF, DF, RT, SM, TB
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
* Compute the eigenvalues
*
SM = A + C
DF = A - C
ADF = ABS( DF )
TB = B + B
AB = ABS( TB )
IF( ABS( A ).GT.ABS( C ) ) THEN
ACMX = A
ACMN = C
ELSE
ACMX = C
ACMN = A
END IF
IF( ADF.GT.AB ) THEN
RT = ADF*SQRT( ONE+( AB / ADF )**2 )
ELSE IF( ADF.LT.AB ) THEN
RT = AB*SQRT( ONE+( ADF / AB )**2 )
ELSE
*
* Includes case AB=ADF=0
*
RT = AB*SQRT( TWO )
END IF
IF( SM.LT.ZERO ) THEN
RT1 = HALF*( SM-RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE IF( SM.GT.ZERO ) THEN
RT1 = HALF*( SM+RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE
*
* Includes case RT1 = RT2 = 0
*
RT1 = HALF*RT
RT2 = -HALF*RT
END IF
RETURN
*
* End of DLAE2
*
END
*> \brief \b DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEBZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaebz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaebz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaebz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL,
* RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT,
* NAB, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
* DOUBLE PRECISION ABSTOL, PIVMIN, RELTOL
* ..
* .. Array Arguments ..
* INTEGER IWORK( * ), NAB( MMAX, * ), NVAL( * )
* DOUBLE PRECISION AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEBZ contains the iteration loops which compute and use the
*> function N(w), which is the count of eigenvalues of a symmetric
*> tridiagonal matrix T less than or equal to its argument w. It
*> performs a choice of two types of loops:
*>
*> IJOB=1, followed by
*> IJOB=2: It takes as input a list of intervals and returns a list of
*> sufficiently small intervals whose union contains the same
*> eigenvalues as the union of the original intervals.
*> The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
*> The output interval (AB(j,1),AB(j,2)] will contain
*> eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.
*>
*> IJOB=3: It performs a binary search in each input interval
*> (AB(j,1),AB(j,2)] for a point w(j) such that
*> N(w(j))=NVAL(j), and uses C(j) as the starting point of
*> the search. If such a w(j) is found, then on output
*> AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output
*> (AB(j,1),AB(j,2)] will be a small interval containing the
*> point where N(w) jumps through NVAL(j), unless that point
*> lies outside the initial interval.
*>
*> Note that the intervals are in all cases half-open intervals,
*> i.e., of the form (a,b] , which includes b but not a .
*>
*> To avoid underflow, the matrix should be scaled so that its largest
*> element is no greater than overflow**(1/2) * underflow**(1/4)
*> in absolute value. To assure the most accurate computation
*> of small eigenvalues, the matrix should be scaled to be
*> not much smaller than that, either.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966
*>
*> Note: the arguments are, in general, *not* checked for unreasonable
*> values.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] IJOB
*> \verbatim
*> IJOB is INTEGER
*> Specifies what is to be done:
*> = 1: Compute NAB for the initial intervals.
*> = 2: Perform bisection iteration to find eigenvalues of T.
*> = 3: Perform bisection iteration to invert N(w), i.e.,
*> to find a point which has a specified number of
*> eigenvalues of T to its left.
*> Other values will cause DLAEBZ to return with INFO=-1.
*> \endverbatim
*>
*> \param[in] NITMAX
*> \verbatim
*> NITMAX is INTEGER
*> The maximum number of "levels" of bisection to be
*> performed, i.e., an interval of width W will not be made
*> smaller than 2^(-NITMAX) * W. If not all intervals
*> have converged after NITMAX iterations, then INFO is set
*> to the number of non-converged intervals.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension n of the tridiagonal matrix T. It must be at
*> least 1.
*> \endverbatim
*>
*> \param[in] MMAX
*> \verbatim
*> MMAX is INTEGER
*> The maximum number of intervals. If more than MMAX intervals
*> are generated, then DLAEBZ will quit with INFO=MMAX+1.
*> \endverbatim
*>
*> \param[in] MINP
*> \verbatim
*> MINP is INTEGER
*> The initial number of intervals. It may not be greater than
*> MMAX.
*> \endverbatim
*>
*> \param[in] NBMIN
*> \verbatim
*> NBMIN is INTEGER
*> The smallest number of intervals that should be processed
*> using a vector loop. If zero, then only the scalar loop
*> will be used.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The minimum (absolute) width of an interval. When an
*> interval is narrower than ABSTOL, or than RELTOL times the
*> larger (in magnitude) endpoint, then it is considered to be
*> sufficiently small, i.e., converged. This must be at least
*> zero.
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*> RELTOL is DOUBLE PRECISION
*> The minimum relative width of an interval. When an interval
*> is narrower than ABSTOL, or than RELTOL times the larger (in
*> magnitude) endpoint, then it is considered to be
*> sufficiently small, i.e., converged. Note: this should
*> always be at least radix*machine epsilon.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum absolute value of a "pivot" in the Sturm
*> sequence loop.
*> This must be at least max |e(j)**2|*safe_min and at
*> least safe_min, where safe_min is at least
*> the smallest number that can divide one without overflow.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> The offdiagonal elements of the tridiagonal matrix T in
*> positions 1 through N-1. E(N) is arbitrary.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*> E2 is DOUBLE PRECISION array, dimension (N)
*> The squares of the offdiagonal elements of the tridiagonal
*> matrix T. E2(N) is ignored.
*> \endverbatim
*>
*> \param[in,out] NVAL
*> \verbatim
*> NVAL is INTEGER array, dimension (MINP)
*> If IJOB=1 or 2, not referenced.
*> If IJOB=3, the desired values of N(w). The elements of NVAL
*> will be reordered to correspond with the intervals in AB.
*> Thus, NVAL(j) on output will not, in general be the same as
*> NVAL(j) on input, but it will correspond with the interval
*> (AB(j,1),AB(j,2)] on output.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (MMAX,2)
*> The endpoints of the intervals. AB(j,1) is a(j), the left
*> endpoint of the j-th interval, and AB(j,2) is b(j), the
*> right endpoint of the j-th interval. The input intervals
*> will, in general, be modified, split, and reordered by the
*> calculation.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (MMAX)
*> If IJOB=1, ignored.
*> If IJOB=2, workspace.
*> If IJOB=3, then on input C(j) should be initialized to the
*> first search point in the binary search.
*> \endverbatim
*>
*> \param[out] MOUT
*> \verbatim
*> MOUT is INTEGER
*> If IJOB=1, the number of eigenvalues in the intervals.
*> If IJOB=2 or 3, the number of intervals output.
*> If IJOB=3, MOUT will equal MINP.
*> \endverbatim
*>
*> \param[in,out] NAB
*> \verbatim
*> NAB is INTEGER array, dimension (MMAX,2)
*> If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).
*> If IJOB=2, then on input, NAB(i,j) should be set. It must
*> satisfy the condition:
*> N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)),
*> which means that in interval i only eigenvalues
*> NAB(i,1)+1,...,NAB(i,2) will be considered. Usually,
*> NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with
*> IJOB=1.
*> On output, NAB(i,j) will contain
*> max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of
*> the input interval that the output interval
*> (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the
*> the input values of NAB(k,1) and NAB(k,2).
*> If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)),
*> unless N(w) > NVAL(i) for all search points w , in which
*> case NAB(i,1) will not be modified, i.e., the output
*> value will be the same as the input value (modulo
*> reorderings -- see NVAL and AB), or unless N(w) < NVAL(i)
*> for all search points w , in which case NAB(i,2) will
*> not be modified. Normally, NAB should be set to some
*> distinctive value(s) before DLAEBZ is called.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MMAX)
*> Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MMAX)
*> Workspace.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: All intervals converged.
*> = 1--MMAX: The last INFO intervals did not converge.
*> = MMAX+1: More than MMAX intervals were generated.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laebz
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> This routine is intended to be called only by other LAPACK
*> routines, thus the interface is less user-friendly. It is intended
*> for two purposes:
*>
*> (a) finding eigenvalues. In this case, DLAEBZ should have one or
*> more initial intervals set up in AB, and DLAEBZ should be called
*> with IJOB=1. This sets up NAB, and also counts the eigenvalues.
*> Intervals with no eigenvalues would usually be thrown out at
*> this point. Also, if not all the eigenvalues in an interval i
*> are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
*> For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
*> eigenvalue. DLAEBZ is then called with IJOB=2 and MMAX
*> no smaller than the value of MOUT returned by the call with
*> IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1
*> through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
*> tolerance specified by ABSTOL and RELTOL.
*>
*> (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
*> In this case, start with a Gershgorin interval (a,b). Set up
*> AB to contain 2 search intervals, both initially (a,b). One
*> NVAL element should contain f-1 and the other should contain l
*> , while C should contain a and b, resp. NAB(i,1) should be -1
*> and NAB(i,2) should be N+1, to flag an error if the desired
*> interval does not lie in (a,b). DLAEBZ is then called with
*> IJOB=3. On exit, if w(f-1) < w(f), then one of the intervals --
*> j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
*> if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
*> >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and
*> N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and
*> w(l-r)=...=w(l+k) are handled similarly.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL,
$ RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT,
$ NAB, WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
DOUBLE PRECISION ABSTOL, PIVMIN, RELTOL
* ..
* .. Array Arguments ..
INTEGER IWORK( * ), NAB( MMAX, * ), NVAL( * )
DOUBLE PRECISION AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, TWO, HALF
PARAMETER ( ZERO = 0.0D0, TWO = 2.0D0,
$ HALF = 1.0D0 / TWO )
* ..
* .. Local Scalars ..
INTEGER ITMP1, ITMP2, J, JI, JIT, JP, KF, KFNEW, KL,
$ KLNEW
DOUBLE PRECISION TMP1, TMP2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Check for Errors
*
INFO = 0
IF( IJOB.LT.1 .OR. IJOB.GT.3 ) THEN
INFO = -1
RETURN
END IF
*
* Initialize NAB
*
IF( IJOB.EQ.1 ) THEN
*
* Compute the number of eigenvalues in the initial intervals.
*
MOUT = 0
DO 30 JI = 1, MINP
DO 20 JP = 1, 2
TMP1 = D( 1 ) - AB( JI, JP )
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
NAB( JI, JP ) = 0
IF( TMP1.LE.ZERO )
$ NAB( JI, JP ) = 1
*
DO 10 J = 2, N
TMP1 = D( J ) - E2( J-1 ) / TMP1 - AB( JI, JP )
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
IF( TMP1.LE.ZERO )
$ NAB( JI, JP ) = NAB( JI, JP ) + 1
10 CONTINUE
20 CONTINUE
MOUT = MOUT + NAB( JI, 2 ) - NAB( JI, 1 )
30 CONTINUE
RETURN
END IF
*
* Initialize for loop
*
* KF and KL have the following meaning:
* Intervals 1,...,KF-1 have converged.
* Intervals KF,...,KL still need to be refined.
*
KF = 1
KL = MINP
*
* If IJOB=2, initialize C.
* If IJOB=3, use the user-supplied starting point.
*
IF( IJOB.EQ.2 ) THEN
DO 40 JI = 1, MINP
C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
40 CONTINUE
END IF
*
* Iteration loop
*
DO 130 JIT = 1, NITMAX
*
* Loop over intervals
*
IF( KL-KF+1.GE.NBMIN .AND. NBMIN.GT.0 ) THEN
*
* Begin of Parallel Version of the loop
*
DO 60 JI = KF, KL
*
* Compute N(c), the number of eigenvalues less than c
*
WORK( JI ) = D( 1 ) - C( JI )
IWORK( JI ) = 0
IF( WORK( JI ).LE.PIVMIN ) THEN
IWORK( JI ) = 1
WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
END IF
*
DO 50 J = 2, N
WORK( JI ) = D( J ) - E2( J-1 ) / WORK( JI ) - C( JI )
IF( WORK( JI ).LE.PIVMIN ) THEN
IWORK( JI ) = IWORK( JI ) + 1
WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
END IF
50 CONTINUE
60 CONTINUE
*
IF( IJOB.LE.2 ) THEN
*
* IJOB=2: Choose all intervals containing eigenvalues.
*
KLNEW = KL
DO 70 JI = KF, KL
*
* Insure that N(w) is monotone
*
IWORK( JI ) = MIN( NAB( JI, 2 ),
$ MAX( NAB( JI, 1 ), IWORK( JI ) ) )
*
* Update the Queue -- add intervals if both halves
* contain eigenvalues.
*
IF( IWORK( JI ).EQ.NAB( JI, 2 ) ) THEN
*
* No eigenvalue in the upper interval:
* just use the lower interval.
*
AB( JI, 2 ) = C( JI )
*
ELSE IF( IWORK( JI ).EQ.NAB( JI, 1 ) ) THEN
*
* No eigenvalue in the lower interval:
* just use the upper interval.
*
AB( JI, 1 ) = C( JI )
ELSE
KLNEW = KLNEW + 1
IF( KLNEW.LE.MMAX ) THEN
*
* Eigenvalue in both intervals -- add upper to
* queue.
*
AB( KLNEW, 2 ) = AB( JI, 2 )
NAB( KLNEW, 2 ) = NAB( JI, 2 )
AB( KLNEW, 1 ) = C( JI )
NAB( KLNEW, 1 ) = IWORK( JI )
AB( JI, 2 ) = C( JI )
NAB( JI, 2 ) = IWORK( JI )
ELSE
INFO = MMAX + 1
END IF
END IF
70 CONTINUE
IF( INFO.NE.0 )
$ RETURN
KL = KLNEW
ELSE
*
* IJOB=3: Binary search. Keep only the interval containing
* w s.t. N(w) = NVAL
*
DO 80 JI = KF, KL
IF( IWORK( JI ).LE.NVAL( JI ) ) THEN
AB( JI, 1 ) = C( JI )
NAB( JI, 1 ) = IWORK( JI )
END IF
IF( IWORK( JI ).GE.NVAL( JI ) ) THEN
AB( JI, 2 ) = C( JI )
NAB( JI, 2 ) = IWORK( JI )
END IF
80 CONTINUE
END IF
*
ELSE
*
* End of Parallel Version of the loop
*
* Begin of Serial Version of the loop
*
KLNEW = KL
DO 100 JI = KF, KL
*
* Compute N(w), the number of eigenvalues less than w
*
TMP1 = C( JI )
TMP2 = D( 1 ) - TMP1
ITMP1 = 0
IF( TMP2.LE.PIVMIN ) THEN
ITMP1 = 1
TMP2 = MIN( TMP2, -PIVMIN )
END IF
*
DO 90 J = 2, N
TMP2 = D( J ) - E2( J-1 ) / TMP2 - TMP1
IF( TMP2.LE.PIVMIN ) THEN
ITMP1 = ITMP1 + 1
TMP2 = MIN( TMP2, -PIVMIN )
END IF
90 CONTINUE
*
IF( IJOB.LE.2 ) THEN
*
* IJOB=2: Choose all intervals containing eigenvalues.
*
* Insure that N(w) is monotone
*
ITMP1 = MIN( NAB( JI, 2 ),
$ MAX( NAB( JI, 1 ), ITMP1 ) )
*
* Update the Queue -- add intervals if both halves
* contain eigenvalues.
*
IF( ITMP1.EQ.NAB( JI, 2 ) ) THEN
*
* No eigenvalue in the upper interval:
* just use the lower interval.
*
AB( JI, 2 ) = TMP1
*
ELSE IF( ITMP1.EQ.NAB( JI, 1 ) ) THEN
*
* No eigenvalue in the lower interval:
* just use the upper interval.
*
AB( JI, 1 ) = TMP1
ELSE IF( KLNEW.LT.MMAX ) THEN
*
* Eigenvalue in both intervals -- add upper to queue.
*
KLNEW = KLNEW + 1
AB( KLNEW, 2 ) = AB( JI, 2 )
NAB( KLNEW, 2 ) = NAB( JI, 2 )
AB( KLNEW, 1 ) = TMP1
NAB( KLNEW, 1 ) = ITMP1
AB( JI, 2 ) = TMP1
NAB( JI, 2 ) = ITMP1
ELSE
INFO = MMAX + 1
RETURN
END IF
ELSE
*
* IJOB=3: Binary search. Keep only the interval
* containing w s.t. N(w) = NVAL
*
IF( ITMP1.LE.NVAL( JI ) ) THEN
AB( JI, 1 ) = TMP1
NAB( JI, 1 ) = ITMP1
END IF
IF( ITMP1.GE.NVAL( JI ) ) THEN
AB( JI, 2 ) = TMP1
NAB( JI, 2 ) = ITMP1
END IF
END IF
100 CONTINUE
KL = KLNEW
*
END IF
*
* Check for convergence
*
KFNEW = KF
DO 110 JI = KF, KL
TMP1 = ABS( AB( JI, 2 )-AB( JI, 1 ) )
TMP2 = MAX( ABS( AB( JI, 2 ) ), ABS( AB( JI, 1 ) ) )
IF( TMP1.LT.MAX( ABSTOL, PIVMIN, RELTOL*TMP2 ) .OR.
$ NAB( JI, 1 ).GE.NAB( JI, 2 ) ) THEN
*
* Converged -- Swap with position KFNEW,
* then increment KFNEW
*
IF( JI.GT.KFNEW ) THEN
TMP1 = AB( JI, 1 )
TMP2 = AB( JI, 2 )
ITMP1 = NAB( JI, 1 )
ITMP2 = NAB( JI, 2 )
AB( JI, 1 ) = AB( KFNEW, 1 )
AB( JI, 2 ) = AB( KFNEW, 2 )
NAB( JI, 1 ) = NAB( KFNEW, 1 )
NAB( JI, 2 ) = NAB( KFNEW, 2 )
AB( KFNEW, 1 ) = TMP1
AB( KFNEW, 2 ) = TMP2
NAB( KFNEW, 1 ) = ITMP1
NAB( KFNEW, 2 ) = ITMP2
IF( IJOB.EQ.3 ) THEN
ITMP1 = NVAL( JI )
NVAL( JI ) = NVAL( KFNEW )
NVAL( KFNEW ) = ITMP1
END IF
END IF
KFNEW = KFNEW + 1
END IF
110 CONTINUE
KF = KFNEW
*
* Choose Midpoints
*
DO 120 JI = KF, KL
C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
120 CONTINUE
*
* If no more intervals to refine, quit.
*
IF( KF.GT.KL )
$ GO TO 140
130 CONTINUE
*
* Converged
*
140 CONTINUE
INFO = MAX( KL+1-KF, 0 )
MOUT = KL
*
RETURN
*
* End of DLAEBZ
*
END
*> \brief \b DLAED0 used by DSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED0 computes all eigenvalues and corresponding eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> = 2: Compute eigenvalues and eigenvectors of tridiagonal
*> matrix.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the main diagonal of the tridiagonal matrix.
*> On exit, its eigenvalues.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, Q must contain an N-by-N orthogonal matrix.
*> If ICOMPQ = 0 Q is not referenced.
*> If ICOMPQ = 1 On entry, Q is a subset of the columns of the
*> orthogonal matrix used to reduce the full
*> matrix to tridiagonal form corresponding to
*> the subset of the full matrix which is being
*> decomposed at this time.
*> If ICOMPQ = 2 On entry, Q will be the identity matrix.
*> On exit, Q contains the eigenvectors of the
*> tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. If eigenvectors are
*> desired, then LDQ >= max(1,N). In any case, LDQ >= 1.
*> \endverbatim
*>
*> \param[out] QSTORE
*> \verbatim
*> QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
*> Referenced only when ICOMPQ = 1. Used to store parts of
*> the eigenvector matrix when the updating matrix multiplies
*> take place.
*> \endverbatim
*>
*> \param[in] LDQS
*> \verbatim
*> LDQS is INTEGER
*> The leading dimension of the array QSTORE. If ICOMPQ = 1,
*> then LDQS >= max(1,N). In any case, LDQS >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> If ICOMPQ = 0 or 1, the dimension of WORK must be at least
*> 1 + 3*N + 2*N*lg N + 3*N**2
*> ( lg( N ) = smallest integer k
*> such that 2^k >= N )
*> If ICOMPQ = 2, the dimension of WORK must be at least
*> 4*N + N**2.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array,
*> If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
*> 6 + 6*N + 5*N*lg N.
*> ( lg( N ) = smallest integer k
*> such that 2^k >= N )
*> If ICOMPQ = 2, the dimension of IWORK must be at least
*> 3 + 5*N.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute an eigenvalue while
*> working on the submatrix lying in rows and columns
*> INFO/(N+1) through mod(INFO,N+1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed0
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
$ WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.D0, ONE = 1.D0, TWO = 2.D0 )
* ..
* .. Local Scalars ..
INTEGER CURLVL, CURPRB, CURR, I, IGIVCL, IGIVNM,
$ IGIVPT, INDXQ, IPERM, IPRMPT, IQ, IQPTR, IWREM,
$ J, K, LGN, MATSIZ, MSD2, SMLSIZ, SMM1, SPM1,
$ SPM2, SUBMAT, SUBPBS, TLVLS
DOUBLE PRECISION TEMP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLAED1, DLAED7,
$ DSTEQR,
$ XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.2 ) THEN
INFO = -1
ELSE IF( ( ICOMPQ.EQ.1 ) .AND. ( QSIZ.LT.MAX( 0, N ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDQS.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED0', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
SMLSIZ = ILAENV( 9, 'DLAED0', ' ', 0, 0, 0, 0 )
*
* Determine the size and placement of the submatrices, and save in
* the leading elements of IWORK.
*
IWORK( 1 ) = N
SUBPBS = 1
TLVLS = 0
10 CONTINUE
IF( IWORK( SUBPBS ).GT.SMLSIZ ) THEN
DO 20 J = SUBPBS, 1, -1
IWORK( 2*J ) = ( IWORK( J )+1 ) / 2
IWORK( 2*J-1 ) = IWORK( J ) / 2
20 CONTINUE
TLVLS = TLVLS + 1
SUBPBS = 2*SUBPBS
GO TO 10
END IF
DO 30 J = 2, SUBPBS
IWORK( J ) = IWORK( J ) + IWORK( J-1 )
30 CONTINUE
*
* Divide the matrix into SUBPBS submatrices of size at most SMLSIZ+1
* using rank-1 modifications (cuts).
*
SPM1 = SUBPBS - 1
DO 40 I = 1, SPM1
SUBMAT = IWORK( I ) + 1
SMM1 = SUBMAT - 1
D( SMM1 ) = D( SMM1 ) - ABS( E( SMM1 ) )
D( SUBMAT ) = D( SUBMAT ) - ABS( E( SMM1 ) )
40 CONTINUE
*
INDXQ = 4*N + 3
IF( ICOMPQ.NE.2 ) THEN
*
* Set up workspaces for eigenvalues only/accumulate new vectors
* routine
*
TEMP = LOG( DBLE( N ) ) / LOG( TWO )
LGN = INT( TEMP )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IPRMPT = INDXQ + N + 1
IPERM = IPRMPT + N*LGN
IQPTR = IPERM + N*LGN
IGIVPT = IQPTR + N + 2
IGIVCL = IGIVPT + N*LGN
*
IGIVNM = 1
IQ = IGIVNM + 2*N*LGN
IWREM = IQ + N**2 + 1
*
* Initialize pointers
*
DO 50 I = 0, SUBPBS
IWORK( IPRMPT+I ) = 1
IWORK( IGIVPT+I ) = 1
50 CONTINUE
IWORK( IQPTR ) = 1
END IF
*
* Solve each submatrix eigenproblem at the bottom of the divide and
* conquer tree.
*
CURR = 0
DO 70 I = 0, SPM1
IF( I.EQ.0 ) THEN
SUBMAT = 1
MATSIZ = IWORK( 1 )
ELSE
SUBMAT = IWORK( I ) + 1
MATSIZ = IWORK( I+1 ) - IWORK( I )
END IF
IF( ICOMPQ.EQ.2 ) THEN
CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
$ Q( SUBMAT, SUBMAT ), LDQ, WORK, INFO )
IF( INFO.NE.0 )
$ GO TO 130
ELSE
CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
$ WORK( IQ-1+IWORK( IQPTR+CURR ) ), MATSIZ, WORK,
$ INFO )
IF( INFO.NE.0 )
$ GO TO 130
IF( ICOMPQ.EQ.1 ) THEN
CALL DGEMM( 'N', 'N', QSIZ, MATSIZ, MATSIZ, ONE,
$ Q( 1, SUBMAT ), LDQ, WORK( IQ-1+IWORK( IQPTR+
$ CURR ) ), MATSIZ, ZERO, QSTORE( 1, SUBMAT ),
$ LDQS )
END IF
IWORK( IQPTR+CURR+1 ) = IWORK( IQPTR+CURR ) + MATSIZ**2
CURR = CURR + 1
END IF
K = 1
DO 60 J = SUBMAT, IWORK( I+1 )
IWORK( INDXQ+J ) = K
K = K + 1
60 CONTINUE
70 CONTINUE
*
* Successively merge eigensystems of adjacent submatrices
* into eigensystem for the corresponding larger matrix.
*
* while ( SUBPBS > 1 )
*
CURLVL = 1
80 CONTINUE
IF( SUBPBS.GT.1 ) THEN
SPM2 = SUBPBS - 2
DO 90 I = 0, SPM2, 2
IF( I.EQ.0 ) THEN
SUBMAT = 1
MATSIZ = IWORK( 2 )
MSD2 = IWORK( 1 )
CURPRB = 0
ELSE
SUBMAT = IWORK( I ) + 1
MATSIZ = IWORK( I+2 ) - IWORK( I )
MSD2 = MATSIZ / 2
CURPRB = CURPRB + 1
END IF
*
* Merge lower order eigensystems (of size MSD2 and MATSIZ - MSD2)
* into an eigensystem of size MATSIZ.
* DLAED1 is used only for the full eigensystem of a tridiagonal
* matrix.
* DLAED7 handles the cases in which eigenvalues only or eigenvalues
* and eigenvectors of a full symmetric matrix (which was reduced to
* tridiagonal form) are desired.
*
IF( ICOMPQ.EQ.2 ) THEN
CALL DLAED1( MATSIZ, D( SUBMAT ), Q( SUBMAT, SUBMAT ),
$ LDQ, IWORK( INDXQ+SUBMAT ),
$ E( SUBMAT+MSD2-1 ), MSD2, WORK,
$ IWORK( SUBPBS+1 ), INFO )
ELSE
CALL DLAED7( ICOMPQ, MATSIZ, QSIZ, TLVLS, CURLVL,
$ CURPRB,
$ D( SUBMAT ), QSTORE( 1, SUBMAT ), LDQS,
$ IWORK( INDXQ+SUBMAT ), E( SUBMAT+MSD2-1 ),
$ MSD2, WORK( IQ ), IWORK( IQPTR ),
$ IWORK( IPRMPT ), IWORK( IPERM ),
$ IWORK( IGIVPT ), IWORK( IGIVCL ),
$ WORK( IGIVNM ), WORK( IWREM ),
$ IWORK( SUBPBS+1 ), INFO )
END IF
IF( INFO.NE.0 )
$ GO TO 130
IWORK( I / 2+1 ) = IWORK( I+2 )
90 CONTINUE
SUBPBS = SUBPBS / 2
CURLVL = CURLVL + 1
GO TO 80
END IF
*
* end while
*
* Re-merge the eigenvalues/vectors which were deflated at the final
* merge step.
*
IF( ICOMPQ.EQ.1 ) THEN
DO 100 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
CALL DCOPY( QSIZ, QSTORE( 1, J ), 1, Q( 1, I ), 1 )
100 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
ELSE IF( ICOMPQ.EQ.2 ) THEN
DO 110 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
CALL DCOPY( N, Q( 1, J ), 1, WORK( N*I+1 ), 1 )
110 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
CALL DLACPY( 'A', N, N, WORK( N+1 ), N, Q, LDQ )
ELSE
DO 120 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
120 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
END IF
GO TO 140
*
130 CONTINUE
INFO = SUBMAT*( N+1 ) + SUBMAT + MATSIZ - 1
*
140 CONTINUE
RETURN
*
* End of DLAED0
*
END
*> \brief \b DLAED1 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER CUTPNT, INFO, LDQ, N
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER INDXQ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED1 computes the updated eigensystem of a diagonal
*> matrix after modification by a rank-one symmetric matrix. This
*> routine is used only for the eigenproblem which requires all
*> eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles
*> the case in which eigenvalues only or eigenvalues and eigenvectors
*> of a full symmetric matrix (which was reduced to tridiagonal form)
*> are desired.
*>
*> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*> where Z = Q**T*u, u is a vector of length N with ones in the
*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*>
*> The eigenvectors of the original matrix are stored in Q, and the
*> eigenvalues are in D. The algorithm consists of three stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple eigenvalues or if there is a zero in
*> the Z vector. For each such occurrence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine DLAED2.
*>
*> The second stage consists of calculating the updated
*> eigenvalues. This is done by finding the roots of the secular
*> equation via the routine DLAED4 (as called by DLAED3).
*> This routine also calculates the eigenvectors of the current
*> problem.
*>
*> The final stage consists of computing the updated eigenvectors
*> directly using the updated eigenvalues. The eigenvectors for
*> the current problem are multiplied with the eigenvectors from
*> the overall problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the eigenvalues of the rank-1-perturbed matrix.
*> On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, the eigenvectors of the rank-1-perturbed matrix.
*> On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> On entry, the permutation which separately sorts the two
*> subproblems in D into ascending order.
*> On exit, the permutation which will reintegrate the
*> subproblems back into sorted order,
*> i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The subdiagonal entry used to create the rank-1 modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> The location of the last eigenvalue in the leading sub-matrix.
*> min(1,N) <= CUTPNT <= N/2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N + N**2)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed1
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK,
$ IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER CUTPNT, INFO, LDQ, N
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER INDXQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS,
$ IW, IZ, K, N1, N2, ZPP1
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAED2, DLAED3, DLAMRG,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED1', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* The following values are integer pointers which indicate
* the portion of the workspace
* used by a particular array in DLAED2 and DLAED3.
*
IZ = 1
IDLMDA = IZ + N
IW = IDLMDA + N
IQ2 = IW + N
*
INDX = 1
INDXC = INDX + N
COLTYP = INDXC + N
INDXP = COLTYP + N
*
*
* Form the z-vector which consists of the last row of Q_1 and the
* first row of Q_2.
*
CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
ZPP1 = CUTPNT + 1
CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ),
$ 1 )
*
* Deflate eigenvalues.
*
CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
$ WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
$ IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
$ IWORK( COLTYP ), INFO )
*
IF( INFO.NE.0 )
$ GO TO 20
*
* Solve Secular Equation.
*
IF( K.NE.0 ) THEN
IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
$ ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
$ WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
$ WORK( IW ), WORK( IS ), INFO )
IF( INFO.NE.0 )
$ GO TO 20
*
* Prepare the INDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
ELSE
DO 10 I = 1, N
INDXQ( I ) = I
10 CONTINUE
END IF
*
20 CONTINUE
RETURN
*
* End of DLAED1
*
END
*> \brief \b DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA, W,
* Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, N, N1
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
* $ INDXQ( * )
* DOUBLE PRECISION D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
* $ W( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED2 merges the two sets of eigenvalues together into a single
*> sorted set. Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur: when two or more
*> eigenvalues are close together or if there is a tiny entry in the
*> Z vector. For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> The number of non-deflated eigenvalues, and the order of the
*> related secular equation. 0 <= K <=N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The location of the last eigenvalue in the leading sub-matrix.
*> min(1,N) <= N1 <= N/2.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D contains the eigenvalues of the two submatrices to
*> be combined.
*> On exit, D contains the trailing (N-K) updated eigenvalues
*> (those which were deflated) sorted into increasing order.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, Q contains the eigenvectors of two submatrices in
*> the two square blocks with corners at (1,1), (N1,N1)
*> and (N1+1, N1+1), (N,N).
*> On exit, Q contains the trailing (N-K) updated eigenvectors
*> (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> The permutation which separately sorts the two sub-problems
*> in D into ascending order. Note that elements in the second
*> half of this permutation must first have N1 added to their
*> values. Destroyed on exit.
*> \endverbatim
*>
*> \param[in,out] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> On entry, the off-diagonal element associated with the rank-1
*> cut which originally split the two submatrices which are now
*> being recombined.
*> On exit, RHO has been modified to the value required by
*> DLAED3.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> On entry, Z contains the updating vector (the last
*> row of the first sub-eigenvector matrix and the first row of
*> the second sub-eigenvector matrix).
*> On exit, the contents of Z have been destroyed by the updating
*> process.
*> \endverbatim
*>
*> \param[out] DLAMBDA
*> \verbatim
*> DLAMBDA is DOUBLE PRECISION array, dimension (N)
*> A copy of the first K eigenvalues which will be used by
*> DLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first k values of the final deflation-altered z-vector
*> which will be passed to DLAED3.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*> Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
*> A copy of the first K eigenvectors which will be used by
*> DLAED3 in a matrix multiply (DGEMM) to solve for the new
*> eigenvectors.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to sort the contents of DLAMBDA into
*> ascending order.
*> \endverbatim
*>
*> \param[out] INDXC
*> \verbatim
*> INDXC is INTEGER array, dimension (N)
*> The permutation used to arrange the columns of the deflated
*> Q matrix into three groups: the first group contains non-zero
*> elements only at and above N1, the second contains
*> non-zero elements only below N1, and the third is dense.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*> INDXP is INTEGER array, dimension (N)
*> The permutation used to place deflated values of D at the end
*> of the array. INDXP(1:K) points to the nondeflated D-values
*> and INDXP(K+1:N) points to the deflated eigenvalues.
*> \endverbatim
*>
*> \param[out] COLTYP
*> \verbatim
*> COLTYP is INTEGER array, dimension (N)
*> During execution, a label which will indicate which of the
*> following types a column in the Q2 matrix is:
*> 1 : non-zero in the upper half only;
*> 2 : dense;
*> 3 : non-zero in the lower half only;
*> 4 : deflated.
*> On exit, COLTYP(i) is the number of columns of type i,
*> for i=1 to 4 only.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed2
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA,
$ W,
$ Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, N, N1
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
$ INDXQ( * )
DOUBLE PRECISION D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
$ W( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, EIGHT = 8.0D0 )
* ..
* .. Local Arrays ..
INTEGER CTOT( 4 ), PSM( 4 )
* ..
* .. Local Scalars ..
INTEGER CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
$ N2, NJ, PJ
DOUBLE PRECISION C, EPS, S, T, TAU, TOL
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL IDAMAX, DLAMCH, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
N2 = N - N1
N1P1 = N1 + 1
*
IF( RHO.LT.ZERO ) THEN
CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
END IF
*
* Normalize z so that norm(z) = 1. Since z is the concatenation of
* two normalized vectors, norm2(z) = sqrt(2).
*
T = ONE / SQRT( TWO )
CALL DSCAL( N, T, Z, 1 )
*
* RHO = ABS( norm(z)**2 * RHO )
*
RHO = ABS( TWO*RHO )
*
* Sort the eigenvalues into increasing order
*
DO 10 I = N1P1, N
INDXQ( I ) = INDXQ( I ) + N1
10 CONTINUE
*
* re-integrate the deflated parts from the last pass
*
DO 20 I = 1, N
DLAMBDA( I ) = D( INDXQ( I ) )
20 CONTINUE
CALL DLAMRG( N1, N2, DLAMBDA, 1, 1, INDXC )
DO 30 I = 1, N
INDX( I ) = INDXQ( INDXC( I ) )
30 CONTINUE
*
* Calculate the allowable deflation tolerance
*
IMAX = IDAMAX( N, Z, 1 )
JMAX = IDAMAX( N, D, 1 )
EPS = DLAMCH( 'Epsilon' )
TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
*
* If the rank-1 modifier is small enough, no more needs to be done
* except to reorganize Q so that its columns correspond with the
* elements in D.
*
IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
K = 0
IQ2 = 1
DO 40 J = 1, N
I = INDX( J )
CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
DLAMBDA( J ) = D( I )
IQ2 = IQ2 + N
40 CONTINUE
CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
CALL DCOPY( N, DLAMBDA, 1, D, 1 )
GO TO 190
END IF
*
* If there are multiple eigenvalues then the problem deflates. Here
* the number of equal eigenvalues are found. As each equal
* eigenvalue is found, an elementary reflector is computed to rotate
* the corresponding eigensubspace so that the corresponding
* components of Z are zero in this new basis.
*
DO 50 I = 1, N1
COLTYP( I ) = 1
50 CONTINUE
DO 60 I = N1P1, N
COLTYP( I ) = 3
60 CONTINUE
*
*
K = 0
K2 = N + 1
DO 70 J = 1, N
NJ = INDX( J )
IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
COLTYP( NJ ) = 4
INDXP( K2 ) = NJ
IF( J.EQ.N )
$ GO TO 100
ELSE
PJ = NJ
GO TO 80
END IF
70 CONTINUE
80 CONTINUE
J = J + 1
NJ = INDX( J )
IF( J.GT.N )
$ GO TO 100
IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
COLTYP( NJ ) = 4
INDXP( K2 ) = NJ
ELSE
*
* Check if eigenvalues are close enough to allow deflation.
*
S = Z( PJ )
C = Z( NJ )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = DLAPY2( C, S )
T = D( NJ ) - D( PJ )
C = C / TAU
S = -S / TAU
IF( ABS( T*C*S ).LE.TOL ) THEN
*
* Deflation is possible.
*
Z( NJ ) = TAU
Z( PJ ) = ZERO
IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
$ COLTYP( NJ ) = 2
COLTYP( PJ ) = 4
CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S )
T = D( PJ )*C**2 + D( NJ )*S**2
D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
D( PJ ) = T
K2 = K2 - 1
I = 1
90 CONTINUE
IF( K2+I.LE.N ) THEN
IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
INDXP( K2+I-1 ) = INDXP( K2+I )
INDXP( K2+I ) = PJ
I = I + 1
GO TO 90
ELSE
INDXP( K2+I-1 ) = PJ
END IF
ELSE
INDXP( K2+I-1 ) = PJ
END IF
PJ = NJ
ELSE
K = K + 1
DLAMBDA( K ) = D( PJ )
W( K ) = Z( PJ )
INDXP( K ) = PJ
PJ = NJ
END IF
END IF
GO TO 80
100 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
DLAMBDA( K ) = D( PJ )
W( K ) = Z( PJ )
INDXP( K ) = PJ
*
* Count up the total number of the various types of columns, then
* form a permutation which positions the four column types into
* four uniform groups (although one or more of these groups may be
* empty).
*
DO 110 J = 1, 4
CTOT( J ) = 0
110 CONTINUE
DO 120 J = 1, N
CT = COLTYP( J )
CTOT( CT ) = CTOT( CT ) + 1
120 CONTINUE
*
* PSM(*) = Position in SubMatrix (of types 1 through 4)
*
PSM( 1 ) = 1
PSM( 2 ) = 1 + CTOT( 1 )
PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
K = N - CTOT( 4 )
*
* Fill out the INDXC array so that the permutation which it induces
* will place all type-1 columns first, all type-2 columns next,
* then all type-3's, and finally all type-4's.
*
DO 130 J = 1, N
JS = INDXP( J )
CT = COLTYP( JS )
INDX( PSM( CT ) ) = JS
INDXC( PSM( CT ) ) = J
PSM( CT ) = PSM( CT ) + 1
130 CONTINUE
*
* Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
* and Q2 respectively. The eigenvalues/vectors which were not
* deflated go into the first K slots of DLAMBDA and Q2 respectively,
* while those which were deflated go into the last N - K slots.
*
I = 1
IQ1 = 1
IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1
DO 140 J = 1, CTOT( 1 )
JS = INDX( I )
CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
Z( I ) = D( JS )
I = I + 1
IQ1 = IQ1 + N1
140 CONTINUE
*
DO 150 J = 1, CTOT( 2 )
JS = INDX( I )
CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
Z( I ) = D( JS )
I = I + 1
IQ1 = IQ1 + N1
IQ2 = IQ2 + N2
150 CONTINUE
*
DO 160 J = 1, CTOT( 3 )
JS = INDX( I )
CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
Z( I ) = D( JS )
I = I + 1
IQ2 = IQ2 + N2
160 CONTINUE
*
IQ1 = IQ2
DO 170 J = 1, CTOT( 4 )
JS = INDX( I )
CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 )
IQ2 = IQ2 + N
Z( I ) = D( JS )
I = I + 1
170 CONTINUE
*
* The deflated eigenvalues and their corresponding vectors go back
* into the last N - K slots of D and Q respectively.
*
IF( K.LT.N ) THEN
CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N,
$ Q( 1, K+1 ), LDQ )
CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
END IF
*
* Copy CTOT into COLTYP for referencing in DLAED3.
*
DO 180 J = 1, 4
COLTYP( J ) = CTOT( J )
180 CONTINUE
*
190 CONTINUE
RETURN
*
* End of DLAED2
*
END
*> \brief \b DLAED3 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMBDA, Q2, INDX,
* CTOT, W, S, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, N, N1
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER CTOT( * ), INDX( * )
* DOUBLE PRECISION D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
* $ S( * ), W( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED3 finds the roots of the secular equation, as defined by the
*> values in D, W, and RHO, between 1 and K. It makes the
*> appropriate calls to DLAED4 and then updates the eigenvectors by
*> multiplying the matrix of eigenvectors of the pair of eigensystems
*> being combined by the matrix of eigenvectors of the K-by-K system
*> which is solved here.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved by
*> DLAED4. K >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the Q matrix.
*> N >= K (deflation may result in N>K).
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The location of the last eigenvalue in the leading submatrix.
*> min(1,N) <= N1 <= N/2.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> D(I) contains the updated eigenvalues for
*> 1 <= I <= K.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> Initially the first K columns are used as workspace.
*> On output the columns 1 to K contain
*> the updated eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The value of the parameter in the rank one update equation.
*> RHO >= 0 required.
*> \endverbatim
*>
*> \param[in] DLAMBDA
*> \verbatim
*> DLAMBDA is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[in] Q2
*> \verbatim
*> Q2 is DOUBLE PRECISION array, dimension (LDQ2*N)
*> The first K columns of this matrix contain the non-deflated
*> eigenvectors for the split problem.
*> \endverbatim
*>
*> \param[in] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to arrange the columns of the deflated
*> Q matrix into three groups (see DLAED2).
*> The rows of the eigenvectors found by DLAED4 must be likewise
*> permuted before the matrix multiply can take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*> CTOT is INTEGER array, dimension (4)
*> A count of the total number of the various types of columns
*> in Q, as described in INDX. The fourth column type is any
*> column which has been deflated.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating vector. Destroyed on
*> output.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N1 + 1)*K
*> Will contain the eigenvectors of the repaired matrix which
*> will be multiplied by the previously accumulated eigenvectors
*> to update the system.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed3
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMBDA, Q2, INDX,
$ CTOT, W, S, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, N, N1
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER CTOT( * ), INDX( * )
DOUBLE PRECISION D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
$ S( * ), W( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, II, IQ2, J, N12, N2, N23
DOUBLE PRECISION TEMP
* ..
* .. External Functions ..
DOUBLE PRECISION DNRM2
EXTERNAL DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLAED4, DLASET,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( K.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.K ) THEN
INFO = -2
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED3', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 )
$ RETURN
*
*
DO 20 J = 1, K
CALL DLAED4( K, J, DLAMBDA, W, Q( 1, J ), RHO, D( J ),
$ INFO )
*
* If the zero finder fails, the computation is terminated.
*
IF( INFO.NE.0 )
$ GO TO 120
20 CONTINUE
*
IF( K.EQ.1 )
$ GO TO 110
IF( K.EQ.2 ) THEN
DO 30 J = 1, K
W( 1 ) = Q( 1, J )
W( 2 ) = Q( 2, J )
II = INDX( 1 )
Q( 1, J ) = W( II )
II = INDX( 2 )
Q( 2, J ) = W( II )
30 CONTINUE
GO TO 110
END IF
*
* Compute updated W.
*
CALL DCOPY( K, W, 1, S, 1 )
*
* Initialize W(I) = Q(I,I)
*
CALL DCOPY( K, Q, LDQ+1, W, 1 )
DO 60 J = 1, K
DO 40 I = 1, J - 1
W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
40 CONTINUE
DO 50 I = J + 1, K
W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
50 CONTINUE
60 CONTINUE
DO 70 I = 1, K
W( I ) = SIGN( SQRT( -W( I ) ), S( I ) )
70 CONTINUE
*
* Compute eigenvectors of the modified rank-1 modification.
*
DO 100 J = 1, K
DO 80 I = 1, K
S( I ) = W( I ) / Q( I, J )
80 CONTINUE
TEMP = DNRM2( K, S, 1 )
DO 90 I = 1, K
II = INDX( I )
Q( I, J ) = S( II ) / TEMP
90 CONTINUE
100 CONTINUE
*
* Compute the updated eigenvectors.
*
110 CONTINUE
*
N2 = N - N1
N12 = CTOT( 1 ) + CTOT( 2 )
N23 = CTOT( 2 ) + CTOT( 3 )
*
CALL DLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 )
IQ2 = N1*N12 + 1
IF( N23.NE.0 ) THEN
CALL DGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S,
$ N23,
$ ZERO, Q( N1+1, 1 ), LDQ )
ELSE
CALL DLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ )
END IF
*
CALL DLACPY( 'A', N12, K, Q, LDQ, S, N12 )
IF( N12.NE.0 ) THEN
CALL DGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO,
$ Q,
$ LDQ )
ELSE
CALL DLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ )
END IF
*
*
120 CONTINUE
RETURN
*
* End of DLAED3
*
END
*> \brief \b DLAED4 used by DSTEDC. Finds a single root of the secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed4.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
*
* .. Scalar Arguments ..
* INTEGER I, INFO, N
* DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DELTA( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the I-th updated eigenvalue of a symmetric
*> rank-one modification to a diagonal matrix whose elements are
*> given in the array d, and that
*>
*> D(i) < D(j) for i < j
*>
*> and that RHO > 0. This is arranged by the calling routine, and is
*> no loss in generality. The rank-one modified system is thus
*>
*> diag( D ) + RHO * Z * Z_transpose.
*>
*> where we assume the Euclidean norm of Z is 1.
*>
*> The method consists of approximating the rational functions in the
*> secular equation by simpler interpolating rational functions.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of all arrays.
*> \endverbatim
*>
*> \param[in] I
*> \verbatim
*> I is INTEGER
*> The index of the eigenvalue to be computed. 1 <= I <= N.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The original eigenvalues. It is assumed that they are in
*> order, D(I) < D(J) for I < J.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension (N)
*> If N > 2, DELTA contains (D(j) - lambda_I) in its j-th
*> component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
*> for detail. The vector DELTA contains the information necessary
*> to construct the eigenvectors by DLAED3 and DLAED9.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DLAM
*> \verbatim
*> DLAM is DOUBLE PRECISION
*> The computed lambda_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = 1, the updating process failed.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> Logical variable ORGATI (origin-at-i?) is used for distinguishing
*> whether D(i) or D(i+1) is treated as the origin.
*>
*> ORGATI = .true. origin at i
*> ORGATI = .false. origin at i+1
*>
*> Logical variable SWTCH3 (switch-for-3-poles?) is for noting
*> if we are working with THREE poles!
*>
*> MAXIT is the maximum number of iterations allowed for each
*> eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed4
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I, INFO, N
DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DELTA( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0,
$ TEN = 10.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ORGATI, SWTCH, SWTCH3
INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER
DOUBLE PRECISION A, B, C, DEL, DLTLB, DLTUB, DPHI, DPSI, DW,
$ EPS, ERRETM, ETA, MIDPT, PHI, PREW, PSI,
$ RHOINV, TAU, TEMP, TEMP1, W
* ..
* .. Local Arrays ..
DOUBLE PRECISION ZZ( 3 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLAED5, DLAED6
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Since this routine is called in an inner loop, we do no argument
* checking.
*
* Quick return for N=1 and 2.
*
INFO = 0
IF( N.EQ.1 ) THEN
*
* Presumably, I=1 upon entry
*
DLAM = D( 1 ) + RHO*Z( 1 )*Z( 1 )
DELTA( 1 ) = ONE
RETURN
END IF
IF( N.EQ.2 ) THEN
CALL DLAED5( I, D, Z, DELTA, RHO, DLAM )
RETURN
END IF
*
* Compute machine epsilon
*
EPS = DLAMCH( 'Epsilon' )
RHOINV = ONE / RHO
*
* The case I = N
*
IF( I.EQ.N ) THEN
*
* Initialize some basic variables
*
II = N - 1
NITER = 1
*
* Calculate initial guess
*
MIDPT = RHO / TWO
*
* If ||Z||_2 is not one, then TEMP should be set to
* RHO * ||Z||_2^2 / TWO
*
DO 10 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
10 CONTINUE
*
PSI = ZERO
DO 20 J = 1, N - 2
PSI = PSI + Z( J )*Z( J ) / DELTA( J )
20 CONTINUE
*
C = RHOINV + PSI
W = C + Z( II )*Z( II ) / DELTA( II ) +
$ Z( N )*Z( N ) / DELTA( N )
*
IF( W.LE.ZERO ) THEN
TEMP = Z( N-1 )*Z( N-1 ) / ( D( N )-D( N-1 )+RHO ) +
$ Z( N )*Z( N ) / RHO
IF( C.LE.TEMP ) THEN
TAU = RHO
ELSE
DEL = D( N ) - D( N-1 )
A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DEL
IF( A.LT.ZERO ) THEN
TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
END IF
*
* It can be proved that
* D(N)+RHO/2 <= LAMBDA(N) < D(N)+TAU <= D(N)+RHO
*
DLTLB = MIDPT
DLTUB = RHO
ELSE
DEL = D( N ) - D( N-1 )
A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DEL
IF( A.LT.ZERO ) THEN
TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
*
* It can be proved that
* D(N) < D(N)+TAU < LAMBDA(N) < D(N)+RHO/2
*
DLTLB = ZERO
DLTUB = MIDPT
END IF
*
DO 30 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - TAU
30 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 40 J = 1, II
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
40 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / DELTA( N )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
$ ABS( TAU )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
DLAM = D( I ) + TAU
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
NITER = NITER + 1
C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
A = ( DELTA( N-1 )+DELTA( N ) )*W -
$ DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
B = DELTA( N-1 )*DELTA( N )*W
IF( C.LT.ZERO )
$ C = ABS( C )
IF( C.EQ.ZERO ) THEN
* ETA = B/A
* ETA = RHO - TAU
* ETA = DLTUB - TAU
*
* Update proposed by Li, Ren-Cang:
ETA = -W / ( DPSI+DPHI )
ELSE IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
DO 50 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
50 CONTINUE
*
TAU = TAU + ETA
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 60 J = 1, II
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
60 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / DELTA( N )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
$ ABS( TAU )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Main loop to update the values of the array DELTA
*
ITER = NITER + 1
*
DO 90 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
DLAM = D( I ) + TAU
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
A = ( DELTA( N-1 )+DELTA( N ) )*W -
$ DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
B = DELTA( N-1 )*DELTA( N )*W
IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
DO 70 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
70 CONTINUE
*
TAU = TAU + ETA
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 80 J = 1, II
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
80 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / DELTA( N )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
$ ABS( TAU )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
90 CONTINUE
*
* Return with INFO = 1, NITER = MAXIT and not converged
*
INFO = 1
DLAM = D( I ) + TAU
GO TO 250
*
* End for the case I = N
*
ELSE
*
* The case for I < N
*
NITER = 1
IP1 = I + 1
*
* Calculate initial guess
*
DEL = D( IP1 ) - D( I )
MIDPT = DEL / TWO
DO 100 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
100 CONTINUE
*
PSI = ZERO
DO 110 J = 1, I - 1
PSI = PSI + Z( J )*Z( J ) / DELTA( J )
110 CONTINUE
*
PHI = ZERO
DO 120 J = N, I + 2, -1
PHI = PHI + Z( J )*Z( J ) / DELTA( J )
120 CONTINUE
C = RHOINV + PSI + PHI
W = C + Z( I )*Z( I ) / DELTA( I ) +
$ Z( IP1 )*Z( IP1 ) / DELTA( IP1 )
*
IF( W.GT.ZERO ) THEN
*
* d(i)< the ith eigenvalue < (d(i)+d(i+1))/2
*
* We choose d(i) as origin.
*
ORGATI = .TRUE.
A = C*DEL + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
B = Z( I )*Z( I )*DEL
IF( A.GT.ZERO ) THEN
TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
ELSE
TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
END IF
DLTLB = ZERO
DLTUB = MIDPT
ELSE
*
* (d(i)+d(i+1))/2 <= the ith eigenvalue < d(i+1)
*
* We choose d(i+1) as origin.
*
ORGATI = .FALSE.
A = C*DEL - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
B = Z( IP1 )*Z( IP1 )*DEL
IF( A.LT.ZERO ) THEN
TAU = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
ELSE
TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
END IF
DLTLB = -MIDPT
DLTUB = ZERO
END IF
*
IF( ORGATI ) THEN
DO 130 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - TAU
130 CONTINUE
ELSE
DO 140 J = 1, N
DELTA( J ) = ( D( J )-D( IP1 ) ) - TAU
140 CONTINUE
END IF
IF( ORGATI ) THEN
II = I
ELSE
II = I + 1
END IF
IIM1 = II - 1
IIP1 = II + 1
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 150 J = 1, IIM1
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
150 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 160 J = N, IIP1, -1
TEMP = Z( J ) / DELTA( J )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
160 CONTINUE
*
W = RHOINV + PHI + PSI
*
* W is the value of the secular function with
* its ii-th element removed.
*
SWTCH3 = .FALSE.
IF( ORGATI ) THEN
IF( W.LT.ZERO )
$ SWTCH3 = .TRUE.
ELSE
IF( W.GT.ZERO )
$ SWTCH3 = .TRUE.
END IF
IF( II.EQ.1 .OR. II.EQ.N )
$ SWTCH3 = .FALSE.
*
TEMP = Z( II ) / DELTA( II )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = W + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
$ THREE*ABS( TEMP ) + ABS( TAU )*DW
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
IF( ORGATI ) THEN
DLAM = D( I ) + TAU
ELSE
DLAM = D( IP1 ) + TAU
END IF
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
NITER = NITER + 1
IF( .NOT.SWTCH3 ) THEN
IF( ORGATI ) THEN
C = W - DELTA( IP1 )*DW - ( D( I )-D( IP1 ) )*
$ ( Z( I ) / DELTA( I ) )**2
ELSE
C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
$ ( Z( IP1 ) / DELTA( IP1 ) )**2
END IF
A = ( DELTA( I )+DELTA( IP1 ) )*W -
$ DELTA( I )*DELTA( IP1 )*DW
B = DELTA( I )*DELTA( IP1 )*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DELTA( IP1 )*DELTA( IP1 )*
$ ( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) + DELTA( I )*DELTA( I )*
$ ( DPSI+DPHI )
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
TEMP = RHOINV + PSI + PHI
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
$ ( D( IIM1 )-D( IIP1 ) )*TEMP1
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
$ ( ( DPSI-TEMP1 )+DPHI )
ELSE
TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
$ ( D( IIP1 )-D( IIM1 ) )*TEMP1
ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
$ ( DPSI+( DPHI-TEMP1 ) )
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
ZZ( 2 ) = Z( II )*Z( II )
CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA,
$ INFO )
IF( INFO.NE.0 )
$ GO TO 250
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
*
PREW = W
*
DO 180 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
180 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 190 J = 1, IIM1
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
190 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 200 J = N, IIP1, -1
TEMP = Z( J ) / DELTA( J )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
200 CONTINUE
*
TEMP = Z( II ) / DELTA( II )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = RHOINV + PHI + PSI + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
$ THREE*ABS( TEMP ) + ABS( TAU+ETA )*DW
*
SWTCH = .FALSE.
IF( ORGATI ) THEN
IF( -W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
ELSE
IF( W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
END IF
*
TAU = TAU + ETA
*
* Main loop to update the values of the array DELTA
*
ITER = NITER + 1
*
DO 240 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
IF( ORGATI ) THEN
DLAM = D( I ) + TAU
ELSE
DLAM = D( IP1 ) + TAU
END IF
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
IF( .NOT.SWTCH3 ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
C = W - DELTA( IP1 )*DW -
$ ( D( I )-D( IP1 ) )*( Z( I ) / DELTA( I ) )**2
ELSE
C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
$ ( Z( IP1 ) / DELTA( IP1 ) )**2
END IF
ELSE
TEMP = Z( II ) / DELTA( II )
IF( ORGATI ) THEN
DPSI = DPSI + TEMP*TEMP
ELSE
DPHI = DPHI + TEMP*TEMP
END IF
C = W - DELTA( I )*DPSI - DELTA( IP1 )*DPHI
END IF
A = ( DELTA( I )+DELTA( IP1 ) )*W -
$ DELTA( I )*DELTA( IP1 )*DW
B = DELTA( I )*DELTA( IP1 )*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DELTA( IP1 )*
$ DELTA( IP1 )*( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) +
$ DELTA( I )*DELTA( I )*( DPSI+DPHI )
END IF
ELSE
A = DELTA( I )*DELTA( I )*DPSI +
$ DELTA( IP1 )*DELTA( IP1 )*DPHI
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
TEMP = RHOINV + PSI + PHI
IF( SWTCH ) THEN
C = TEMP - DELTA( IIM1 )*DPSI - DELTA( IIP1 )*DPHI
ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*DPSI
ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*DPHI
ELSE
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
$ ( D( IIM1 )-D( IIP1 ) )*TEMP1
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
$ ( ( DPSI-TEMP1 )+DPHI )
ELSE
TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
$ ( D( IIP1 )-D( IIM1 ) )*TEMP1
ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
$ ( DPSI+( DPHI-TEMP1 ) )
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
END IF
CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W,
$ ETA,
$ INFO )
IF( INFO.NE.0 )
$ GO TO 250
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
*
DO 210 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
210 CONTINUE
*
TAU = TAU + ETA
PREW = W
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 220 J = 1, IIM1
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
220 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 230 J = N, IIP1, -1
TEMP = Z( J ) / DELTA( J )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
230 CONTINUE
*
TEMP = Z( II ) / DELTA( II )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = RHOINV + PHI + PSI + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
$ THREE*ABS( TEMP ) + ABS( TAU )*DW
IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN )
$ SWTCH = .NOT.SWTCH
*
240 CONTINUE
*
* Return with INFO = 1, NITER = MAXIT and not converged
*
INFO = 1
IF( ORGATI ) THEN
DLAM = D( I ) + TAU
ELSE
DLAM = D( IP1 ) + TAU
END IF
*
END IF
*
250 CONTINUE
*
RETURN
*
* End of DLAED4
*
END
*> \brief \b DLAED5 used by DSTEDC. Solves the 2-by-2 secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
* .. Scalar Arguments ..
* INTEGER I
* DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( 2 ), DELTA( 2 ), Z( 2 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the I-th eigenvalue of a symmetric rank-one
*> modification of a 2-by-2 diagonal matrix
*>
*> diag( D ) + RHO * Z * transpose(Z) .
*>
*> The diagonal elements in the array D are assumed to satisfy
*>
*> D(i) < D(j) for i < j .
*>
*> We also assume RHO > 0 and that the Euclidean norm of the vector
*> Z is one.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I
*> \verbatim
*> I is INTEGER
*> The index of the eigenvalue to be computed. I = 1 or I = 2.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (2)
*> The original eigenvalues. We assume D(1) < D(2).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (2)
*> The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension (2)
*> The vector DELTA contains the information necessary
*> to construct the eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DLAM
*> \verbatim
*> DLAM is DOUBLE PRECISION
*> The computed lambda_I, the I-th updated eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed5
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I
DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( 2 ), DELTA( 2 ), Z( 2 )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, FOUR
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION B, C, DEL, TAU, TEMP, W
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
DEL = D( 2 ) - D( 1 )
IF( I.EQ.1 ) THEN
W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
IF( W.GT.ZERO ) THEN
B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 1 )*Z( 1 )*DEL
*
* B > ZERO, always
*
TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
DLAM = D( 1 ) + TAU
DELTA( 1 ) = -Z( 1 ) / TAU
DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
ELSE
B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DEL
IF( B.GT.ZERO ) THEN
TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
ELSE
TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
END IF
DLAM = D( 2 ) + TAU
DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
DELTA( 2 ) = -Z( 2 ) / TAU
END IF
TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
DELTA( 1 ) = DELTA( 1 ) / TEMP
DELTA( 2 ) = DELTA( 2 ) / TEMP
ELSE
*
* Now I=2
*
B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DEL
IF( B.GT.ZERO ) THEN
TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
ELSE
TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
END IF
DLAM = D( 2 ) + TAU
DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
DELTA( 2 ) = -Z( 2 ) / TAU
TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
DELTA( 1 ) = DELTA( 1 ) / TEMP
DELTA( 2 ) = DELTA( 2 ) / TEMP
END IF
RETURN
*
* End of DLAED5
*
END
*> \brief \b DLAED6 used by DSTEDC. Computes one Newton step in solution of the secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed6.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
* .. Scalar Arguments ..
* LOGICAL ORGATI
* INTEGER INFO, KNITER
* DOUBLE PRECISION FINIT, RHO, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( 3 ), Z( 3 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED6 computes the positive or negative root (closest to the origin)
*> of
*> z(1) z(2) z(3)
*> f(x) = rho + --------- + ---------- + ---------
*> d(1)-x d(2)-x d(3)-x
*>
*> It is assumed that
*>
*> if ORGATI = .true. the root is between d(2) and d(3);
*> otherwise it is between d(1) and d(2)
*>
*> This routine will be called by DLAED4 when necessary. In most cases,
*> the root sought is the smallest in magnitude, though it might not be
*> in some extremely rare situations.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] KNITER
*> \verbatim
*> KNITER is INTEGER
*> Refer to DLAED4 for its significance.
*> \endverbatim
*>
*> \param[in] ORGATI
*> \verbatim
*> ORGATI is LOGICAL
*> If ORGATI is true, the needed root is between d(2) and
*> d(3); otherwise it is between d(1) and d(2). See
*> DLAED4 for further details.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> Refer to the equation f(x) above.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (3)
*> D satisfies d(1) < d(2) < d(3).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (3)
*> Each of the elements in z must be positive.
*> \endverbatim
*>
*> \param[in] FINIT
*> \verbatim
*> FINIT is DOUBLE PRECISION
*> The value of f at 0. It is more accurate than the one
*> evaluated inside this routine (if someone wants to do
*> so).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The root of the equation f(x).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = 1, failure to converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed6
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 10/02/03: This version has a few statements commented out for thread
*> safety (machine parameters are computed on each entry). SJH.
*>
*> 05/10/06: Modified from a new version of Ren-Cang Li, use
*> Gragg-Thornton-Warner cubic convergent scheme for better stability.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL ORGATI
INTEGER INFO, KNITER
DOUBLE PRECISION FINIT, RHO, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( 3 ), Z( 3 )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 40 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Local Arrays ..
DOUBLE PRECISION DSCALE( 3 ), ZSCALE( 3 )
* ..
* .. Local Scalars ..
LOGICAL SCALE
INTEGER I, ITER, NITER
DOUBLE PRECISION A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F,
$ FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1,
$ SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4,
$ LBD, UBD
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
IF( ORGATI ) THEN
LBD = D(2)
UBD = D(3)
ELSE
LBD = D(1)
UBD = D(2)
END IF
IF( FINIT .LT. ZERO )THEN
LBD = ZERO
ELSE
UBD = ZERO
END IF
*
NITER = 1
TAU = ZERO
IF( KNITER.EQ.2 ) THEN
IF( ORGATI ) THEN
TEMP = ( D( 3 )-D( 2 ) ) / TWO
C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP )
A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 )
B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 )
ELSE
TEMP = ( D( 1 )-D( 2 ) ) / TWO
C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP )
A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 )
B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 )
END IF
TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
A = A / TEMP
B = B / TEMP
C = C / TEMP
IF( C.EQ.ZERO ) THEN
TAU = B / A
ELSE IF( A.LE.ZERO ) THEN
TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
IF( TAU .LT. LBD .OR. TAU .GT. UBD )
$ TAU = ( LBD+UBD )/TWO
IF( D(1).EQ.TAU .OR. D(2).EQ.TAU .OR. D(3).EQ.TAU ) THEN
TAU = ZERO
ELSE
TEMP = FINIT + TAU*Z(1)/( D(1)*( D( 1 )-TAU ) ) +
$ TAU*Z(2)/( D(2)*( D( 2 )-TAU ) ) +
$ TAU*Z(3)/( D(3)*( D( 3 )-TAU ) )
IF( TEMP .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
IF( ABS( FINIT ).LE.ABS( TEMP ) )
$ TAU = ZERO
END IF
END IF
*
* get machine parameters for possible scaling to avoid overflow
*
* modified by Sven: parameters SMALL1, SMINV1, SMALL2,
* SMINV2, EPS are not SAVEd anymore between one call to the
* others but recomputed at each call
*
EPS = DLAMCH( 'Epsilon' )
BASE = DLAMCH( 'Base' )
SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) /
$ THREE ) )
SMINV1 = ONE / SMALL1
SMALL2 = SMALL1*SMALL1
SMINV2 = SMINV1*SMINV1
*
* Determine if scaling of inputs necessary to avoid overflow
* when computing 1/TEMP**3
*
IF( ORGATI ) THEN
TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) )
ELSE
TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) )
END IF
SCALE = .FALSE.
IF( TEMP.LE.SMALL1 ) THEN
SCALE = .TRUE.
IF( TEMP.LE.SMALL2 ) THEN
*
* Scale up by power of radix nearest 1/SAFMIN**(2/3)
*
SCLFAC = SMINV2
SCLINV = SMALL2
ELSE
*
* Scale up by power of radix nearest 1/SAFMIN**(1/3)
*
SCLFAC = SMINV1
SCLINV = SMALL1
END IF
*
* Scaling up safe because D, Z, TAU scaled elsewhere to be O(1)
*
DO 10 I = 1, 3
DSCALE( I ) = D( I )*SCLFAC
ZSCALE( I ) = Z( I )*SCLFAC
10 CONTINUE
TAU = TAU*SCLFAC
LBD = LBD*SCLFAC
UBD = UBD*SCLFAC
ELSE
*
* Copy D and Z to DSCALE and ZSCALE
*
DO 20 I = 1, 3
DSCALE( I ) = D( I )
ZSCALE( I ) = Z( I )
20 CONTINUE
END IF
*
FC = ZERO
DF = ZERO
DDF = ZERO
DO 30 I = 1, 3
TEMP = ONE / ( DSCALE( I )-TAU )
TEMP1 = ZSCALE( I )*TEMP
TEMP2 = TEMP1*TEMP
TEMP3 = TEMP2*TEMP
FC = FC + TEMP1 / DSCALE( I )
DF = DF + TEMP2
DDF = DDF + TEMP3
30 CONTINUE
F = FINIT + TAU*FC
*
IF( ABS( F ).LE.ZERO )
$ GO TO 60
IF( F .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
*
* Iteration begins -- Use Gragg-Thornton-Warner cubic convergent
* scheme
*
* It is not hard to see that
*
* 1) Iterations will go up monotonically
* if FINIT < 0;
*
* 2) Iterations will go down monotonically
* if FINIT > 0.
*
ITER = NITER + 1
*
DO 50 NITER = ITER, MAXIT
*
IF( ORGATI ) THEN
TEMP1 = DSCALE( 2 ) - TAU
TEMP2 = DSCALE( 3 ) - TAU
ELSE
TEMP1 = DSCALE( 1 ) - TAU
TEMP2 = DSCALE( 2 ) - TAU
END IF
A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF
B = TEMP1*TEMP2*F
C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF
TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
A = A / TEMP
B = B / TEMP
C = C / TEMP
IF( C.EQ.ZERO ) THEN
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
IF( F*ETA.GE.ZERO ) THEN
ETA = -F / DF
END IF
*
TAU = TAU + ETA
IF( TAU .LT. LBD .OR. TAU .GT. UBD )
$ TAU = ( LBD + UBD )/TWO
*
FC = ZERO
ERRETM = ZERO
DF = ZERO
DDF = ZERO
DO 40 I = 1, 3
IF ( ( DSCALE( I )-TAU ).NE.ZERO ) THEN
TEMP = ONE / ( DSCALE( I )-TAU )
TEMP1 = ZSCALE( I )*TEMP
TEMP2 = TEMP1*TEMP
TEMP3 = TEMP2*TEMP
TEMP4 = TEMP1 / DSCALE( I )
FC = FC + TEMP4
ERRETM = ERRETM + ABS( TEMP4 )
DF = DF + TEMP2
DDF = DDF + TEMP3
ELSE
GO TO 60
END IF
40 CONTINUE
F = FINIT + TAU*FC
ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) +
$ ABS( TAU )*DF
IF( ( ABS( F ).LE.FOUR*EPS*ERRETM ) .OR.
$ ( (UBD-LBD).LE.FOUR*EPS*ABS(TAU) ) )
$ GO TO 60
IF( F .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
50 CONTINUE
INFO = 1
60 CONTINUE
*
* Undo scaling
*
IF( SCALE )
$ TAU = TAU*SCLINV
RETURN
*
* End of DLAED6
*
END
*> \brief \b DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED7 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed7.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed7.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed7.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
* LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
* PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
* $ QSIZ, TLVLS
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
* $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
* DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
* $ QSTORE( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED7 computes the updated eigensystem of a diagonal
*> matrix after modification by a rank-one symmetric matrix. This
*> routine is used only for the eigenproblem which requires all
*> eigenvalues and optionally eigenvectors of a dense symmetric matrix
*> that has been reduced to tridiagonal form. DLAED1 handles
*> the case in which all eigenvalues and eigenvectors of a symmetric
*> tridiagonal matrix are desired.
*>
*> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*> where Z = Q**Tu, u is a vector of length N with ones in the
*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*>
*> The eigenvectors of the original matrix are stored in Q, and the
*> eigenvalues are in D. The algorithm consists of three stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple eigenvalues or if there is a zero in
*> the Z vector. For each such occurrence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine DLAED8.
*>
*> The second stage consists of calculating the updated
*> eigenvalues. This is done by finding the roots of the secular
*> equation via the routine DLAED4 (as called by DLAED9).
*> This routine also calculates the eigenvectors of the current
*> problem.
*>
*> The final stage consists of computing the updated eigenvectors
*> directly using the updated eigenvalues. The eigenvectors for
*> the current problem are multiplied with the eigenvectors from
*> the overall problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in] TLVLS
*> \verbatim
*> TLVLS is INTEGER
*> The total number of merging levels in the overall divide and
*> conquer tree.
*> \endverbatim
*>
*> \param[in] CURLVL
*> \verbatim
*> CURLVL is INTEGER
*> The current level in the overall merge routine,
*> 0 <= CURLVL <= TLVLS.
*> \endverbatim
*>
*> \param[in] CURPBM
*> \verbatim
*> CURPBM is INTEGER
*> The current problem in the current level in the overall
*> merge routine (counting from upper left to lower right).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the eigenvalues of the rank-1-perturbed matrix.
*> On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, the eigenvectors of the rank-1-perturbed matrix.
*> On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> The permutation which will reintegrate the subproblem just
*> solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
*> will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The subdiagonal element used to create the rank-1
*> modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> Contains the location of the last eigenvalue in the leading
*> sub-matrix. min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in,out] QSTORE
*> \verbatim
*> QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
*> Stores eigenvectors of submatrices encountered during
*> divide and conquer, packed together. QPTR points to
*> beginning of the submatrices.
*> \endverbatim
*>
*> \param[in,out] QPTR
*> \verbatim
*> QPTR is INTEGER array, dimension (N+2)
*> List of indices pointing to beginning of submatrices stored
*> in QSTORE. The submatrices are numbered starting at the
*> bottom left of the divide and conquer tree, from left to
*> right and bottom to top.
*> \endverbatim
*>
*> \param[in] PRMPTR
*> \verbatim
*> PRMPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in PERM a
*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
*> indicates the size of the permutation and also the size of
*> the full, non-deflated problem.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N lg N)
*> Contains the permutations (from deflation and sorting) to be
*> applied to each eigenblock.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in GIVCOL a
*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
*> indicates the number of Givens rotations.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N lg N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed7
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D,
$ Q,
$ LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
$ PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
$ QSIZ, TLVLS
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
$ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
$ QSTORE( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
$ IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLAED8, DLAED9, DLAEDA, DLAMRG,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
INFO = -3
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED7', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* The following values are for bookkeeping purposes only. They are
* integer pointers which indicate the portion of the workspace
* used by a particular array in DLAED8 and DLAED9.
*
IF( ICOMPQ.EQ.1 ) THEN
LDQ2 = QSIZ
ELSE
LDQ2 = N
END IF
*
IZ = 1
IDLMDA = IZ + N
IW = IDLMDA + N
IQ2 = IW + N
IS = IQ2 + N*LDQ2
*
INDX = 1
INDXC = INDX + N
COLTYP = INDXC + N
INDXP = COLTYP + N
*
* Form the z-vector which consists of the last row of Q_1 and the
* first row of Q_2.
*
PTR = 1 + 2**TLVLS
DO 10 I = 1, CURLVL - 1
PTR = PTR + 2**( TLVLS-I )
10 CONTINUE
CURR = PTR + CURPBM
CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
$ GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ),
$ WORK( IZ+N ), INFO )
*
* When solving the final problem, we no longer need the stored data,
* so we will overwrite the data from this level onto the previously
* used storage space.
*
IF( CURLVL.EQ.TLVLS ) THEN
QPTR( CURR ) = 1
PRMPTR( CURR ) = 1
GIVPTR( CURR ) = 1
END IF
*
* Sort and Deflate eigenvalues.
*
CALL DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
$ WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2,
$ WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
$ GIVCOL( 1, GIVPTR( CURR ) ),
$ GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ),
$ IWORK( INDX ), INFO )
PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
*
* Solve Secular Equation.
*
IF( K.NE.0 ) THEN
CALL DLAED9( K, 1, K, N, D, WORK( IS ), K, RHO,
$ WORK( IDLMDA ),
$ WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( ICOMPQ.EQ.1 ) THEN
CALL DGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2,
$ QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ )
END IF
QPTR( CURR+1 ) = QPTR( CURR ) + K**2
*
* Prepare the INDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
ELSE
QPTR( CURR+1 ) = QPTR( CURR )
DO 20 I = 1, N
INDXQ( I ) = I
20 CONTINUE
END IF
*
30 CONTINUE
RETURN
*
* End of DLAED7
*
END
*> \brief \b DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED8 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed8.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed8.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed8.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
* CUTPNT, Z, DLAMBDA, Q2, LDQ2, W, PERM, GIVPTR,
* GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
* .. Scalar Arguments ..
* INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
* $ QSIZ
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
* $ INDXQ( * ), PERM( * )
* DOUBLE PRECISION D( * ), DLAMBDA( * ), GIVNUM( 2, * ),
* $ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED8 merges the two sets of eigenvalues together into a single
*> sorted set. Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur: when two or more
*> eigenvalues are close together or if there is a tiny element in the
*> Z vector. For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> The number of non-deflated eigenvalues, and the order of the
*> related secular equation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the eigenvalues of the two submatrices to be
*> combined. On exit, the trailing (N-K) updated eigenvalues
*> (those which were deflated) sorted into increasing order.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> If ICOMPQ = 0, Q is not referenced. Otherwise,
*> on entry, Q contains the eigenvectors of the partially solved
*> system which has been previously updated in matrix
*> multiplies with other partially solved eigensystems.
*> On exit, Q contains the trailing (N-K) updated eigenvectors
*> (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> The permutation which separately sorts the two sub-problems
*> in D into ascending order. Note that elements in the second
*> half of this permutation must first have CUTPNT added to
*> their values in order to be accurate.
*> \endverbatim
*>
*> \param[in,out] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> On entry, the off-diagonal element associated with the rank-1
*> cut which originally split the two submatrices which are now
*> being recombined.
*> On exit, RHO has been modified to the value required by
*> DLAED3.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> The location of the last eigenvalue in the leading
*> sub-matrix. min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> On entry, Z contains the updating vector (the last row of
*> the first sub-eigenvector matrix and the first row of the
*> second sub-eigenvector matrix).
*> On exit, the contents of Z are destroyed by the updating
*> process.
*> \endverbatim
*>
*> \param[out] DLAMBDA
*> \verbatim
*> DLAMBDA is DOUBLE PRECISION array, dimension (N)
*> A copy of the first K eigenvalues which will be used by
*> DLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*> Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
*> If ICOMPQ = 0, Q2 is not referenced. Otherwise,
*> a copy of the first K eigenvectors which will be used by
*> DLAED7 in a matrix multiply (DGEMM) to update the new
*> eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ2
*> \verbatim
*> LDQ2 is INTEGER
*> The leading dimension of the array Q2. LDQ2 >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first k values of the final deflation-altered z-vector and
*> will be passed to DLAED3.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N)
*> The permutations (from deflation and sorting) to be applied
*> to each eigenblock.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> The number of Givens rotations which took place in this
*> subproblem.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension (2, N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*> INDXP is INTEGER array, dimension (N)
*> The permutation used to place deflated values of D at the end
*> of the array. INDXP(1:K) points to the nondeflated D-values
*> and INDXP(K+1:N) points to the deflated eigenvalues.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to sort the contents of D into ascending
*> order.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed8
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
$ CUTPNT, Z, DLAMBDA, Q2, LDQ2, W, PERM, GIVPTR,
$ GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
$ QSIZ
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
$ INDXQ( * ), PERM( * )
DOUBLE PRECISION D( * ), DLAMBDA( * ), GIVNUM( 2, * ),
$ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, EIGHT = 8.0D0 )
* ..
* .. Local Scalars ..
*
INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
DOUBLE PRECISION C, EPS, S, T, TAU, TOL
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL IDAMAX, DLAMCH, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
INFO = -10
ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED8', -INFO )
RETURN
END IF
*
* Need to initialize GIVPTR to O here in case of quick exit
* to prevent an unspecified code behavior (usually sigfault)
* when IWORK array on entry to *stedc is not zeroed
* (or at least some IWORK entries which used in *laed7 for GIVPTR).
*
GIVPTR = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
N1 = CUTPNT
N2 = N - N1
N1P1 = N1 + 1
*
IF( RHO.LT.ZERO ) THEN
CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
END IF
*
* Normalize z so that norm(z) = 1
*
T = ONE / SQRT( TWO )
DO 10 J = 1, N
INDX( J ) = J
10 CONTINUE
CALL DSCAL( N, T, Z, 1 )
RHO = ABS( TWO*RHO )
*
* Sort the eigenvalues into increasing order
*
DO 20 I = CUTPNT + 1, N
INDXQ( I ) = INDXQ( I ) + CUTPNT
20 CONTINUE
DO 30 I = 1, N
DLAMBDA( I ) = D( INDXQ( I ) )
W( I ) = Z( INDXQ( I ) )
30 CONTINUE
I = 1
J = CUTPNT + 1
CALL DLAMRG( N1, N2, DLAMBDA, 1, 1, INDX )
DO 40 I = 1, N
D( I ) = DLAMBDA( INDX( I ) )
Z( I ) = W( INDX( I ) )
40 CONTINUE
*
* Calculate the allowable deflation tolerance
*
IMAX = IDAMAX( N, Z, 1 )
JMAX = IDAMAX( N, D, 1 )
EPS = DLAMCH( 'Epsilon' )
TOL = EIGHT*EPS*ABS( D( JMAX ) )
*
* If the rank-1 modifier is small enough, no more needs to be done
* except to reorganize Q so that its columns correspond with the
* elements in D.
*
IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
K = 0
IF( ICOMPQ.EQ.0 ) THEN
DO 50 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
50 CONTINUE
ELSE
DO 60 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ),
$ 1 )
60 CONTINUE
CALL DLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ),
$ LDQ )
END IF
RETURN
END IF
*
* If there are multiple eigenvalues then the problem deflates. Here
* the number of equal eigenvalues are found. As each equal
* eigenvalue is found, an elementary reflector is computed to rotate
* the corresponding eigensubspace so that the corresponding
* components of Z are zero in this new basis.
*
K = 0
K2 = N + 1
DO 70 J = 1, N
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
IF( J.EQ.N )
$ GO TO 110
ELSE
JLAM = J
GO TO 80
END IF
70 CONTINUE
80 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 100
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
ELSE
*
* Check if eigenvalues are close enough to allow deflation.
*
S = Z( JLAM )
C = Z( J )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = DLAPY2( C, S )
T = D( J ) - D( JLAM )
C = C / TAU
S = -S / TAU
IF( ABS( T*C*S ).LE.TOL ) THEN
*
* Deflation is possible.
*
Z( J ) = TAU
Z( JLAM ) = ZERO
*
* Record the appropriate Givens rotation
*
GIVPTR = GIVPTR + 1
GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
GIVNUM( 1, GIVPTR ) = C
GIVNUM( 2, GIVPTR ) = S
IF( ICOMPQ.EQ.1 ) THEN
CALL DROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
$ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
END IF
T = D( JLAM )*C*C + D( J )*S*S
D( J ) = D( JLAM )*S*S + D( J )*C*C
D( JLAM ) = T
K2 = K2 - 1
I = 1
90 CONTINUE
IF( K2+I.LE.N ) THEN
IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
INDXP( K2+I-1 ) = INDXP( K2+I )
INDXP( K2+I ) = JLAM
I = I + 1
GO TO 90
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
JLAM = J
ELSE
K = K + 1
W( K ) = Z( JLAM )
DLAMBDA( K ) = D( JLAM )
INDXP( K ) = JLAM
JLAM = J
END IF
END IF
GO TO 80
100 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
W( K ) = Z( JLAM )
DLAMBDA( K ) = D( JLAM )
INDXP( K ) = JLAM
*
110 CONTINUE
*
* Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
* and Q2 respectively. The eigenvalues/vectors which were not
* deflated go into the first K slots of DLAMBDA and Q2 respectively,
* while those which were deflated go into the last N - K slots.
*
IF( ICOMPQ.EQ.0 ) THEN
DO 120 J = 1, N
JP = INDXP( J )
DLAMBDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
120 CONTINUE
ELSE
DO 130 J = 1, N
JP = INDXP( J )
DLAMBDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
130 CONTINUE
END IF
*
* The deflated eigenvalues and their corresponding vectors go back
* into the last N - K slots of D and Q respectively.
*
IF( K.LT.N ) THEN
IF( ICOMPQ.EQ.0 ) THEN
CALL DCOPY( N-K, DLAMBDA( K+1 ), 1, D( K+1 ), 1 )
ELSE
CALL DCOPY( N-K, DLAMBDA( K+1 ), 1, D( K+1 ), 1 )
CALL DLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2,
$ Q( 1, K+1 ), LDQ )
END IF
END IF
*
RETURN
*
* End of DLAED8
*
END
*> \brief \b DLAED9 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED9 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed9.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed9.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed9.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMBDA,
* W, S, LDS, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DLAMBDA( * ), Q( LDQ, * ), S( LDS, * ),
* $ W( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED9 finds the roots of the secular equation, as defined by the
*> values in D, Z, and RHO, between KSTART and KSTOP. It makes the
*> appropriate calls to DLAED4 and then stores the new matrix of
*> eigenvectors for use in calculating the next level of Z vectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved by
*> DLAED4. K >= 0.
*> \endverbatim
*>
*> \param[in] KSTART
*> \verbatim
*> KSTART is INTEGER
*> \endverbatim
*>
*> \param[in] KSTOP
*> \verbatim
*> KSTOP is INTEGER
*> The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
*> are to be computed. 1 <= KSTART <= KSTOP <= K.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the Q matrix.
*> N >= K (delation may result in N > K).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> D(I) contains the updated eigenvalues
*> for KSTART <= I <= KSTOP.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max( 1, N ).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The value of the parameter in the rank one update equation.
*> RHO >= 0 required.
*> \endverbatim
*>
*> \param[in] DLAMBDA
*> \verbatim
*> DLAMBDA is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating vector.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (LDS, K)
*> Will contain the eigenvectors of the repaired matrix which
*> will be stored for subsequent Z vector calculation and
*> multiplied by the previously accumulated eigenvectors
*> to update the system.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*> LDS is INTEGER
*> The leading dimension of S. LDS >= max( 1, K ).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed9
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO,
$ DLAMBDA,
$ W, S, LDS, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DLAMBDA( * ), Q( LDQ, * ), S( LDS, * ),
$ W( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION TEMP
* ..
* .. External Functions ..
DOUBLE PRECISION DNRM2
EXTERNAL DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAED4, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( K.LT.0 ) THEN
INFO = -1
ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN
INFO = -2
ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) )
$ THEN
INFO = -3
ELSE IF( N.LT.K ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDS.LT.MAX( 1, K ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED9', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 )
$ RETURN
*
DO 20 J = KSTART, KSTOP
CALL DLAED4( K, J, DLAMBDA, W, Q( 1, J ), RHO, D( J ),
$ INFO )
*
* If the zero finder fails, the computation is terminated.
*
IF( INFO.NE.0 )
$ GO TO 120
20 CONTINUE
*
IF( K.EQ.1 .OR. K.EQ.2 ) THEN
DO 40 I = 1, K
DO 30 J = 1, K
S( J, I ) = Q( J, I )
30 CONTINUE
40 CONTINUE
GO TO 120
END IF
*
* Compute updated W.
*
CALL DCOPY( K, W, 1, S, 1 )
*
* Initialize W(I) = Q(I,I)
*
CALL DCOPY( K, Q, LDQ+1, W, 1 )
DO 70 J = 1, K
DO 50 I = 1, J - 1
W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
50 CONTINUE
DO 60 I = J + 1, K
W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
60 CONTINUE
70 CONTINUE
DO 80 I = 1, K
W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) )
80 CONTINUE
*
* Compute eigenvectors of the modified rank-1 modification.
*
DO 110 J = 1, K
DO 90 I = 1, K
Q( I, J ) = W( I ) / Q( I, J )
90 CONTINUE
TEMP = DNRM2( K, Q( 1, J ), 1 )
DO 100 I = 1, K
S( I, J ) = Q( I, J ) / TEMP
100 CONTINUE
110 CONTINUE
*
120 CONTINUE
RETURN
*
* End of DLAED9
*
END
*> \brief \b DLAEDA used by DSTEDC. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEDA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaeda.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaeda.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaeda.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
* GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO )
*
* .. Scalar Arguments ..
* INTEGER CURLVL, CURPBM, INFO, N, TLVLS
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), GIVPTR( * ), PERM( * ),
* $ PRMPTR( * ), QPTR( * )
* DOUBLE PRECISION GIVNUM( 2, * ), Q( * ), Z( * ), ZTEMP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEDA computes the Z vector corresponding to the merge step in the
*> CURLVLth step of the merge process with TLVLS steps for the CURPBMth
*> problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] TLVLS
*> \verbatim
*> TLVLS is INTEGER
*> The total number of merging levels in the overall divide and
*> conquer tree.
*> \endverbatim
*>
*> \param[in] CURLVL
*> \verbatim
*> CURLVL is INTEGER
*> The current level in the overall merge routine,
*> 0 <= curlvl <= tlvls.
*> \endverbatim
*>
*> \param[in] CURPBM
*> \verbatim
*> CURPBM is INTEGER
*> The current problem in the current level in the overall
*> merge routine (counting from upper left to lower right).
*> \endverbatim
*>
*> \param[in] PRMPTR
*> \verbatim
*> PRMPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in PERM a
*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
*> indicates the size of the permutation and incidentally the
*> size of the full, non-deflated problem.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N lg N)
*> Contains the permutations (from deflation and sorting) to be
*> applied to each eigenblock.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in GIVCOL a
*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
*> indicates the number of Givens rotations.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N lg N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (N**2)
*> Contains the square eigenblocks from previous levels, the
*> starting positions for blocks are given by QPTR.
*> \endverbatim
*>
*> \param[in] QPTR
*> \verbatim
*> QPTR is INTEGER array, dimension (N+2)
*> Contains a list of pointers which indicate where in Q an
*> eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates
*> the size of the block.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> On output this vector contains the updating vector (the last
*> row of the first sub-eigenvector matrix and the first row of
*> the second sub-eigenvector matrix).
*> \endverbatim
*>
*> \param[out] ZTEMP
*> \verbatim
*> ZTEMP is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laeda
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM,
$ GIVPTR,
$ GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER CURLVL, CURPBM, INFO, N, TLVLS
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), GIVPTR( * ), PERM( * ),
$ PRMPTR( * ), QPTR( * )
DOUBLE PRECISION GIVNUM( 2, * ), Q( * ), Z( * ), ZTEMP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER BSIZ1, BSIZ2, CURR, I, K, MID, PSIZ1, PSIZ2,
$ PTR, ZPTR1
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DROT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, INT, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAEDA', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine location of first number in second half.
*
MID = N / 2 + 1
*
* Gather last/first rows of appropriate eigenblocks into center of Z
*
PTR = 1
*
* Determine location of lowest level subproblem in the full storage
* scheme
*
CURR = PTR + CURPBM*2**CURLVL + 2**( CURLVL-1 ) - 1
*
* Determine size of these matrices. We add HALF to the value of
* the SQRT in case the machine underestimates one of these square
* roots.
*
BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) )
BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+1 ) ) ) )
DO 10 K = 1, MID - BSIZ1 - 1
Z( K ) = ZERO
10 CONTINUE
CALL DCOPY( BSIZ1, Q( QPTR( CURR )+BSIZ1-1 ), BSIZ1,
$ Z( MID-BSIZ1 ), 1 )
CALL DCOPY( BSIZ2, Q( QPTR( CURR+1 ) ), BSIZ2, Z( MID ), 1 )
DO 20 K = MID + BSIZ2, N
Z( K ) = ZERO
20 CONTINUE
*
* Loop through remaining levels 1 -> CURLVL applying the Givens
* rotations and permutation and then multiplying the center matrices
* against the current Z.
*
PTR = 2**TLVLS + 1
DO 70 K = 1, CURLVL - 1
CURR = PTR + CURPBM*2**( CURLVL-K ) + 2**( CURLVL-K-1 ) - 1
PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR )
PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 )
ZPTR1 = MID - PSIZ1
*
* Apply Givens at CURR and CURR+1
*
DO 30 I = GIVPTR( CURR ), GIVPTR( CURR+1 ) - 1
CALL DROT( 1, Z( ZPTR1+GIVCOL( 1, I )-1 ), 1,
$ Z( ZPTR1+GIVCOL( 2, I )-1 ), 1, GIVNUM( 1, I ),
$ GIVNUM( 2, I ) )
30 CONTINUE
DO 40 I = GIVPTR( CURR+1 ), GIVPTR( CURR+2 ) - 1
CALL DROT( 1, Z( MID-1+GIVCOL( 1, I ) ), 1,
$ Z( MID-1+GIVCOL( 2, I ) ), 1, GIVNUM( 1, I ),
$ GIVNUM( 2, I ) )
40 CONTINUE
PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR )
PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 )
DO 50 I = 0, PSIZ1 - 1
ZTEMP( I+1 ) = Z( ZPTR1+PERM( PRMPTR( CURR )+I )-1 )
50 CONTINUE
DO 60 I = 0, PSIZ2 - 1
ZTEMP( PSIZ1+I+1 ) = Z( MID+PERM( PRMPTR( CURR+1 )+I )-1 )
60 CONTINUE
*
* Multiply Blocks at CURR and CURR+1
*
* Determine size of these matrices. We add HALF to the value of
* the SQRT in case the machine underestimates one of these
* square roots.
*
BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) )
BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+
$ 1 ) ) ) )
IF( BSIZ1.GT.0 ) THEN
CALL DGEMV( 'T', BSIZ1, BSIZ1, ONE, Q( QPTR( CURR ) ),
$ BSIZ1, ZTEMP( 1 ), 1, ZERO, Z( ZPTR1 ), 1 )
END IF
CALL DCOPY( PSIZ1-BSIZ1, ZTEMP( BSIZ1+1 ), 1,
$ Z( ZPTR1+BSIZ1 ),
$ 1 )
IF( BSIZ2.GT.0 ) THEN
CALL DGEMV( 'T', BSIZ2, BSIZ2, ONE, Q( QPTR( CURR+1 ) ),
$ BSIZ2, ZTEMP( PSIZ1+1 ), 1, ZERO, Z( MID ), 1 )
END IF
CALL DCOPY( PSIZ2-BSIZ2, ZTEMP( PSIZ1+BSIZ2+1 ), 1,
$ Z( MID+BSIZ2 ), 1 )
*
PTR = PTR + 2**( TLVLS-K )
70 CONTINUE
*
RETURN
*
* End of DLAEDA
*
END
*> \brief \b DLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaein.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B,
* LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO )
*
* .. Scalar Arguments ..
* LOGICAL NOINIT, RIGHTV
* INTEGER INFO, LDB, LDH, N
* DOUBLE PRECISION BIGNUM, EPS3, SMLNUM, WI, WR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), H( LDH, * ), VI( * ), VR( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEIN uses inverse iteration to find a right or left eigenvector
*> corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg
*> matrix H.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] RIGHTV
*> \verbatim
*> RIGHTV is LOGICAL
*> = .TRUE. : compute right eigenvector;
*> = .FALSE.: compute left eigenvector.
*> \endverbatim
*>
*> \param[in] NOINIT
*> \verbatim
*> NOINIT is LOGICAL
*> = .TRUE. : no initial vector supplied in (VR,VI).
*> = .FALSE.: initial vector supplied in (VR,VI).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> The upper Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*> WR is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is DOUBLE PRECISION
*> The real and imaginary parts of the eigenvalue of H whose
*> corresponding right or left eigenvector is to be computed.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in,out] VI
*> \verbatim
*> VI is DOUBLE PRECISION array, dimension (N)
*> On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
*> a real starting vector for inverse iteration using the real
*> eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
*> must contain the real and imaginary parts of a complex
*> starting vector for inverse iteration using the complex
*> eigenvalue (WR,WI); otherwise VR and VI need not be set.
*> On exit, if WI = 0.0 (real eigenvalue), VR contains the
*> computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
*> VR and VI contain the real and imaginary parts of the
*> computed complex eigenvector. The eigenvector is normalized
*> so that the component of largest magnitude has magnitude 1;
*> here the magnitude of a complex number (x,y) is taken to be
*> |x| + |y|.
*> VI is not referenced if WI = 0.0.
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= N+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in] EPS3
*> \verbatim
*> EPS3 is DOUBLE PRECISION
*> A small machine-dependent value which is used to perturb
*> close eigenvalues, and to replace zero pivots.
*> \endverbatim
*>
*> \param[in] SMLNUM
*> \verbatim
*> SMLNUM is DOUBLE PRECISION
*> A machine-dependent value close to the underflow threshold.
*> \endverbatim
*>
*> \param[in] BIGNUM
*> \verbatim
*> BIGNUM is DOUBLE PRECISION
*> A machine-dependent value close to the overflow threshold.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> = 1: inverse iteration did not converge; VR is set to the
*> last iterate, and so is VI if WI.ne.0.0.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laein
*
* =====================================================================
SUBROUTINE DLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI,
$ B,
$ LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL NOINIT, RIGHTV
INTEGER INFO, LDB, LDH, N
DOUBLE PRECISION BIGNUM, EPS3, SMLNUM, WI, WR
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), H( LDH, * ), VI( * ), VR( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TENTH
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TENTH = 1.0D-1 )
* ..
* .. Local Scalars ..
CHARACTER NORMIN, TRANS
INTEGER I, I1, I2, I3, IERR, ITS, J
DOUBLE PRECISION ABSBII, ABSBJJ, EI, EJ, GROWTO, NORM, NRMSML,
$ REC, ROOTN, SCALE, TEMP, VCRIT, VMAX, VNORM, W,
$ W1, X, XI, XR, Y
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DLAPY2, DNRM2
EXTERNAL IDAMAX, DASUM, DLAPY2, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DLADIV, DLATRS, DSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* GROWTO is the threshold used in the acceptance test for an
* eigenvector.
*
ROOTN = SQRT( DBLE( N ) )
GROWTO = TENTH / ROOTN
NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
*
* Form B = H - (WR,WI)*I (except that the subdiagonal elements and
* the imaginary parts of the diagonal elements are not stored).
*
DO 20 J = 1, N
DO 10 I = 1, J - 1
B( I, J ) = H( I, J )
10 CONTINUE
B( J, J ) = H( J, J ) - WR
20 CONTINUE
*
IF( WI.EQ.ZERO ) THEN
*
* Real eigenvalue.
*
IF( NOINIT ) THEN
*
* Set initial vector.
*
DO 30 I = 1, N
VR( I ) = EPS3
30 CONTINUE
ELSE
*
* Scale supplied initial vector.
*
VNORM = DNRM2( N, VR, 1 )
CALL DSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), VR,
$ 1 )
END IF
*
IF( RIGHTV ) THEN
*
* LU decomposition with partial pivoting of B, replacing zero
* pivots by EPS3.
*
DO 60 I = 1, N - 1
EI = H( I+1, I )
IF( ABS( B( I, I ) ).LT.ABS( EI ) ) THEN
*
* Interchange rows and eliminate.
*
X = B( I, I ) / EI
B( I, I ) = EI
DO 40 J = I + 1, N
TEMP = B( I+1, J )
B( I+1, J ) = B( I, J ) - X*TEMP
B( I, J ) = TEMP
40 CONTINUE
ELSE
*
* Eliminate without interchange.
*
IF( B( I, I ).EQ.ZERO )
$ B( I, I ) = EPS3
X = EI / B( I, I )
IF( X.NE.ZERO ) THEN
DO 50 J = I + 1, N
B( I+1, J ) = B( I+1, J ) - X*B( I, J )
50 CONTINUE
END IF
END IF
60 CONTINUE
IF( B( N, N ).EQ.ZERO )
$ B( N, N ) = EPS3
*
TRANS = 'N'
*
ELSE
*
* UL decomposition with partial pivoting of B, replacing zero
* pivots by EPS3.
*
DO 90 J = N, 2, -1
EJ = H( J, J-1 )
IF( ABS( B( J, J ) ).LT.ABS( EJ ) ) THEN
*
* Interchange columns and eliminate.
*
X = B( J, J ) / EJ
B( J, J ) = EJ
DO 70 I = 1, J - 1
TEMP = B( I, J-1 )
B( I, J-1 ) = B( I, J ) - X*TEMP
B( I, J ) = TEMP
70 CONTINUE
ELSE
*
* Eliminate without interchange.
*
IF( B( J, J ).EQ.ZERO )
$ B( J, J ) = EPS3
X = EJ / B( J, J )
IF( X.NE.ZERO ) THEN
DO 80 I = 1, J - 1
B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
80 CONTINUE
END IF
END IF
90 CONTINUE
IF( B( 1, 1 ).EQ.ZERO )
$ B( 1, 1 ) = EPS3
*
TRANS = 'T'
*
END IF
*
NORMIN = 'N'
DO 110 ITS = 1, N
*
* Solve U*x = scale*v for a right eigenvector
* or U**T*x = scale*v for a left eigenvector,
* overwriting x on v.
*
CALL DLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B,
$ LDB,
$ VR, SCALE, WORK, IERR )
NORMIN = 'Y'
*
* Test for sufficient growth in the norm of v.
*
VNORM = DASUM( N, VR, 1 )
IF( VNORM.GE.GROWTO*SCALE )
$ GO TO 120
*
* Choose new orthogonal starting vector and try again.
*
TEMP = EPS3 / ( ROOTN+ONE )
VR( 1 ) = EPS3
DO 100 I = 2, N
VR( I ) = TEMP
100 CONTINUE
VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN
110 CONTINUE
*
* Failure to find eigenvector in N iterations.
*
INFO = 1
*
120 CONTINUE
*
* Normalize eigenvector.
*
I = IDAMAX( N, VR, 1 )
CALL DSCAL( N, ONE / ABS( VR( I ) ), VR, 1 )
ELSE
*
* Complex eigenvalue.
*
IF( NOINIT ) THEN
*
* Set initial vector.
*
DO 130 I = 1, N
VR( I ) = EPS3
VI( I ) = ZERO
130 CONTINUE
ELSE
*
* Scale supplied initial vector.
*
NORM = DLAPY2( DNRM2( N, VR, 1 ),
$ DNRM2( N, VI, 1 ) )
REC = ( EPS3*ROOTN ) / MAX( NORM, NRMSML )
CALL DSCAL( N, REC, VR, 1 )
CALL DSCAL( N, REC, VI, 1 )
END IF
*
IF( RIGHTV ) THEN
*
* LU decomposition with partial pivoting of B, replacing zero
* pivots by EPS3.
*
* The imaginary part of the (i,j)-th element of U is stored in
* B(j+1,i).
*
B( 2, 1 ) = -WI
DO 140 I = 2, N
B( I+1, 1 ) = ZERO
140 CONTINUE
*
DO 170 I = 1, N - 1
ABSBII = DLAPY2( B( I, I ), B( I+1, I ) )
EI = H( I+1, I )
IF( ABSBII.LT.ABS( EI ) ) THEN
*
* Interchange rows and eliminate.
*
XR = B( I, I ) / EI
XI = B( I+1, I ) / EI
B( I, I ) = EI
B( I+1, I ) = ZERO
DO 150 J = I + 1, N
TEMP = B( I+1, J )
B( I+1, J ) = B( I, J ) - XR*TEMP
B( J+1, I+1 ) = B( J+1, I ) - XI*TEMP
B( I, J ) = TEMP
B( J+1, I ) = ZERO
150 CONTINUE
B( I+2, I ) = -WI
B( I+1, I+1 ) = B( I+1, I+1 ) - XI*WI
B( I+2, I+1 ) = B( I+2, I+1 ) + XR*WI
ELSE
*
* Eliminate without interchanging rows.
*
IF( ABSBII.EQ.ZERO ) THEN
B( I, I ) = EPS3
B( I+1, I ) = ZERO
ABSBII = EPS3
END IF
EI = ( EI / ABSBII ) / ABSBII
XR = B( I, I )*EI
XI = -B( I+1, I )*EI
DO 160 J = I + 1, N
B( I+1, J ) = B( I+1, J ) - XR*B( I, J ) +
$ XI*B( J+1, I )
B( J+1, I+1 ) = -XR*B( J+1, I ) - XI*B( I, J )
160 CONTINUE
B( I+2, I+1 ) = B( I+2, I+1 ) - WI
END IF
*
* Compute 1-norm of offdiagonal elements of i-th row.
*
WORK( I ) = DASUM( N-I, B( I, I+1 ), LDB ) +
$ DASUM( N-I, B( I+2, I ), 1 )
170 CONTINUE
IF( B( N, N ).EQ.ZERO .AND. B( N+1, N ).EQ.ZERO )
$ B( N, N ) = EPS3
WORK( N ) = ZERO
*
I1 = N
I2 = 1
I3 = -1
ELSE
*
* UL decomposition with partial pivoting of conjg(B),
* replacing zero pivots by EPS3.
*
* The imaginary part of the (i,j)-th element of U is stored in
* B(j+1,i).
*
B( N+1, N ) = WI
DO 180 J = 1, N - 1
B( N+1, J ) = ZERO
180 CONTINUE
*
DO 210 J = N, 2, -1
EJ = H( J, J-1 )
ABSBJJ = DLAPY2( B( J, J ), B( J+1, J ) )
IF( ABSBJJ.LT.ABS( EJ ) ) THEN
*
* Interchange columns and eliminate
*
XR = B( J, J ) / EJ
XI = B( J+1, J ) / EJ
B( J, J ) = EJ
B( J+1, J ) = ZERO
DO 190 I = 1, J - 1
TEMP = B( I, J-1 )
B( I, J-1 ) = B( I, J ) - XR*TEMP
B( J, I ) = B( J+1, I ) - XI*TEMP
B( I, J ) = TEMP
B( J+1, I ) = ZERO
190 CONTINUE
B( J+1, J-1 ) = WI
B( J-1, J-1 ) = B( J-1, J-1 ) + XI*WI
B( J, J-1 ) = B( J, J-1 ) - XR*WI
ELSE
*
* Eliminate without interchange.
*
IF( ABSBJJ.EQ.ZERO ) THEN
B( J, J ) = EPS3
B( J+1, J ) = ZERO
ABSBJJ = EPS3
END IF
EJ = ( EJ / ABSBJJ ) / ABSBJJ
XR = B( J, J )*EJ
XI = -B( J+1, J )*EJ
DO 200 I = 1, J - 1
B( I, J-1 ) = B( I, J-1 ) - XR*B( I, J ) +
$ XI*B( J+1, I )
B( J, I ) = -XR*B( J+1, I ) - XI*B( I, J )
200 CONTINUE
B( J, J-1 ) = B( J, J-1 ) + WI
END IF
*
* Compute 1-norm of offdiagonal elements of j-th column.
*
WORK( J ) = DASUM( J-1, B( 1, J ), 1 ) +
$ DASUM( J-1, B( J+1, 1 ), LDB )
210 CONTINUE
IF( B( 1, 1 ).EQ.ZERO .AND. B( 2, 1 ).EQ.ZERO )
$ B( 1, 1 ) = EPS3
WORK( 1 ) = ZERO
*
I1 = 1
I2 = N
I3 = 1
END IF
*
DO 270 ITS = 1, N
SCALE = ONE
VMAX = ONE
VCRIT = BIGNUM
*
* Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector,
* or U**T*(xr,xi) = scale*(vr,vi) for a left eigenvector,
* overwriting (xr,xi) on (vr,vi).
*
DO 250 I = I1, I2, I3
*
IF( WORK( I ).GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N, REC, VR, 1 )
CALL DSCAL( N, REC, VI, 1 )
SCALE = SCALE*REC
VMAX = ONE
VCRIT = BIGNUM
END IF
*
XR = VR( I )
XI = VI( I )
IF( RIGHTV ) THEN
DO 220 J = I + 1, N
XR = XR - B( I, J )*VR( J ) + B( J+1, I )*VI( J )
XI = XI - B( I, J )*VI( J ) - B( J+1, I )*VR( J )
220 CONTINUE
ELSE
DO 230 J = 1, I - 1
XR = XR - B( J, I )*VR( J ) + B( I+1, J )*VI( J )
XI = XI - B( J, I )*VI( J ) - B( I+1, J )*VR( J )
230 CONTINUE
END IF
*
W = ABS( B( I, I ) ) + ABS( B( I+1, I ) )
IF( W.GT.SMLNUM ) THEN
IF( W.LT.ONE ) THEN
W1 = ABS( XR ) + ABS( XI )
IF( W1.GT.W*BIGNUM ) THEN
REC = ONE / W1
CALL DSCAL( N, REC, VR, 1 )
CALL DSCAL( N, REC, VI, 1 )
XR = VR( I )
XI = VI( I )
SCALE = SCALE*REC
VMAX = VMAX*REC
END IF
END IF
*
* Divide by diagonal element of B.
*
CALL DLADIV( XR, XI, B( I, I ), B( I+1, I ),
$ VR( I ),
$ VI( I ) )
VMAX = MAX( ABS( VR( I ) )+ABS( VI( I ) ), VMAX )
VCRIT = BIGNUM / VMAX
ELSE
DO 240 J = 1, N
VR( J ) = ZERO
VI( J ) = ZERO
240 CONTINUE
VR( I ) = ONE
VI( I ) = ONE
SCALE = ZERO
VMAX = ONE
VCRIT = BIGNUM
END IF
250 CONTINUE
*
* Test for sufficient growth in the norm of (VR,VI).
*
VNORM = DASUM( N, VR, 1 ) + DASUM( N, VI, 1 )
IF( VNORM.GE.GROWTO*SCALE )
$ GO TO 280
*
* Choose a new orthogonal starting vector and try again.
*
Y = EPS3 / ( ROOTN+ONE )
VR( 1 ) = EPS3
VI( 1 ) = ZERO
*
DO 260 I = 2, N
VR( I ) = Y
VI( I ) = ZERO
260 CONTINUE
VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN
270 CONTINUE
*
* Failure to find eigenvector in N iterations
*
INFO = 1
*
280 CONTINUE
*
* Normalize eigenvector.
*
VNORM = ZERO
DO 290 I = 1, N
VNORM = MAX( VNORM, ABS( VR( I ) )+ABS( VI( I ) ) )
290 CONTINUE
CALL DSCAL( N, ONE / VNORM, VR, 1 )
CALL DSCAL( N, ONE / VNORM, VI, 1 )
*
END IF
*
RETURN
*
* End of DLAEIN
*
END
*> \brief \b DLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaev2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaev2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaev2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, CS1, RT1, RT2, SN1
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
*> [ A B ]
*> [ B C ].
*> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
*> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
*> eigenvector for RT1, giving the decomposition
*>
*> [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ]
*> [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION
*> The (1,2) element and the conjugate of the (2,1) element of
*> the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*> RT1 is DOUBLE PRECISION
*> The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*> RT2 is DOUBLE PRECISION
*> The eigenvalue of smaller absolute value.
*> \endverbatim
*>
*> \param[out] CS1
*> \verbatim
*> CS1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SN1
*> \verbatim
*> SN1 is DOUBLE PRECISION
*> The vector (CS1, SN1) is a unit right eigenvector for RT1.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laev2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> RT1 is accurate to a few ulps barring over/underflow.
*>
*> RT2 may be inaccurate if there is massive cancellation in the
*> determinant A*C-B*B; higher precision or correctly rounded or
*> correctly truncated arithmetic would be needed to compute RT2
*> accurately in all cases.
*>
*> CS1 and SN1 are accurate to a few ulps barring over/underflow.
*>
*> Overflow is possible only if RT1 is within a factor of 5 of overflow.
*> Underflow is harmless if the input data is 0 or exceeds
*> underflow_threshold / macheps.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, CS1, RT1, RT2, SN1
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
* ..
* .. Local Scalars ..
INTEGER SGN1, SGN2
DOUBLE PRECISION AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
$ TB, TN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
* Compute the eigenvalues
*
SM = A + C
DF = A - C
ADF = ABS( DF )
TB = B + B
AB = ABS( TB )
IF( ABS( A ).GT.ABS( C ) ) THEN
ACMX = A
ACMN = C
ELSE
ACMX = C
ACMN = A
END IF
IF( ADF.GT.AB ) THEN
RT = ADF*SQRT( ONE+( AB / ADF )**2 )
ELSE IF( ADF.LT.AB ) THEN
RT = AB*SQRT( ONE+( ADF / AB )**2 )
ELSE
*
* Includes case AB=ADF=0
*
RT = AB*SQRT( TWO )
END IF
IF( SM.LT.ZERO ) THEN
RT1 = HALF*( SM-RT )
SGN1 = -1
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE IF( SM.GT.ZERO ) THEN
RT1 = HALF*( SM+RT )
SGN1 = 1
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE
*
* Includes case RT1 = RT2 = 0
*
RT1 = HALF*RT
RT2 = -HALF*RT
SGN1 = 1
END IF
*
* Compute the eigenvector
*
IF( DF.GE.ZERO ) THEN
CS = DF + RT
SGN2 = 1
ELSE
CS = DF - RT
SGN2 = -1
END IF
ACS = ABS( CS )
IF( ACS.GT.AB ) THEN
CT = -TB / CS
SN1 = ONE / SQRT( ONE+CT*CT )
CS1 = CT*SN1
ELSE
IF( AB.EQ.ZERO ) THEN
CS1 = ONE
SN1 = ZERO
ELSE
TN = -CS / TB
CS1 = ONE / SQRT( ONE+TN*TN )
SN1 = TN*CS1
END IF
END IF
IF( SGN1.EQ.SGN2 ) THEN
TN = CS1
CS1 = -SN1
SN1 = TN
END IF
RETURN
*
* End of DLAEV2
*
END
*> \brief \b DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEXC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaexc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaexc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaexc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
* INFO )
*
* .. Scalar Arguments ..
* LOGICAL WANTQ
* INTEGER INFO, J1, LDQ, LDT, N, N1, N2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
*> an upper quasi-triangular matrix T by an orthogonal similarity
*> transformation.
*>
*> T must be in Schur canonical form, that is, block upper triangular
*> with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
*> has its diagonal elements equal and its off-diagonal elements of
*> opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTQ
*> \verbatim
*> WANTQ is LOGICAL
*> = .TRUE. : accumulate the transformation in the matrix Q;
*> = .FALSE.: do not accumulate the transformation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> On entry, the upper quasi-triangular matrix T, in Schur
*> canonical form.
*> On exit, the updated matrix T, again in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
*> On exit, if WANTQ is .TRUE., the updated matrix Q.
*> If WANTQ is .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*> J1 is INTEGER
*> The index of the first row of the first block T11.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The order of the first block T11. N1 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> The order of the second block T22. N2 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> = 1: the transformed matrix T would be too far from Schur
*> form; the blocks are not swapped and T and Q are
*> unchanged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laexc
*
* =====================================================================
SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
$ INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL WANTQ
INTEGER INFO, J1, LDQ, LDT, N, N1, N2
* ..
* .. Array Arguments ..
DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TEN
PARAMETER ( TEN = 1.0D+1 )
INTEGER LDD, LDX
PARAMETER ( LDD = 4, LDX = 2 )
* ..
* .. Local Scalars ..
INTEGER IERR, J2, J3, J4, K, ND
DOUBLE PRECISION CS, DNORM, EPS, SCALE, SMLNUM, SN, T11, T22,
$ T33, TAU, TAU1, TAU2, TEMP, THRESH, WI1, WI2,
$ WR1, WR2, XNORM
* ..
* .. Local Arrays ..
DOUBLE PRECISION D( LDD, 4 ), U( 3 ), U1( 3 ), U2( 3 ),
$ X( LDX, 2 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLANV2, DLARFG, DLARFX, DLARTG,
$ DLASY2,
$ DROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 .OR. N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
IF( J1+N1.GT.N )
$ RETURN
*
J2 = J1 + 1
J3 = J1 + 2
J4 = J1 + 3
*
IF( N1.EQ.1 .AND. N2.EQ.1 ) THEN
*
* Swap two 1-by-1 blocks.
*
T11 = T( J1, J1 )
T22 = T( J2, J2 )
*
* Determine the transformation to perform the interchange.
*
CALL DLARTG( T( J1, J2 ), T22-T11, CS, SN, TEMP )
*
* Apply transformation to the matrix T.
*
IF( J3.LE.N )
$ CALL DROT( N-J1-1, T( J1, J3 ), LDT, T( J2, J3 ), LDT,
$ CS,
$ SN )
CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
*
T( J1, J1 ) = T22
T( J2, J2 ) = T11
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
END IF
*
ELSE
*
* Swapping involves at least one 2-by-2 block.
*
* Copy the diagonal block of order N1+N2 to the local array D
* and compute its norm.
*
ND = N1 + N2
CALL DLACPY( 'Full', ND, ND, T( J1, J1 ), LDT, D, LDD )
DNORM = DLANGE( 'Max', ND, ND, D, LDD, WORK )
*
* Compute machine-dependent threshold for test for accepting
* swap.
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
THRESH = MAX( TEN*EPS*DNORM, SMLNUM )
*
* Solve T11*X - X*T22 = scale*T12 for X.
*
CALL DLASY2( .FALSE., .FALSE., -1, N1, N2, D, LDD,
$ D( N1+1, N1+1 ), LDD, D( 1, N1+1 ), LDD, SCALE, X,
$ LDX, XNORM, IERR )
*
* Swap the adjacent diagonal blocks.
*
K = N1 + N1 + N2 - 3
GO TO ( 10, 20, 30 )K
*
10 CONTINUE
*
* N1 = 1, N2 = 2: generate elementary reflector H so that:
*
* ( scale, X11, X12 ) H = ( 0, 0, * )
*
U( 1 ) = SCALE
U( 2 ) = X( 1, 1 )
U( 3 ) = X( 1, 2 )
CALL DLARFG( 3, U( 3 ), U, 1, TAU )
U( 3 ) = ONE
T11 = T( J1, J1 )
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 3,
$ 3 )-T11 ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'L', 3, N-J1+1, U, TAU, T( J1, J1 ), LDT,
$ WORK )
CALL DLARFX( 'R', J2, 3, U, TAU, T( 1, J1 ), LDT, WORK )
*
T( J3, J1 ) = ZERO
T( J3, J2 ) = ZERO
T( J3, J3 ) = T11
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
END IF
GO TO 40
*
20 CONTINUE
*
* N1 = 2, N2 = 1: generate elementary reflector H so that:
*
* H ( -X11 ) = ( * )
* ( -X21 ) = ( 0 )
* ( scale ) = ( 0 )
*
U( 1 ) = -X( 1, 1 )
U( 2 ) = -X( 2, 1 )
U( 3 ) = SCALE
CALL DLARFG( 3, U( 1 ), U( 2 ), 1, TAU )
U( 1 ) = ONE
T33 = T( J3, J3 )
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 2, 1 ) ), ABS( D( 3, 1 ) ), ABS( D( 1,
$ 1 )-T33 ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'R', J3, 3, U, TAU, T( 1, J1 ), LDT, WORK )
CALL DLARFX( 'L', 3, N-J1, U, TAU, T( J1, J2 ), LDT, WORK )
*
T( J1, J1 ) = T33
T( J2, J1 ) = ZERO
T( J3, J1 ) = ZERO
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
END IF
GO TO 40
*
30 CONTINUE
*
* N1 = 2, N2 = 2: generate elementary reflectors H(1) and H(2) so
* that:
*
* H(2) H(1) ( -X11 -X12 ) = ( * * )
* ( -X21 -X22 ) ( 0 * )
* ( scale 0 ) ( 0 0 )
* ( 0 scale ) ( 0 0 )
*
U1( 1 ) = -X( 1, 1 )
U1( 2 ) = -X( 2, 1 )
U1( 3 ) = SCALE
CALL DLARFG( 3, U1( 1 ), U1( 2 ), 1, TAU1 )
U1( 1 ) = ONE
*
TEMP = -TAU1*( X( 1, 2 )+U1( 2 )*X( 2, 2 ) )
U2( 1 ) = -TEMP*U1( 2 ) - X( 2, 2 )
U2( 2 ) = -TEMP*U1( 3 )
U2( 3 ) = SCALE
CALL DLARFG( 3, U2( 1 ), U2( 2 ), 1, TAU2 )
U2( 1 ) = ONE
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 4, U1, TAU1, D, LDD, WORK )
CALL DLARFX( 'R', 4, 3, U1, TAU1, D, LDD, WORK )
CALL DLARFX( 'L', 3, 4, U2, TAU2, D( 2, 1 ), LDD, WORK )
CALL DLARFX( 'R', 4, 3, U2, TAU2, D( 1, 2 ), LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 4, 1 ) ),
$ ABS( D( 4, 2 ) ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'L', 3, N-J1+1, U1, TAU1, T( J1, J1 ), LDT,
$ WORK )
CALL DLARFX( 'R', J4, 3, U1, TAU1, T( 1, J1 ), LDT, WORK )
CALL DLARFX( 'L', 3, N-J1+1, U2, TAU2, T( J2, J1 ), LDT,
$ WORK )
CALL DLARFX( 'R', J4, 3, U2, TAU2, T( 1, J2 ), LDT, WORK )
*
T( J3, J1 ) = ZERO
T( J3, J2 ) = ZERO
T( J4, J1 ) = ZERO
T( J4, J2 ) = ZERO
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U1, TAU1, Q( 1, J1 ), LDQ, WORK )
CALL DLARFX( 'R', N, 3, U2, TAU2, Q( 1, J2 ), LDQ, WORK )
END IF
*
40 CONTINUE
*
IF( N2.EQ.2 ) THEN
*
* Standardize new 2-by-2 block T11
*
CALL DLANV2( T( J1, J1 ), T( J1, J2 ), T( J2, J1 ),
$ T( J2, J2 ), WR1, WI1, WR2, WI2, CS, SN )
CALL DROT( N-J1-1, T( J1, J1+2 ), LDT, T( J2, J1+2 ),
$ LDT,
$ CS, SN )
CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
IF( WANTQ )
$ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
END IF
*
IF( N1.EQ.2 ) THEN
*
* Standardize new 2-by-2 block T22
*
J3 = J1 + N2
J4 = J3 + 1
CALL DLANV2( T( J3, J3 ), T( J3, J4 ), T( J4, J3 ),
$ T( J4, J4 ), WR1, WI1, WR2, WI2, CS, SN )
IF( J3+2.LE.N )
$ CALL DROT( N-J3-1, T( J3, J3+2 ), LDT, T( J4, J3+2 ),
$ LDT, CS, SN )
CALL DROT( J3-1, T( 1, J3 ), 1, T( 1, J4 ), 1, CS, SN )
IF( WANTQ )
$ CALL DROT( N, Q( 1, J3 ), 1, Q( 1, J4 ), 1, CS, SN )
END IF
*
END IF
RETURN
*
* Exit with INFO = 1 if swap was rejected.
*
50 CONTINUE
INFO = 1
RETURN
*
* End of DLAEXC
*
END
*> \brief \b DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAG2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlag2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlag2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlag2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
* WR2, WI )
*
* .. Scalar Arguments ..
* INTEGER LDA, LDB
* DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
*> problem A - w B, with scaling as necessary to avoid over-/underflow.
*>
*> The scaling factor "s" results in a modified eigenvalue equation
*>
*> s A - w B
*>
*> where s is a non-negative scaling factor chosen so that w, w B,
*> and s A do not overflow and, if possible, do not underflow, either.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, 2)
*> On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
*> is less than 1/SAFMIN. Entries less than
*> sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= 2.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, 2)
*> On entry, the 2 x 2 upper triangular matrix B. It is
*> assumed that the one-norm of B is less than 1/SAFMIN. The
*> diagonals should be at least sqrt(SAFMIN) times the largest
*> element of B (in absolute value); if a diagonal is smaller
*> than that, then +/- sqrt(SAFMIN) will be used instead of
*> that diagonal.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= 2.
*> \endverbatim
*>
*> \param[in] SAFMIN
*> \verbatim
*> SAFMIN is DOUBLE PRECISION
*> The smallest positive number s.t. 1/SAFMIN does not
*> overflow. (This should always be DLAMCH('S') -- it is an
*> argument in order to avoid having to call DLAMCH frequently.)
*> \endverbatim
*>
*> \param[out] SCALE1
*> \verbatim
*> SCALE1 is DOUBLE PRECISION
*> A scaling factor used to avoid over-/underflow in the
*> eigenvalue equation which defines the first eigenvalue. If
*> the eigenvalues are complex, then the eigenvalues are
*> ( WR1 +/- WI i ) / SCALE1 (which may lie outside the
*> exponent range of the machine), SCALE1=SCALE2, and SCALE1
*> will always be positive. If the eigenvalues are real, then
*> the first (real) eigenvalue is WR1 / SCALE1 , but this may
*> overflow or underflow, and in fact, SCALE1 may be zero or
*> less than the underflow threshold if the exact eigenvalue
*> is sufficiently large.
*> \endverbatim
*>
*> \param[out] SCALE2
*> \verbatim
*> SCALE2 is DOUBLE PRECISION
*> A scaling factor used to avoid over-/underflow in the
*> eigenvalue equation which defines the second eigenvalue. If
*> the eigenvalues are complex, then SCALE2=SCALE1. If the
*> eigenvalues are real, then the second (real) eigenvalue is
*> WR2 / SCALE2 , but this may overflow or underflow, and in
*> fact, SCALE2 may be zero or less than the underflow
*> threshold if the exact eigenvalue is sufficiently large.
*> \endverbatim
*>
*> \param[out] WR1
*> \verbatim
*> WR1 is DOUBLE PRECISION
*> If the eigenvalue is real, then WR1 is SCALE1 times the
*> eigenvalue closest to the (2,2) element of A B**(-1). If the
*> eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
*> part of the eigenvalues.
*> \endverbatim
*>
*> \param[out] WR2
*> \verbatim
*> WR2 is DOUBLE PRECISION
*> If the eigenvalue is real, then WR2 is SCALE2 times the
*> other eigenvalue. If the eigenvalue is complex, then
*> WR1=WR2 is SCALE1 times the real part of the eigenvalues.
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION
*> If the eigenvalue is real, then WI is zero. If the
*> eigenvalue is complex, then WI is SCALE1 times the imaginary
*> part of the eigenvalues. WI will always be non-negative.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lag2
*
* =====================================================================
SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
$ WR2, WI )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = ONE / TWO )
DOUBLE PRECISION FUZZY1
PARAMETER ( FUZZY1 = ONE+1.0D-5 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
$ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
$ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
$ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
$ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
$ WSCALE, WSIZE, WSMALL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
RTMIN = SQRT( SAFMIN )
RTMAX = ONE / RTMIN
SAFMAX = ONE / SAFMIN
*
* Scale A
*
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
$ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
ASCALE = ONE / ANORM
A11 = ASCALE*A( 1, 1 )
A21 = ASCALE*A( 2, 1 )
A12 = ASCALE*A( 1, 2 )
A22 = ASCALE*A( 2, 2 )
*
* Perturb B if necessary to insure non-singularity
*
B11 = B( 1, 1 )
B12 = B( 1, 2 )
B22 = B( 2, 2 )
BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
IF( ABS( B11 ).LT.BMIN )
$ B11 = SIGN( BMIN, B11 )
IF( ABS( B22 ).LT.BMIN )
$ B22 = SIGN( BMIN, B22 )
*
* Scale B
*
BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
BSCALE = ONE / BSIZE
B11 = B11*BSCALE
B12 = B12*BSCALE
B22 = B22*BSCALE
*
* Compute larger eigenvalue by method described by C. van Loan
*
* ( AS is A shifted by -SHIFT*B )
*
BINV11 = ONE / B11
BINV22 = ONE / B22
S1 = A11*BINV11
S2 = A22*BINV22
IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
AS12 = A12 - S1*B12
AS22 = A22 - S1*B22
SS = A21*( BINV11*BINV22 )
ABI22 = AS22*BINV22 - SS*B12
PP = HALF*ABI22
SHIFT = S1
ELSE
AS12 = A12 - S2*B12
AS11 = A11 - S2*B11
SS = A21*( BINV11*BINV22 )
ABI22 = -SS*B12
PP = HALF*( AS11*BINV11+ABI22 )
SHIFT = S2
END IF
QQ = SS*AS12
IF( ABS( PP*RTMIN ).GE.ONE ) THEN
DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
R = SQRT( ABS( DISCR ) )*RTMAX
ELSE
IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
R = SQRT( ABS( DISCR ) )*RTMIN
ELSE
DISCR = PP**2 + QQ
R = SQRT( ABS( DISCR ) )
END IF
END IF
*
* Note: the test of R in the following IF is to cover the case when
* DISCR is small and negative and is flushed to zero during
* the calculation of R. On machines which have a consistent
* flush-to-zero threshold and handle numbers above that
* threshold correctly, it would not be necessary.
*
IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
SUM = PP + SIGN( R, PP )
DIFF = PP - SIGN( R, PP )
WBIG = SHIFT + SUM
*
* Compute smaller eigenvalue
*
WSMALL = SHIFT + DIFF
IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
WSMALL = WDET / WBIG
END IF
*
* Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
* for WR1.
*
IF( PP.GT.ABI22 ) THEN
WR1 = MIN( WBIG, WSMALL )
WR2 = MAX( WBIG, WSMALL )
ELSE
WR1 = MAX( WBIG, WSMALL )
WR2 = MIN( WBIG, WSMALL )
END IF
WI = ZERO
ELSE
*
* Complex eigenvalues
*
WR1 = SHIFT + PP
WR2 = WR1
WI = R
END IF
*
* Further scaling to avoid underflow and overflow in computing
* SCALE1 and overflow in computing w*B.
*
* This scale factor (WSCALE) is bounded from above using C1 and C2,
* and from below using C3 and C4.
* C1 implements the condition s A must never overflow.
* C2 implements the condition w B must never overflow.
* C3, with C2,
* implement the condition that s A - w B must never overflow.
* C4 implements the condition s should not underflow.
* C5 implements the condition max(s,|w|) should be at least 2.
*
C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
C2 = SAFMIN*MAX( ONE, BNORM )
C3 = BSIZE*SAFMIN
IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
ELSE
C4 = ONE
END IF
IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
C5 = MIN( ONE, ASCALE*BSIZE )
ELSE
C5 = ONE
END IF
*
* Scale first eigenvalue
*
WABS = ABS( WR1 ) + ABS( WI )
WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
$ MIN( C4, HALF*MAX( WABS, C5 ) ) )
IF( WSIZE.NE.ONE ) THEN
WSCALE = ONE / WSIZE
IF( WSIZE.GT.ONE ) THEN
SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
$ MIN( ASCALE, BSIZE )
ELSE
SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
$ MAX( ASCALE, BSIZE )
END IF
WR1 = WR1*WSCALE
IF( WI.NE.ZERO ) THEN
WI = WI*WSCALE
WR2 = WR1
SCALE2 = SCALE1
END IF
ELSE
SCALE1 = ASCALE*BSIZE
SCALE2 = SCALE1
END IF
*
* Scale second eigenvalue (if real)
*
IF( WI.EQ.ZERO ) THEN
WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
$ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
IF( WSIZE.NE.ONE ) THEN
WSCALE = ONE / WSIZE
IF( WSIZE.GT.ONE ) THEN
SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
$ MIN( ASCALE, BSIZE )
ELSE
SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
$ MAX( ASCALE, BSIZE )
END IF
WR2 = WR2*WSCALE
ELSE
SCALE2 = ASCALE*BSIZE
END IF
END IF
*
* End of DLAG2
*
RETURN
END
*> \brief \b DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlags2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlags2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlags2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
* SNV, CSQ, SNQ )
*
* .. Scalar Arguments ..
* LOGICAL UPPER
* DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
* $ SNU, SNV
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
*> that if ( UPPER ) then
*>
*> U**T *A*Q = U**T *( A1 A2 )*Q = ( x 0 )
*> ( 0 A3 ) ( x x )
*> and
*> V**T*B*Q = V**T *( B1 B2 )*Q = ( x 0 )
*> ( 0 B3 ) ( x x )
*>
*> or if ( .NOT.UPPER ) then
*>
*> U**T *A*Q = U**T *( A1 0 )*Q = ( x x )
*> ( A2 A3 ) ( 0 x )
*> and
*> V**T*B*Q = V**T*( B1 0 )*Q = ( x x )
*> ( B2 B3 ) ( 0 x )
*>
*> The rows of the transformed A and B are parallel, where
*>
*> U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ )
*> ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ )
*>
*> Z**T denotes the transpose of Z.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPPER
*> \verbatim
*> UPPER is LOGICAL
*> = .TRUE.: the input matrices A and B are upper triangular.
*> = .FALSE.: the input matrices A and B are lower triangular.
*> \endverbatim
*>
*> \param[in] A1
*> \verbatim
*> A1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] A2
*> \verbatim
*> A2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] A3
*> \verbatim
*> A3 is DOUBLE PRECISION
*> On entry, A1, A2 and A3 are elements of the input 2-by-2
*> upper (lower) triangular matrix A.
*> \endverbatim
*>
*> \param[in] B1
*> \verbatim
*> B1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B2
*> \verbatim
*> B2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B3
*> \verbatim
*> B3 is DOUBLE PRECISION
*> On entry, B1, B2 and B3 are elements of the input 2-by-2
*> upper (lower) triangular matrix B.
*> \endverbatim
*>
*> \param[out] CSU
*> \verbatim
*> CSU is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNU
*> \verbatim
*> SNU is DOUBLE PRECISION
*> The desired orthogonal matrix U.
*> \endverbatim
*>
*> \param[out] CSV
*> \verbatim
*> CSV is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNV
*> \verbatim
*> SNV is DOUBLE PRECISION
*> The desired orthogonal matrix V.
*> \endverbatim
*>
*> \param[out] CSQ
*> \verbatim
*> CSQ is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNQ
*> \verbatim
*> SNQ is DOUBLE PRECISION
*> The desired orthogonal matrix Q.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lags2
*
* =====================================================================
SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
$ CSV,
$ SNV, CSQ, SNQ )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL UPPER
DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
$ SNU, SNV
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12,
$ AVB21, AVB22, B, C, CSL, CSR, D, R, S1, S2,
$ SNL, SNR, UA11, UA11R, UA12, UA21, UA22, UA22R,
$ VB11, VB11R, VB12, VB21, VB22, VB22R
* ..
* .. External Subroutines ..
EXTERNAL DLARTG, DLASV2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
IF( UPPER ) THEN
*
* Input matrices A and B are upper triangular matrices
*
* Form matrix C = A*adj(B) = ( a b )
* ( 0 d )
*
A = A1*B3
D = A3*B1
B = A2*B1 - A1*B2
*
* The SVD of real 2-by-2 triangular C
*
* ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 )
* ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T )
*
CALL DLASV2( A, B, D, S1, S2, SNR, CSR, SNL, CSL )
*
IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
$ THEN
*
* Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
* and (1,2) element of |U|**T *|A| and |V|**T *|B|.
*
UA11R = CSL*A1
UA12 = CSL*A2 + SNL*A3
*
VB11R = CSR*B1
VB12 = CSR*B2 + SNR*B3
*
AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 )
AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 )
*
* zero (1,2) elements of U**T *A and V**T *B
*
IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN
IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 /
$ ( ABS( VB11R )+ABS( VB12 ) ) ) THEN
CALL DLARTG( -UA11R, UA12, CSQ, SNQ, R )
ELSE
CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R )
END IF
*
CSU = CSL
SNU = -SNL
CSV = CSR
SNV = -SNR
*
ELSE
*
* Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
* and (2,2) element of |U|**T *|A| and |V|**T *|B|.
*
UA21 = -SNL*A1
UA22 = -SNL*A2 + CSL*A3
*
VB21 = -SNR*B1
VB22 = -SNR*B2 + CSR*B3
*
AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 )
AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 )
*
* zero (2,2) elements of U**T*A and V**T*B, and then swap.
*
IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN
IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 /
$ ( ABS( VB21 )+ABS( VB22 ) ) ) THEN
CALL DLARTG( -UA21, UA22, CSQ, SNQ, R )
ELSE
CALL DLARTG( -VB21, VB22, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( -VB21, VB22, CSQ, SNQ, R )
END IF
*
CSU = SNL
SNU = CSL
CSV = SNR
SNV = CSR
*
END IF
*
ELSE
*
* Input matrices A and B are lower triangular matrices
*
* Form matrix C = A*adj(B) = ( a 0 )
* ( c d )
*
A = A1*B3
D = A3*B1
C = A2*B3 - A3*B2
*
* The SVD of real 2-by-2 triangular C
*
* ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 )
* ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T )
*
CALL DLASV2( A, C, D, S1, S2, SNR, CSR, SNL, CSL )
*
IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
$ THEN
*
* Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
* and (2,1) element of |U|**T *|A| and |V|**T *|B|.
*
UA21 = -SNR*A1 + CSR*A2
UA22R = CSR*A3
*
VB21 = -SNL*B1 + CSL*B2
VB22R = CSL*B3
*
AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 )
AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 )
*
* zero (2,1) elements of U**T *A and V**T *B.
*
IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN
IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 /
$ ( ABS( VB21 )+ABS( VB22R ) ) ) THEN
CALL DLARTG( UA22R, UA21, CSQ, SNQ, R )
ELSE
CALL DLARTG( VB22R, VB21, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( VB22R, VB21, CSQ, SNQ, R )
END IF
*
CSU = CSR
SNU = -SNR
CSV = CSL
SNV = -SNL
*
ELSE
*
* Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
* and (1,1) element of |U|**T *|A| and |V|**T *|B|.
*
UA11 = CSR*A1 + SNR*A2
UA12 = SNR*A3
*
VB11 = CSL*B1 + SNL*B2
VB12 = SNL*B3
*
AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 )
AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 )
*
* zero (1,1) elements of U**T*A and V**T*B, and then swap.
*
IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN
IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 /
$ ( ABS( VB11 )+ABS( VB12 ) ) ) THEN
CALL DLARTG( UA12, UA11, CSQ, SNQ, R )
ELSE
CALL DLARTG( VB12, VB11, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( VB12, VB11, CSQ, SNQ, R )
END IF
*
CSU = SNR
SNU = CSR
CSV = SNL
SNV = CSL
*
END IF
*
END IF
*
RETURN
*
* End of DLAGS2
*
END
*> \brief \b DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagtf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagtf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagtf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* DOUBLE PRECISION LAMBDA, TOL
* ..
* .. Array Arguments ..
* INTEGER IN( * )
* DOUBLE PRECISION A( * ), B( * ), C( * ), D( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
*> tridiagonal matrix and lambda is a scalar, as
*>
*> T - lambda*I = PLU,
*>
*> where P is a permutation matrix, L is a unit lower tridiagonal matrix
*> with at most one non-zero sub-diagonal elements per column and U is
*> an upper triangular matrix with at most two non-zero super-diagonal
*> elements per column.
*>
*> The factorization is obtained by Gaussian elimination with partial
*> pivoting and implicit row scaling.
*>
*> The parameter LAMBDA is included in the routine so that DLAGTF may
*> be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
*> inverse iteration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (N)
*> On entry, A must contain the diagonal elements of T.
*>
*> On exit, A is overwritten by the n diagonal elements of the
*> upper triangular matrix U of the factorization of T.
*> \endverbatim
*>
*> \param[in] LAMBDA
*> \verbatim
*> LAMBDA is DOUBLE PRECISION
*> On entry, the scalar lambda.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (N-1)
*> On entry, B must contain the (n-1) super-diagonal elements of
*> T.
*>
*> On exit, B is overwritten by the (n-1) super-diagonal
*> elements of the matrix U of the factorization of T.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N-1)
*> On entry, C must contain the (n-1) sub-diagonal elements of
*> T.
*>
*> On exit, C is overwritten by the (n-1) sub-diagonal elements
*> of the matrix L of the factorization of T.
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> On entry, a relative tolerance used to indicate whether or
*> not the matrix (T - lambda*I) is nearly singular. TOL should
*> normally be chose as approximately the largest relative error
*> in the elements of T. For example, if the elements of T are
*> correct to about 4 significant figures, then TOL should be
*> set to about 5*10**(-4). If TOL is supplied as less than eps,
*> where eps is the relative machine precision, then the value
*> eps is used in place of TOL.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N-2)
*> On exit, D is overwritten by the (n-2) second super-diagonal
*> elements of the matrix U of the factorization of T.
*> \endverbatim
*>
*> \param[out] IN
*> \verbatim
*> IN is INTEGER array, dimension (N)
*> On exit, IN contains details of the permutation matrix P. If
*> an interchange occurred at the kth step of the elimination,
*> then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
*> returns the smallest positive integer j such that
*>
*> abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
*>
*> where norm( A(j) ) denotes the sum of the absolute values of
*> the jth row of the matrix A. If no such j exists then IN(n)
*> is returned as zero. If IN(n) is returned as positive, then a
*> diagonal element of U is small, indicating that
*> (T - lambda*I) is singular or nearly singular,
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the kth argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagtf
*
* =====================================================================
SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
DOUBLE PRECISION LAMBDA, TOL
* ..
* .. Array Arguments ..
INTEGER IN( * )
DOUBLE PRECISION A( * ), B( * ), C( * ), D( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER K
DOUBLE PRECISION EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DLAGTF', -INFO )
RETURN
END IF
*
IF( N.EQ.0 )
$ RETURN
*
A( 1 ) = A( 1 ) - LAMBDA
IN( N ) = 0
IF( N.EQ.1 ) THEN
IF( A( 1 ).EQ.ZERO )
$ IN( 1 ) = 1
RETURN
END IF
*
EPS = DLAMCH( 'Epsilon' )
*
TL = MAX( TOL, EPS )
SCALE1 = ABS( A( 1 ) ) + ABS( B( 1 ) )
DO 10 K = 1, N - 1
A( K+1 ) = A( K+1 ) - LAMBDA
SCALE2 = ABS( C( K ) ) + ABS( A( K+1 ) )
IF( K.LT.( N-1 ) )
$ SCALE2 = SCALE2 + ABS( B( K+1 ) )
IF( A( K ).EQ.ZERO ) THEN
PIV1 = ZERO
ELSE
PIV1 = ABS( A( K ) ) / SCALE1
END IF
IF( C( K ).EQ.ZERO ) THEN
IN( K ) = 0
PIV2 = ZERO
SCALE1 = SCALE2
IF( K.LT.( N-1 ) )
$ D( K ) = ZERO
ELSE
PIV2 = ABS( C( K ) ) / SCALE2
IF( PIV2.LE.PIV1 ) THEN
IN( K ) = 0
SCALE1 = SCALE2
C( K ) = C( K ) / A( K )
A( K+1 ) = A( K+1 ) - C( K )*B( K )
IF( K.LT.( N-1 ) )
$ D( K ) = ZERO
ELSE
IN( K ) = 1
MULT = A( K ) / C( K )
A( K ) = C( K )
TEMP = A( K+1 )
A( K+1 ) = B( K ) - MULT*TEMP
IF( K.LT.( N-1 ) ) THEN
D( K ) = B( K+1 )
B( K+1 ) = -MULT*D( K )
END IF
B( K ) = TEMP
C( K ) = MULT
END IF
END IF
IF( ( MAX( PIV1, PIV2 ).LE.TL ) .AND. ( IN( N ).EQ.0 ) )
$ IN( N ) = K
10 CONTINUE
IF( ( ABS( A( N ) ).LE.SCALE1*TL ) .AND. ( IN( N ).EQ.0 ) )
$ IN( N ) = N
*
RETURN
*
* End of DLAGTF
*
END
*> \brief \b DLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGTM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagtm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagtm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagtm.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAGTM( TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA,
* B, LDB )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER LDB, LDX, N, NRHS
* DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAGTM performs a matrix-matrix product of the form
*>
*> B := alpha * A * X + beta * B
*>
*> where A is a tridiagonal matrix of order N, B and X are N by NRHS
*> matrices, and alpha and beta are real scalars, each of which may be
*> 0., 1., or -1.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': No transpose, B := alpha * A * X + beta * B
*> = 'T': Transpose, B := alpha * A'* X + beta * B
*> = 'C': Conjugate transpose = Transpose
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices X and B.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> The scalar alpha. ALPHA must be 0., 1., or -1.; otherwise,
*> it is assumed to be 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) sub-diagonal elements of T.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of T.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) super-diagonal elements of T.
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> The N by NRHS matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(N,1).
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> The scalar beta. BETA must be 0., 1., or -1.; otherwise,
*> it is assumed to be 1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N by NRHS matrix B.
*> On exit, B is overwritten by the matrix expression
*> B := alpha * A * X + beta * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(N,1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagtm
*
* =====================================================================
SUBROUTINE DLAGTM( TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX,
$ BETA,
$ B, LDB )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER LDB, LDX, N, NRHS
DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 )
$ RETURN
*
* Multiply B by BETA if BETA.NE.1.
*
IF( BETA.EQ.ZERO ) THEN
DO 20 J = 1, NRHS
DO 10 I = 1, N
B( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE IF( BETA.EQ.-ONE ) THEN
DO 40 J = 1, NRHS
DO 30 I = 1, N
B( I, J ) = -B( I, J )
30 CONTINUE
40 CONTINUE
END IF
*
IF( ALPHA.EQ.ONE ) THEN
IF( LSAME( TRANS, 'N' ) ) THEN
*
* Compute B := B + A*X
*
DO 60 J = 1, NRHS
IF( N.EQ.1 ) THEN
B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J )
ELSE
B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) +
$ DU( 1 )*X( 2, J )
B( N, J ) = B( N, J ) + DL( N-1 )*X( N-1, J ) +
$ D( N )*X( N, J )
DO 50 I = 2, N - 1
B( I, J ) = B( I, J ) + DL( I-1 )*X( I-1, J ) +
$ D( I )*X( I, J ) + DU( I )*X( I+1, J )
50 CONTINUE
END IF
60 CONTINUE
ELSE
*
* Compute B := B + A**T*X
*
DO 80 J = 1, NRHS
IF( N.EQ.1 ) THEN
B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J )
ELSE
B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) +
$ DL( 1 )*X( 2, J )
B( N, J ) = B( N, J ) + DU( N-1 )*X( N-1, J ) +
$ D( N )*X( N, J )
DO 70 I = 2, N - 1
B( I, J ) = B( I, J ) + DU( I-1 )*X( I-1, J ) +
$ D( I )*X( I, J ) + DL( I )*X( I+1, J )
70 CONTINUE
END IF
80 CONTINUE
END IF
ELSE IF( ALPHA.EQ.-ONE ) THEN
IF( LSAME( TRANS, 'N' ) ) THEN
*
* Compute B := B - A*X
*
DO 100 J = 1, NRHS
IF( N.EQ.1 ) THEN
B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J )
ELSE
B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) -
$ DU( 1 )*X( 2, J )
B( N, J ) = B( N, J ) - DL( N-1 )*X( N-1, J ) -
$ D( N )*X( N, J )
DO 90 I = 2, N - 1
B( I, J ) = B( I, J ) - DL( I-1 )*X( I-1, J ) -
$ D( I )*X( I, J ) - DU( I )*X( I+1, J )
90 CONTINUE
END IF
100 CONTINUE
ELSE
*
* Compute B := B - A**T*X
*
DO 120 J = 1, NRHS
IF( N.EQ.1 ) THEN
B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J )
ELSE
B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) -
$ DL( 1 )*X( 2, J )
B( N, J ) = B( N, J ) - DU( N-1 )*X( N-1, J ) -
$ D( N )*X( N, J )
DO 110 I = 2, N - 1
B( I, J ) = B( I, J ) - DU( I-1 )*X( I-1, J ) -
$ D( I )*X( I, J ) - DL( I )*X( I+1, J )
110 CONTINUE
END IF
120 CONTINUE
END IF
END IF
RETURN
*
* End of DLAGTM
*
END
*> \brief \b DLAGTS solves the system of equations (T-λI)x = y
*> or (T-λI)^Tx = y, where T is a general tridiagonal matrix
*> and λ a scalar, using the LU factorization computed by slagtf.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGTS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagts.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagts.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagts.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, JOB, N
* DOUBLE PRECISION TOL
* ..
* .. Array Arguments ..
* INTEGER IN( * )
* DOUBLE PRECISION A( * ), B( * ), C( * ), D( * ), Y( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAGTS may be used to solve one of the systems of equations
*>
*> (T - lambda*I)*x = y or (T - lambda*I)**T*x = y,
*>
*> where T is an n by n tridiagonal matrix, for x, following the
*> factorization of (T - lambda*I) as
*>
*> (T - lambda*I) = P*L*U ,
*>
*> by routine DLAGTF. The choice of equation to be solved is
*> controlled by the argument JOB, and in each case there is an option
*> to perturb zero or very small diagonal elements of U, this option
*> being intended for use in applications such as inverse iteration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is INTEGER
*> Specifies the job to be performed by DLAGTS as follows:
*> = 1: The equations (T - lambda*I)x = y are to be solved,
*> but diagonal elements of U are not to be perturbed.
*> = -1: The equations (T - lambda*I)x = y are to be solved
*> and, if overflow would otherwise occur, the diagonal
*> elements of U are to be perturbed. See argument TOL
*> below.
*> = 2: The equations (T - lambda*I)**Tx = y are to be solved,
*> but diagonal elements of U are not to be perturbed.
*> = -2: The equations (T - lambda*I)**Tx = y are to be solved
*> and, if overflow would otherwise occur, the diagonal
*> elements of U are to be perturbed. See argument TOL
*> below.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (N)
*> On entry, A must contain the diagonal elements of U as
*> returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (N-1)
*> On entry, B must contain the first super-diagonal elements of
*> U as returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N-1)
*> On entry, C must contain the sub-diagonal elements of L as
*> returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N-2)
*> On entry, D must contain the second super-diagonal elements
*> of U as returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] IN
*> \verbatim
*> IN is INTEGER array, dimension (N)
*> On entry, IN must contain details of the matrix P as returned
*> from DLAGTF.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (N)
*> On entry, the right hand side vector y.
*> On exit, Y is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[in,out] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> On entry, with JOB < 0, TOL should be the minimum
*> perturbation to be made to very small diagonal elements of U.
*> TOL should normally be chosen as about eps*norm(U), where eps
*> is the relative machine precision, but if TOL is supplied as
*> non-positive, then it is reset to eps*max( abs( u(i,j) ) ).
*> If JOB > 0 then TOL is not referenced.
*>
*> On exit, TOL is changed as described above, only if TOL is
*> non-positive on entry. Otherwise TOL is unchanged.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: overflow would occur when computing the INFO(th)
*> element of the solution vector x. This can only occur
*> when JOB is supplied as positive and either means
*> that a diagonal element of U is very small, or that
*> the elements of the right-hand side vector y are very
*> large.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagts
*
* =====================================================================
SUBROUTINE DLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, JOB, N
DOUBLE PRECISION TOL
* ..
* .. Array Arguments ..
INTEGER IN( * )
DOUBLE PRECISION A( * ), B( * ), C( * ), D( * ), Y( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER K
DOUBLE PRECISION ABSAK, AK, BIGNUM, EPS, PERT, SFMIN, TEMP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( ( ABS( JOB ).GT.2 ) .OR. ( JOB.EQ.0 ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAGTS', -INFO )
RETURN
END IF
*
IF( N.EQ.0 )
$ RETURN
*
EPS = DLAMCH( 'Epsilon' )
SFMIN = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SFMIN
*
IF( JOB.LT.0 ) THEN
IF( TOL.LE.ZERO ) THEN
TOL = ABS( A( 1 ) )
IF( N.GT.1 )
$ TOL = MAX( TOL, ABS( A( 2 ) ), ABS( B( 1 ) ) )
DO 10 K = 3, N
TOL = MAX( TOL, ABS( A( K ) ), ABS( B( K-1 ) ),
$ ABS( D( K-2 ) ) )
10 CONTINUE
TOL = TOL*EPS
IF( TOL.EQ.ZERO )
$ TOL = EPS
END IF
END IF
*
IF( ABS( JOB ).EQ.1 ) THEN
DO 20 K = 2, N
IF( IN( K-1 ).EQ.0 ) THEN
Y( K ) = Y( K ) - C( K-1 )*Y( K-1 )
ELSE
TEMP = Y( K-1 )
Y( K-1 ) = Y( K )
Y( K ) = TEMP - C( K-1 )*Y( K )
END IF
20 CONTINUE
IF( JOB.EQ.1 ) THEN
DO 30 K = N, 1, -1
IF( K.LE.N-2 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 )
ELSE IF( K.EQ.N-1 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
INFO = K
RETURN
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
INFO = K
RETURN
END IF
END IF
Y( K ) = TEMP / AK
30 CONTINUE
ELSE
DO 50 K = N, 1, -1
IF( K.LE.N-2 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 )
ELSE IF( K.EQ.N-1 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
PERT = SIGN( TOL, AK )
40 CONTINUE
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 40
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 40
END IF
END IF
Y( K ) = TEMP / AK
50 CONTINUE
END IF
ELSE
*
* Come to here if JOB = 2 or -2
*
IF( JOB.EQ.2 ) THEN
DO 60 K = 1, N
IF( K.GE.3 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 )
ELSE IF( K.EQ.2 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
INFO = K
RETURN
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
INFO = K
RETURN
END IF
END IF
Y( K ) = TEMP / AK
60 CONTINUE
ELSE
DO 80 K = 1, N
IF( K.GE.3 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 )
ELSE IF( K.EQ.2 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
PERT = SIGN( TOL, AK )
70 CONTINUE
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 70
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 70
END IF
END IF
Y( K ) = TEMP / AK
80 CONTINUE
END IF
*
DO 90 K = N, 2, -1
IF( IN( K-1 ).EQ.0 ) THEN
Y( K-1 ) = Y( K-1 ) - C( K-1 )*Y( K )
ELSE
TEMP = Y( K-1 )
Y( K-1 ) = Y( K )
Y( K ) = TEMP - C( K-1 )*Y( K )
END IF
90 CONTINUE
END IF
*
* End of DLAGTS
*
END
*> \brief \b DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
* CSR, SNR )
*
* .. Scalar Arguments ..
* INTEGER LDA, LDB
* DOUBLE PRECISION CSL, CSR, SNL, SNR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
* $ B( LDB, * ), BETA( 2 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
*> matrix pencil (A,B) where B is upper triangular. This routine
*> computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
*> SNR such that
*>
*> 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
*> types), then
*>
*> [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
*> [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
*>
*> [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
*> [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ],
*>
*> 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
*> then
*>
*> [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
*> [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
*>
*> [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
*> [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ]
*>
*> where b11 >= b22 > 0.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, 2)
*> On entry, the 2 x 2 matrix A.
*> On exit, A is overwritten by the ``A-part'' of the
*> generalized Schur form.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> THe leading dimension of the array A. LDA >= 2.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, 2)
*> On entry, the upper triangular 2 x 2 matrix B.
*> On exit, B is overwritten by the ``B-part'' of the
*> generalized Schur form.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> THe leading dimension of the array B. LDB >= 2.
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (2)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (2)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (2)
*> (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
*> pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may
*> be zero.
*> \endverbatim
*>
*> \param[out] CSL
*> \verbatim
*> CSL is DOUBLE PRECISION
*> The cosine of the left rotation matrix.
*> \endverbatim
*>
*> \param[out] SNL
*> \verbatim
*> SNL is DOUBLE PRECISION
*> The sine of the left rotation matrix.
*> \endverbatim
*>
*> \param[out] CSR
*> \verbatim
*> CSR is DOUBLE PRECISION
*> The cosine of the right rotation matrix.
*> \endverbatim
*>
*> \param[out] SNR
*> \verbatim
*> SNR is DOUBLE PRECISION
*> The sine of the right rotation matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagv2
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL,
$ SNL,
$ CSR, SNR )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION CSL, CSR, SNL, SNR
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
$ B( LDB, * ), BETA( 2 )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ,
$ R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1,
$ WR2
* ..
* .. External Subroutines ..
EXTERNAL DLAG2, DLARTG, DLASV2, DROT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
SAFMIN = DLAMCH( 'S' )
ULP = DLAMCH( 'P' )
*
* Scale A
*
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
$ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
ASCALE = ONE / ANORM
A( 1, 1 ) = ASCALE*A( 1, 1 )
A( 1, 2 ) = ASCALE*A( 1, 2 )
A( 2, 1 ) = ASCALE*A( 2, 1 )
A( 2, 2 ) = ASCALE*A( 2, 2 )
*
* Scale B
*
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
$ SAFMIN )
BSCALE = ONE / BNORM
B( 1, 1 ) = BSCALE*B( 1, 1 )
B( 1, 2 ) = BSCALE*B( 1, 2 )
B( 2, 2 ) = BSCALE*B( 2, 2 )
*
* Check if A can be deflated
*
IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN
CSL = ONE
SNL = ZERO
CSR = ONE
SNR = ZERO
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
WI = ZERO
*
* Check if B is singular
*
ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN
CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
CSR = ONE
SNR = ZERO
CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
A( 2, 1 ) = ZERO
B( 1, 1 ) = ZERO
B( 2, 1 ) = ZERO
WI = ZERO
*
ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN
CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T )
SNR = -SNR
CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
CSL = ONE
SNL = ZERO
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
B( 2, 2 ) = ZERO
WI = ZERO
*
ELSE
*
* B is nonsingular, first compute the eigenvalues of (A,B)
*
CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
$ WR2,
$ WI )
*
IF( WI.EQ.ZERO ) THEN
*
* two real eigenvalues, compute s*A-w*B
*
H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 )
H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 )
H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 )
*
RR = DLAPY2( H1, H2 )
QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 )
*
IF( RR.GT.QQ ) THEN
*
* find right rotation matrix to zero 1,1 element of
* (sA - wB)
*
CALL DLARTG( H2, H1, CSR, SNR, T )
*
ELSE
*
* find right rotation matrix to zero 2,1 element of
* (sA - wB)
*
CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T )
*
END IF
*
SNR = -SNR
CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
*
* compute inf norms of A and B
*
H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ),
$ ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) )
H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
$ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
*
IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
*
* find left rotation matrix Q to zero out B(2,1)
*
CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R )
*
ELSE
*
* find left rotation matrix Q to zero out A(2,1)
*
CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
*
END IF
*
CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
*
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
*
ELSE
*
* a pair of complex conjugate eigenvalues
* first compute the SVD of the matrix B
*
CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR,
$ CSR, SNL, CSL )
*
* Form (A,B) := Q(A,B)Z**T where Q is left rotation matrix and
* Z is right rotation matrix computed from DLASV2
*
CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
*
B( 2, 1 ) = ZERO
B( 1, 2 ) = ZERO
*
END IF
*
END IF
*
* Unscaling
*
A( 1, 1 ) = ANORM*A( 1, 1 )
A( 2, 1 ) = ANORM*A( 2, 1 )
A( 1, 2 ) = ANORM*A( 1, 2 )
A( 2, 2 ) = ANORM*A( 2, 2 )
B( 1, 1 ) = BNORM*B( 1, 1 )
B( 2, 1 ) = BNORM*B( 2, 1 )
B( 1, 2 ) = BNORM*B( 1, 2 )
B( 2, 2 ) = BNORM*B( 2, 2 )
*
IF( WI.EQ.ZERO ) THEN
ALPHAR( 1 ) = A( 1, 1 )
ALPHAR( 2 ) = A( 2, 2 )
ALPHAI( 1 ) = ZERO
ALPHAI( 2 ) = ZERO
BETA( 1 ) = B( 1, 1 )
BETA( 2 ) = B( 2, 2 )
ELSE
ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM
ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM
ALPHAR( 2 ) = ALPHAR( 1 )
ALPHAI( 2 ) = -ALPHAI( 1 )
BETA( 1 ) = ONE
BETA( 2 ) = ONE
END IF
*
RETURN
*
* End of DLAGV2
*
END
*> \brief \b DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAHQR is an auxiliary routine called by DHSEQR to update the
*> eigenvalues and Schur decomposition already computed by DHSEQR, by
*> dealing with the Hessenberg submatrix in rows and columns ILO to
*> IHI.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper quasi-triangular in
*> rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
*> ILO = 1). DLAHQR works primarily with the Hessenberg
*> submatrix in rows and columns ILO to IHI, but applies
*> transformations to all of H if WANTT is .TRUE..
*> 1 <= ILO <= max(1,IHI); IHI <= N.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO is zero and if WANTT is .TRUE., H is upper
*> quasi-triangular in rows and columns ILO:IHI, with any
*> 2-by-2 diagonal blocks in standard form. If INFO is zero
*> and WANTT is .FALSE., the contents of H are unspecified on
*> exit. The output state of H if INFO is nonzero is given
*> below under the description of INFO.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues ILO to IHI are stored in the corresponding
*> elements of WR and WI. If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
*> eigenvalues are stored in the same order as on the diagonal
*> of the Schur form returned in H, with WR(i) = H(i,i), and, if
*> H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
*> WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> If WANTZ is .TRUE., on entry Z must contain the current
*> matrix Z of transformations accumulated by DHSEQR, and on
*> exit Z has been updated; transformations are applied only to
*> the submatrix Z(ILOZ:IHIZ,ILO:IHI).
*> If WANTZ is .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: If INFO = i, DLAHQR failed to compute all the
*> eigenvalues ILO to IHI in a total of 30 iterations
*> per eigenvalue; elements i+1:ihi of WR and WI
*> contain those eigenvalues which have been
*> successfully computed.
*>
*> If INFO > 0 and WANTT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the
*> eigenvalues of the upper Hessenberg matrix rows
*> and columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO > 0 and WANTT is .TRUE., then on exit
*> (*) (initial value of H)*U = U*(final value of H)
*> where U is an orthogonal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO > 0 and WANTZ is .TRUE., then on exit
*> (final value of Z) = (initial value of Z)*U
*> where U is the orthogonal matrix in (*)
*> (regardless of the value of WANTT.)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahqr
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 02-96 Based on modifications by
*> David Day, Sandia National Laboratory, USA
*>
*> 12-04 Further modifications by
*> Ralph Byers, University of Kansas, USA
*> This is a modified version of DLAHQR from LAPACK version 3.0.
*> It is (1) more robust against overflow and underflow and
*> (2) adopts the more conservative Ahues & Tisseur stopping
*> criterion (LAWN 122, 1997).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
IMPLICIT NONE
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
* =========================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0, TWO = 2.0d0 )
DOUBLE PRECISION DAT1, DAT2
PARAMETER ( DAT1 = 3.0d0 / 4.0d0, DAT2 = -0.4375d0 )
INTEGER KEXSH
PARAMETER ( KEXSH = 10 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S,
$ H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX,
$ SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST,
$ ULP, V2, V3
INTEGER I, I1, I2, ITS, ITMAX, J, K, L, M, NH, NR, NZ,
$ KDEFL
* ..
* .. Local Arrays ..
DOUBLE PRECISION V( 3 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLANV2, DLARFG, DROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( ILO.EQ.IHI ) THEN
WR( ILO ) = H( ILO, ILO )
WI( ILO ) = ZERO
RETURN
END IF
*
* ==== clear out the trash ====
DO 10 J = ILO, IHI - 3
H( J+2, J ) = ZERO
H( J+3, J ) = ZERO
10 CONTINUE
IF( ILO.LE.IHI-2 )
$ H( IHI, IHI-2 ) = ZERO
*
NH = IHI - ILO + 1
NZ = IHIZ - ILOZ + 1
*
* Set machine-dependent constants for the stopping criterion.
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( NH ) / ULP )
*
* I1 and I2 are the indices of the first row and last column of H
* to which transformations must be applied. If eigenvalues only are
* being computed, I1 and I2 are set inside the main loop.
*
IF( WANTT ) THEN
I1 = 1
I2 = N
END IF
*
* ITMAX is the total number of QR iterations allowed.
*
ITMAX = 30 * MAX( 10, NH )
*
* KDEFL counts the number of iterations since a deflation
*
KDEFL = 0
*
* The main loop begins here. I is the loop index and decreases from
* IHI to ILO in steps of 1 or 2. Each iteration of the loop works
* with the active submatrix in rows and columns L to I.
* Eigenvalues I+1 to IHI have already converged. Either L = ILO or
* H(L,L-1) is negligible so that the matrix splits.
*
I = IHI
20 CONTINUE
L = ILO
IF( I.LT.ILO )
$ GO TO 160
*
* Perform QR iterations on rows and columns ILO to I until a
* submatrix of order 1 or 2 splits off at the bottom because a
* subdiagonal element has become negligible.
*
DO 140 ITS = 0, ITMAX
*
* Look for a single small subdiagonal element.
*
DO 30 K = I, L + 1, -1
IF( ABS( H( K, K-1 ) ).LE.SMLNUM )
$ GO TO 40
TST = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
IF( TST.EQ.ZERO ) THEN
IF( K-2.GE.ILO )
$ TST = TST + ABS( H( K-1, K-2 ) )
IF( K+1.LE.IHI )
$ TST = TST + ABS( H( K+1, K ) )
END IF
* ==== The following is a conservative small subdiagonal
* . deflation criterion due to Ahues & Tisseur (LAWN 122,
* . 1997). It has better mathematical foundation and
* . improves accuracy in some cases. ====
IF( ABS( H( K, K-1 ) ).LE.ULP*TST ) THEN
AB = MAX( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
BA = MIN( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
AA = MAX( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
BB = MIN( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
S = AA + AB
IF( BA*( AB / S ).LE.MAX( SMLNUM,
$ ULP*( BB*( AA / S ) ) ) )GO TO 40
END IF
30 CONTINUE
40 CONTINUE
L = K
IF( L.GT.ILO ) THEN
*
* H(L,L-1) is negligible
*
H( L, L-1 ) = ZERO
END IF
*
* Exit from loop if a submatrix of order 1 or 2 has split off.
*
IF( L.GE.I-1 )
$ GO TO 150
KDEFL = KDEFL + 1
*
* Now the active submatrix is in rows and columns L to I. If
* eigenvalues only are being computed, only the active submatrix
* need be transformed.
*
IF( .NOT.WANTT ) THEN
I1 = L
I2 = I
END IF
*
IF( MOD(KDEFL,2*KEXSH).EQ.0 ) THEN
*
* Exceptional shift.
*
S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
H11 = DAT1*S + H( I, I )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE IF( MOD(KDEFL,KEXSH).EQ.0 ) THEN
*
* Exceptional shift.
*
S = ABS( H( L+1, L ) ) + ABS( H( L+2, L+1 ) )
H11 = DAT1*S + H( L, L )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE
*
* Prepare to use Francis' double shift
* (i.e. 2nd degree generalized Rayleigh quotient)
*
H11 = H( I-1, I-1 )
H21 = H( I, I-1 )
H12 = H( I-1, I )
H22 = H( I, I )
END IF
S = ABS( H11 ) + ABS( H12 ) + ABS( H21 ) + ABS( H22 )
IF( S.EQ.ZERO ) THEN
RT1R = ZERO
RT1I = ZERO
RT2R = ZERO
RT2I = ZERO
ELSE
H11 = H11 / S
H21 = H21 / S
H12 = H12 / S
H22 = H22 / S
TR = ( H11+H22 ) / TWO
DET = ( H11-TR )*( H22-TR ) - H12*H21
RTDISC = SQRT( ABS( DET ) )
IF( DET.GE.ZERO ) THEN
*
* ==== complex conjugate shifts ====
*
RT1R = TR*S
RT2R = RT1R
RT1I = RTDISC*S
RT2I = -RT1I
ELSE
*
* ==== real shifts (use only one of them) ====
*
RT1R = TR + RTDISC
RT2R = TR - RTDISC
IF( ABS( RT1R-H22 ).LE.ABS( RT2R-H22 ) ) THEN
RT1R = RT1R*S
RT2R = RT1R
ELSE
RT2R = RT2R*S
RT1R = RT2R
END IF
RT1I = ZERO
RT2I = ZERO
END IF
END IF
*
* Look for two consecutive small subdiagonal elements.
*
DO 50 M = I - 2, L, -1
* Determine the effect of starting the double-shift QR
* iteration at row M, and see if this would make H(M,M-1)
* negligible. (The following uses scaling to avoid
* overflows and most underflows.)
*
H21S = H( M+1, M )
S = ABS( H( M, M )-RT2R ) + ABS( RT2I ) + ABS( H21S )
H21S = H( M+1, M ) / S
V( 1 ) = H21S*H( M, M+1 ) + ( H( M, M )-RT1R )*
$ ( ( H( M, M )-RT2R ) / S ) - RT1I*( RT2I / S )
V( 2 ) = H21S*( H( M, M )+H( M+1, M+1 )-RT1R-RT2R )
V( 3 ) = H21S*H( M+2, M+1 )
S = ABS( V( 1 ) ) + ABS( V( 2 ) ) + ABS( V( 3 ) )
V( 1 ) = V( 1 ) / S
V( 2 ) = V( 2 ) / S
V( 3 ) = V( 3 ) / S
IF( M.EQ.L )
$ GO TO 60
IF( ABS( H( M, M-1 ) )*( ABS( V( 2 ) )+ABS( V( 3 ) ) ).LE.
$ ULP*ABS( V( 1 ) )*( ABS( H( M-1, M-1 ) )+ABS( H( M,
$ M ) )+ABS( H( M+1, M+1 ) ) ) )GO TO 60
50 CONTINUE
60 CONTINUE
*
* Double-shift QR step
*
DO 130 K = M, I - 1
*
* The first iteration of this loop determines a reflection G
* from the vector V and applies it from left and right to H,
* thus creating a nonzero bulge below the subdiagonal.
*
* Each subsequent iteration determines a reflection G to
* restore the Hessenberg form in the (K-1)th column, and thus
* chases the bulge one step toward the bottom of the active
* submatrix. NR is the order of G.
*
NR = MIN( 3, I-K+1 )
IF( K.GT.M )
$ CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 )
CALL DLARFG( NR, V( 1 ), V( 2 ), 1, T1 )
IF( K.GT.M ) THEN
H( K, K-1 ) = V( 1 )
H( K+1, K-1 ) = ZERO
IF( K.LT.I-1 )
$ H( K+2, K-1 ) = ZERO
ELSE IF( M.GT.L ) THEN
* ==== Use the following instead of
* . H( K, K-1 ) = -H( K, K-1 ) to
* . avoid a bug when v(2) and v(3)
* . underflow. ====
H( K, K-1 ) = H( K, K-1 )*( ONE-T1 )
END IF
V2 = V( 2 )
T2 = T1*V2
IF( NR.EQ.3 ) THEN
V3 = V( 3 )
T3 = T1*V3
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 70 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
H( K+2, J ) = H( K+2, J ) - SUM*T3
70 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 80 J = I1, MIN( K+3, I )
SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
H( J, K+2 ) = H( J, K+2 ) - SUM*T3
80 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 90 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3
90 CONTINUE
END IF
ELSE IF( NR.EQ.2 ) THEN
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 100 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
100 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 110 J = I1, I
SUM = H( J, K ) + V2*H( J, K+1 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
110 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 120 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
120 CONTINUE
END IF
END IF
130 CONTINUE
*
140 CONTINUE
*
* Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
150 CONTINUE
*
IF( L.EQ.I ) THEN
*
* H(I,I-1) is negligible: one eigenvalue has converged.
*
WR( I ) = H( I, I )
WI( I ) = ZERO
ELSE IF( L.EQ.I-1 ) THEN
*
* H(I-1,I-2) is negligible: a pair of eigenvalues have converged.
*
* Transform the 2-by-2 submatrix to standard Schur form,
* and compute and store the eigenvalues.
*
CALL DLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ),
$ H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ),
$ CS, SN )
*
IF( WANTT ) THEN
*
* Apply the transformation to the rest of H.
*
IF( I2.GT.I )
$ CALL DROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH,
$ CS, SN )
CALL DROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS,
$ SN )
END IF
IF( WANTZ ) THEN
*
* Apply the transformation to Z.
*
CALL DROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS,
$ SN )
END IF
END IF
* reset deflation counter
KDEFL = 0
*
* return to start of the main loop with new value of I.
*
I = L - 1
GO TO 20
*
160 CONTINUE
RETURN
*
* End of DLAHQR
*
END
*> \brief \b DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* .. Scalar Arguments ..
* INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
* $ Y( LDY, NB )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
*> matrix A so that elements below the k-th subdiagonal are zero. The
*> reduction is performed by an orthogonal similarity transformation
*> Q**T * A * Q. The routine returns the matrices V and T which determine
*> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
*>
*> This is an auxiliary routine called by DGEHRD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The offset for the reduction. Elements below the k-th
*> subdiagonal in the first NB columns are reduced to zero.
*> K < N.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
*> On entry, the n-by-(n-k+1) general matrix A.
*> On exit, the elements on and above the k-th subdiagonal in
*> the first NB columns are overwritten with the corresponding
*> elements of the reduced matrix; the elements below the k-th
*> subdiagonal, with the array TAU, represent the matrix Q as a
*> product of elementary reflectors. The other columns of A are
*> unchanged. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (NB)
*> The scalar factors of the elementary reflectors. See Further
*> Details.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NB)
*> The upper triangular matrix T.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (LDY,NB)
*> The n-by-nb matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*> LDY is INTEGER
*> The leading dimension of the array Y. LDY >= N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahr2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of nb elementary reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*> A(i+k+1:n,i), and tau in TAU(i).
*>
*> The elements of the vectors v together form the (n-k+1)-by-nb matrix
*> V which is needed, with T and Y, to apply the transformation to the
*> unreduced part of the matrix, using an update of the form:
*> A := (I - V*T*V**T) * (A - Y*V**T).
*>
*> The contents of A on exit are illustrated by the following example
*> with n = 7, k = 3 and nb = 2:
*>
*> ( a a a a a )
*> ( a a a a a )
*> ( a a a a a )
*> ( h h a a a )
*> ( v1 h a a a )
*> ( v1 v2 a a a )
*> ( v1 v2 a a a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*>
*> This subroutine is a slight modification of LAPACK-3.0's DLAHRD
*> incorporating improvements proposed by Quintana-Orti and Van de
*> Gejin. Note that the entries of A(1:K,2:NB) differ from those
*> returned by the original LAPACK-3.0's DLAHRD routine. (This
*> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
*> \endverbatim
*
*> \par References:
* ================
*>
*> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
*> performance of reduction to Hessenberg form," ACM Transactions on
*> Mathematical Software, 32(2):180-194, June 2006.
*>
* =====================================================================
SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
$ Y( LDY, NB )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0,
$ ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
$ DLARFG, DSCAL, DTRMM, DTRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
DO 10 I = 1, NB
IF( I.GT.1 ) THEN
*
* Update A(K+1:N,I)
*
* Update I-th column of A - Y * V**T
*
CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1),
$ LDY,
$ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
*
* Apply I - V * T**T * V**T to this column (call it b) from the
* left, using the last column of T as workspace
*
* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
* ( V2 ) ( b2 )
*
* where V1 is unit lower triangular
*
* w := V1**T * b1
*
CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
$ I-1, A( K+1, 1 ),
$ LDA, T( 1, NB ), 1 )
*
* w := w + V2**T * b2
*
CALL DGEMV( 'Transpose', N-K-I+1, I-1,
$ ONE, A( K+I, 1 ),
$ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
*
* w := T**T * w
*
CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
$ I-1, T, LDT,
$ T( 1, NB ), 1 )
*
* b2 := b2 - V2*w
*
CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
$ A( K+I, 1 ),
$ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
*
* b1 := b1 - V1*w
*
CALL DTRMV( 'Lower', 'NO TRANSPOSE',
$ 'UNIT', I-1,
$ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
A( K+I-1, I-1 ) = EI
END IF
*
* Generate the elementary reflector H(I) to annihilate
* A(K+I+1:N,I)
*
CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ),
$ 1,
$ TAU( I ) )
EI = A( K+I, I )
A( K+I, I ) = ONE
*
* Compute Y(K+1:N,I)
*
CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
$ ONE, A( K+1, I+1 ),
$ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
CALL DGEMV( 'Transpose', N-K-I+1, I-1,
$ ONE, A( K+I, 1 ), LDA,
$ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
$ Y( K+1, 1 ), LDY,
$ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
*
* Compute T(1:I,I)
*
CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
$ I-1, T, LDT,
$ T( 1, I ), 1 )
T( I, I ) = TAU( I )
*
10 CONTINUE
A( K+NB, NB ) = EI
*
* Compute Y(1:K,1:NB)
*
CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
$ 'UNIT', K, NB,
$ ONE, A( K+1, 1 ), LDA, Y, LDY )
IF( N.GT.K+NB )
$ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
$ NB, N-K-NB, ONE,
$ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
$ LDY )
CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
$ 'NON-UNIT', K, NB,
$ ONE, T, LDT, Y, LDY )
*
RETURN
*
* End of DLAHR2
*
END
*> \brief \b DLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* .. Scalar Arguments ..
* INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
* $ Y( LDY, NB )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DLAHR2.
*>
*> DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
*> matrix A so that elements below the k-th subdiagonal are zero. The
*> reduction is performed by an orthogonal similarity transformation
*> Q**T * A * Q. The routine returns the matrices V and T which determine
*> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The offset for the reduction. Elements below the k-th
*> subdiagonal in the first NB columns are reduced to zero.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
*> On entry, the n-by-(n-k+1) general matrix A.
*> On exit, the elements on and above the k-th subdiagonal in
*> the first NB columns are overwritten with the corresponding
*> elements of the reduced matrix; the elements below the k-th
*> subdiagonal, with the array TAU, represent the matrix Q as a
*> product of elementary reflectors. The other columns of A are
*> unchanged. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (NB)
*> The scalar factors of the elementary reflectors. See Further
*> Details.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NB)
*> The upper triangular matrix T.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (LDY,NB)
*> The n-by-nb matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*> LDY is INTEGER
*> The leading dimension of the array Y. LDY >= N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of nb elementary reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*> A(i+k+1:n,i), and tau in TAU(i).
*>
*> The elements of the vectors v together form the (n-k+1)-by-nb matrix
*> V which is needed, with T and Y, to apply the transformation to the
*> unreduced part of the matrix, using an update of the form:
*> A := (I - V*T*V**T) * (A - Y*V**T).
*>
*> The contents of A on exit are illustrated by the following example
*> with n = 7, k = 3 and nb = 2:
*>
*> ( a h a a a )
*> ( a h a a a )
*> ( a h a a a )
*> ( h h a a a )
*> ( v1 h a a a )
*> ( v1 v2 a a a )
*> ( v1 v2 a a a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
$ Y( LDY, NB )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
DO 10 I = 1, NB
IF( I.GT.1 ) THEN
*
* Update A(1:n,i)
*
* Compute i-th column of A - Y * V**T
*
CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
$ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
*
* Apply I - V * T**T * V**T to this column (call it b) from the
* left, using the last column of T as workspace
*
* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
* ( V2 ) ( b2 )
*
* where V1 is unit lower triangular
*
* w := V1**T * b1
*
CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1,
$ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
*
* w := w + V2**T *b2
*
CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
$ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
*
* w := T**T *w
*
CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
$ T( 1, NB ), 1 )
*
* b2 := b2 - V2*w
*
CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE,
$ A( K+I, 1 ), LDA, T( 1, NB ), 1, ONE,
$ A( K+I, I ), 1 )
*
* b1 := b1 - V1*w
*
CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
$ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
A( K+I-1, I-1 ) = EI
END IF
*
* Generate the elementary reflector H(i) to annihilate
* A(k+i+1:n,i)
*
CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
$ TAU( I ) )
EI = A( K+I, I )
A( K+I, I ) = ONE
*
* Compute Y(1:n,i)
*
CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ),
$ LDA, A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
$ LDA, A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ),
$ 1, ONE, Y( 1, I ), 1 )
CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
*
* Compute T(1:i,i)
*
CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
$ T( 1, I ), 1 )
T( I, I ) = TAU( I )
*
10 CONTINUE
A( K+NB, NB ) = EI
*
RETURN
*
* End of DLAHRD
*
END
*> \brief \b DLAIC1 applies one step of incremental condition estimation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAIC1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaic1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaic1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaic1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
*
* .. Scalar Arguments ..
* INTEGER J, JOB
* DOUBLE PRECISION C, GAMMA, S, SEST, SESTPR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION W( J ), X( J )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAIC1 applies one step of incremental condition estimation in
*> its simplest version:
*>
*> Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
*> lower triangular matrix L, such that
*> twonorm(L*x) = sest
*> Then DLAIC1 computes sestpr, s, c such that
*> the vector
*> [ s*x ]
*> xhat = [ c ]
*> is an approximate singular vector of
*> [ L 0 ]
*> Lhat = [ w**T gamma ]
*> in the sense that
*> twonorm(Lhat*xhat) = sestpr.
*>
*> Depending on JOB, an estimate for the largest or smallest singular
*> value is computed.
*>
*> Note that [s c]**T and sestpr**2 is an eigenpair of the system
*>
*> diag(sest*sest, 0) + [alpha gamma] * [ alpha ]
*> [ gamma ]
*>
*> where alpha = x**T*w.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is INTEGER
*> = 1: an estimate for the largest singular value is computed.
*> = 2: an estimate for the smallest singular value is computed.
*> \endverbatim
*>
*> \param[in] J
*> \verbatim
*> J is INTEGER
*> Length of X and W
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (J)
*> The j-vector x.
*> \endverbatim
*>
*> \param[in] SEST
*> \verbatim
*> SEST is DOUBLE PRECISION
*> Estimated singular value of j by j matrix L
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (J)
*> The j-vector w.
*> \endverbatim
*>
*> \param[in] GAMMA
*> \verbatim
*> GAMMA is DOUBLE PRECISION
*> The diagonal element gamma.
*> \endverbatim
*>
*> \param[out] SESTPR
*> \verbatim
*> SESTPR is DOUBLE PRECISION
*> Estimated singular value of (j+1) by (j+1) matrix Lhat.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION
*> Sine needed in forming xhat.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> Cosine needed in forming xhat.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laic1
*
* =====================================================================
SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER J, JOB
DOUBLE PRECISION C, GAMMA, S, SEST, SESTPR
* ..
* .. Array Arguments ..
DOUBLE PRECISION W( J ), X( J )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
DOUBLE PRECISION HALF, FOUR
PARAMETER ( HALF = 0.5D0, FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS,
$ NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN, SQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL DDOT, DLAMCH
* ..
* .. Executable Statements ..
*
EPS = DLAMCH( 'Epsilon' )
ALPHA = DDOT( J, X, 1, W, 1 )
*
ABSALP = ABS( ALPHA )
ABSGAM = ABS( GAMMA )
ABSEST = ABS( SEST )
*
IF( JOB.EQ.1 ) THEN
*
* Estimating largest singular value
*
* special cases
*
IF( SEST.EQ.ZERO ) THEN
S1 = MAX( ABSGAM, ABSALP )
IF( S1.EQ.ZERO ) THEN
S = ZERO
C = ONE
SESTPR = ZERO
ELSE
S = ALPHA / S1
C = GAMMA / S1
TMP = SQRT( S*S+C*C )
S = S / TMP
C = C / TMP
SESTPR = S1*TMP
END IF
RETURN
ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
S = ONE
C = ZERO
TMP = MAX( ABSEST, ABSALP )
S1 = ABSEST / TMP
S2 = ABSALP / TMP
SESTPR = TMP*SQRT( S1*S1+S2*S2 )
RETURN
ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
S1 = ABSGAM
S2 = ABSEST
IF( S1.LE.S2 ) THEN
S = ONE
C = ZERO
SESTPR = S2
ELSE
S = ZERO
C = ONE
SESTPR = S1
END IF
RETURN
ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
S1 = ABSGAM
S2 = ABSALP
IF( S1.LE.S2 ) THEN
TMP = S1 / S2
S = SQRT( ONE+TMP*TMP )
SESTPR = S2*S
C = ( GAMMA / S2 ) / S
S = SIGN( ONE, ALPHA ) / S
ELSE
TMP = S2 / S1
C = SQRT( ONE+TMP*TMP )
SESTPR = S1*C
S = ( ALPHA / S1 ) / C
C = SIGN( ONE, GAMMA ) / C
END IF
RETURN
ELSE
*
* normal case
*
ZETA1 = ALPHA / ABSEST
ZETA2 = GAMMA / ABSEST
*
B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF
C = ZETA1*ZETA1
IF( B.GT.ZERO ) THEN
T = C / ( B+SQRT( B*B+C ) )
ELSE
T = SQRT( B*B+C ) - B
END IF
*
SINE = -ZETA1 / T
COSINE = -ZETA2 / ( ONE+T )
TMP = SQRT( SINE*SINE+COSINE*COSINE )
S = SINE / TMP
C = COSINE / TMP
SESTPR = SQRT( T+ONE )*ABSEST
RETURN
END IF
*
ELSE IF( JOB.EQ.2 ) THEN
*
* Estimating smallest singular value
*
* special cases
*
IF( SEST.EQ.ZERO ) THEN
SESTPR = ZERO
IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN
SINE = ONE
COSINE = ZERO
ELSE
SINE = -GAMMA
COSINE = ALPHA
END IF
S1 = MAX( ABS( SINE ), ABS( COSINE ) )
S = SINE / S1
C = COSINE / S1
TMP = SQRT( S*S+C*C )
S = S / TMP
C = C / TMP
RETURN
ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
S = ZERO
C = ONE
SESTPR = ABSGAM
RETURN
ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
S1 = ABSGAM
S2 = ABSEST
IF( S1.LE.S2 ) THEN
S = ZERO
C = ONE
SESTPR = S1
ELSE
S = ONE
C = ZERO
SESTPR = S2
END IF
RETURN
ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
S1 = ABSGAM
S2 = ABSALP
IF( S1.LE.S2 ) THEN
TMP = S1 / S2
C = SQRT( ONE+TMP*TMP )
SESTPR = ABSEST*( TMP / C )
S = -( GAMMA / S2 ) / C
C = SIGN( ONE, ALPHA ) / C
ELSE
TMP = S2 / S1
S = SQRT( ONE+TMP*TMP )
SESTPR = ABSEST / S
C = ( ALPHA / S1 ) / S
S = -SIGN( ONE, GAMMA ) / S
END IF
RETURN
ELSE
*
* normal case
*
ZETA1 = ALPHA / ABSEST
ZETA2 = GAMMA / ABSEST
*
NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ),
$ ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 )
*
* See if root is closer to zero or to ONE
*
TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 )
IF( TEST.GE.ZERO ) THEN
*
* root is close to zero, compute directly
*
B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF
C = ZETA2*ZETA2
T = C / ( B+SQRT( ABS( B*B-C ) ) )
SINE = ZETA1 / ( ONE-T )
COSINE = -ZETA2 / T
SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST
ELSE
*
* root is closer to ONE, shift by that amount
*
B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF
C = ZETA1*ZETA1
IF( B.GE.ZERO ) THEN
T = -C / ( B+SQRT( B*B+C ) )
ELSE
T = B - SQRT( B*B+C )
END IF
SINE = -ZETA1 / T
COSINE = -ZETA2 / ( ONE+T )
SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST
END IF
TMP = SQRT( SINE*SINE+COSINE*COSINE )
S = SINE / TMP
C = COSINE / TMP
RETURN
*
END IF
END IF
RETURN
*
* End of DLAIC1
*
END
*> \brief \b DLAISNAN tests input for NaN by comparing two arguments for inequality.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAISNAN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaisnan.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaisnan.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaisnan.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION, INTENT(IN) :: DIN1, DIN2
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is not for general use. It exists solely to avoid
*> over-optimization in DISNAN.
*>
*> DLAISNAN checks for NaNs by comparing its two arguments for
*> inequality. NaN is the only floating-point value where NaN != NaN
*> returns .TRUE. To check for NaNs, pass the same variable as both
*> arguments.
*>
*> A compiler must assume that the two arguments are
*> not the same variable, and the test will not be optimized away.
*> Interprocedural or whole-program optimization may delete this
*> test. The ISNAN functions will be replaced by the correct
*> Fortran 03 intrinsic once the intrinsic is widely available.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIN1
*> \verbatim
*> DIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] DIN2
*> \verbatim
*> DIN2 is DOUBLE PRECISION
*> Two numbers to compare for inequality.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laisnan
*
* =====================================================================
LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION, INTENT(IN) :: DIN1, DIN2
* ..
*
* =====================================================================
*
* .. Executable Statements ..
DLAISNAN = (DIN1.NE.DIN2)
RETURN
END
*> \brief \b DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALN2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaln2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaln2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaln2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
* LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
* .. Scalar Arguments ..
* LOGICAL LTRANS
* INTEGER INFO, LDA, LDB, LDX, NA, NW
* DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLALN2 solves a system of the form (ca A - w D ) X = s B
*> or (ca A**T - w D) X = s B with possible scaling ("s") and
*> perturbation of A. (A**T means A-transpose.)
*>
*> A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
*> real diagonal matrix, w is a real or complex value, and X and B are
*> NA x 1 matrices -- real if w is real, complex if w is complex. NA
*> may be 1 or 2.
*>
*> If w is complex, X and B are represented as NA x 2 matrices,
*> the first column of each being the real part and the second
*> being the imaginary part.
*>
*> "s" is a scaling factor (<= 1), computed by DLALN2, which is
*> so chosen that X can be computed without overflow. X is further
*> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
*> than overflow.
*>
*> If both singular values of (ca A - w D) are less than SMIN,
*> SMIN*identity will be used instead of (ca A - w D). If only one
*> singular value is less than SMIN, one element of (ca A - w D) will be
*> perturbed enough to make the smallest singular value roughly SMIN.
*> If both singular values are at least SMIN, (ca A - w D) will not be
*> perturbed. In any case, the perturbation will be at most some small
*> multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values
*> are computed by infinity-norm approximations, and thus will only be
*> correct to a factor of 2 or so.
*>
*> Note: all input quantities are assumed to be smaller than overflow
*> by a reasonable factor. (See BIGNUM.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] LTRANS
*> \verbatim
*> LTRANS is LOGICAL
*> =.TRUE.: A-transpose will be used.
*> =.FALSE.: A will be used (not transposed.)
*> \endverbatim
*>
*> \param[in] NA
*> \verbatim
*> NA is INTEGER
*> The size of the matrix A. It may (only) be 1 or 2.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> 1 if "w" is real, 2 if "w" is complex. It may only be 1
*> or 2.
*> \endverbatim
*>
*> \param[in] SMIN
*> \verbatim
*> SMIN is DOUBLE PRECISION
*> The desired lower bound on the singular values of A. This
*> should be a safe distance away from underflow or overflow,
*> say, between (underflow/machine precision) and (machine
*> precision * overflow ). (See BIGNUM and ULP.)
*> \endverbatim
*>
*> \param[in] CA
*> \verbatim
*> CA is DOUBLE PRECISION
*> The coefficient c, which A is multiplied by.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,NA)
*> The NA x NA matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. It must be at least NA.
*> \endverbatim
*>
*> \param[in] D1
*> \verbatim
*> D1 is DOUBLE PRECISION
*> The 1,1 element in the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] D2
*> \verbatim
*> D2 is DOUBLE PRECISION
*> The 2,2 element in the diagonal matrix D. Not used if NA=1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NW)
*> The NA x NW matrix B (right-hand side). If NW=2 ("w" is
*> complex), column 1 contains the real part of B and column 2
*> contains the imaginary part.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. It must be at least NA.
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*> WR is DOUBLE PRECISION
*> The real part of the scalar "w".
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is DOUBLE PRECISION
*> The imaginary part of the scalar "w". Not used if NW=1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NW)
*> The NA x NW matrix X (unknowns), as computed by DLALN2.
*> If NW=2 ("w" is complex), on exit, column 1 will contain
*> the real part of X and column 2 will contain the imaginary
*> part.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of X. It must be at least NA.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The scale factor that B must be multiplied by to insure
*> that overflow does not occur when computing X. Thus,
*> (ca A - w D) X will be SCALE*B, not B (ignoring
*> perturbations of A.) It will be at most 1.
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*> XNORM is DOUBLE PRECISION
*> The infinity-norm of X, when X is regarded as an NA x NW
*> real matrix.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> An error flag. It will be set to zero if no error occurs,
*> a negative number if an argument is in error, or a positive
*> number if ca A - w D had to be perturbed.
*> The possible values are:
*> = 0: No error occurred, and (ca A - w D) did not have to be
*> perturbed.
*> = 1: (ca A - w D) had to be perturbed to make its smallest
*> (or only) singular value greater than SMIN.
*> NOTE: In the interests of speed, this routine does not
*> check the inputs for errors.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laln2
*
* =====================================================================
SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
$ LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL LTRANS
INTEGER INFO, LDA, LDB, LDX, NA, NW
DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
INTEGER ICMAX, J
DOUBLE PRECISION BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21,
$ CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21,
$ LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R,
$ UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S,
$ UR22, XI1, XI2, XR1, XR2
* ..
* .. Local Arrays ..
LOGICAL RSWAP( 4 ), ZSWAP( 4 )
INTEGER IPIVOT( 4, 4 )
DOUBLE PRECISION CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLADIV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Equivalences ..
EQUIVALENCE ( CI( 1, 1 ), CIV( 1 ) ),
$ ( CR( 1, 1 ), CRV( 1 ) )
* ..
* .. Data statements ..
DATA ZSWAP / .FALSE., .FALSE., .TRUE., .TRUE. /
DATA RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. /
DATA IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4,
$ 3, 2, 1 /
* ..
* .. Executable Statements ..
*
* Compute BIGNUM
*
SMLNUM = TWO*DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
SMINI = MAX( SMIN, SMLNUM )
*
* Don't check for input errors
*
INFO = 0
*
* Standard Initializations
*
SCALE = ONE
*
IF( NA.EQ.1 ) THEN
*
* 1 x 1 (i.e., scalar) system C X = B
*
IF( NW.EQ.1 ) THEN
*
* Real 1x1 system.
*
* C = ca A - w D
*
CSR = CA*A( 1, 1 ) - WR*D1
CNORM = ABS( CSR )
*
* If | C | < SMINI, use C = SMINI
*
IF( CNORM.LT.SMINI ) THEN
CSR = SMINI
CNORM = SMINI
INFO = 1
END IF
*
* Check scaling for X = B / C
*
BNORM = ABS( B( 1, 1 ) )
IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*CNORM )
$ SCALE = ONE / BNORM
END IF
*
* Compute X
*
X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR
XNORM = ABS( X( 1, 1 ) )
ELSE
*
* Complex 1x1 system (w is complex)
*
* C = ca A - w D
*
CSR = CA*A( 1, 1 ) - WR*D1
CSI = -WI*D1
CNORM = ABS( CSR ) + ABS( CSI )
*
* If | C | < SMINI, use C = SMINI
*
IF( CNORM.LT.SMINI ) THEN
CSR = SMINI
CSI = ZERO
CNORM = SMINI
INFO = 1
END IF
*
* Check scaling for X = B / C
*
BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) )
IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*CNORM )
$ SCALE = ONE / BNORM
END IF
*
* Compute X
*
CALL DLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI,
$ X( 1, 1 ), X( 1, 2 ) )
XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
END IF
*
ELSE
*
* 2x2 System
*
* Compute the real part of C = ca A - w D (or ca A**T - w D )
*
CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1
CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2
IF( LTRANS ) THEN
CR( 1, 2 ) = CA*A( 2, 1 )
CR( 2, 1 ) = CA*A( 1, 2 )
ELSE
CR( 2, 1 ) = CA*A( 2, 1 )
CR( 1, 2 ) = CA*A( 1, 2 )
END IF
*
IF( NW.EQ.1 ) THEN
*
* Real 2x2 system (w is real)
*
* Find the largest element in C
*
CMAX = ZERO
ICMAX = 0
*
DO 10 J = 1, 4
IF( ABS( CRV( J ) ).GT.CMAX ) THEN
CMAX = ABS( CRV( J ) )
ICMAX = J
END IF
10 CONTINUE
*
* If norm(C) < SMINI, use SMINI*identity.
*
IF( CMAX.LT.SMINI ) THEN
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) )
IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*SMINI )
$ SCALE = ONE / BNORM
END IF
TEMP = SCALE / SMINI
X( 1, 1 ) = TEMP*B( 1, 1 )
X( 2, 1 ) = TEMP*B( 2, 1 )
XNORM = TEMP*BNORM
INFO = 1
RETURN
END IF
*
* Gaussian elimination with complete pivoting.
*
UR11 = CRV( ICMAX )
CR21 = CRV( IPIVOT( 2, ICMAX ) )
UR12 = CRV( IPIVOT( 3, ICMAX ) )
CR22 = CRV( IPIVOT( 4, ICMAX ) )
UR11R = ONE / UR11
LR21 = UR11R*CR21
UR22 = CR22 - UR12*LR21
*
* If smaller pivot < SMINI, use SMINI
*
IF( ABS( UR22 ).LT.SMINI ) THEN
UR22 = SMINI
INFO = 1
END IF
IF( RSWAP( ICMAX ) ) THEN
BR1 = B( 2, 1 )
BR2 = B( 1, 1 )
ELSE
BR1 = B( 1, 1 )
BR2 = B( 2, 1 )
END IF
BR2 = BR2 - LR21*BR1
BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) )
IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN
IF( BBND.GE.BIGNUM*ABS( UR22 ) )
$ SCALE = ONE / BBND
END IF
*
XR2 = ( BR2*SCALE ) / UR22
XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 )
IF( ZSWAP( ICMAX ) ) THEN
X( 1, 1 ) = XR2
X( 2, 1 ) = XR1
ELSE
X( 1, 1 ) = XR1
X( 2, 1 ) = XR2
END IF
XNORM = MAX( ABS( XR1 ), ABS( XR2 ) )
*
* Further scaling if norm(A) norm(X) > overflow
*
IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
IF( XNORM.GT.BIGNUM / CMAX ) THEN
TEMP = CMAX / BIGNUM
X( 1, 1 ) = TEMP*X( 1, 1 )
X( 2, 1 ) = TEMP*X( 2, 1 )
XNORM = TEMP*XNORM
SCALE = TEMP*SCALE
END IF
END IF
ELSE
*
* Complex 2x2 system (w is complex)
*
* Find the largest element in C
*
CI( 1, 1 ) = -WI*D1
CI( 2, 1 ) = ZERO
CI( 1, 2 ) = ZERO
CI( 2, 2 ) = -WI*D2
CMAX = ZERO
ICMAX = 0
*
DO 20 J = 1, 4
IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN
CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) )
ICMAX = J
END IF
20 CONTINUE
*
* If norm(C) < SMINI, use SMINI*identity.
*
IF( CMAX.LT.SMINI ) THEN
BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
$ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*SMINI )
$ SCALE = ONE / BNORM
END IF
TEMP = SCALE / SMINI
X( 1, 1 ) = TEMP*B( 1, 1 )
X( 2, 1 ) = TEMP*B( 2, 1 )
X( 1, 2 ) = TEMP*B( 1, 2 )
X( 2, 2 ) = TEMP*B( 2, 2 )
XNORM = TEMP*BNORM
INFO = 1
RETURN
END IF
*
* Gaussian elimination with complete pivoting.
*
UR11 = CRV( ICMAX )
UI11 = CIV( ICMAX )
CR21 = CRV( IPIVOT( 2, ICMAX ) )
CI21 = CIV( IPIVOT( 2, ICMAX ) )
UR12 = CRV( IPIVOT( 3, ICMAX ) )
UI12 = CIV( IPIVOT( 3, ICMAX ) )
CR22 = CRV( IPIVOT( 4, ICMAX ) )
CI22 = CIV( IPIVOT( 4, ICMAX ) )
IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN
*
* Code when off-diagonals of pivoted C are real
*
IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN
TEMP = UI11 / UR11
UR11R = ONE / ( UR11*( ONE+TEMP**2 ) )
UI11R = -TEMP*UR11R
ELSE
TEMP = UR11 / UI11
UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) )
UR11R = -TEMP*UI11R
END IF
LR21 = CR21*UR11R
LI21 = CR21*UI11R
UR12S = UR12*UR11R
UI12S = UR12*UI11R
UR22 = CR22 - UR12*LR21
UI22 = CI22 - UR12*LI21
ELSE
*
* Code when diagonals of pivoted C are real
*
UR11R = ONE / UR11
UI11R = ZERO
LR21 = CR21*UR11R
LI21 = CI21*UR11R
UR12S = UR12*UR11R
UI12S = UI12*UR11R
UR22 = CR22 - UR12*LR21 + UI12*LI21
UI22 = -UR12*LI21 - UI12*LR21
END IF
U22ABS = ABS( UR22 ) + ABS( UI22 )
*
* If smaller pivot < SMINI, use SMINI
*
IF( U22ABS.LT.SMINI ) THEN
UR22 = SMINI
UI22 = ZERO
INFO = 1
END IF
IF( RSWAP( ICMAX ) ) THEN
BR2 = B( 1, 1 )
BR1 = B( 2, 1 )
BI2 = B( 1, 2 )
BI1 = B( 2, 2 )
ELSE
BR1 = B( 1, 1 )
BR2 = B( 2, 1 )
BI1 = B( 1, 2 )
BI2 = B( 2, 2 )
END IF
BR2 = BR2 - LR21*BR1 + LI21*BI1
BI2 = BI2 - LI21*BR1 - LR21*BI1
BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )*
$ ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ),
$ ABS( BR2 )+ABS( BI2 ) )
IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN
IF( BBND.GE.BIGNUM*U22ABS ) THEN
SCALE = ONE / BBND
BR1 = SCALE*BR1
BI1 = SCALE*BI1
BR2 = SCALE*BR2
BI2 = SCALE*BI2
END IF
END IF
*
CALL DLADIV( BR2, BI2, UR22, UI22, XR2, XI2 )
XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2
XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2
IF( ZSWAP( ICMAX ) ) THEN
X( 1, 1 ) = XR2
X( 2, 1 ) = XR1
X( 1, 2 ) = XI2
X( 2, 2 ) = XI1
ELSE
X( 1, 1 ) = XR1
X( 2, 1 ) = XR2
X( 1, 2 ) = XI1
X( 2, 2 ) = XI2
END IF
XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) )
*
* Further scaling if norm(A) norm(X) > overflow
*
IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
IF( XNORM.GT.BIGNUM / CMAX ) THEN
TEMP = CMAX / BIGNUM
X( 1, 1 ) = TEMP*X( 1, 1 )
X( 2, 1 ) = TEMP*X( 2, 1 )
X( 1, 2 ) = TEMP*X( 1, 2 )
X( 2, 2 ) = TEMP*X( 2, 2 )
XNORM = TEMP*XNORM
SCALE = TEMP*SCALE
END IF
END IF
END IF
END IF
*
RETURN
*
* End of DLALN2
*
END
*> \brief \b DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALS0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlals0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlals0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlals0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
* PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
* POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
* $ LDGNUM, NL, NR, NRHS, SQRE
* DOUBLE PRECISION C, S
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), PERM( * )
* DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
* $ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
* $ POLES( LDGNUM, * ), WORK( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLALS0 applies back the multiplying factors of either the left or the
*> right singular vector matrix of a diagonal matrix appended by a row
*> to the right hand side matrix B in solving the least squares problem
*> using the divide-and-conquer SVD approach.
*>
*> For the left singular vector matrix, three types of orthogonal
*> matrices are involved:
*>
*> (1L) Givens rotations: the number of such rotations is GIVPTR; the
*> pairs of columns/rows they were applied to are stored in GIVCOL;
*> and the C- and S-values of these rotations are stored in GIVNUM.
*>
*> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
*> row, and for J=2:N, PERM(J)-th row of B is to be moved to the
*> J-th row.
*>
*> (3L) The left singular vector matrix of the remaining matrix.
*>
*> For the right singular vector matrix, four types of orthogonal
*> matrices are involved:
*>
*> (1R) The right singular vector matrix of the remaining matrix.
*>
*> (2R) If SQRE = 1, one extra Givens rotation to generate the right
*> null space.
*>
*> (3R) The inverse transformation of (2L).
*>
*> (4R) The inverse transformation of (1L).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether singular vectors are to be computed in
*> factored form:
*> = 0: Left singular vector matrix.
*> = 1: Right singular vector matrix.
*> \endverbatim
*>
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: the lower block is an NR-by-NR square matrix.
*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*> The bidiagonal matrix has row dimension N = NL + NR + 1,
*> and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of columns of B and BX. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
*> On input, B contains the right hand sides of the least
*> squares problem in rows 1 through M. On output, B contains
*> the solution X in rows 1 through N.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB must be at least
*> max(1,MAX( M, N ) ).
*> \endverbatim
*>
*> \param[out] BX
*> \verbatim
*> BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
*> \endverbatim
*>
*> \param[in] LDBX
*> \verbatim
*> LDBX is INTEGER
*> The leading dimension of BX.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension ( N )
*> The permutations (from deflation and sorting) applied
*> to the two blocks.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> The number of Givens rotations which took place in this
*> subproblem.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*> Each pair of numbers indicates a pair of rows/columns
*> involved in a Givens rotation.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER
*> The leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*> Each number indicates the C or S value used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*> LDGNUM is INTEGER
*> The leading dimension of arrays DIFR, POLES and
*> GIVNUM, must be at least K.
*> \endverbatim
*>
*> \param[in] POLES
*> \verbatim
*> POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*> On entry, POLES(1:K, 1) contains the new singular
*> values obtained from solving the secular equation, and
*> POLES(1:K, 2) is an array containing the poles in the secular
*> equation.
*> \endverbatim
*>
*> \param[in] DIFL
*> \verbatim
*> DIFL is DOUBLE PRECISION array, dimension ( K ).
*> On entry, DIFL(I) is the distance between I-th updated
*> (undeflated) singular value and the I-th (undeflated) old
*> singular value.
*> \endverbatim
*>
*> \param[in] DIFR
*> \verbatim
*> DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
*> On entry, DIFR(I, 1) contains the distances between I-th
*> updated (undeflated) singular value and the I+1-th
*> (undeflated) old singular value. And DIFR(I, 2) is the
*> normalizing factor for the I-th right singular vector.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( K )
*> Contain the components of the deflation-adjusted updating row
*> vector.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> Contains the dimension of the non-deflated matrix,
*> This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> C contains garbage if SQRE =0 and the C-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION
*> S contains garbage if SQRE =0 and the S-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension ( K )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lals0
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
*> California at Berkeley, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*
* =====================================================================
SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX,
$ LDBX,
$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
$ POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
$ LDGNUM, NL, NR, NRHS, SQRE
DOUBLE PRECISION C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), PERM( * )
DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
$ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
$ POLES( LDGNUM, * ), WORK( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, NEGONE
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, NEGONE = -1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, J, M, N, NLP1
DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DROT,
$ DSCAL,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3, DNRM2
EXTERNAL DLAMC3, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
N = NL + NR + 1
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( NL.LT.1 ) THEN
INFO = -2
ELSE IF( NR.LT.1 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( NRHS.LT.1 ) THEN
INFO = -5
ELSE IF( LDB.LT.N ) THEN
INFO = -7
ELSE IF( LDBX.LT.N ) THEN
INFO = -9
ELSE IF( GIVPTR.LT.0 ) THEN
INFO = -11
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -13
ELSE IF( LDGNUM.LT.N ) THEN
INFO = -15
ELSE IF( K.LT.1 ) THEN
INFO = -20
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLALS0', -INFO )
RETURN
END IF
*
M = N + SQRE
NLP1 = NL + 1
*
IF( ICOMPQ.EQ.0 ) THEN
*
* Apply back orthogonal transformations from the left.
*
* Step (1L): apply back the Givens rotations performed.
*
DO 10 I = 1, GIVPTR
CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
$ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
$ GIVNUM( I, 1 ) )
10 CONTINUE
*
* Step (2L): permute rows of B.
*
CALL DCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX )
DO 20 I = 2, N
CALL DCOPY( NRHS, B( PERM( I ), 1 ), LDB, BX( I, 1 ),
$ LDBX )
20 CONTINUE
*
* Step (3L): apply the inverse of the left singular vector
* matrix to BX.
*
IF( K.EQ.1 ) THEN
CALL DCOPY( NRHS, BX, LDBX, B, LDB )
IF( Z( 1 ).LT.ZERO ) THEN
CALL DSCAL( NRHS, NEGONE, B, LDB )
END IF
ELSE
DO 50 J = 1, K
DIFLJ = DIFL( J )
DJ = POLES( J, 1 )
DSIGJ = -POLES( J, 2 )
IF( J.LT.K ) THEN
DIFRJ = -DIFR( J, 1 )
DSIGJP = -POLES( J+1, 2 )
END IF
IF( ( Z( J ).EQ.ZERO ) .OR. ( POLES( J, 2 ).EQ.ZERO ) )
$ THEN
WORK( J ) = ZERO
ELSE
WORK( J ) = -POLES( J, 2 )*Z( J ) / DIFLJ /
$ ( POLES( J, 2 )+DJ )
END IF
DO 30 I = 1, J - 1
IF( ( Z( I ).EQ.ZERO ) .OR.
$ ( POLES( I, 2 ).EQ.ZERO ) ) THEN
WORK( I ) = ZERO
ELSE
*
* Use calls to the subroutine DLAMC3 to enforce the
* parentheses (x+y)+z. The goal is to prevent
* optimizing compilers from doing x+(y+z).
*
WORK( I ) = POLES( I, 2 )*Z( I ) /
$ ( DLAMC3( POLES( I, 2 ), DSIGJ )-
$ DIFLJ ) / ( POLES( I, 2 )+DJ )
END IF
30 CONTINUE
DO 40 I = J + 1, K
IF( ( Z( I ).EQ.ZERO ) .OR.
$ ( POLES( I, 2 ).EQ.ZERO ) ) THEN
WORK( I ) = ZERO
ELSE
WORK( I ) = POLES( I, 2 )*Z( I ) /
$ ( DLAMC3( POLES( I, 2 ), DSIGJP )+
$ DIFRJ ) / ( POLES( I, 2 )+DJ )
END IF
40 CONTINUE
WORK( 1 ) = NEGONE
TEMP = DNRM2( K, WORK, 1 )
CALL DGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1,
$ ZERO,
$ B( J, 1 ), LDB )
CALL DLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ),
$ LDB, INFO )
50 CONTINUE
END IF
*
* Move the deflated rows of BX to B also.
*
IF( K.LT.MAX( M, N ) )
$ CALL DLACPY( 'A', N-K, NRHS, BX( K+1, 1 ), LDBX,
$ B( K+1, 1 ), LDB )
ELSE
*
* Apply back the right orthogonal transformations.
*
* Step (1R): apply back the new right singular vector matrix
* to B.
*
IF( K.EQ.1 ) THEN
CALL DCOPY( NRHS, B, LDB, BX, LDBX )
ELSE
DO 80 J = 1, K
DSIGJ = POLES( J, 2 )
IF( Z( J ).EQ.ZERO ) THEN
WORK( J ) = ZERO
ELSE
WORK( J ) = -Z( J ) / DIFL( J ) /
$ ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 )
END IF
DO 60 I = 1, J - 1
IF( Z( J ).EQ.ZERO ) THEN
WORK( I ) = ZERO
ELSE
*
* Use calls to the subroutine DLAMC3 to enforce the
* parentheses (x+y)+z. The goal is to prevent
* optimizing compilers from doing x+(y+z).
*
WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ,
$ -POLES( I+1,
$ 2 ) )-DIFR( I, 1 ) ) /
$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
END IF
60 CONTINUE
DO 70 I = J + 1, K
IF( Z( J ).EQ.ZERO ) THEN
WORK( I ) = ZERO
ELSE
WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I,
$ 2 ) )-DIFL( I ) ) /
$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
END IF
70 CONTINUE
CALL DGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO,
$ BX( J, 1 ), LDBX )
80 CONTINUE
END IF
*
* Step (2R): if SQRE = 1, apply back the rotation that is
* related to the right null space of the subproblem.
*
IF( SQRE.EQ.1 ) THEN
CALL DCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX )
CALL DROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C,
$ S )
END IF
IF( K.LT.MAX( M, N ) )
$ CALL DLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB, BX( K+1,
$ 1 ),
$ LDBX )
*
* Step (3R): permute rows of B.
*
CALL DCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB )
IF( SQRE.EQ.1 ) THEN
CALL DCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB )
END IF
DO 90 I = 2, N
CALL DCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ),
$ LDB )
90 CONTINUE
*
* Step (4R): apply back the Givens rotations performed.
*
DO 100 I = GIVPTR, 1, -1
CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
$ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
$ -GIVNUM( I, 1 ) )
100 CONTINUE
END IF
*
RETURN
*
* End of DLALS0
*
END
*> \brief \b DLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALSA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlalsa.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlalsa.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlalsa.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
* LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
* GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
* $ SMLSIZ
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
* $ K( * ), PERM( LDGCOL, * )
* DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), C( * ),
* $ DIFL( LDU, * ), DIFR( LDU, * ),
* $ GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
* $ U( LDU, * ), VT( LDU, * ), WORK( * ),
* $ Z( LDU, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLALSA is an intermediate step in solving the least squares problem
*> by computing the SVD of the coefficient matrix in compact form (The
*> singular vectors are computed as products of simple orthogonal
*> matrices.).
*>
*> If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
*> matrix of an upper bidiagonal matrix to the right hand side; and if
*> ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
*> right hand side. The singular vector matrices were generated in
*> compact form by DLALSA.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether the left or the right singular vector
*> matrix is involved.
*> = 0: Left singular vector matrix
*> = 1: Right singular vector matrix
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> The maximum size of the subproblems at the bottom of the
*> computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The row and column dimensions of the upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of columns of B and BX. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
*> On input, B contains the right hand sides of the least
*> squares problem in rows 1 through M.
*> On output, B contains the solution X in rows 1 through N.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B in the calling subprogram.
*> LDB must be at least max(1,MAX( M, N ) ).
*> \endverbatim
*>
*> \param[out] BX
*> \verbatim
*> BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
*> On exit, the result of applying the left or right singular
*> vector matrix to B.
*> \endverbatim
*>
*> \param[in] LDBX
*> \verbatim
*> LDBX is INTEGER
*> The leading dimension of BX.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
*> On entry, U contains the left singular vector matrices of all
*> subproblems at the bottom level.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER, LDU = > N.
*> The leading dimension of arrays U, VT, DIFL, DIFR,
*> POLES, GIVNUM, and Z.
*> \endverbatim
*>
*> \param[in] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
*> On entry, VT**T contains the right singular vector matrices of
*> all subproblems at the bottom level.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER array, dimension ( N ).
*> \endverbatim
*>
*> \param[in] DIFL
*> \verbatim
*> DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
*> where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
*> \endverbatim
*>
*> \param[in] DIFR
*> \verbatim
*> DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*> On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
*> distances between singular values on the I-th level and
*> singular values on the (I -1)-th level, and DIFR(*, 2 * I)
*> record the normalizing factors of the right singular vectors
*> matrices of subproblems on I-th level.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
*> On entry, Z(1, I) contains the components of the deflation-
*> adjusted updating row vector for subproblems on the I-th
*> level.
*> \endverbatim
*>
*> \param[in] POLES
*> \verbatim
*> POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*> On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
*> singular values involved in the secular equations on the I-th
*> level.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array, dimension ( N ).
*> On entry, GIVPTR( I ) records the number of Givens
*> rotations performed on the I-th problem on the computation
*> tree.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
*> On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
*> locations of Givens rotations performed on the I-th level on
*> the computation tree.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER, LDGCOL = > N.
*> The leading dimension of arrays GIVCOL and PERM.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
*> On entry, PERM(*, I) records permutations done on the I-th
*> level of the computation tree.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*> On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
*> values of Givens rotations performed on the I-th level on the
*> computation tree.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension ( N ).
*> On entry, if the I-th subproblem is not square,
*> C( I ) contains the C-value of a Givens rotation related to
*> the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension ( N ).
*> On entry, if the I-th subproblem is not square,
*> S( I ) contains the S-value of a Givens rotation related to
*> the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lalsa
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
*> California at Berkeley, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*
* =====================================================================
SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX,
$ U,
$ LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
$ GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
$ SMLSIZ
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
$ K( * ), PERM( LDGCOL, * )
DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), C( * ),
$ DIFL( LDU, * ), DIFR( LDU, * ),
$ GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
$ U( LDU, * ), VT( LDU, * ), WORK( * ),
$ Z( LDU, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2,
$ ND, NDB1, NDIML, NDIMR, NL, NLF, NLP1, NLVL,
$ NR, NRF, NRP1, SQRE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLALS0, DLASDT,
$ XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( SMLSIZ.LT.3 ) THEN
INFO = -2
ELSE IF( N.LT.SMLSIZ ) THEN
INFO = -3
ELSE IF( NRHS.LT.1 ) THEN
INFO = -4
ELSE IF( LDB.LT.N ) THEN
INFO = -6
ELSE IF( LDBX.LT.N ) THEN
INFO = -8
ELSE IF( LDU.LT.N ) THEN
INFO = -10
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -19
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLALSA', -INFO )
RETURN
END IF
*
* Book-keeping and setting up the computation tree.
*
INODE = 1
NDIML = INODE + N
NDIMR = NDIML + N
*
CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
$ IWORK( NDIMR ), SMLSIZ )
*
* The following code applies back the left singular vector factors.
* For applying back the right singular vector factors, go to 50.
*
IF( ICOMPQ.EQ.1 ) THEN
GO TO 50
END IF
*
* The nodes on the bottom level of the tree were solved
* by DLASDQ. The corresponding left and right singular vector
* matrices are in explicit form. First apply back the left
* singular vector matrices.
*
NDB1 = ( ND+1 ) / 2
DO 10 I = NDB1, ND
*
* IC : center row of each node
* NL : number of rows of left subproblem
* NR : number of rows of right subproblem
* NLF: starting row of the left subproblem
* NRF: starting row of the right subproblem
*
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NR = IWORK( NDIMR+I1 )
NLF = IC - NL
NRF = IC + 1
CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
10 CONTINUE
*
* Next copy the rows of B that correspond to unchanged rows
* in the bidiagonal matrix to BX.
*
DO 20 I = 1, ND
IC = IWORK( INODE+I-1 )
CALL DCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX )
20 CONTINUE
*
* Finally go through the left singular vector matrices of all
* the other subproblems bottom-up on the tree.
*
J = 2**NLVL
SQRE = 0
*
DO 40 LVL = NLVL, 1, -1
LVL2 = 2*LVL - 1
*
* find the first node LF and last node LL on
* the current level LVL
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 30 I = LF, LL
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
NRF = IC + 1
J = J - 1
CALL DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, BX( NLF, 1 ),
$ LDBX,
$ B( NLF, 1 ), LDB, PERM( NLF, LVL ),
$ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
$ GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
$ DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
$ Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK,
$ INFO )
30 CONTINUE
40 CONTINUE
GO TO 90
*
* ICOMPQ = 1: applying back the right singular vector factors.
*
50 CONTINUE
*
* First now go through the right singular vector matrices of all
* the tree nodes top-down.
*
J = 0
DO 70 LVL = 1, NLVL
LVL2 = 2*LVL - 1
*
* Find the first node LF and last node LL on
* the current level LVL.
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 60 I = LL, LF, -1
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
NRF = IC + 1
IF( I.EQ.LL ) THEN
SQRE = 0
ELSE
SQRE = 1
END IF
J = J + 1
CALL DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B( NLF, 1 ),
$ LDB,
$ BX( NLF, 1 ), LDBX, PERM( NLF, LVL ),
$ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
$ GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
$ DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
$ Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK,
$ INFO )
60 CONTINUE
70 CONTINUE
*
* The nodes on the bottom level of the tree were solved
* by DLASDQ. The corresponding right singular vector
* matrices are in explicit form. Apply them back.
*
NDB1 = ( ND+1 ) / 2
DO 80 I = NDB1, ND
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NR = IWORK( NDIMR+I1 )
NLP1 = NL + 1
IF( I.EQ.ND ) THEN
NRP1 = NR
ELSE
NRP1 = NR + 1
END IF
NLF = IC - NL
NRF = IC + 1
CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ),
$ LDU,
$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ),
$ LDU,
$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
80 CONTINUE
*
90 CONTINUE
*
RETURN
*
* End of DLALSA
*
END
*> \brief \b DLALSD uses the singular value decomposition of A to solve the least squares problem.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlalsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlalsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlalsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
* RANK, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION B( LDB, * ), D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLALSD uses the singular value decomposition of A to solve the least
*> squares problem of finding X to minimize the Euclidean norm of each
*> column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
*> are N-by-NRHS. The solution X overwrites B.
*>
*> The singular values of A smaller than RCOND times the largest
*> singular value are treated as zero in solving the least squares
*> problem; in this case a minimum norm solution is returned.
*> The actual singular values are returned in D in ascending order.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': D and E define an upper bidiagonal matrix.
*> = 'L': D and E define a lower bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> The maximum size of the subproblems at the bottom of the
*> computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the bidiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of columns of B. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry D contains the main diagonal of the bidiagonal
*> matrix. On exit, if INFO = 0, D contains its singular values.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> Contains the super-diagonal entries of the bidiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On input, B contains the right hand sides of the least
*> squares problem. On output, B contains the solution X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B in the calling subprogram.
*> LDB must be at least max(1,N).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The singular values of A less than or equal to RCOND times
*> the largest singular value are treated as zero in solving
*> the least squares problem. If RCOND is negative,
*> machine precision is used instead.
*> For example, if diag(S)*X=B were the least squares problem,
*> where diag(S) is a diagonal matrix of singular values, the
*> solution would be X(i) = B(i) / S(i) if S(i) is greater than
*> RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
*> RCOND*max(S).
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The number of singular values of A greater than RCOND times
*> the largest singular value.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension at least
*> (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
*> where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension at least
*> (3*N*NLVL + 11*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute a singular value while
*> working on the submatrix lying in rows and columns
*> INFO/(N+1) through MOD(INFO,N+1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lalsd
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
*> California at Berkeley, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*
* =====================================================================
SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
$ RANK, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION B( LDB, * ), D( * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
* ..
* .. Local Scalars ..
INTEGER BX, BXST, C, DIFL, DIFR, GIVCOL, GIVNUM,
$ GIVPTR, I, ICMPQ1, ICMPQ2, IWK, J, K, NLVL,
$ NM1, NSIZE, NSUB, NWORK, PERM, POLES, S, SIZEI,
$ SMLSZP, SQRE, ST, ST1, U, VT, Z
DOUBLE PRECISION CS, EPS, ORGNRM, R, RCND, SN, TOL
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL IDAMAX, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLALSA, DLARTG,
$ DLASCL,
$ DLASDA, DLASDQ, DLASET, DLASRT, DROT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, SIGN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.1 ) THEN
INFO = -4
ELSE IF( ( LDB.LT.1 ) .OR. ( LDB.LT.N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLALSD', -INFO )
RETURN
END IF
*
EPS = DLAMCH( 'Epsilon' )
*
* Set up the tolerance.
*
IF( ( RCOND.LE.ZERO ) .OR. ( RCOND.GE.ONE ) ) THEN
RCND = EPS
ELSE
RCND = RCOND
END IF
*
RANK = 0
*
* Quick return if possible.
*
IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
IF( D( 1 ).EQ.ZERO ) THEN
CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B, LDB )
ELSE
RANK = 1
CALL DLASCL( 'G', 0, 0, D( 1 ), ONE, 1, NRHS, B, LDB,
$ INFO )
D( 1 ) = ABS( D( 1 ) )
END IF
RETURN
END IF
*
* Rotate the matrix if it is lower bidiagonal.
*
IF( UPLO.EQ.'L' ) THEN
DO 10 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
IF( NRHS.EQ.1 ) THEN
CALL DROT( 1, B( I, 1 ), 1, B( I+1, 1 ), 1, CS, SN )
ELSE
WORK( I*2-1 ) = CS
WORK( I*2 ) = SN
END IF
10 CONTINUE
IF( NRHS.GT.1 ) THEN
DO 30 I = 1, NRHS
DO 20 J = 1, N - 1
CS = WORK( J*2-1 )
SN = WORK( J*2 )
CALL DROT( 1, B( J, I ), 1, B( J+1, I ), 1, CS,
$ SN )
20 CONTINUE
30 CONTINUE
END IF
END IF
*
* Scale.
*
NM1 = N - 1
ORGNRM = DLANST( 'M', N, D, E )
IF( ORGNRM.EQ.ZERO ) THEN
CALL DLASET( 'A', N, NRHS, ZERO, ZERO, B, LDB )
RETURN
END IF
*
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, INFO )
*
* If N is smaller than the minimum divide size SMLSIZ, then solve
* the problem with another solver.
*
IF( N.LE.SMLSIZ ) THEN
NWORK = 1 + N*N
CALL DLASET( 'A', N, N, ZERO, ONE, WORK, N )
CALL DLASDQ( 'U', 0, N, N, 0, NRHS, D, E, WORK, N, WORK, N,
$ B,
$ LDB, WORK( NWORK ), INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
DO 40 I = 1, N
IF( D( I ).LE.TOL ) THEN
CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B( I, 1 ),
$ LDB )
ELSE
CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, B( I,
$ 1 ),
$ LDB, INFO )
RANK = RANK + 1
END IF
40 CONTINUE
CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB,
$ ZERO,
$ WORK( NWORK ), N )
CALL DLACPY( 'A', N, NRHS, WORK( NWORK ), N, B, LDB )
*
* Unscale.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
CALL DLASRT( 'D', N, D, INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
*
RETURN
END IF
*
* Book-keeping and setting up some constants.
*
NLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
*
SMLSZP = SMLSIZ + 1
*
U = 1
VT = 1 + SMLSIZ*N
DIFL = VT + SMLSZP*N
DIFR = DIFL + NLVL*N
Z = DIFR + NLVL*N*2
C = Z + NLVL*N
S = C + N
POLES = S + N
GIVNUM = POLES + 2*NLVL*N
BX = GIVNUM + 2*NLVL*N
NWORK = BX + N*NRHS
*
SIZEI = 1 + N
K = SIZEI + N
GIVPTR = K + N
PERM = GIVPTR + N
GIVCOL = PERM + NLVL*N
IWK = GIVCOL + NLVL*N*2
*
ST = 1
SQRE = 0
ICMPQ1 = 1
ICMPQ2 = 0
NSUB = 0
*
DO 50 I = 1, N
IF( ABS( D( I ) ).LT.EPS ) THEN
D( I ) = SIGN( EPS, D( I ) )
END IF
50 CONTINUE
*
DO 60 I = 1, NM1
IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
NSUB = NSUB + 1
IWORK( NSUB ) = ST
*
* Subproblem found. First determine its size and then
* apply divide and conquer on it.
*
IF( I.LT.NM1 ) THEN
*
* A subproblem with E(I) small for I < NM1.
*
NSIZE = I - ST + 1
IWORK( SIZEI+NSUB-1 ) = NSIZE
ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
*
* A subproblem with E(NM1) not too small but I = NM1.
*
NSIZE = N - ST + 1
IWORK( SIZEI+NSUB-1 ) = NSIZE
ELSE
*
* A subproblem with E(NM1) small. This implies an
* 1-by-1 subproblem at D(N), which is not solved
* explicitly.
*
NSIZE = I - ST + 1
IWORK( SIZEI+NSUB-1 ) = NSIZE
NSUB = NSUB + 1
IWORK( NSUB ) = N
IWORK( SIZEI+NSUB-1 ) = 1
CALL DCOPY( NRHS, B( N, 1 ), LDB, WORK( BX+NM1 ), N )
END IF
ST1 = ST - 1
IF( NSIZE.EQ.1 ) THEN
*
* This is a 1-by-1 subproblem and is not solved
* explicitly.
*
CALL DCOPY( NRHS, B( ST, 1 ), LDB, WORK( BX+ST1 ), N )
ELSE IF( NSIZE.LE.SMLSIZ ) THEN
*
* This is a small subproblem and is solved by DLASDQ.
*
CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE,
$ WORK( VT+ST1 ), N )
CALL DLASDQ( 'U', 0, NSIZE, NSIZE, 0, NRHS, D( ST ),
$ E( ST ), WORK( VT+ST1 ), N, WORK( NWORK ),
$ N, B( ST, 1 ), LDB, WORK( NWORK ), INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
CALL DLACPY( 'A', NSIZE, NRHS, B( ST, 1 ), LDB,
$ WORK( BX+ST1 ), N )
ELSE
*
* A large problem. Solve it using divide and conquer.
*
CALL DLASDA( ICMPQ1, SMLSIZ, NSIZE, SQRE, D( ST ),
$ E( ST ), WORK( U+ST1 ), N, WORK( VT+ST1 ),
$ IWORK( K+ST1 ), WORK( DIFL+ST1 ),
$ WORK( DIFR+ST1 ), WORK( Z+ST1 ),
$ WORK( POLES+ST1 ), IWORK( GIVPTR+ST1 ),
$ IWORK( GIVCOL+ST1 ), N, IWORK( PERM+ST1 ),
$ WORK( GIVNUM+ST1 ), WORK( C+ST1 ),
$ WORK( S+ST1 ), WORK( NWORK ), IWORK( IWK ),
$ INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
BXST = BX + ST1
CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, B( ST, 1 ),
$ LDB, WORK( BXST ), N, WORK( U+ST1 ), N,
$ WORK( VT+ST1 ), IWORK( K+ST1 ),
$ WORK( DIFL+ST1 ), WORK( DIFR+ST1 ),
$ WORK( Z+ST1 ), WORK( POLES+ST1 ),
$ IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
$ IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ),
$ WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ),
$ IWORK( IWK ), INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
END IF
ST = I + 1
END IF
60 CONTINUE
*
* Apply the singular values and treat the tiny ones as zero.
*
TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
*
DO 70 I = 1, N
*
* Some of the elements in D can be negative because 1-by-1
* subproblems were not solved explicitly.
*
IF( ABS( D( I ) ).LE.TOL ) THEN
CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, WORK( BX+I-1 ),
$ N )
ELSE
RANK = RANK + 1
CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS,
$ WORK( BX+I-1 ), N, INFO )
END IF
D( I ) = ABS( D( I ) )
70 CONTINUE
*
* Now apply back the right singular vectors.
*
ICMPQ2 = 1
DO 80 I = 1, NSUB
ST = IWORK( I )
ST1 = ST - 1
NSIZE = IWORK( SIZEI+I-1 )
BXST = BX + ST1
IF( NSIZE.EQ.1 ) THEN
CALL DCOPY( NRHS, WORK( BXST ), N, B( ST, 1 ), LDB )
ELSE IF( NSIZE.LE.SMLSIZ ) THEN
CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
$ WORK( VT+ST1 ), N, WORK( BXST ), N, ZERO,
$ B( ST, 1 ), LDB )
ELSE
CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, WORK( BXST ),
$ N,
$ B( ST, 1 ), LDB, WORK( U+ST1 ), N,
$ WORK( VT+ST1 ), IWORK( K+ST1 ),
$ WORK( DIFL+ST1 ), WORK( DIFR+ST1 ),
$ WORK( Z+ST1 ), WORK( POLES+ST1 ),
$ IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
$ IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ),
$ WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ),
$ IWORK( IWK ), INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
END IF
80 CONTINUE
*
* Unscale and sort the singular values.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
CALL DLASRT( 'D', N, D, INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
*
RETURN
*
* End of DLALSD
*
END
*> \brief \b DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAMRG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlamrg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlamrg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlamrg.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX )
*
* .. Scalar Arguments ..
* INTEGER DTRD1, DTRD2, N1, N2
* ..
* .. Array Arguments ..
* INTEGER INDEX( * )
* DOUBLE PRECISION A( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAMRG will create a permutation list which will merge the elements
*> of A (which is composed of two independently sorted sets) into a
*> single set which is sorted in ascending order.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> These arguments contain the respective lengths of the two
*> sorted lists to be merged.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (N1+N2)
*> The first N1 elements of A contain a list of numbers which
*> are sorted in either ascending or descending order. Likewise
*> for the final N2 elements.
*> \endverbatim
*>
*> \param[in] DTRD1
*> \verbatim
*> DTRD1 is INTEGER
*> \endverbatim
*>
*> \param[in] DTRD2
*> \verbatim
*> DTRD2 is INTEGER
*> These are the strides to be taken through the array A.
*> Allowable strides are 1 and -1. They indicate whether a
*> subset of A is sorted in ascending (DTRDx = 1) or descending
*> (DTRDx = -1) order.
*> \endverbatim
*>
*> \param[out] INDEX
*> \verbatim
*> INDEX is INTEGER array, dimension (N1+N2)
*> On exit this array will contain a permutation such that
*> if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
*> sorted in ascending order.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lamrg
*
* =====================================================================
SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER DTRD1, DTRD2, N1, N2
* ..
* .. Array Arguments ..
INTEGER INDEX( * )
DOUBLE PRECISION A( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IND1, IND2, N1SV, N2SV
* ..
* .. Executable Statements ..
*
N1SV = N1
N2SV = N2
IF( DTRD1.GT.0 ) THEN
IND1 = 1
ELSE
IND1 = N1
END IF
IF( DTRD2.GT.0 ) THEN
IND2 = 1 + N1
ELSE
IND2 = N1 + N2
END IF
I = 1
* while ( (N1SV > 0) & (N2SV > 0) )
10 CONTINUE
IF( N1SV.GT.0 .AND. N2SV.GT.0 ) THEN
IF( A( IND1 ).LE.A( IND2 ) ) THEN
INDEX( I ) = IND1
I = I + 1
IND1 = IND1 + DTRD1
N1SV = N1SV - 1
ELSE
INDEX( I ) = IND2
I = I + 1
IND2 = IND2 + DTRD2
N2SV = N2SV - 1
END IF
GO TO 10
END IF
* end while
IF( N1SV.EQ.0 ) THEN
DO 20 N1SV = 1, N2SV
INDEX( I ) = IND2
I = I + 1
IND2 = IND2 + DTRD2
20 CONTINUE
ELSE
* N2SV .EQ. 0
DO 30 N2SV = 1, N1SV
INDEX( I ) = IND1
I = I + 1
IND1 = IND1 + DTRD1
30 CONTINUE
END IF
*
RETURN
*
* End of DLAMRG
*
END
*> \brief \b DLANEG computes the Sturm count.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANEG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaneg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaneg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaneg.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION DLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
* .. Scalar Arguments ..
* INTEGER N, R
* DOUBLE PRECISION PIVMIN, SIGMA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), LLD( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANEG computes the Sturm count, the number of negative pivots
*> encountered while factoring tridiagonal T - sigma I = L D L^T.
*> This implementation works directly on the factors without forming
*> the tridiagonal matrix T. The Sturm count is also the number of
*> eigenvalues of T less than sigma.
*>
*> This routine is called from DLARRB.
*>
*> The current routine does not use the PIVMIN parameter but rather
*> requires IEEE-754 propagation of Infinities and NaNs. This
*> routine also has no input range restrictions but does require
*> default exception handling such that x/0 produces Inf when x is
*> non-zero, and Inf/Inf produces NaN. For more information, see:
*>
*> Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
*> Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on
*> Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624
*> (Tech report version in LAWN 172 with the same title.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*> LLD is DOUBLE PRECISION array, dimension (N-1)
*> The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> Shift amount in T - sigma I = L D L^T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot in the Sturm sequence. May be used
*> when zero pivots are encountered on non-IEEE-754
*> architectures.
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*> R is INTEGER
*> The twist index for the twisted factorization that is used
*> for the negcount.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laneg
*
*> \par Contributors:
* ==================
*>
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA \n
*> Jason Riedy, University of California, Berkeley, USA \n
*>
* =====================================================================
INTEGER FUNCTION DLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER N, R
DOUBLE PRECISION PIVMIN, SIGMA
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), LLD( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* Some architectures propagate Infinities and NaNs very slowly, so
* the code computes counts in BLKLEN chunks. Then a NaN can
* propagate at most BLKLEN columns before being detected. This is
* not a general tuning parameter; it needs only to be just large
* enough that the overhead is tiny in common cases.
INTEGER BLKLEN
PARAMETER ( BLKLEN = 128 )
* ..
* .. Local Scalars ..
INTEGER BJ, J, NEG1, NEG2, NEGCNT
DOUBLE PRECISION BSAV, DMINUS, DPLUS, GAMMA, P, T, TMP
LOGICAL SAWNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN, MAX
* ..
* .. External Functions ..
LOGICAL DISNAN
EXTERNAL DISNAN
* ..
* .. Executable Statements ..
NEGCNT = 0
* I) upper part: L D L^T - SIGMA I = L+ D+ L+^T
T = -SIGMA
DO 210 BJ = 1, R-1, BLKLEN
NEG1 = 0
BSAV = T
DO 21 J = BJ, MIN(BJ+BLKLEN-1, R-1)
DPLUS = D( J ) + T
IF( DPLUS.LT.ZERO ) NEG1 = NEG1 + 1
TMP = T / DPLUS
T = TMP * LLD( J ) - SIGMA
21 CONTINUE
SAWNAN = DISNAN( T )
* Run a slower version of the above loop if a NaN is detected.
* A NaN should occur only with a zero pivot after an infinite
* pivot. In that case, substituting 1 for T/DPLUS is the
* correct limit.
IF( SAWNAN ) THEN
NEG1 = 0
T = BSAV
DO 22 J = BJ, MIN(BJ+BLKLEN-1, R-1)
DPLUS = D( J ) + T
IF( DPLUS.LT.ZERO ) NEG1 = NEG1 + 1
TMP = T / DPLUS
IF (DISNAN(TMP)) TMP = ONE
T = TMP * LLD(J) - SIGMA
22 CONTINUE
END IF
NEGCNT = NEGCNT + NEG1
210 CONTINUE
*
* II) lower part: L D L^T - SIGMA I = U- D- U-^T
P = D( N ) - SIGMA
DO 230 BJ = N-1, R, -BLKLEN
NEG2 = 0
BSAV = P
DO 23 J = BJ, MAX(BJ-BLKLEN+1, R), -1
DMINUS = LLD( J ) + P
IF( DMINUS.LT.ZERO ) NEG2 = NEG2 + 1
TMP = P / DMINUS
P = TMP * D( J ) - SIGMA
23 CONTINUE
SAWNAN = DISNAN( P )
* As above, run a slower version that substitutes 1 for Inf/Inf.
*
IF( SAWNAN ) THEN
NEG2 = 0
P = BSAV
DO 24 J = BJ, MAX(BJ-BLKLEN+1, R), -1
DMINUS = LLD( J ) + P
IF( DMINUS.LT.ZERO ) NEG2 = NEG2 + 1
TMP = P / DMINUS
IF (DISNAN(TMP)) TMP = ONE
P = TMP * D(J) - SIGMA
24 CONTINUE
END IF
NEGCNT = NEGCNT + NEG2
230 CONTINUE
*
* III) Twist index
* T was shifted by SIGMA initially.
GAMMA = (T + SIGMA) + P
IF( GAMMA.LT.ZERO ) NEGCNT = NEGCNT+1
DLANEG = NEGCNT
END
*> \brief \b DLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANGB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlangb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlangb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlangb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANGB( NORM, N, KL, KU, AB, LDAB,
* WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER KL, KU, LDAB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANGB returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of an
*> n by n band matrix A, with kl sub-diagonals and ku super-diagonals.
*> \endverbatim
*>
*> \return DLANGB
*> \verbatim
*>
*> DLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANGB as described
*> above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANGB is
*> set to zero.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of sub-diagonals of the matrix A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of super-diagonals of the matrix A. KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The band matrix A, stored in rows 1 to KL+KU+1. The j-th
*> column of A is stored in the j-th column of the array AB as
*> follows:
*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*> referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup langb
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANGB( NORM, N, KL, KU, AB, LDAB,
$ WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER KL, KU, LDAB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
* =====================================================================
*
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, K, L
DOUBLE PRECISION SCALE, SUM, VALUE, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
DO 20 J = 1, N
DO 10 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
TEMP = ABS( AB( I, J ) )
IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
10 CONTINUE
20 CONTINUE
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
* Find norm1(A).
*
VALUE = ZERO
DO 40 J = 1, N
SUM = ZERO
DO 30 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
SUM = SUM + ABS( AB( I, J ) )
30 CONTINUE
IF( VALUE.LT.SUM .OR. DISNAN( SUM ) ) VALUE = SUM
40 CONTINUE
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
DO 50 I = 1, N
WORK( I ) = ZERO
50 CONTINUE
DO 70 J = 1, N
K = KU + 1 - J
DO 60 I = MAX( 1, J-KU ), MIN( N, J+KL )
WORK( I ) = WORK( I ) + ABS( AB( K+I, J ) )
60 CONTINUE
70 CONTINUE
VALUE = ZERO
DO 80 I = 1, N
TEMP = WORK( I )
IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
80 CONTINUE
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
DO 90 J = 1, N
L = MAX( 1, J-KU )
K = KU + 1 - J + L
CALL DLASSQ( MIN( N, J+KL )-L+1, AB( K, J ), 1, SCALE,
$ SUM )
90 CONTINUE
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANGB = VALUE
RETURN
*
* End of DLANGB
*
END
*> \brief \b DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANGE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlange.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlange.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlange.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANGE returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real matrix A.
*> \endverbatim
*>
*> \return DLANGE
*> \verbatim
*>
*> DLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANGE as described
*> above.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0. When M = 0,
*> DLANGE is set to zero.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0. When N = 0,
*> DLANGE is set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(M,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= M when NORM = 'I'; otherwise, WORK is not
*> referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lange
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION SCALE, SUM, VALUE, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN, SQRT
* ..
* .. Executable Statements ..
*
IF( MIN( M, N ).EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
DO 20 J = 1, N
DO 10 I = 1, M
TEMP = ABS( A( I, J ) )
IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
10 CONTINUE
20 CONTINUE
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
* Find norm1(A).
*
VALUE = ZERO
DO 40 J = 1, N
SUM = ZERO
DO 30 I = 1, M
SUM = SUM + ABS( A( I, J ) )
30 CONTINUE
IF( VALUE.LT.SUM .OR. DISNAN( SUM ) ) VALUE = SUM
40 CONTINUE
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
DO 50 I = 1, M
WORK( I ) = ZERO
50 CONTINUE
DO 70 J = 1, N
DO 60 I = 1, M
WORK( I ) = WORK( I ) + ABS( A( I, J ) )
60 CONTINUE
70 CONTINUE
VALUE = ZERO
DO 80 I = 1, M
TEMP = WORK( I )
IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
80 CONTINUE
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
DO 90 J = 1, N
CALL DLASSQ( M, A( 1, J ), 1, SCALE, SUM )
90 CONTINUE
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANGE = VALUE
RETURN
*
* End of DLANGE
*
END
*> \brief \b DLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANGT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlangt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlangt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlangt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANGT( NORM, N, DL, D, DU )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DL( * ), DU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANGT returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real tridiagonal matrix A.
*> \endverbatim
*>
*> \return DLANGT
*> \verbatim
*>
*> DLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANGT as described
*> above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANGT is
*> set to zero.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) sub-diagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) super-diagonal elements of A.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup langt
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANGT( NORM, N, DL, D, DU )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DL( * ), DU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION ANORM, SCALE, SUM, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( N.LE.0 ) THEN
ANORM = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
ANORM = ABS( D( N ) )
DO 10 I = 1, N - 1
IF( ANORM.LT.ABS( DL( I ) ) .OR.
$ DISNAN( ABS( DL( I ) ) ) )
$ ANORM = ABS(DL(I))
IF( ANORM.LT.ABS( D( I ) ) .OR. DISNAN( ABS( D( I ) ) ) )
$ ANORM = ABS(D(I))
IF( ANORM.LT.ABS( DU( I ) ) .OR.
$ DISNAN (ABS( DU( I ) ) ) )
$ ANORM = ABS(DU(I))
10 CONTINUE
ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' ) THEN
*
* Find norm1(A).
*
IF( N.EQ.1 ) THEN
ANORM = ABS( D( 1 ) )
ELSE
ANORM = ABS( D( 1 ) )+ABS( DL( 1 ) )
TEMP = ABS( D( N ) )+ABS( DU( N-1 ) )
IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
DO 20 I = 2, N - 1
TEMP = ABS( D( I ) )+ABS( DL( I ) )+ABS( DU( I-1 ) )
IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
20 CONTINUE
END IF
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
IF( N.EQ.1 ) THEN
ANORM = ABS( D( 1 ) )
ELSE
ANORM = ABS( D( 1 ) )+ABS( DU( 1 ) )
TEMP = ABS( D( N ) )+ABS( DL( N-1 ) )
IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
DO 30 I = 2, N - 1
TEMP = ABS( D( I ) )+ABS( DU( I ) )+ABS( DL( I-1 ) )
IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
30 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
CALL DLASSQ( N, D, 1, SCALE, SUM )
IF( N.GT.1 ) THEN
CALL DLASSQ( N-1, DL, 1, SCALE, SUM )
CALL DLASSQ( N-1, DU, 1, SCALE, SUM )
END IF
ANORM = SCALE*SQRT( SUM )
END IF
*
DLANGT = ANORM
RETURN
*
* End of DLANGT
*
END
*> \brief \b DLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANHS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanhs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanhs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanhs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANHS( NORM, N, A, LDA, WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANHS returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> Hessenberg matrix A.
*> \endverbatim
*>
*> \return DLANHS
*> \verbatim
*>
*> DLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANHS as described
*> above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANHS is
*> set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The n by n upper Hessenberg matrix A; the part of A below the
*> first sub-diagonal is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(N,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*> referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhs
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANHS( NORM, N, A, LDA, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
DO 20 J = 1, N
DO 10 I = 1, MIN( N, J+1 )
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
20 CONTINUE
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
* Find norm1(A).
*
VALUE = ZERO
DO 40 J = 1, N
SUM = ZERO
DO 30 I = 1, MIN( N, J+1 )
SUM = SUM + ABS( A( I, J ) )
30 CONTINUE
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
40 CONTINUE
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
DO 50 I = 1, N
WORK( I ) = ZERO
50 CONTINUE
DO 70 J = 1, N
DO 60 I = 1, MIN( N, J+1 )
WORK( I ) = WORK( I ) + ABS( A( I, J ) )
60 CONTINUE
70 CONTINUE
VALUE = ZERO
DO 80 I = 1, N
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
80 CONTINUE
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
DO 90 J = 1, N
CALL DLASSQ( MIN( N, J+1 ), A( 1, J ), 1, SCALE, SUM )
90 CONTINUE
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANHS = VALUE
RETURN
*
* End of DLANHS
*
END
*> \brief \b DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
* WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM, UPLO
* INTEGER K, LDAB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANSB returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of an
*> n by n symmetric band matrix A, with k super-diagonals.
*> \endverbatim
*>
*> \return DLANSB
*> \verbatim
*>
*> DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANSB as described
*> above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> band matrix A is supplied.
*> = 'U': Upper triangular part is supplied
*> = 'L': Lower triangular part is supplied
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANSB is
*> set to zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of super-diagonals or sub-diagonals of the
*> band matrix A. K >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangle of the symmetric band matrix A,
*> stored in the first K+1 rows of AB. The j-th column of A is
*> stored in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= K+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*> WORK is not referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhb
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
$ WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER K, LDAB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, L
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = MAX( K+2-J, 1 ), K + 1
SUM = ABS( AB( I, J ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = 1, MIN( N+1-J, K+1 )
SUM = ABS( AB( I, J ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
$ ( LSAME( NORM, 'O' ) ) .OR.
$ ( NORM.EQ.'1' ) ) THEN
*
* Find normI(A) ( = norm1(A), since A is symmetric).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
SUM = ZERO
L = K + 1 - J
DO 50 I = MAX( 1, J-K ), J - 1
ABSA = ABS( AB( L+I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
50 CONTINUE
WORK( J ) = SUM + ABS( AB( K+1, J ) )
60 CONTINUE
DO 70 I = 1, N
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
ELSE
DO 80 I = 1, N
WORK( I ) = ZERO
80 CONTINUE
DO 100 J = 1, N
SUM = WORK( J ) + ABS( AB( 1, J ) )
L = 1 - J
DO 90 I = J + 1, MIN( N, J+K )
ABSA = ABS( AB( L+I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
90 CONTINUE
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
100 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( K.GT.0 ) THEN
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL DLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ),
$ J ),
$ 1, SCALE, SUM )
110 CONTINUE
L = K + 1
ELSE
DO 120 J = 1, N - 1
CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
$ SUM )
120 CONTINUE
L = 1
END IF
SUM = 2*SUM
ELSE
L = 1
END IF
CALL DLASSQ( N, AB( L, 1 ), LDAB, SCALE, SUM )
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANSB = VALUE
RETURN
*
* End of DLANSB
*
END
*> \brief \b DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANSF( NORM, TRANSR, UPLO, N, A, WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM, TRANSR, UPLO
* INTEGER N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: * ), WORK( 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANSF returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real symmetric matrix A in RFP format.
*> \endverbatim
*>
*> \return DLANSF
*> \verbatim
*>
*> DLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANSF as described
*> above.
*> \endverbatim
*>
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> Specifies whether the RFP format of A is normal or
*> transposed format.
*> = 'N': RFP format is Normal;
*> = 'T': RFP format is Transpose.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the RFP matrix A came from
*> an upper or lower triangular matrix as follows:
*> = 'U': RFP A came from an upper triangular matrix;
*> = 'L': RFP A came from a lower triangular matrix.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANSF is
*> set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
*> On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
*> part of the symmetric matrix A stored in RFP format. See the
*> "Notes" below for more details.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*> WORK is not referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANSF( NORM, TRANSR, UPLO, N, A,
$ WORK )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM, TRANSR, UPLO
INTEGER N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( 0: * ), WORK( 0: * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, IFM, ILU, NOE, N1, K, L, LDA
DOUBLE PRECISION SCALE, S, VALUE, AA, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
DLANSF = ZERO
RETURN
ELSE IF( N.EQ.1 ) THEN
DLANSF = ABS( A(0) )
RETURN
END IF
*
* set noe = 1 if n is odd. if n is even set noe=0
*
NOE = 1
IF( MOD( N, 2 ).EQ.0 )
$ NOE = 0
*
* set ifm = 0 when form='T or 't' and 1 otherwise
*
IFM = 1
IF( LSAME( TRANSR, 'T' ) )
$ IFM = 0
*
* set ilu = 0 when uplo='U or 'u' and 1 otherwise
*
ILU = 1
IF( LSAME( UPLO, 'U' ) )
$ ILU = 0
*
* set lda = (n+1)/2 when ifm = 0
* set lda = n when ifm = 1 and noe = 1
* set lda = n+1 when ifm = 1 and noe = 0
*
IF( IFM.EQ.1 ) THEN
IF( NOE.EQ.1 ) THEN
LDA = N
ELSE
* noe=0
LDA = N + 1
END IF
ELSE
* ifm=0
LDA = ( N+1 ) / 2
END IF
*
IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
K = ( N+1 ) / 2
VALUE = ZERO
IF( NOE.EQ.1 ) THEN
* n is odd
IF( IFM.EQ.1 ) THEN
* A is n by k
DO J = 0, K - 1
DO I = 0, N - 1
TEMP = ABS( A( I+J*LDA ) )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END DO
ELSE
* xpose case; A is k by n
DO J = 0, N - 1
DO I = 0, K - 1
TEMP = ABS( A( I+J*LDA ) )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END DO
END IF
ELSE
* n is even
IF( IFM.EQ.1 ) THEN
* A is n+1 by k
DO J = 0, K - 1
DO I = 0, N
TEMP = ABS( A( I+J*LDA ) )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END DO
ELSE
* xpose case; A is k by n+1
DO J = 0, N
DO I = 0, K - 1
TEMP = ABS( A( I+J*LDA ) )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END DO
END IF
END IF
ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
$ ( LSAME( NORM, 'O' ) ) .OR.
$ ( NORM.EQ.'1' ) ) THEN
*
* Find normI(A) ( = norm1(A), since A is symmetric).
*
IF( IFM.EQ.1 ) THEN
K = N / 2
IF( NOE.EQ.1 ) THEN
* n is odd
IF( ILU.EQ.0 ) THEN
DO I = 0, K - 1
WORK( I ) = ZERO
END DO
DO J = 0, K
S = ZERO
DO I = 0, K + J - 1
AA = ABS( A( I+J*LDA ) )
* -> A(i,j+k)
S = S + AA
WORK( I ) = WORK( I ) + AA
END DO
AA = ABS( A( I+J*LDA ) )
* -> A(j+k,j+k)
WORK( J+K ) = S + AA
IF( I.EQ.K+K )
$ GO TO 10
I = I + 1
AA = ABS( A( I+J*LDA ) )
* -> A(j,j)
WORK( J ) = WORK( J ) + AA
S = ZERO
DO L = J + 1, K - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* -> A(l,j)
S = S + AA
WORK( L ) = WORK( L ) + AA
END DO
WORK( J ) = WORK( J ) + S
END DO
10 CONTINUE
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
ELSE
* ilu = 1
K = K + 1
* k=(n+1)/2 for n odd and ilu=1
DO I = K, N - 1
WORK( I ) = ZERO
END DO
DO J = K - 1, 0, -1
S = ZERO
DO I = 0, J - 2
AA = ABS( A( I+J*LDA ) )
* -> A(j+k,i+k)
S = S + AA
WORK( I+K ) = WORK( I+K ) + AA
END DO
IF( J.GT.0 ) THEN
AA = ABS( A( I+J*LDA ) )
* -> A(j+k,j+k)
S = S + AA
WORK( I+K ) = WORK( I+K ) + S
* i=j
I = I + 1
END IF
AA = ABS( A( I+J*LDA ) )
* -> A(j,j)
WORK( J ) = AA
S = ZERO
DO L = J + 1, N - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* -> A(l,j)
S = S + AA
WORK( L ) = WORK( L ) + AA
END DO
WORK( J ) = WORK( J ) + S
END DO
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END IF
ELSE
* n is even
IF( ILU.EQ.0 ) THEN
DO I = 0, K - 1
WORK( I ) = ZERO
END DO
DO J = 0, K - 1
S = ZERO
DO I = 0, K + J - 1
AA = ABS( A( I+J*LDA ) )
* -> A(i,j+k)
S = S + AA
WORK( I ) = WORK( I ) + AA
END DO
AA = ABS( A( I+J*LDA ) )
* -> A(j+k,j+k)
WORK( J+K ) = S + AA
I = I + 1
AA = ABS( A( I+J*LDA ) )
* -> A(j,j)
WORK( J ) = WORK( J ) + AA
S = ZERO
DO L = J + 1, K - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* -> A(l,j)
S = S + AA
WORK( L ) = WORK( L ) + AA
END DO
WORK( J ) = WORK( J ) + S
END DO
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
ELSE
* ilu = 1
DO I = K, N - 1
WORK( I ) = ZERO
END DO
DO J = K - 1, 0, -1
S = ZERO
DO I = 0, J - 1
AA = ABS( A( I+J*LDA ) )
* -> A(j+k,i+k)
S = S + AA
WORK( I+K ) = WORK( I+K ) + AA
END DO
AA = ABS( A( I+J*LDA ) )
* -> A(j+k,j+k)
S = S + AA
WORK( I+K ) = WORK( I+K ) + S
* i=j
I = I + 1
AA = ABS( A( I+J*LDA ) )
* -> A(j,j)
WORK( J ) = AA
S = ZERO
DO L = J + 1, N - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* -> A(l,j)
S = S + AA
WORK( L ) = WORK( L ) + AA
END DO
WORK( J ) = WORK( J ) + S
END DO
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END IF
END IF
ELSE
* ifm=0
K = N / 2
IF( NOE.EQ.1 ) THEN
* n is odd
IF( ILU.EQ.0 ) THEN
N1 = K
* n/2
K = K + 1
* k is the row size and lda
DO I = N1, N - 1
WORK( I ) = ZERO
END DO
DO J = 0, N1 - 1
S = ZERO
DO I = 0, K - 1
AA = ABS( A( I+J*LDA ) )
* A(j,n1+i)
WORK( I+N1 ) = WORK( I+N1 ) + AA
S = S + AA
END DO
WORK( J ) = S
END DO
* j=n1=k-1 is special
S = ABS( A( 0+J*LDA ) )
* A(k-1,k-1)
DO I = 1, K - 1
AA = ABS( A( I+J*LDA ) )
* A(k-1,i+n1)
WORK( I+N1 ) = WORK( I+N1 ) + AA
S = S + AA
END DO
WORK( J ) = WORK( J ) + S
DO J = K, N - 1
S = ZERO
DO I = 0, J - K - 1
AA = ABS( A( I+J*LDA ) )
* A(i,j-k)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
* i=j-k
AA = ABS( A( I+J*LDA ) )
* A(j-k,j-k)
S = S + AA
WORK( J-K ) = WORK( J-K ) + S
I = I + 1
S = ABS( A( I+J*LDA ) )
* A(j,j)
DO L = J + 1, N - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* A(j,l)
WORK( L ) = WORK( L ) + AA
S = S + AA
END DO
WORK( J ) = WORK( J ) + S
END DO
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
ELSE
* ilu=1
K = K + 1
* k=(n+1)/2 for n odd and ilu=1
DO I = K, N - 1
WORK( I ) = ZERO
END DO
DO J = 0, K - 2
* process
S = ZERO
DO I = 0, J - 1
AA = ABS( A( I+J*LDA ) )
* A(j,i)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
AA = ABS( A( I+J*LDA ) )
* i=j so process of A(j,j)
S = S + AA
WORK( J ) = S
* is initialised here
I = I + 1
* i=j process A(j+k,j+k)
AA = ABS( A( I+J*LDA ) )
S = AA
DO L = K + J + 1, N - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* A(l,k+j)
S = S + AA
WORK( L ) = WORK( L ) + AA
END DO
WORK( K+J ) = WORK( K+J ) + S
END DO
* j=k-1 is special :process col A(k-1,0:k-1)
S = ZERO
DO I = 0, K - 2
AA = ABS( A( I+J*LDA ) )
* A(k,i)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
* i=k-1
AA = ABS( A( I+J*LDA ) )
* A(k-1,k-1)
S = S + AA
WORK( I ) = S
* done with col j=k+1
DO J = K, N - 1
* process col j of A = A(j,0:k-1)
S = ZERO
DO I = 0, K - 1
AA = ABS( A( I+J*LDA ) )
* A(j,i)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
WORK( J ) = WORK( J ) + S
END DO
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END IF
ELSE
* n is even
IF( ILU.EQ.0 ) THEN
DO I = K, N - 1
WORK( I ) = ZERO
END DO
DO J = 0, K - 1
S = ZERO
DO I = 0, K - 1
AA = ABS( A( I+J*LDA ) )
* A(j,i+k)
WORK( I+K ) = WORK( I+K ) + AA
S = S + AA
END DO
WORK( J ) = S
END DO
* j=k
AA = ABS( A( 0+J*LDA ) )
* A(k,k)
S = AA
DO I = 1, K - 1
AA = ABS( A( I+J*LDA ) )
* A(k,k+i)
WORK( I+K ) = WORK( I+K ) + AA
S = S + AA
END DO
WORK( J ) = WORK( J ) + S
DO J = K + 1, N - 1
S = ZERO
DO I = 0, J - 2 - K
AA = ABS( A( I+J*LDA ) )
* A(i,j-k-1)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
* i=j-1-k
AA = ABS( A( I+J*LDA ) )
* A(j-k-1,j-k-1)
S = S + AA
WORK( J-K-1 ) = WORK( J-K-1 ) + S
I = I + 1
AA = ABS( A( I+J*LDA ) )
* A(j,j)
S = AA
DO L = J + 1, N - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* A(j,l)
WORK( L ) = WORK( L ) + AA
S = S + AA
END DO
WORK( J ) = WORK( J ) + S
END DO
* j=n
S = ZERO
DO I = 0, K - 2
AA = ABS( A( I+J*LDA ) )
* A(i,k-1)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
* i=k-1
AA = ABS( A( I+J*LDA ) )
* A(k-1,k-1)
S = S + AA
WORK( I ) = WORK( I ) + S
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
ELSE
* ilu=1
DO I = K, N - 1
WORK( I ) = ZERO
END DO
* j=0 is special :process col A(k:n-1,k)
S = ABS( A( 0 ) )
* A(k,k)
DO I = 1, K - 1
AA = ABS( A( I ) )
* A(k+i,k)
WORK( I+K ) = WORK( I+K ) + AA
S = S + AA
END DO
WORK( K ) = WORK( K ) + S
DO J = 1, K - 1
* process
S = ZERO
DO I = 0, J - 2
AA = ABS( A( I+J*LDA ) )
* A(j-1,i)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
AA = ABS( A( I+J*LDA ) )
* i=j-1 so process of A(j-1,j-1)
S = S + AA
WORK( J-1 ) = S
* is initialised here
I = I + 1
* i=j process A(j+k,j+k)
AA = ABS( A( I+J*LDA ) )
S = AA
DO L = K + J + 1, N - 1
I = I + 1
AA = ABS( A( I+J*LDA ) )
* A(l,k+j)
S = S + AA
WORK( L ) = WORK( L ) + AA
END DO
WORK( K+J ) = WORK( K+J ) + S
END DO
* j=k is special :process col A(k,0:k-1)
S = ZERO
DO I = 0, K - 2
AA = ABS( A( I+J*LDA ) )
* A(k,i)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
* i=k-1
AA = ABS( A( I+J*LDA ) )
* A(k-1,k-1)
S = S + AA
WORK( I ) = S
* done with col j=k+1
DO J = K + 1, N
* process col j-1 of A = A(j-1,0:k-1)
S = ZERO
DO I = 0, K - 1
AA = ABS( A( I+J*LDA ) )
* A(j-1,i)
WORK( I ) = WORK( I ) + AA
S = S + AA
END DO
WORK( J-1 ) = WORK( J-1 ) + S
END DO
VALUE = WORK( 0 )
DO I = 1, N-1
TEMP = WORK( I )
IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
$ VALUE = TEMP
END DO
END IF
END IF
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
K = ( N+1 ) / 2
SCALE = ZERO
S = ONE
IF( NOE.EQ.1 ) THEN
* n is odd
IF( IFM.EQ.1 ) THEN
* A is normal
IF( ILU.EQ.0 ) THEN
* A is upper
DO J = 0, K - 3
CALL DLASSQ( K-J-2, A( K+J+1+J*LDA ), 1, SCALE,
$ S )
* L at A(k,0)
END DO
DO J = 0, K - 1
CALL DLASSQ( K+J-1, A( 0+J*LDA ), 1, SCALE, S )
* trap U at A(0,0)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K-1, A( K ), LDA+1, SCALE, S )
* tri L at A(k,0)
CALL DLASSQ( K, A( K-1 ), LDA+1, SCALE, S )
* tri U at A(k-1,0)
ELSE
* ilu=1 & A is lower
DO J = 0, K - 1
CALL DLASSQ( N-J-1, A( J+1+J*LDA ), 1, SCALE,
$ S )
* trap L at A(0,0)
END DO
DO J = 0, K - 2
CALL DLASSQ( J, A( 0+( 1+J )*LDA ), 1, SCALE,
$ S )
* U at A(0,1)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
* tri L at A(0,0)
CALL DLASSQ( K-1, A( 0+LDA ), LDA+1, SCALE, S )
* tri U at A(0,1)
END IF
ELSE
* A is xpose
IF( ILU.EQ.0 ) THEN
* A**T is upper
DO J = 1, K - 2
CALL DLASSQ( J, A( 0+( K+J )*LDA ), 1, SCALE,
$ S )
* U at A(0,k)
END DO
DO J = 0, K - 2
CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
* k by k-1 rect. at A(0,0)
END DO
DO J = 0, K - 2
CALL DLASSQ( K-J-1, A( J+1+( J+K-1 )*LDA ), 1,
$ SCALE, S )
* L at A(0,k-1)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K-1, A( 0+K*LDA ), LDA+1, SCALE, S )
* tri U at A(0,k)
CALL DLASSQ( K, A( 0+( K-1 )*LDA ), LDA+1, SCALE,
$ S )
* tri L at A(0,k-1)
ELSE
* A**T is lower
DO J = 1, K - 1
CALL DLASSQ( J, A( 0+J*LDA ), 1, SCALE, S )
* U at A(0,0)
END DO
DO J = K, N - 1
CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
* k by k-1 rect. at A(0,k)
END DO
DO J = 0, K - 3
CALL DLASSQ( K-J-2, A( J+2+J*LDA ), 1, SCALE,
$ S )
* L at A(1,0)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
* tri U at A(0,0)
CALL DLASSQ( K-1, A( 1 ), LDA+1, SCALE, S )
* tri L at A(1,0)
END IF
END IF
ELSE
* n is even
IF( IFM.EQ.1 ) THEN
* A is normal
IF( ILU.EQ.0 ) THEN
* A is upper
DO J = 0, K - 2
CALL DLASSQ( K-J-1, A( K+J+2+J*LDA ), 1, SCALE,
$ S )
* L at A(k+1,0)
END DO
DO J = 0, K - 1
CALL DLASSQ( K+J, A( 0+J*LDA ), 1, SCALE, S )
* trap U at A(0,0)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K, A( K+1 ), LDA+1, SCALE, S )
* tri L at A(k+1,0)
CALL DLASSQ( K, A( K ), LDA+1, SCALE, S )
* tri U at A(k,0)
ELSE
* ilu=1 & A is lower
DO J = 0, K - 1
CALL DLASSQ( N-J-1, A( J+2+J*LDA ), 1, SCALE,
$ S )
* trap L at A(1,0)
END DO
DO J = 1, K - 1
CALL DLASSQ( J, A( 0+J*LDA ), 1, SCALE, S )
* U at A(0,0)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K, A( 1 ), LDA+1, SCALE, S )
* tri L at A(1,0)
CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
* tri U at A(0,0)
END IF
ELSE
* A is xpose
IF( ILU.EQ.0 ) THEN
* A**T is upper
DO J = 1, K - 1
CALL DLASSQ( J, A( 0+( K+1+J )*LDA ), 1, SCALE,
$ S )
* U at A(0,k+1)
END DO
DO J = 0, K - 1
CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
* k by k rect. at A(0,0)
END DO
DO J = 0, K - 2
CALL DLASSQ( K-J-1, A( J+1+( J+K )*LDA ), 1,
$ SCALE,
$ S )
* L at A(0,k)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K, A( 0+( K+1 )*LDA ), LDA+1, SCALE,
$ S )
* tri U at A(0,k+1)
CALL DLASSQ( K, A( 0+K*LDA ), LDA+1, SCALE, S )
* tri L at A(0,k)
ELSE
* A**T is lower
DO J = 1, K - 1
CALL DLASSQ( J, A( 0+( J+1 )*LDA ), 1, SCALE,
$ S )
* U at A(0,1)
END DO
DO J = K + 1, N
CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
* k by k rect. at A(0,k+1)
END DO
DO J = 0, K - 2
CALL DLASSQ( K-J-1, A( J+1+J*LDA ), 1, SCALE,
$ S )
* L at A(0,0)
END DO
S = S + S
* double s for the off diagonal elements
CALL DLASSQ( K, A( LDA ), LDA+1, SCALE, S )
* tri L at A(0,1)
CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
* tri U at A(0,0)
END IF
END IF
END IF
VALUE = SCALE*SQRT( S )
END IF
*
DLANSF = VALUE
RETURN
*
* End of DLANSF
*
END
*> \brief \b DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANSP( NORM, UPLO, N, AP, WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM, UPLO
* INTEGER N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANSP returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real symmetric matrix A, supplied in packed form.
*> \endverbatim
*>
*> \return DLANSP
*> \verbatim
*>
*> DLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANSP as described
*> above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is supplied.
*> = 'U': Upper triangular part of A is supplied
*> = 'L': Lower triangular part of A is supplied
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANSP is
*> set to zero.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangle of the symmetric matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*> WORK is not referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhp
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANSP( NORM, UPLO, N, AP, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, K
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
K = 1
DO 20 J = 1, N
DO 10 I = K, K + J - 1
SUM = ABS( AP( I ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
K = K + J
20 CONTINUE
ELSE
K = 1
DO 40 J = 1, N
DO 30 I = K, K + N - J
SUM = ABS( AP( I ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
K = K + N - J + 1
40 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
$ ( LSAME( NORM, 'O' ) ) .OR.
$ ( NORM.EQ.'1' ) ) THEN
*
* Find normI(A) ( = norm1(A), since A is symmetric).
*
VALUE = ZERO
K = 1
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
SUM = ZERO
DO 50 I = 1, J - 1
ABSA = ABS( AP( K ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
K = K + 1
50 CONTINUE
WORK( J ) = SUM + ABS( AP( K ) )
K = K + 1
60 CONTINUE
DO 70 I = 1, N
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
ELSE
DO 80 I = 1, N
WORK( I ) = ZERO
80 CONTINUE
DO 100 J = 1, N
SUM = WORK( J ) + ABS( AP( K ) )
K = K + 1
DO 90 I = J + 1, N
ABSA = ABS( AP( K ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
K = K + 1
90 CONTINUE
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
100 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
K = 2
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM )
K = K + J
110 CONTINUE
ELSE
DO 120 J = 1, N - 1
CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM )
K = K + N - J + 1
120 CONTINUE
END IF
SUM = 2*SUM
K = 1
DO 130 I = 1, N
IF( AP( K ).NE.ZERO ) THEN
ABSA = ABS( AP( K ) )
IF( SCALE.LT.ABSA ) THEN
SUM = ONE + SUM*( SCALE / ABSA )**2
SCALE = ABSA
ELSE
SUM = SUM + ( ABSA / SCALE )**2
END IF
END IF
IF( LSAME( UPLO, 'U' ) ) THEN
K = K + I + 1
ELSE
K = K + N - I + 1
END IF
130 CONTINUE
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANSP = VALUE
RETURN
*
* End of DLANSP
*
END
*> \brief \b DLANST returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanst.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANST returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real symmetric tridiagonal matrix A.
*> \endverbatim
*>
*> \return DLANST
*> \verbatim
*>
*> DLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANST as described
*> above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANST is
*> set to zero.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) sub-diagonal or super-diagonal elements of A.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanht
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION ANORM, SCALE, SUM
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( N.LE.0 ) THEN
ANORM = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
ANORM = ABS( D( N ) )
DO 10 I = 1, N - 1
SUM = ABS( D( I ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
SUM = ABS( E( I ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
10 CONTINUE
ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' .OR.
$ LSAME( NORM, 'I' ) ) THEN
*
* Find norm1(A).
*
IF( N.EQ.1 ) THEN
ANORM = ABS( D( 1 ) )
ELSE
ANORM = ABS( D( 1 ) )+ABS( E( 1 ) )
SUM = ABS( E( N-1 ) )+ABS( D( N ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
DO 20 I = 2, N - 1
SUM = ABS( D( I ) )+ABS( E( I ) )+ABS( E( I-1 ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
20 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( N.GT.1 ) THEN
CALL DLASSQ( N-1, E, 1, SCALE, SUM )
SUM = 2*SUM
END IF
CALL DLASSQ( N, D, 1, SCALE, SUM )
ANORM = SCALE*SQRT( SUM )
END IF
*
DLANST = ANORM
RETURN
*
* End of DLANST
*
END
*> \brief \b DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansy.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM, UPLO
* INTEGER LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANSY returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real symmetric matrix A.
*> \endverbatim
*>
*> \return DLANSY
*> \verbatim
*>
*> DLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANSY as described
*> above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is to be referenced.
*> = 'U': Upper triangular part of A is referenced
*> = 'L': Lower triangular part of A is referenced
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANSY is
*> set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The symmetric matrix A. If UPLO = 'U', the leading n by n
*> upper triangular part of A contains the upper triangular part
*> of the matrix A, and the strictly lower triangular part of A
*> is not referenced. If UPLO = 'L', the leading n by n lower
*> triangular part of A contains the lower triangular part of
*> the matrix A, and the strictly upper triangular part of A is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(N,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*> WORK is not referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhe
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, J
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = J, N
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
$ ( LSAME( NORM, 'O' ) ) .OR.
$ ( NORM.EQ.'1' ) ) THEN
*
* Find normI(A) ( = norm1(A), since A is symmetric).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
SUM = ZERO
DO 50 I = 1, J - 1
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
50 CONTINUE
WORK( J ) = SUM + ABS( A( J, J ) )
60 CONTINUE
DO 70 I = 1, N
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
ELSE
DO 80 I = 1, N
WORK( I ) = ZERO
80 CONTINUE
DO 100 J = 1, N
SUM = WORK( J ) + ABS( A( J, J ) )
DO 90 I = J + 1, N
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
90 CONTINUE
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
100 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL DLASSQ( J-1, A( 1, J ), 1, SCALE, SUM )
110 CONTINUE
ELSE
DO 120 J = 1, N - 1
CALL DLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM )
120 CONTINUE
END IF
SUM = 2*SUM
CALL DLASSQ( N, A, LDA+1, SCALE, SUM )
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANSY = VALUE
RETURN
*
* End of DLANSY
*
END
*> \brief \b DLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANTB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlantb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlantb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlantb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB,
* LDAB, WORK )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORM, UPLO
* INTEGER K, LDAB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANTB returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of an
*> n by n triangular band matrix A, with ( k + 1 ) diagonals.
*> \endverbatim
*>
*> \return DLANTB
*> \verbatim
*>
*> DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANTB as described
*> above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANTB is
*> set to zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of super-diagonals of the matrix A if UPLO = 'U',
*> or the number of sub-diagonals of the matrix A if UPLO = 'L'.
*> K >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangular band matrix A, stored in the
*> first k+1 rows of AB. The j-th column of A is stored
*> in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
*> Note that when DIAG = 'U', the elements of the array AB
*> corresponding to the diagonal elements of the matrix A are
*> not referenced, but are assumed to be one.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= K+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*> referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lantb
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB,
$ LDAB, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER K, LDAB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UDIAG
INTEGER I, J, L
DOUBLE PRECISION SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
IF( LSAME( DIAG, 'U' ) ) THEN
VALUE = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = MAX( K+2-J, 1 ), K
SUM = ABS( AB( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = 2, MIN( N+1-J, K+1 )
SUM = ABS( AB( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
40 CONTINUE
END IF
ELSE
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
DO 50 I = MAX( K+2-J, 1 ), K + 1
SUM = ABS( AB( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
50 CONTINUE
60 CONTINUE
ELSE
DO 80 J = 1, N
DO 70 I = 1, MIN( N+1-J, K+1 )
SUM = ABS( AB( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
80 CONTINUE
END IF
END IF
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
* Find norm1(A).
*
VALUE = ZERO
UDIAG = LSAME( DIAG, 'U' )
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 1, N
IF( UDIAG ) THEN
SUM = ONE
DO 90 I = MAX( K+2-J, 1 ), K
SUM = SUM + ABS( AB( I, J ) )
90 CONTINUE
ELSE
SUM = ZERO
DO 100 I = MAX( K+2-J, 1 ), K + 1
SUM = SUM + ABS( AB( I, J ) )
100 CONTINUE
END IF
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
110 CONTINUE
ELSE
DO 140 J = 1, N
IF( UDIAG ) THEN
SUM = ONE
DO 120 I = 2, MIN( N+1-J, K+1 )
SUM = SUM + ABS( AB( I, J ) )
120 CONTINUE
ELSE
SUM = ZERO
DO 130 I = 1, MIN( N+1-J, K+1 )
SUM = SUM + ABS( AB( I, J ) )
130 CONTINUE
END IF
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
140 CONTINUE
END IF
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
DO 150 I = 1, N
WORK( I ) = ONE
150 CONTINUE
DO 170 J = 1, N
L = K + 1 - J
DO 160 I = MAX( 1, J-K ), J - 1
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
160 CONTINUE
170 CONTINUE
ELSE
DO 180 I = 1, N
WORK( I ) = ZERO
180 CONTINUE
DO 200 J = 1, N
L = K + 1 - J
DO 190 I = MAX( 1, J-K ), J
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
190 CONTINUE
200 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
DO 210 I = 1, N
WORK( I ) = ONE
210 CONTINUE
DO 230 J = 1, N
L = 1 - J
DO 220 I = J + 1, MIN( N, J+K )
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
220 CONTINUE
230 CONTINUE
ELSE
DO 240 I = 1, N
WORK( I ) = ZERO
240 CONTINUE
DO 260 J = 1, N
L = 1 - J
DO 250 I = J, MIN( N, J+K )
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
250 CONTINUE
260 CONTINUE
END IF
END IF
DO 270 I = 1, N
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
270 CONTINUE
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
IF( K.GT.0 ) THEN
DO 280 J = 2, N
CALL DLASSQ( MIN( J-1, K ),
$ AB( MAX( K+2-J, 1 ), J ), 1, SCALE,
$ SUM )
280 CONTINUE
END IF
ELSE
SCALE = ZERO
SUM = ONE
DO 290 J = 1, N
CALL DLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ),
$ J ),
$ 1, SCALE, SUM )
290 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
IF( K.GT.0 ) THEN
DO 300 J = 1, N - 1
CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1,
$ SCALE,
$ SUM )
300 CONTINUE
END IF
ELSE
SCALE = ZERO
SUM = ONE
DO 310 J = 1, N
CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1,
$ SCALE,
$ SUM )
310 CONTINUE
END IF
END IF
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANTB = VALUE
RETURN
*
* End of DLANTB
*
END
*> \brief \b DLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANTP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlantp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlantp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlantp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANTP( NORM, UPLO, DIAG, N, AP, WORK )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORM, UPLO
* INTEGER N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANTP returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> triangular matrix A, supplied in packed form.
*> \endverbatim
*>
*> \return DLANTP
*> \verbatim
*>
*> DLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANTP as described
*> above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANTP is
*> set to zero.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangular matrix A, packed columnwise in
*> a linear array. The j-th column of A is stored in the array
*> AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> Note that when DIAG = 'U', the elements of the array AP
*> corresponding to the diagonal elements of the matrix A are
*> not referenced, but are assumed to be one.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*> referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lantp
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANTP( NORM, UPLO, DIAG, N, AP,
$ WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UDIAG
INTEGER I, J, K
DOUBLE PRECISION SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
K = 1
IF( LSAME( DIAG, 'U' ) ) THEN
VALUE = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = K, K + J - 2
SUM = ABS( AP( I ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
K = K + J
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = K + 1, K + N - J
SUM = ABS( AP( I ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
K = K + N - J + 1
40 CONTINUE
END IF
ELSE
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
DO 50 I = K, K + J - 1
SUM = ABS( AP( I ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
50 CONTINUE
K = K + J
60 CONTINUE
ELSE
DO 80 J = 1, N
DO 70 I = K, K + N - J
SUM = ABS( AP( I ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
K = K + N - J + 1
80 CONTINUE
END IF
END IF
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
* Find norm1(A).
*
VALUE = ZERO
K = 1
UDIAG = LSAME( DIAG, 'U' )
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 1, N
IF( UDIAG ) THEN
SUM = ONE
DO 90 I = K, K + J - 2
SUM = SUM + ABS( AP( I ) )
90 CONTINUE
ELSE
SUM = ZERO
DO 100 I = K, K + J - 1
SUM = SUM + ABS( AP( I ) )
100 CONTINUE
END IF
K = K + J
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
110 CONTINUE
ELSE
DO 140 J = 1, N
IF( UDIAG ) THEN
SUM = ONE
DO 120 I = K + 1, K + N - J
SUM = SUM + ABS( AP( I ) )
120 CONTINUE
ELSE
SUM = ZERO
DO 130 I = K, K + N - J
SUM = SUM + ABS( AP( I ) )
130 CONTINUE
END IF
K = K + N - J + 1
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
140 CONTINUE
END IF
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
K = 1
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
DO 150 I = 1, N
WORK( I ) = ONE
150 CONTINUE
DO 170 J = 1, N
DO 160 I = 1, J - 1
WORK( I ) = WORK( I ) + ABS( AP( K ) )
K = K + 1
160 CONTINUE
K = K + 1
170 CONTINUE
ELSE
DO 180 I = 1, N
WORK( I ) = ZERO
180 CONTINUE
DO 200 J = 1, N
DO 190 I = 1, J
WORK( I ) = WORK( I ) + ABS( AP( K ) )
K = K + 1
190 CONTINUE
200 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
DO 210 I = 1, N
WORK( I ) = ONE
210 CONTINUE
DO 230 J = 1, N
K = K + 1
DO 220 I = J + 1, N
WORK( I ) = WORK( I ) + ABS( AP( K ) )
K = K + 1
220 CONTINUE
230 CONTINUE
ELSE
DO 240 I = 1, N
WORK( I ) = ZERO
240 CONTINUE
DO 260 J = 1, N
DO 250 I = J, N
WORK( I ) = WORK( I ) + ABS( AP( K ) )
K = K + 1
250 CONTINUE
260 CONTINUE
END IF
END IF
VALUE = ZERO
DO 270 I = 1, N
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
270 CONTINUE
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
K = 2
DO 280 J = 2, N
CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM )
K = K + J
280 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
K = 1
DO 290 J = 1, N
CALL DLASSQ( J, AP( K ), 1, SCALE, SUM )
K = K + J
290 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
K = 2
DO 300 J = 1, N - 1
CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM )
K = K + N - J + 1
300 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
K = 1
DO 310 J = 1, N
CALL DLASSQ( N-J+1, AP( K ), 1, SCALE, SUM )
K = K + N - J + 1
310 CONTINUE
END IF
END IF
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANTP = VALUE
RETURN
*
* End of DLANTP
*
END
*> \brief \b DLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlantr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlantr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlantr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANTR( NORM, UPLO, DIAG, M, N, A, LDA,
* WORK )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORM, UPLO
* INTEGER LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANTR returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> trapezoidal or triangular matrix A.
*> \endverbatim
*>
*> \return DLANTR
*> \verbatim
*>
*> DLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANTR as described
*> above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower trapezoidal.
*> = 'U': Upper trapezoidal
*> = 'L': Lower trapezoidal
*> Note that A is triangular instead of trapezoidal if M = N.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A has unit diagonal.
*> = 'N': Non-unit diagonal
*> = 'U': Unit diagonal
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0, and if
*> UPLO = 'U', M <= N. When M = 0, DLANTR is set to zero.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0, and if
*> UPLO = 'L', N <= M. When N = 0, DLANTR is set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The trapezoidal matrix A (A is triangular if M = N).
*> If UPLO = 'U', the leading m by n upper trapezoidal part of
*> the array A contains the upper trapezoidal matrix, and the
*> strictly lower triangular part of A is not referenced.
*> If UPLO = 'L', the leading m by n lower trapezoidal part of
*> the array A contains the lower trapezoidal matrix, and the
*> strictly upper triangular part of A is not referenced. Note
*> that when DIAG = 'U', the diagonal elements of A are not
*> referenced and are assumed to be one.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(M,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= M when NORM = 'I'; otherwise, WORK is not
*> referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lantr
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANTR( NORM, UPLO, DIAG, M, N, A,
$ LDA,
$ WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UDIAG
INTEGER I, J
DOUBLE PRECISION SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN, SQRT
* ..
* .. Executable Statements ..
*
IF( MIN( M, N ).EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
IF( LSAME( DIAG, 'U' ) ) THEN
VALUE = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, MIN( M, J-1 )
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = J + 1, M
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
40 CONTINUE
END IF
ELSE
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
DO 50 I = 1, MIN( M, J )
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
50 CONTINUE
60 CONTINUE
ELSE
DO 80 J = 1, N
DO 70 I = J, M
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR.
$ DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
80 CONTINUE
END IF
END IF
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
* Find norm1(A).
*
VALUE = ZERO
UDIAG = LSAME( DIAG, 'U' )
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 1, N
IF( ( UDIAG ) .AND. ( J.LE.M ) ) THEN
SUM = ONE
DO 90 I = 1, J - 1
SUM = SUM + ABS( A( I, J ) )
90 CONTINUE
ELSE
SUM = ZERO
DO 100 I = 1, MIN( M, J )
SUM = SUM + ABS( A( I, J ) )
100 CONTINUE
END IF
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
110 CONTINUE
ELSE
DO 140 J = 1, N
IF( UDIAG ) THEN
SUM = ONE
DO 120 I = J + 1, M
SUM = SUM + ABS( A( I, J ) )
120 CONTINUE
ELSE
SUM = ZERO
DO 130 I = J, M
SUM = SUM + ABS( A( I, J ) )
130 CONTINUE
END IF
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
140 CONTINUE
END IF
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
DO 150 I = 1, M
WORK( I ) = ONE
150 CONTINUE
DO 170 J = 1, N
DO 160 I = 1, MIN( M, J-1 )
WORK( I ) = WORK( I ) + ABS( A( I, J ) )
160 CONTINUE
170 CONTINUE
ELSE
DO 180 I = 1, M
WORK( I ) = ZERO
180 CONTINUE
DO 200 J = 1, N
DO 190 I = 1, MIN( M, J )
WORK( I ) = WORK( I ) + ABS( A( I, J ) )
190 CONTINUE
200 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
DO 210 I = 1, MIN( M, N )
WORK( I ) = ONE
210 CONTINUE
DO 220 I = N + 1, M
WORK( I ) = ZERO
220 CONTINUE
DO 240 J = 1, N
DO 230 I = J + 1, M
WORK( I ) = WORK( I ) + ABS( A( I, J ) )
230 CONTINUE
240 CONTINUE
ELSE
DO 250 I = 1, M
WORK( I ) = ZERO
250 CONTINUE
DO 270 J = 1, N
DO 260 I = J, M
WORK( I ) = WORK( I ) + ABS( A( I, J ) )
260 CONTINUE
270 CONTINUE
END IF
END IF
VALUE = ZERO
DO 280 I = 1, M
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
280 CONTINUE
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
$ ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = MIN( M, N )
DO 290 J = 2, N
CALL DLASSQ( MIN( M, J-1 ), A( 1, J ), 1, SCALE,
$ SUM )
290 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
DO 300 J = 1, N
CALL DLASSQ( MIN( M, J ), A( 1, J ), 1, SCALE,
$ SUM )
300 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = MIN( M, N )
DO 310 J = 1, N
CALL DLASSQ( M-J, A( MIN( M, J+1 ), J ), 1, SCALE,
$ SUM )
310 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
DO 320 J = 1, N
CALL DLASSQ( M-J+1, A( J, J ), 1, SCALE, SUM )
320 CONTINUE
END IF
END IF
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANTR = VALUE
RETURN
*
* End of DLANTR
*
END
*> \brief \b DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
*> matrix in standard form:
*>
*> [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
*> [ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
*>
*> where either
*> 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
*> 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
*> conjugate eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION
*> On entry, the elements of the input matrix.
*> On exit, they are overwritten by the elements of the
*> standardised Schur form.
*> \endverbatim
*>
*> \param[out] RT1R
*> \verbatim
*> RT1R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT1I
*> \verbatim
*> RT1I is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2R
*> \verbatim
*> RT2R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2I
*> \verbatim
*> RT2I is DOUBLE PRECISION
*> The real and imaginary parts of the eigenvalues. If the
*> eigenvalues are a complex conjugate pair, RT1I > 0.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> Parameters of the rotation matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanv2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified by V. Sima, Research Institute for Informatics, Bucharest,
*> Romania, to reduce the risk of cancellation errors,
*> when computing real eigenvalues, and to ensure, if possible, that
*> abs(RT1R) >= abs(RT2R).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
$ TWO = 2.0D0 )
DOUBLE PRECISION MULTPL
PARAMETER ( MULTPL = 4.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
$ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z, SAFMIN,
$ SAFMN2, SAFMX2
INTEGER COUNT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
SAFMIN = DLAMCH( 'S' )
EPS = DLAMCH( 'P' )
SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
$ LOG( DLAMCH( 'B' ) ) / TWO )
SAFMX2 = ONE / SAFMN2
IF( C.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
*
ELSE IF( B.EQ.ZERO ) THEN
*
* Swap rows and columns
*
CS = ZERO
SN = ONE
TEMP = D
D = A
A = TEMP
B = -C
C = ZERO
*
ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
$ THEN
CS = ONE
SN = ZERO
*
ELSE
*
TEMP = A - D
P = HALF*TEMP
BCMAX = MAX( ABS( B ), ABS( C ) )
BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
SCALE = MAX( ABS( P ), BCMAX )
Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
*
* If Z is of the order of the machine accuracy, postpone the
* decision on the nature of eigenvalues
*
IF( Z.GE.MULTPL*EPS ) THEN
*
* Real eigenvalues. Compute A and D.
*
Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
A = D + Z
D = D - ( BCMAX / Z )*BCMIS
*
* Compute B and the rotation matrix
*
TAU = DLAPY2( C, Z )
CS = Z / TAU
SN = C / TAU
B = B - C
C = ZERO
*
ELSE
*
* Complex eigenvalues, or real (almost) equal eigenvalues.
* Make diagonal elements equal.
*
COUNT = 0
SIGMA = B + C
10 CONTINUE
COUNT = COUNT + 1
SCALE = MAX( ABS(TEMP), ABS(SIGMA) )
IF( SCALE.GE.SAFMX2 ) THEN
SIGMA = SIGMA * SAFMN2
TEMP = TEMP * SAFMN2
IF (COUNT .LE. 20)
$ GOTO 10
END IF
IF( SCALE.LE.SAFMN2 ) THEN
SIGMA = SIGMA * SAFMX2
TEMP = TEMP * SAFMX2
IF (COUNT .LE. 20)
$ GOTO 10
END IF
P = HALF*TEMP
TAU = DLAPY2( SIGMA, TEMP )
CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
*
* Compute [ AA BB ] = [ A B ] [ CS -SN ]
* [ CC DD ] [ C D ] [ SN CS ]
*
AA = A*CS + B*SN
BB = -A*SN + B*CS
CC = C*CS + D*SN
DD = -C*SN + D*CS
*
* Compute [ A B ] = [ CS SN ] [ AA BB ]
* [ C D ] [-SN CS ] [ CC DD ]
*
* Note: Some of the multiplications are wrapped in parentheses to
* prevent compilers from using FMA instructions. See
* https://github.com/Reference-LAPACK/lapack/issues/1031.
*
A = AA*CS + CC*SN
B = ( BB*CS ) + ( DD*SN )
C = -( AA*SN ) + ( CC*CS )
D = -BB*SN + DD*CS
*
TEMP = HALF*( A+D )
A = TEMP
D = TEMP
*
IF( C.NE.ZERO ) THEN
IF( B.NE.ZERO ) THEN
IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
*
* Real eigenvalues: reduce to upper triangular form
*
SAB = SQRT( ABS( B ) )
SAC = SQRT( ABS( C ) )
P = SIGN( SAB*SAC, C )
TAU = ONE / SQRT( ABS( B+C ) )
A = TEMP + P
D = TEMP - P
B = B - C
C = ZERO
CS1 = SAB*TAU
SN1 = SAC*TAU
TEMP = CS*CS1 - SN*SN1
SN = CS*SN1 + SN*CS1
CS = TEMP
END IF
ELSE
B = -C
C = ZERO
TEMP = CS
CS = -SN
SN = TEMP
END IF
END IF
END IF
*
END IF
*
* Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
*
RT1R = A
RT2R = D
IF( C.EQ.ZERO ) THEN
RT1I = ZERO
RT2I = ZERO
ELSE
RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
RT2I = -RT1I
END IF
RETURN
*
* End of DLANV2
*
END
*> \brief \b DLAPLL measures the linear dependence of two vectors.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPLL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapll.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapll.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapll.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
* .. Scalar Arguments ..
* INTEGER INCX, INCY, N
* DOUBLE PRECISION SSMIN
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X( * ), Y( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given two column vectors X and Y, let
*>
*> A = ( X Y ).
*>
*> The subroutine first computes the QR factorization of A = Q*R,
*> and then computes the SVD of the 2-by-2 upper triangular matrix R.
*> The smaller singular value of R is returned in SSMIN, which is used
*> as the measurement of the linear dependency of the vectors X and Y.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of the vectors X and Y.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCX)
*> On entry, X contains the N-vector X.
*> On exit, X is overwritten.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between successive elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCY)
*> On entry, Y contains the N-vector Y.
*> On exit, Y is overwritten.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> The increment between successive elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is DOUBLE PRECISION
*> The smallest singular value of the N-by-2 matrix A = ( X Y ).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapll
*
* =====================================================================
SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX, INCY, N
DOUBLE PRECISION SSMIN
* ..
* .. Array Arguments ..
DOUBLE PRECISION X( * ), Y( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION A11, A12, A22, C, SSMAX, TAU
* ..
* .. External Functions ..
DOUBLE PRECISION DDOT
EXTERNAL DDOT
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLARFG, DLAS2
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.1 ) THEN
SSMIN = ZERO
RETURN
END IF
*
* Compute the QR factorization of the N-by-2 matrix ( X Y )
*
CALL DLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU )
A11 = X( 1 )
X( 1 ) = ONE
*
C = -TAU*DDOT( N, X, INCX, Y, INCY )
CALL DAXPY( N, C, X, INCX, Y, INCY )
*
CALL DLARFG( N-1, Y( 1+INCY ), Y( 1+2*INCY ), INCY, TAU )
*
A12 = Y( 1 )
A22 = Y( 1+INCY )
*
* Compute the SVD of 2-by-2 Upper triangular matrix.
*
CALL DLAS2( A11, A12, A22, SSMIN, SSMAX )
*
RETURN
*
* End of DLAPLL
*
END
*> \brief \b DLAPMR rearranges rows of a matrix as specified by a permutation vector.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPMR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapmr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapmr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapmr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAPMR( FORWRD, M, N, X, LDX, K )
*
* .. Scalar Arguments ..
* LOGICAL FORWRD
* INTEGER LDX, M, N
* ..
* .. Array Arguments ..
* INTEGER K( * )
* DOUBLE PRECISION X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAPMR rearranges the rows of the M by N matrix X as specified
*> by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
*> If FORWRD = .TRUE., forward permutation:
*>
*> X(K(I),*) is moved X(I,*) for I = 1,2,...,M.
*>
*> If FORWRD = .FALSE., backward permutation:
*>
*> X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FORWRD
*> \verbatim
*> FORWRD is LOGICAL
*> = .TRUE., forward permutation
*> = .FALSE., backward permutation
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix X. N >= 0.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,N)
*> On entry, the M by N matrix X.
*> On exit, X contains the permuted matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X, LDX >= MAX(1,M).
*> \endverbatim
*>
*> \param[in,out] K
*> \verbatim
*> K is INTEGER array, dimension (M)
*> On entry, K contains the permutation vector. K is used as
*> internal workspace, but reset to its original value on
*> output.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapmr
*
* =====================================================================
SUBROUTINE DLAPMR( FORWRD, M, N, X, LDX, K )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL FORWRD
INTEGER LDX, M, N
* ..
* .. Array Arguments ..
INTEGER K( * )
DOUBLE PRECISION X( LDX, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IN, J, JJ
DOUBLE PRECISION TEMP
* ..
* .. Executable Statements ..
*
IF( M.LE.1 )
$ RETURN
*
DO 10 I = 1, M
K( I ) = -K( I )
10 CONTINUE
*
IF( FORWRD ) THEN
*
* Forward permutation
*
DO 50 I = 1, M
*
IF( K( I ).GT.0 )
$ GO TO 40
*
J = I
K( J ) = -K( J )
IN = K( J )
*
20 CONTINUE
IF( K( IN ).GT.0 )
$ GO TO 40
*
DO 30 JJ = 1, N
TEMP = X( J, JJ )
X( J, JJ ) = X( IN, JJ )
X( IN, JJ ) = TEMP
30 CONTINUE
*
K( IN ) = -K( IN )
J = IN
IN = K( IN )
GO TO 20
*
40 CONTINUE
*
50 CONTINUE
*
ELSE
*
* Backward permutation
*
DO 90 I = 1, M
*
IF( K( I ).GT.0 )
$ GO TO 80
*
K( I ) = -K( I )
J = K( I )
60 CONTINUE
IF( J.EQ.I )
$ GO TO 80
*
DO 70 JJ = 1, N
TEMP = X( I, JJ )
X( I, JJ ) = X( J, JJ )
X( J, JJ ) = TEMP
70 CONTINUE
*
K( J ) = -K( J )
J = K( J )
GO TO 60
*
80 CONTINUE
*
90 CONTINUE
*
END IF
*
RETURN
*
* End of DLAPMR
*
END
*> \brief \b DLAPMT performs a forward or backward permutation of the columns of a matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPMT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapmt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapmt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapmt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAPMT( FORWRD, M, N, X, LDX, K )
*
* .. Scalar Arguments ..
* LOGICAL FORWRD
* INTEGER LDX, M, N
* ..
* .. Array Arguments ..
* INTEGER K( * )
* DOUBLE PRECISION X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAPMT rearranges the columns of the M by N matrix X as specified
*> by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
*> If FORWRD = .TRUE., forward permutation:
*>
*> X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.
*>
*> If FORWRD = .FALSE., backward permutation:
*>
*> X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FORWRD
*> \verbatim
*> FORWRD is LOGICAL
*> = .TRUE., forward permutation
*> = .FALSE., backward permutation
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix X. N >= 0.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,N)
*> On entry, the M by N matrix X.
*> On exit, X contains the permuted matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X, LDX >= MAX(1,M).
*> \endverbatim
*>
*> \param[in,out] K
*> \verbatim
*> K is INTEGER array, dimension (N)
*> On entry, K contains the permutation vector. K is used as
*> internal workspace, but reset to its original value on
*> output.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapmt
*
* =====================================================================
SUBROUTINE DLAPMT( FORWRD, M, N, X, LDX, K )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL FORWRD
INTEGER LDX, M, N
* ..
* .. Array Arguments ..
INTEGER K( * )
DOUBLE PRECISION X( LDX, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, II, IN, J
DOUBLE PRECISION TEMP
* ..
* .. Executable Statements ..
*
IF( N.LE.1 )
$ RETURN
*
DO 10 I = 1, N
K( I ) = -K( I )
10 CONTINUE
*
IF( FORWRD ) THEN
*
* Forward permutation
*
DO 50 I = 1, N
*
IF( K( I ).GT.0 )
$ GO TO 40
*
J = I
K( J ) = -K( J )
IN = K( J )
*
20 CONTINUE
IF( K( IN ).GT.0 )
$ GO TO 40
*
DO 30 II = 1, M
TEMP = X( II, J )
X( II, J ) = X( II, IN )
X( II, IN ) = TEMP
30 CONTINUE
*
K( IN ) = -K( IN )
J = IN
IN = K( IN )
GO TO 20
*
40 CONTINUE
*
50 CONTINUE
*
ELSE
*
* Backward permutation
*
DO 90 I = 1, N
*
IF( K( I ).GT.0 )
$ GO TO 80
*
K( I ) = -K( I )
J = K( I )
60 CONTINUE
IF( J.EQ.I )
$ GO TO 80
*
DO 70 II = 1, M
TEMP = X( II, I )
X( II, I ) = X( II, J )
X( II, J ) = TEMP
70 CONTINUE
*
K( J ) = -K( J )
J = K( J )
GO TO 60
*
80 CONTINUE
*
90 CONTINUE
*
END IF
*
RETURN
*
* End of DLAPMT
*
END
*> \brief \b DLAPY2 returns sqrt(x2+y2).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPY2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapy2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapy2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapy2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION X, Y
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary
*> overflow and unnecessary underflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*> Y is DOUBLE PRECISION
*> X and Y specify the values x and y.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapy2
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION X, Y
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION W, XABS, YABS, Z, HUGEVAL
LOGICAL X_IS_NAN, Y_IS_NAN
* ..
* .. External Functions ..
LOGICAL DISNAN
EXTERNAL DISNAN
* ..
* .. External Subroutines ..
DOUBLE PRECISION DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
X_IS_NAN = DISNAN( X )
Y_IS_NAN = DISNAN( Y )
IF ( X_IS_NAN ) DLAPY2 = X
IF ( Y_IS_NAN ) DLAPY2 = Y
HUGEVAL = DLAMCH( 'Overflow' )
*
IF ( .NOT.( X_IS_NAN.OR.Y_IS_NAN ) ) THEN
XABS = ABS( X )
YABS = ABS( Y )
W = MAX( XABS, YABS )
Z = MIN( XABS, YABS )
IF( Z.EQ.ZERO .OR. W.GT.HUGEVAL ) THEN
DLAPY2 = W
ELSE
DLAPY2 = W*SQRT( ONE+( Z / W )**2 )
END IF
END IF
RETURN
*
* End of DLAPY2
*
END
*> \brief \b DLAPY3 returns sqrt(x2+y2+z2).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPY3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapy3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapy3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapy3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLAPY3( X, Y, Z )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION X, Y, Z
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause
*> unnecessary overflow and unnecessary underflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*> Y is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION
*> X, Y and Z specify the values x, y and z.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapy3
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLAPY3( X, Y, Z )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION X, Y, Z
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION W, XABS, YABS, ZABS, HUGEVAL
* ..
* .. External Subroutines ..
DOUBLE PRECISION DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
HUGEVAL = DLAMCH( 'Overflow' )
XABS = ABS( X )
YABS = ABS( Y )
ZABS = ABS( Z )
W = MAX( XABS, YABS, ZABS )
IF( W.EQ.ZERO .OR. W.GT.HUGEVAL ) THEN
* W can be zero for max(0,nan,0)
* adding all three entries together will make sure
* NaN will not disappear.
DLAPY3 = XABS + YABS + ZABS
ELSE
DLAPY3 = W*SQRT( ( XABS / W )**2+( YABS / W )**2+
$ ( ZABS / W )**2 )
END IF
RETURN
*
* End of DLAPY3
*
END
*> \brief \b DLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQGB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqgb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqgb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqgb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQGB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
* AMAX, EQUED )
*
* .. Scalar Arguments ..
* CHARACTER EQUED
* INTEGER KL, KU, LDAB, M, N
* DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQGB equilibrates a general M by N band matrix A with KL
*> subdiagonals and KU superdiagonals using the row and scaling factors
*> in the vectors R and C.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
*>
*> On exit, the equilibrated matrix, in the same storage format
*> as A. See EQUED for the form of the equilibrated matrix.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDA >= KL+KU+1.
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (M)
*> The row scale factors for A.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> The column scale factors for A.
*> \endverbatim
*>
*> \param[in] ROWCND
*> \verbatim
*> ROWCND is DOUBLE PRECISION
*> Ratio of the smallest R(i) to the largest R(i).
*> \endverbatim
*>
*> \param[in] COLCND
*> \verbatim
*> COLCND is DOUBLE PRECISION
*> Ratio of the smallest C(i) to the largest C(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration
*> = 'R': Row equilibration, i.e., A has been premultiplied by
*> diag(R).
*> = 'C': Column equilibration, i.e., A has been postmultiplied
*> by diag(C).
*> = 'B': Both row and column equilibration, i.e., A has been
*> replaced by diag(R) * A * diag(C).
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> THRESH is a threshold value used to decide if row or column scaling
*> should be done based on the ratio of the row or column scaling
*> factors. If ROWCND < THRESH, row scaling is done, and if
*> COLCND < THRESH, column scaling is done.
*>
*> LARGE and SMALL are threshold values used to decide if row scaling
*> should be done based on the absolute size of the largest matrix
*> element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqgb
*
* =====================================================================
SUBROUTINE DLAQGB( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
$ COLCND,
$ AMAX, EQUED )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED
INTEGER KL, KU, LDAB, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, THRESH
PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION CJ, LARGE, SMALL
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 ) THEN
EQUED = 'N'
RETURN
END IF
*
* Initialize LARGE and SMALL.
*
SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
LARGE = ONE / SMALL
*
IF( ROWCND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE )
$ THEN
*
* No row scaling
*
IF( COLCND.GE.THRESH ) THEN
*
* No column scaling
*
EQUED = 'N'
ELSE
*
* Column scaling
*
DO 20 J = 1, N
CJ = C( J )
DO 10 I = MAX( 1, J-KU ), MIN( M, J+KL )
AB( KU+1+I-J, J ) = CJ*AB( KU+1+I-J, J )
10 CONTINUE
20 CONTINUE
EQUED = 'C'
END IF
ELSE IF( COLCND.GE.THRESH ) THEN
*
* Row scaling, no column scaling
*
DO 40 J = 1, N
DO 30 I = MAX( 1, J-KU ), MIN( M, J+KL )
AB( KU+1+I-J, J ) = R( I )*AB( KU+1+I-J, J )
30 CONTINUE
40 CONTINUE
EQUED = 'R'
ELSE
*
* Row and column scaling
*
DO 60 J = 1, N
CJ = C( J )
DO 50 I = MAX( 1, J-KU ), MIN( M, J+KL )
AB( KU+1+I-J, J ) = CJ*R( I )*AB( KU+1+I-J, J )
50 CONTINUE
60 CONTINUE
EQUED = 'B'
END IF
*
RETURN
*
* End of DLAQGB
*
END
*> \brief \b DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQGE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqge.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqge.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqge.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQGE( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
* EQUED )
*
* .. Scalar Arguments ..
* CHARACTER EQUED
* INTEGER LDA, M, N
* DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQGE equilibrates a general M by N matrix A using the row and
*> column scaling factors in the vectors R and C.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M by N matrix A.
*> On exit, the equilibrated matrix. See EQUED for the form of
*> the equilibrated matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(M,1).
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*> R is DOUBLE PRECISION array, dimension (M)
*> The row scale factors for A.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (N)
*> The column scale factors for A.
*> \endverbatim
*>
*> \param[in] ROWCND
*> \verbatim
*> ROWCND is DOUBLE PRECISION
*> Ratio of the smallest R(i) to the largest R(i).
*> \endverbatim
*>
*> \param[in] COLCND
*> \verbatim
*> COLCND is DOUBLE PRECISION
*> Ratio of the smallest C(i) to the largest C(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration
*> = 'R': Row equilibration, i.e., A has been premultiplied by
*> diag(R).
*> = 'C': Column equilibration, i.e., A has been postmultiplied
*> by diag(C).
*> = 'B': Both row and column equilibration, i.e., A has been
*> replaced by diag(R) * A * diag(C).
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> THRESH is a threshold value used to decide if row or column scaling
*> should be done based on the ratio of the row or column scaling
*> factors. If ROWCND < THRESH, row scaling is done, and if
*> COLCND < THRESH, column scaling is done.
*>
*> LARGE and SMALL are threshold values used to decide if row scaling
*> should be done based on the absolute size of the largest matrix
*> element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqge
*
* =====================================================================
SUBROUTINE DLAQGE( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
$ EQUED )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED
INTEGER LDA, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, THRESH
PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION CJ, LARGE, SMALL
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 ) THEN
EQUED = 'N'
RETURN
END IF
*
* Initialize LARGE and SMALL.
*
SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
LARGE = ONE / SMALL
*
IF( ROWCND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE )
$ THEN
*
* No row scaling
*
IF( COLCND.GE.THRESH ) THEN
*
* No column scaling
*
EQUED = 'N'
ELSE
*
* Column scaling
*
DO 20 J = 1, N
CJ = C( J )
DO 10 I = 1, M
A( I, J ) = CJ*A( I, J )
10 CONTINUE
20 CONTINUE
EQUED = 'C'
END IF
ELSE IF( COLCND.GE.THRESH ) THEN
*
* Row scaling, no column scaling
*
DO 40 J = 1, N
DO 30 I = 1, M
A( I, J ) = R( I )*A( I, J )
30 CONTINUE
40 CONTINUE
EQUED = 'R'
ELSE
*
* Row and column scaling
*
DO 60 J = 1, N
CJ = C( J )
DO 50 I = 1, M
A( I, J ) = CJ*R( I )*A( I, J )
50 CONTINUE
60 CONTINUE
EQUED = 'B'
END IF
*
RETURN
*
* End of DLAQGE
*
END
*> \brief \b DLAQP2 computes a QR factorization with column pivoting of the matrix block.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQP2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqp2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqp2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqp2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
* WORK )
*
* .. Scalar Arguments ..
* INTEGER LDA, M, N, OFFSET
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* DOUBLE PRECISION A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQP2 computes a QR factorization with column pivoting of
*> the block A(OFFSET+1:M,1:N).
*> The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*> OFFSET is INTEGER
*> The number of rows of the matrix A that must be pivoted
*> but no factorized. OFFSET >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
*> the triangular factor obtained; the elements in block
*> A(OFFSET+1:M,1:N) below the diagonal, together with the
*> array TAU, represent the orthogonal matrix Q as a product of
*> elementary reflectors. Block A(1:OFFSET,1:N) has been
*> accordingly pivoted, but no factorized.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*> to the front of A*P (a leading column); if JPVT(i) = 0,
*> the i-th column of A is a free column.
*> On exit, if JPVT(i) = k, then the i-th column of A*P
*> was the k-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[in,out] VN1
*> \verbatim
*> VN1 is DOUBLE PRECISION array, dimension (N)
*> The vector with the partial column norms.
*> \endverbatim
*>
*> \param[in,out] VN2
*> \verbatim
*> VN2 is DOUBLE PRECISION array, dimension (N)
*> The vector with the exact column norms.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqp2
*
*> \par Contributors:
* ==================
*>
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*> X. Sun, Computer Science Dept., Duke University, USA
*> \n
*> Partial column norm updating strategy modified on April 2011
*> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*> University of Zagreb, Croatia.
*
*> \par References:
* ================
*>
*> LAPACK Working Note 176
*
*> \htmlonly
*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
*> \endhtmlonly
*
* =====================================================================
SUBROUTINE DLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
$ WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LDA, M, N, OFFSET
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, ITEMP, J, MN, OFFPI, PVT
DOUBLE PRECISION TEMP, TEMP2, TOL3Z
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DNRM2
EXTERNAL IDAMAX, DLAMCH, DNRM2
* ..
* .. Executable Statements ..
*
MN = MIN( M-OFFSET, N )
TOL3Z = SQRT(DLAMCH('Epsilon'))
*
* Compute factorization.
*
DO 20 I = 1, MN
*
OFFPI = OFFSET + I
*
* Determine ith pivot column and swap if necessary.
*
PVT = ( I-1 ) + IDAMAX( N-I+1, VN1( I ), 1 )
*
IF( PVT.NE.I ) THEN
CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
ITEMP = JPVT( PVT )
JPVT( PVT ) = JPVT( I )
JPVT( I ) = ITEMP
VN1( PVT ) = VN1( I )
VN2( PVT ) = VN2( I )
END IF
*
* Generate elementary reflector H(i).
*
IF( OFFPI.LT.M ) THEN
CALL DLARFG( M-OFFPI+1, A( OFFPI, I ), A( OFFPI+1, I ),
$ 1,
$ TAU( I ) )
ELSE
CALL DLARFG( 1, A( M, I ), A( M, I ), 1, TAU( I ) )
END IF
*
IF( I.LT.N ) THEN
*
* Apply H(i)**T to A(offset+i:m,i+1:n) from the left.
*
CALL DLARF1F( 'Left', M-OFFPI+1, N-I, A( OFFPI, I ), 1,
$ TAU( I ), A( OFFPI, I+1 ), LDA, WORK( 1 ) )
END IF
*
* Update partial column norms.
*
DO 10 J = I + 1, N
IF( VN1( J ).NE.ZERO ) THEN
*
* NOTE: The following 4 lines follow from the analysis in
* Lapack Working Note 176.
*
TEMP = ONE - ( ABS( A( OFFPI, J ) ) / VN1( J ) )**2
TEMP = MAX( TEMP, ZERO )
TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
IF( TEMP2 .LE. TOL3Z ) THEN
IF( OFFPI.LT.M ) THEN
VN1( J ) = DNRM2( M-OFFPI, A( OFFPI+1, J ), 1 )
VN2( J ) = VN1( J )
ELSE
VN1( J ) = ZERO
VN2( J ) = ZERO
END IF
ELSE
VN1( J ) = VN1( J )*SQRT( TEMP )
END IF
END IF
10 CONTINUE
*
20 CONTINUE
*
RETURN
*
* End of DLAQP2
*
END
*> \brief \b DLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQPS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqps.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqps.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqps.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
* VN2, AUXV, F, LDF )
*
* .. Scalar Arguments ..
* INTEGER KB, LDA, LDF, M, N, NB, OFFSET
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
* $ VN1( * ), VN2( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQPS computes a step of QR factorization with column pivoting
*> of a real M-by-N matrix A by using Blas-3. It tries to factorize
*> NB columns from A starting from the row OFFSET+1, and updates all
*> of the matrix with Blas-3 xGEMM.
*>
*> In some cases, due to catastrophic cancellations, it cannot
*> factorize NB columns. Hence, the actual number of factorized
*> columns is returned in KB.
*>
*> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*> OFFSET is INTEGER
*> The number of rows of A that have been factorized in
*> previous steps.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of columns to factorize.
*> \endverbatim
*>
*> \param[out] KB
*> \verbatim
*> KB is INTEGER
*> The number of columns actually factorized.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, block A(OFFSET+1:M,1:KB) is the triangular
*> factor obtained and block A(1:OFFSET,1:N) has been
*> accordingly pivoted, but no factorized.
*> The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
*> been updated.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> JPVT(I) = K <==> Column K of the full matrix A has been
*> permuted into position I in AP.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (KB)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[in,out] VN1
*> \verbatim
*> VN1 is DOUBLE PRECISION array, dimension (N)
*> The vector with the partial column norms.
*> \endverbatim
*>
*> \param[in,out] VN2
*> \verbatim
*> VN2 is DOUBLE PRECISION array, dimension (N)
*> The vector with the exact column norms.
*> \endverbatim
*>
*> \param[in,out] AUXV
*> \verbatim
*> AUXV is DOUBLE PRECISION array, dimension (NB)
*> Auxiliary vector.
*> \endverbatim
*>
*> \param[in,out] F
*> \verbatim
*> F is DOUBLE PRECISION array, dimension (LDF,NB)
*> Matrix F**T = L*Y**T*A.
*> \endverbatim
*>
*> \param[in] LDF
*> \verbatim
*> LDF is INTEGER
*> The leading dimension of the array F. LDF >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqps
*
*> \par Contributors:
* ==================
*>
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*> X. Sun, Computer Science Dept., Duke University, USA
*> \n
*> Partial column norm updating strategy modified on April 2011
*> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*> University of Zagreb, Croatia.
*
*> \par References:
* ================
*>
*> LAPACK Working Note 176
*
*> \htmlonly
*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
*> \endhtmlonly
*
* =====================================================================
SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU,
$ VN1,
$ VN2, AUXV, F, LDF )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER KB, LDA, LDF, M, N, NB, OFFSET
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
$ VN1( * ), VN2( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
DOUBLE PRECISION AKK, TEMP, TEMP2, TOL3Z
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DGEMV, DLARFG, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, NINT, SQRT
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DNRM2
EXTERNAL IDAMAX, DLAMCH, DNRM2
* ..
* .. Executable Statements ..
*
LASTRK = MIN( M, N+OFFSET )
LSTICC = 0
K = 0
TOL3Z = SQRT(DLAMCH('Epsilon'))
*
* Beginning of while loop.
*
10 CONTINUE
IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
K = K + 1
RK = OFFSET + K
*
* Determine ith pivot column and swap if necessary
*
PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 )
IF( PVT.NE.K ) THEN
CALL DSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
CALL DSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
ITEMP = JPVT( PVT )
JPVT( PVT ) = JPVT( K )
JPVT( K ) = ITEMP
VN1( PVT ) = VN1( K )
VN2( PVT ) = VN2( K )
END IF
*
* Apply previous Householder reflectors to column K:
* A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T.
*
IF( K.GT.1 ) THEN
CALL DGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK,
$ 1 ),
$ LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 )
END IF
*
* Generate elementary reflector H(k).
*
IF( RK.LT.M ) THEN
CALL DLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1,
$ TAU( K ) )
ELSE
CALL DLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
END IF
*
AKK = A( RK, K )
A( RK, K ) = ONE
*
* Compute Kth column of F:
*
* Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K).
*
IF( K.LT.N ) THEN
CALL DGEMV( 'Transpose', M-RK+1, N-K, TAU( K ),
$ A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO,
$ F( K+1, K ), 1 )
END IF
*
* Padding F(1:K,K) with zeros.
*
DO 20 J = 1, K
F( J, K ) = ZERO
20 CONTINUE
*
* Incremental updating of F:
* F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T
* *A(RK:M,K).
*
IF( K.GT.1 ) THEN
CALL DGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK,
$ 1 ),
$ LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 )
*
CALL DGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF,
$ AUXV( 1 ), 1, ONE, F( 1, K ), 1 )
END IF
*
* Update the current row of A:
* A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T.
*
IF( K.LT.N ) THEN
CALL DGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ),
$ LDF,
$ A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA )
END IF
*
* Update partial column norms.
*
IF( RK.LT.LASTRK ) THEN
DO 30 J = K + 1, N
IF( VN1( J ).NE.ZERO ) THEN
*
* NOTE: The following 4 lines follow from the analysis in
* Lapack Working Note 176.
*
TEMP = ABS( A( RK, J ) ) / VN1( J )
TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
IF( TEMP2 .LE. TOL3Z ) THEN
VN2( J ) = DBLE( LSTICC )
LSTICC = J
ELSE
VN1( J ) = VN1( J )*SQRT( TEMP )
END IF
END IF
30 CONTINUE
END IF
*
A( RK, K ) = AKK
*
* End of while loop.
*
GO TO 10
END IF
KB = K
RK = OFFSET + KB
*
* Apply the block reflector to the rest of the matrix:
* A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
* A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T.
*
IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
CALL DGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB,
$ -ONE,
$ A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE,
$ A( RK+1, KB+1 ), LDA )
END IF
*
* Recomputation of difficult columns.
*
40 CONTINUE
IF( LSTICC.GT.0 ) THEN
ITEMP = NINT( VN2( LSTICC ) )
VN1( LSTICC ) = DNRM2( M-RK, A( RK+1, LSTICC ), 1 )
*
* NOTE: The computation of VN1( LSTICC ) relies on the fact that
* SNRM2 does not fail on vectors with norm below the value of
* SQRT(DLAMCH('S'))
*
VN2( LSTICC ) = VN1( LSTICC )
LSTICC = ITEMP
GO TO 40
END IF
*
RETURN
*
* End of DLAQPS
*
END
*> \brief \b DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR0 computes the eigenvalues of a Hessenberg matrix H
*> and, optionally, the matrices T and Z from the Schur decomposition
*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*> Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*> Optionally Z may be postmultiplied into an input orthogonal
*> matrix Q so that this routine can give the Schur factorization
*> of a matrix A which has been reduced to the Hessenberg form H
*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*> previous call to DGEBAL, and then passed to DGEHRD when the
*> matrix output by DGEBAL is reduced to Hessenberg form.
*> Otherwise, ILO and IHI should be set to 1 and N,
*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO = 0 and WANTT is .TRUE., then H contains
*> the upper quasi-triangular matrix T from the Schur
*> decomposition (the Schur form); 2-by-2 diagonal blocks
*> (corresponding to complex conjugate pairs of eigenvalues)
*> are returned in standard form, with H(i,i) = H(i+1,i+1)
*> and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
*> .FALSE., then the contents of H are unspecified on exit.
*> (The output value of H when INFO > 0 is given under the
*> description of INFO below.)
*>
*> This subroutine may explicitly set H(i,j) = 0 for i > j and
*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (IHI)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (IHI)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
*> and WI(ILO:IHI). If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
*> the eigenvalues are stored in the same order as on the
*> diagonal of the Schur form returned in H, with
*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
*> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*> WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
*> If WANTZ is .FALSE., then Z is not referenced.
*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*> (The output value of Z when INFO > 0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if WANTZ is .TRUE.
*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension LWORK
*> On exit, if LWORK = -1, WORK(1) returns an estimate of
*> the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N)
*> is sufficient, but LWORK typically as large as 6*N may
*> be required for optimal performance. A workspace query
*> to determine the optimal workspace size is recommended.
*>
*> If LWORK = -1, then DLAQR0 does a workspace query.
*> In this case, DLAQR0 checks the input parameters and
*> estimates the optimal workspace size for the given
*> values of N, ILO and IHI. The estimate is returned
*> in WORK(1). No error message related to LWORK is
*> issued by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = i, DLAQR0 failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO > 0 and WANT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO > 0 and WANTT is .TRUE., then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
*> where U is an orthogonal matrix. The final
*> value of H is upper Hessenberg and quasi-triangular
*> in rows and columns INFO+1 through IHI.
*>
*> If INFO > 0 and WANTZ is .TRUE., then on exit
*>
*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
*>
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of WANTT.)
*>
*> If INFO > 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*> \n
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*> of Matrix Analysis, volume 23, pages 948--973, 2002.
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr0
*
* =====================================================================
SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
*
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
* . deflation window after KEXNW iterations. ====
INTEGER KEXNW
PARAMETER ( KEXNW = 5 )
*
* ==== Exceptional shifts: try to cure rare slow convergence
* . with ad-hoc exceptional shifts every KEXSH iterations.
* . ====
INTEGER KEXSH
PARAMETER ( KEXSH = 6 )
*
* ==== The constants WILK1 and WILK2 are used to form the
* . exceptional shifts. ====
DOUBLE PRECISION WILK1, WILK2
PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
$ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
$ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
LOGICAL SORTED
CHARACTER JBCMPZ*2
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Local Arrays ..
DOUBLE PRECISION ZDUM( 1, 1 )
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4,
$ DLAQR5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
* ..
* .. Executable Statements ..
INFO = 0
*
* ==== Quick return for N = 0: nothing to do. ====
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = ONE
RETURN
END IF
*
IF( N.LE.NTINY ) THEN
*
* ==== Tiny matrices must use DLAHQR. ====
*
LWKOPT = 1
IF( LWORK.NE.-1 )
$ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
ELSE
*
* ==== Use small bulge multi-shift QR with aggressive early
* . deflation on larger-than-tiny matrices. ====
*
* ==== Hope for the best. ====
*
INFO = 0
*
* ==== Set up job flags for ILAENV. ====
*
IF( WANTT ) THEN
JBCMPZ( 1: 1 ) = 'S'
ELSE
JBCMPZ( 1: 1 ) = 'E'
END IF
IF( WANTZ ) THEN
JBCMPZ( 2: 2 ) = 'V'
ELSE
JBCMPZ( 2: 2 ) = 'N'
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
*
* ==== Workspace query call to DLAQR3 ====
*
CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
$ N, H, LDH, WORK, -1 )
*
* ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
*
LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== DLAHQR/DLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== Nibble crossover point ====
*
NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NIBBLE = MAX( 0, NIBBLE )
*
* ==== Accumulate reflections during ttswp? Use block
* . 2-by-2 structure during matrix-matrix multiply? ====
*
KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
KACC22 = MAX( 0, KACC22 )
KACC22 = MIN( 2, KACC22 )
*
* ==== NWMAX = the largest possible deflation window for
* . which there is sufficient workspace. ====
*
NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
NW = NWMAX
*
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
*
NDFL = 1
*
* ==== ITMAX = iteration limit ====
*
ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
*
* ==== Last row and column in the active block ====
*
KBOT = IHI
*
* ==== Main Loop ====
*
DO 80 IT = 1, ITMAX
*
* ==== Done when KBOT falls below ILO ====
*
IF( KBOT.LT.ILO )
$ GO TO 90
*
* ==== Locate active block ====
*
DO 10 K = KBOT, ILO + 1, -1
IF( H( K, K-1 ).EQ.ZERO )
$ GO TO 20
10 CONTINUE
K = ILO
20 CONTINUE
KTOP = K
*
* ==== Select deflation window size:
* . Typical Case:
* . If possible and advisable, nibble the entire
* . active block. If not, use size MIN(NWR,NWMAX)
* . or MIN(NWR+1,NWMAX) depending upon which has
* . the smaller corresponding subdiagonal entry
* . (a heuristic).
* .
* . Exceptional Case:
* . If there have been no deflations in KEXNW or
* . more iterations, then vary the deflation window
* . size. At first, because, larger windows are,
* . in general, more powerful than smaller ones,
* . rapidly increase the window to the maximum possible.
* . Then, gradually reduce the window size. ====
*
NH = KBOT - KTOP + 1
NWUPBD = MIN( NH, NWMAX )
IF( NDFL.LT.KEXNW ) THEN
NW = MIN( NWUPBD, NWR )
ELSE
NW = MIN( NWUPBD, 2*NW )
END IF
IF( NW.LT.NWMAX ) THEN
IF( NW.GE.NH-1 ) THEN
NW = NH
ELSE
KWTOP = KBOT - NW + 1
IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
$ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
END IF
END IF
IF( NDFL.LT.KEXNW ) THEN
NDEC = -1
ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
NDEC = NDEC + 1
IF( NW-NDEC.LT.2 )
$ NDEC = 0
NW = NW - NDEC
END IF
*
* ==== Aggressive early deflation:
* . split workspace under the subdiagonal into
* . - an nw-by-nw work array V in the lower
* . left-hand-corner,
* . - an NW-by-at-least-NW-but-more-is-better
* . (NW-by-NHO) horizontal work array along
* . the bottom edge,
* . - an at-least-NW-but-more-is-better (NHV-by-NW)
* . vertical work array along the left-hand-edge.
* . ====
*
KV = N - NW + 1
KT = NW + 1
NHO = ( N-NW-1 ) - KT + 1
KWV = NW + 2
NVE = ( N-NW ) - KWV + 1
*
* ==== Aggressive early deflation ====
*
CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
$ ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
$ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
$ WORK, LWORK )
*
* ==== Adjust KBOT accounting for new deflations. ====
*
KBOT = KBOT - LD
*
* ==== KS points to the shifts. ====
*
KS = KBOT - LS + 1
*
* ==== Skip an expensive QR sweep if there is a (partly
* . heuristic) reason to expect that many eigenvalues
* . will deflate without it. Here, the QR sweep is
* . skipped if many eigenvalues have just been deflated
* . or if the remaining active block is small.
*
IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
$ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
*
* ==== NS = nominal number of simultaneous shifts.
* . This may be lowered (slightly) if DLAQR3
* . did not provide that many shifts. ====
*
NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
NS = NS - MOD( NS, 2 )
*
* ==== If there have been no deflations
* . in a multiple of KEXSH iterations,
* . then try exceptional shifts.
* . Otherwise use shifts provided by
* . DLAQR3 above or from the eigenvalues
* . of a trailing principal submatrix. ====
*
IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
KS = KBOT - NS + 1
DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
AA = WILK1*SS + H( I, I )
BB = SS
CC = WILK2*SS
DD = AA
CALL DLANV2( AA, BB, CC, DD, WR( I-1 ),
$ WI( I-1 ),
$ WR( I ), WI( I ), CS, SN )
30 CONTINUE
IF( KS.EQ.KTOP ) THEN
WR( KS+1 ) = H( KS+1, KS+1 )
WI( KS+1 ) = ZERO
WR( KS ) = WR( KS+1 )
WI( KS ) = WI( KS+1 )
END IF
ELSE
*
* ==== Got NS/2 or fewer shifts? Use DLAQR4 or
* . DLAHQR on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
IF( KBOT-KS+1.LE.NS / 2 ) THEN
KS = KBOT - NS + 1
KT = N - NS + 1
CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
$ H( KT, 1 ), LDH )
IF( NS.GT.NMIN ) THEN
CALL DLAQR4( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, WR( KS ),
$ WI( KS ), 1, 1, ZDUM, 1, WORK,
$ LWORK, INF )
ELSE
CALL DLAHQR( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, WR( KS ),
$ WI( KS ), 1, 1, ZDUM, 1, INF )
END IF
KS = KS + INF
*
* ==== In case of a rare QR failure use
* . eigenvalues of the trailing 2-by-2
* . principal submatrix. ====
*
IF( KS.GE.KBOT ) THEN
AA = H( KBOT-1, KBOT-1 )
CC = H( KBOT, KBOT-1 )
BB = H( KBOT-1, KBOT )
DD = H( KBOT, KBOT )
CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
$ WI( KBOT-1 ), WR( KBOT ),
$ WI( KBOT ), CS, SN )
KS = KBOT - 1
END IF
END IF
*
IF( KBOT-KS+1.GT.NS ) THEN
*
* ==== Sort the shifts (Helps a little)
* . Bubble sort keeps complex conjugate
* . pairs together. ====
*
SORTED = .false.
DO 50 K = KBOT, KS + 1, -1
IF( SORTED )
$ GO TO 60
SORTED = .true.
DO 40 I = KS, K - 1
IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
$ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
SORTED = .false.
*
SWAP = WR( I )
WR( I ) = WR( I+1 )
WR( I+1 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I+1 )
WI( I+1 ) = SWAP
END IF
40 CONTINUE
50 CONTINUE
60 CONTINUE
END IF
*
* ==== Shuffle shifts into pairs of real shifts
* . and pairs of complex conjugate shifts
* . assuming complex conjugate shifts are
* . already adjacent to one another. (Yes,
* . they are.) ====
*
DO 70 I = KBOT, KS + 2, -2
IF( WI( I ).NE.-WI( I-1 ) ) THEN
*
SWAP = WR( I )
WR( I ) = WR( I-1 )
WR( I-1 ) = WR( I-2 )
WR( I-2 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I-1 )
WI( I-1 ) = WI( I-2 )
WI( I-2 ) = SWAP
END IF
70 CONTINUE
END IF
*
* ==== If there are only two shifts and both are
* . real, then use only one. ====
*
IF( KBOT-KS+1.EQ.2 ) THEN
IF( WI( KBOT ).EQ.ZERO ) THEN
IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
$ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
WR( KBOT-1 ) = WR( KBOT )
ELSE
WR( KBOT ) = WR( KBOT-1 )
END IF
END IF
END IF
*
* ==== Use up to NS of the the smallest magnitude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====
*
NS = MIN( NS, KBOT-KS+1 )
NS = NS - MOD( NS, 2 )
KS = KBOT - NS + 1
*
* ==== Small-bulge multi-shift QR sweep:
* . split workspace under the subdiagonal into
* . - a KDU-by-KDU work array U in the lower
* . left-hand-corner,
* . - a KDU-by-at-least-KDU-but-more-is-better
* . (KDU-by-NHo) horizontal work array WH along
* . the bottom edge,
* . - and an at-least-KDU-but-more-is-better-by-KDU
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
KWV = KDU + 4
NVE = N - KDU - KWV + 1
*
* ==== Small-bulge multi-shift QR sweep ====
*
CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
$ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
$ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
$ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
END IF
*
* ==== Note progress (or the lack of it). ====
*
IF( LD.GT.0 ) THEN
NDFL = 1
ELSE
NDFL = NDFL + 1
END IF
*
* ==== End of main loop ====
80 CONTINUE
*
* ==== Iteration limit exceeded. Set INFO to show where
* . the problem occurred and exit. ====
*
INFO = KBOT
90 CONTINUE
END IF
*
* ==== Return the optimal value of LWORK. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR0 ====
*
END
*> \brief \b DLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION SI1, SI2, SR1, SR2
* INTEGER LDH, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), V( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given a 2-by-2 or 3-by-3 matrix H, DLAQR1 sets v to a
*> scalar multiple of the first column of the product
*>
*> (*) K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
*>
*> scaling to avoid overflows and most underflows. It
*> is assumed that either
*>
*> 1) sr1 = sr2 and si1 = -si2
*> or
*> 2) si1 = si2 = 0.
*>
*> This is useful for starting double implicit shift bulges
*> in the QR algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> Order of the matrix H. N must be either 2 or 3.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> The 2-by-2 or 3-by-3 matrix H in (*).
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of H as declared in
*> the calling procedure. LDH >= N
*> \endverbatim
*>
*> \param[in] SR1
*> \verbatim
*> SR1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SI1
*> \verbatim
*> SI1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SR2
*> \verbatim
*> SR2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SI2
*> \verbatim
*> SI2 is DOUBLE PRECISION
*> The shifts in (*).
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (N)
*> A scalar multiple of the first column of the
*> matrix K in (*).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr1
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION SI1, SI2, SR1, SR2
INTEGER LDH, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), V( * )
* ..
*
* ================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION H21S, H31S, S
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.NE.2 .AND. N.NE.3 ) THEN
RETURN
END IF
*
IF( N.EQ.2 ) THEN
S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) )
IF( S.EQ.ZERO ) THEN
V( 1 ) = ZERO
V( 2 ) = ZERO
ELSE
H21S = H( 2, 1 ) / S
V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-SR1 )*
$ ( ( H( 1, 1 )-SR2 ) / S ) - SI1*( SI2 / S )
V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 )
END IF
ELSE
S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) ) +
$ ABS( H( 3, 1 ) )
IF( S.EQ.ZERO ) THEN
V( 1 ) = ZERO
V( 2 ) = ZERO
V( 3 ) = ZERO
ELSE
H21S = H( 2, 1 ) / S
H31S = H( 3, 1 ) / S
V( 1 ) = ( H( 1, 1 )-SR1 )*( ( H( 1, 1 )-SR2 ) / S ) -
$ SI1*( SI2 / S ) + H( 1, 2 )*H21S + H( 1, 3 )*H31S
V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 ) +
$ H( 2, 3 )*H31S
V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-SR1-SR2 ) +
$ H21S*H( 3, 2 )
END IF
END IF
END
*> \brief \b DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
* IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
* LDT, NV, WV, LDWV, WORK, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
* $ LDZ, LWORK, N, ND, NH, NS, NV, NW
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
* $ V( LDV, * ), WORK( * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR2 is identical to DLAQR3 except that it avoids
*> recursion by calling DLAHQR instead of DLAQR4.
*>
*> Aggressive early deflation:
*>
*> This subroutine accepts as input an upper Hessenberg matrix
*> H and performs an orthogonal similarity transformation
*> designed to detect and deflate fully converged eigenvalues from
*> a trailing principal submatrix. On output H has been over-
*> written by a new Hessenberg matrix that is a perturbation of
*> an orthogonal similarity transformation of H. It is to be
*> hoped that the final version of H has many zero subdiagonal
*> entries.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> If .TRUE., then the Hessenberg matrix H is fully updated
*> so that the quasi-triangular Schur factor may be
*> computed (in cooperation with the calling subroutine).
*> If .FALSE., then only enough of H is updated to preserve
*> the eigenvalues.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> If .TRUE., then the orthogonal matrix Z is updated so
*> so that the orthogonal Schur factor may be computed
*> (in cooperation with the calling subroutine).
*> If .FALSE., then Z is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H and (if WANTZ is .TRUE.) the
*> order of the orthogonal matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is INTEGER
*> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
*> KBOT and KTOP together determine an isolated block
*> along the diagonal of the Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is INTEGER
*> It is assumed without a check that either
*> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
*> determine an isolated block along the diagonal of the
*> Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On input the initial N-by-N section of H stores the
*> Hessenberg matrix undergoing aggressive early deflation.
*> On output H has been transformed by an orthogonal
*> similarity transformation, perturbed, and the returned
*> to Hessenberg form that (it is to be hoped) has some
*> zero subdiagonal entries.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> Leading dimension of H just as declared in the calling
*> subroutine. N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> IF WANTZ is .TRUE., then on output, the orthogonal
*> similarity transformation mentioned above has been
*> accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*> If WANTZ is .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of Z just as declared in the
*> calling subroutine. 1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*> NS is INTEGER
*> The number of unconverged (ie approximate) eigenvalues
*> returned in SR and SI that may be used as shifts by the
*> calling subroutine.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*> ND is INTEGER
*> The number of converged eigenvalues uncovered by this
*> subroutine.
*> \endverbatim
*>
*> \param[out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array, dimension (KBOT)
*> \endverbatim
*>
*> \param[out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array, dimension (KBOT)
*> On output, the real and imaginary parts of approximate
*> eigenvalues that may be used for shifts are stored in
*> SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
*> SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
*> The real and imaginary parts of converged eigenvalues
*> are stored in SR(KBOT-ND+1) through SR(KBOT) and
*> SI(KBOT-ND+1) through SI(KBOT), respectively.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NW)
*> An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of V just as declared in the
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> The number of columns of T. NH >= NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of T just as declared in the
*> calling subroutine. NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> The number of rows of work array WV available for
*> workspace. NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is INTEGER
*> The leading dimension of W just as declared in the
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, WORK(1) is set to an estimate of the optimal value
*> of LWORK for the given values of N, NW, KTOP and KBOT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the work array WORK. LWORK = 2*NW
*> suffices, but greater efficiency may result from larger
*> values of LWORK.
*>
*> If LWORK = -1, then a workspace query is assumed; DLAQR2
*> only estimates the optimal workspace size for the given
*> values of N, NW, KTOP and KBOT. The estimate is returned
*> in WORK(1). No error message related to LWORK is issued
*> by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr2
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
$ ILOZ,
$ IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
$ LDT, NV, WV, LDWV, WORK, LWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
$ LDZ, LWORK, N, ND, NH, NS, NV, NW
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
$ V( LDV, * ), WORK( * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
$ SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
INTEGER I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
$ KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2,
$ LWKOPT
LOGICAL BULGE, SORTED
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEHRD, DGEMM, DLACPY,
$ DLAHQR,
$ DLANV2, DLARF1F, DLARFG, DLASET, DORMHR, DTREXC
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* ==== Estimate optimal workspace. ====
*
JW = MIN( NW, KBOT-KTOP+1 )
IF( JW.LE.2 ) THEN
LWKOPT = 1
ELSE
*
* ==== Workspace query call to DGEHRD ====
*
CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
LWK1 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DORMHR ====
*
CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V,
$ LDV,
$ WORK, -1, INFO )
LWK2 = INT( WORK( 1 ) )
*
* ==== Optimal workspace ====
*
LWKOPT = JW + MAX( LWK1, LWK2 )
END IF
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== Nothing to do ...
* ... for an empty active block ... ====
NS = 0
ND = 0
WORK( 1 ) = ONE
IF( KTOP.GT.KBOT )
$ RETURN
* ... nor for an empty deflation window. ====
IF( NW.LT.1 )
$ RETURN
*
* ==== Machine constants ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Setup deflation window ====
*
JW = MIN( NW, KBOT-KTOP+1 )
KWTOP = KBOT - JW + 1
IF( KWTOP.EQ.KTOP ) THEN
S = ZERO
ELSE
S = H( KWTOP, KWTOP-1 )
END IF
*
IF( KBOT.EQ.KWTOP ) THEN
*
* ==== 1-by-1 deflation window: not much to do ====
*
SR( KWTOP ) = H( KWTOP, KWTOP )
SI( KWTOP ) = ZERO
NS = 1
ND = 0
IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
$ THEN
NS = 0
ND = 1
IF( KWTOP.GT.KTOP )
$ H( KWTOP, KWTOP-1 ) = ZERO
END IF
WORK( 1 ) = ONE
RETURN
END IF
*
* ==== Convert to spike-triangular form. (In case of a
* . rare QR failure, this routine continues to do
* . aggressive early deflation using that part of
* . the deflation window that converged using INFQR
* . here and there to keep track.) ====
*
CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ),
$ LDT+1 )
*
CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, INFQR )
*
* ==== DTREXC needs a clean margin near the diagonal ====
*
DO 10 J = 1, JW - 3
T( J+2, J ) = ZERO
T( J+3, J ) = ZERO
10 CONTINUE
IF( JW.GT.2 )
$ T( JW, JW-2 ) = ZERO
*
* ==== Deflation detection loop ====
*
NS = JW
ILST = INFQR + 1
20 CONTINUE
IF( ILST.LE.NS ) THEN
IF( NS.EQ.1 ) THEN
BULGE = .FALSE.
ELSE
BULGE = T( NS, NS-1 ).NE.ZERO
END IF
*
* ==== Small spike tip test for deflation ====
*
IF( .NOT.BULGE ) THEN
*
* ==== Real eigenvalue ====
*
FOO = ABS( T( NS, NS ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 1
ELSE
*
* ==== Undeflatable. Move it up out of the way.
* . (DTREXC can not fail in this case.) ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
$ WORK,
$ INFO )
ILST = ILST + 1
END IF
ELSE
*
* ==== Complex conjugate pair ====
*
FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
$ SQRT( ABS( T( NS-1, NS ) ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
$ MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 2
ELSE
*
* ==== Undeflatable. Move them up out of the way.
* . Fortunately, DTREXC does the right thing with
* . ILST in case of a rare exchange failure. ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
$ WORK,
$ INFO )
ILST = ILST + 2
END IF
END IF
*
* ==== End deflation detection loop ====
*
GO TO 20
END IF
*
* ==== Return to Hessenberg form ====
*
IF( NS.EQ.0 )
$ S = ZERO
*
IF( NS.LT.JW ) THEN
*
* ==== sorting diagonal blocks of T improves accuracy for
* . graded matrices. Bubble sort deals well with
* . exchange failures. ====
*
SORTED = .false.
I = NS + 1
30 CONTINUE
IF( SORTED )
$ GO TO 50
SORTED = .true.
*
KEND = I - 1
I = INFQR + 1
IF( I.EQ.NS ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
40 CONTINUE
IF( K.LE.KEND ) THEN
IF( K.EQ.I+1 ) THEN
EVI = ABS( T( I, I ) )
ELSE
EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
$ SQRT( ABS( T( I, I+1 ) ) )
END IF
*
IF( K.EQ.KEND ) THEN
EVK = ABS( T( K, K ) )
ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
EVK = ABS( T( K, K ) )
ELSE
EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
$ SQRT( ABS( T( K, K+1 ) ) )
END IF
*
IF( EVI.GE.EVK ) THEN
I = K
ELSE
SORTED = .false.
IFST = I
ILST = K
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
$ WORK,
$ INFO )
IF( INFO.EQ.0 ) THEN
I = ILST
ELSE
I = K
END IF
END IF
IF( I.EQ.KEND ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
GO TO 40
END IF
GO TO 30
50 CONTINUE
END IF
*
* ==== Restore shift/eigenvalue array from T ====
*
I = JW
60 CONTINUE
IF( I.GE.INFQR+1 ) THEN
IF( I.EQ.INFQR+1 ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE
AA = T( I-1, I-1 )
CC = T( I, I-1 )
BB = T( I-1, I )
DD = T( I, I )
CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
$ SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
$ SI( KWTOP+I-1 ), CS, SN )
I = I - 2
END IF
GO TO 60
END IF
*
IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
* ==== Reflect spike back into lower triangle ====
*
CALL DCOPY( NS, V, LDV, WORK, 1 )
BETA = WORK( 1 )
CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
*
CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ),
$ LDT )
*
CALL DLARF1F( 'L', NS, JW, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF1F( 'R', NS, NS, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF1F( 'R', JW, NS, WORK, 1, TAU, V, LDV,
$ WORK( JW+1 ) )
*
CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
$ LWORK-JW, INFO )
END IF
*
* ==== Copy updated reduced window into place ====
*
IF( KWTOP.GT.1 )
$ H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
$ LDH+1 )
*
* ==== Accumulate orthogonal matrix in order update
* . H and Z, if requested. ====
*
IF( NS.GT.1 .AND. S.NE.ZERO )
$ CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V,
$ LDV,
$ WORK( JW+1 ), LWORK-JW, INFO )
*
* ==== Update vertical slab in H ====
*
IF( WANTT ) THEN
LTOP = 1
ELSE
LTOP = KTOP
END IF
DO 70 KROW = LTOP, KWTOP - 1, NV
KLN = MIN( NV, KWTOP-KROW )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
$ LDH, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ),
$ LDH )
70 CONTINUE
*
* ==== Update horizontal slab in H ====
*
IF( WANTT ) THEN
DO 80 KCOL = KBOT + 1, N, NH
KLN = MIN( NH, N-KCOL+1 )
CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
$ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
$ LDH )
80 CONTINUE
END IF
*
* ==== Update vertical slab in Z ====
*
IF( WANTZ ) THEN
DO 90 KROW = ILOZ, IHIZ, NV
KLN = MIN( NV, IHIZ-KROW+1 )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW,
$ KWTOP ),
$ LDZ, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
$ LDZ )
90 CONTINUE
END IF
END IF
*
* ==== Return the number of deflations ... ====
*
ND = JW - NS
*
* ==== ... and the number of shifts. (Subtracting
* . INFQR from the spike length takes care
* . of the case of a rare QR failure while
* . calculating eigenvalues of the deflation
* . window.) ====
*
NS = NS - INFQR
*
* ==== Return optimal workspace. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR2 ====
*
END
*> \brief \b DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
* IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
* LDT, NV, WV, LDWV, WORK, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
* $ LDZ, LWORK, N, ND, NH, NS, NV, NW
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
* $ V( LDV, * ), WORK( * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Aggressive early deflation:
*>
*> DLAQR3 accepts as input an upper Hessenberg matrix
*> H and performs an orthogonal similarity transformation
*> designed to detect and deflate fully converged eigenvalues from
*> a trailing principal submatrix. On output H has been over-
*> written by a new Hessenberg matrix that is a perturbation of
*> an orthogonal similarity transformation of H. It is to be
*> hoped that the final version of H has many zero subdiagonal
*> entries.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> If .TRUE., then the Hessenberg matrix H is fully updated
*> so that the quasi-triangular Schur factor may be
*> computed (in cooperation with the calling subroutine).
*> If .FALSE., then only enough of H is updated to preserve
*> the eigenvalues.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> If .TRUE., then the orthogonal matrix Z is updated so
*> so that the orthogonal Schur factor may be computed
*> (in cooperation with the calling subroutine).
*> If .FALSE., then Z is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H and (if WANTZ is .TRUE.) the
*> order of the orthogonal matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is INTEGER
*> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
*> KBOT and KTOP together determine an isolated block
*> along the diagonal of the Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is INTEGER
*> It is assumed without a check that either
*> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
*> determine an isolated block along the diagonal of the
*> Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On input the initial N-by-N section of H stores the
*> Hessenberg matrix undergoing aggressive early deflation.
*> On output H has been transformed by an orthogonal
*> similarity transformation, perturbed, and the returned
*> to Hessenberg form that (it is to be hoped) has some
*> zero subdiagonal entries.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> Leading dimension of H just as declared in the calling
*> subroutine. N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> IF WANTZ is .TRUE., then on output, the orthogonal
*> similarity transformation mentioned above has been
*> accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*> If WANTZ is .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of Z just as declared in the
*> calling subroutine. 1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*> NS is INTEGER
*> The number of unconverged (ie approximate) eigenvalues
*> returned in SR and SI that may be used as shifts by the
*> calling subroutine.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*> ND is INTEGER
*> The number of converged eigenvalues uncovered by this
*> subroutine.
*> \endverbatim
*>
*> \param[out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array, dimension (KBOT)
*> \endverbatim
*>
*> \param[out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array, dimension (KBOT)
*> On output, the real and imaginary parts of approximate
*> eigenvalues that may be used for shifts are stored in
*> SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
*> SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
*> The real and imaginary parts of converged eigenvalues
*> are stored in SR(KBOT-ND+1) through SR(KBOT) and
*> SI(KBOT-ND+1) through SI(KBOT), respectively.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NW)
*> An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of V just as declared in the
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> The number of columns of T. NH >= NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of T just as declared in the
*> calling subroutine. NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> The number of rows of work array WV available for
*> workspace. NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is INTEGER
*> The leading dimension of W just as declared in the
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, WORK(1) is set to an estimate of the optimal value
*> of LWORK for the given values of N, NW, KTOP and KBOT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the work array WORK. LWORK = 2*NW
*> suffices, but greater efficiency may result from larger
*> values of LWORK.
*>
*> If LWORK = -1, then a workspace query is assumed; DLAQR3
*> only estimates the optimal workspace size for the given
*> values of N, NW, KTOP and KBOT. The estimate is returned
*> in WORK(1). No error message related to LWORK is issued
*> by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr3
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
$ ILOZ,
$ IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
$ LDT, NV, WV, LDWV, WORK, LWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
$ LDZ, LWORK, N, ND, NH, NS, NV, NW
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
$ V( LDV, * ), WORK( * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
$ SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
INTEGER I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
$ KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
$ LWKOPT, NMIN
LOGICAL BULGE, SORTED
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
INTEGER ILAENV
EXTERNAL DLAMCH, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEHRD, DGEMM, DLACPY, DLAHQR,
$ DLANV2,
$ DLAQR4, DLARF1F, DLARFG, DLASET, DORMHR, DTREXC
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* ==== Estimate optimal workspace. ====
*
JW = MIN( NW, KBOT-KTOP+1 )
IF( JW.LE.2 ) THEN
LWKOPT = 1
ELSE
*
* ==== Workspace query call to DGEHRD ====
*
CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
LWK1 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DORMHR ====
*
CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V,
$ LDV,
$ WORK, -1, INFO )
LWK2 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DLAQR4 ====
*
CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR, SI, 1,
$ JW,
$ V, LDV, WORK, -1, INFQR )
LWK3 = INT( WORK( 1 ) )
*
* ==== Optimal workspace ====
*
LWKOPT = MAX( JW+MAX( LWK1, LWK2 ), LWK3 )
END IF
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== Nothing to do ...
* ... for an empty active block ... ====
NS = 0
ND = 0
WORK( 1 ) = ONE
IF( KTOP.GT.KBOT )
$ RETURN
* ... nor for an empty deflation window. ====
IF( NW.LT.1 )
$ RETURN
*
* ==== Machine constants ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Setup deflation window ====
*
JW = MIN( NW, KBOT-KTOP+1 )
KWTOP = KBOT - JW + 1
IF( KWTOP.EQ.KTOP ) THEN
S = ZERO
ELSE
S = H( KWTOP, KWTOP-1 )
END IF
*
IF( KBOT.EQ.KWTOP ) THEN
*
* ==== 1-by-1 deflation window: not much to do ====
*
SR( KWTOP ) = H( KWTOP, KWTOP )
SI( KWTOP ) = ZERO
NS = 1
ND = 0
IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
$ THEN
NS = 0
ND = 1
IF( KWTOP.GT.KTOP )
$ H( KWTOP, KWTOP-1 ) = ZERO
END IF
WORK( 1 ) = ONE
RETURN
END IF
*
* ==== Convert to spike-triangular form. (In case of a
* . rare QR failure, this routine continues to do
* . aggressive early deflation using that part of
* . the deflation window that converged using INFQR
* . here and there to keep track.) ====
*
CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ),
$ LDT+1 )
*
CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
NMIN = ILAENV( 12, 'DLAQR3', 'SV', JW, 1, JW, LWORK )
IF( JW.GT.NMIN ) THEN
CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, WORK, LWORK, INFQR )
ELSE
CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, INFQR )
END IF
*
* ==== DTREXC needs a clean margin near the diagonal ====
*
DO 10 J = 1, JW - 3
T( J+2, J ) = ZERO
T( J+3, J ) = ZERO
10 CONTINUE
IF( JW.GT.2 )
$ T( JW, JW-2 ) = ZERO
*
* ==== Deflation detection loop ====
*
NS = JW
ILST = INFQR + 1
20 CONTINUE
IF( ILST.LE.NS ) THEN
IF( NS.EQ.1 ) THEN
BULGE = .FALSE.
ELSE
BULGE = T( NS, NS-1 ).NE.ZERO
END IF
*
* ==== Small spike tip test for deflation ====
*
IF( .NOT. BULGE ) THEN
*
* ==== Real eigenvalue ====
*
FOO = ABS( T( NS, NS ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 1
ELSE
*
* ==== Undeflatable. Move it up out of the way.
* . (DTREXC can not fail in this case.) ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
$ WORK,
$ INFO )
ILST = ILST + 1
END IF
ELSE
*
* ==== Complex conjugate pair ====
*
FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
$ SQRT( ABS( T( NS-1, NS ) ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
$ MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 2
ELSE
*
* ==== Undeflatable. Move them up out of the way.
* . Fortunately, DTREXC does the right thing with
* . ILST in case of a rare exchange failure. ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
$ WORK,
$ INFO )
ILST = ILST + 2
END IF
END IF
*
* ==== End deflation detection loop ====
*
GO TO 20
END IF
*
* ==== Return to Hessenberg form ====
*
IF( NS.EQ.0 )
$ S = ZERO
*
IF( NS.LT.JW ) THEN
*
* ==== sorting diagonal blocks of T improves accuracy for
* . graded matrices. Bubble sort deals well with
* . exchange failures. ====
*
SORTED = .false.
I = NS + 1
30 CONTINUE
IF( SORTED )
$ GO TO 50
SORTED = .true.
*
KEND = I - 1
I = INFQR + 1
IF( I.EQ.NS ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
40 CONTINUE
IF( K.LE.KEND ) THEN
IF( K.EQ.I+1 ) THEN
EVI = ABS( T( I, I ) )
ELSE
EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
$ SQRT( ABS( T( I, I+1 ) ) )
END IF
*
IF( K.EQ.KEND ) THEN
EVK = ABS( T( K, K ) )
ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
EVK = ABS( T( K, K ) )
ELSE
EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
$ SQRT( ABS( T( K, K+1 ) ) )
END IF
*
IF( EVI.GE.EVK ) THEN
I = K
ELSE
SORTED = .false.
IFST = I
ILST = K
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
$ WORK,
$ INFO )
IF( INFO.EQ.0 ) THEN
I = ILST
ELSE
I = K
END IF
END IF
IF( I.EQ.KEND ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
GO TO 40
END IF
GO TO 30
50 CONTINUE
END IF
*
* ==== Restore shift/eigenvalue array from T ====
*
I = JW
60 CONTINUE
IF( I.GE.INFQR+1 ) THEN
IF( I.EQ.INFQR+1 ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE
AA = T( I-1, I-1 )
CC = T( I, I-1 )
BB = T( I-1, I )
DD = T( I, I )
CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
$ SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
$ SI( KWTOP+I-1 ), CS, SN )
I = I - 2
END IF
GO TO 60
END IF
*
IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
* ==== Reflect spike back into lower triangle ====
*
CALL DCOPY( NS, V, LDV, WORK, 1 )
BETA = WORK( 1 )
CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
*
CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ),
$ LDT )
*
CALL DLARF1F( 'L', NS, JW, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF1F( 'R', NS, NS, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF1F( 'R', JW, NS, WORK, 1, TAU, V, LDV,
$ WORK( JW+1 ) )
*
CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
$ LWORK-JW, INFO )
END IF
*
* ==== Copy updated reduced window into place ====
*
IF( KWTOP.GT.1 )
$ H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
$ LDH+1 )
*
* ==== Accumulate orthogonal matrix in order update
* . H and Z, if requested. ====
*
IF( NS.GT.1 .AND. S.NE.ZERO )
$ CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V,
$ LDV,
$ WORK( JW+1 ), LWORK-JW, INFO )
*
* ==== Update vertical slab in H ====
*
IF( WANTT ) THEN
LTOP = 1
ELSE
LTOP = KTOP
END IF
DO 70 KROW = LTOP, KWTOP - 1, NV
KLN = MIN( NV, KWTOP-KROW )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
$ LDH, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ),
$ LDH )
70 CONTINUE
*
* ==== Update horizontal slab in H ====
*
IF( WANTT ) THEN
DO 80 KCOL = KBOT + 1, N, NH
KLN = MIN( NH, N-KCOL+1 )
CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
$ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
$ LDH )
80 CONTINUE
END IF
*
* ==== Update vertical slab in Z ====
*
IF( WANTZ ) THEN
DO 90 KROW = ILOZ, IHIZ, NV
KLN = MIN( NV, IHIZ-KROW+1 )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW,
$ KWTOP ),
$ LDZ, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
$ LDZ )
90 CONTINUE
END IF
END IF
*
* ==== Return the number of deflations ... ====
*
ND = JW - NS
*
* ==== ... and the number of shifts. (Subtracting
* . INFQR from the spike length takes care
* . of the case of a rare QR failure while
* . calculating eigenvalues of the deflation
* . window.) ====
*
NS = NS - INFQR
*
* ==== Return optimal workspace. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR3 ====
*
END
*> \brief \b DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr4.f">
*> [TGZ]</a>
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*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR4 implements one level of recursion for DLAQR0.
*> It is a complete implementation of the small bulge multi-shift
*> QR algorithm. It may be called by DLAQR0 and, for large enough
*> deflation window size, it may be called by DLAQR3. This
*> subroutine is identical to DLAQR0 except that it calls DLAQR2
*> instead of DLAQR3.
*>
*> DLAQR4 computes the eigenvalues of a Hessenberg matrix H
*> and, optionally, the matrices T and Z from the Schur decomposition
*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*> Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*> Optionally Z may be postmultiplied into an input orthogonal
*> matrix Q so that this routine can give the Schur factorization
*> of a matrix A which has been reduced to the Hessenberg form H
*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*> previous call to DGEBAL, and then passed to DGEHRD when the
*> matrix output by DGEBAL is reduced to Hessenberg form.
*> Otherwise, ILO and IHI should be set to 1 and N,
*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO = 0 and WANTT is .TRUE., then H contains
*> the upper quasi-triangular matrix T from the Schur
*> decomposition (the Schur form); 2-by-2 diagonal blocks
*> (corresponding to complex conjugate pairs of eigenvalues)
*> are returned in standard form, with H(i,i) = H(i+1,i+1)
*> and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
*> .FALSE., then the contents of H are unspecified on exit.
*> (The output value of H when INFO > 0 is given under the
*> description of INFO below.)
*>
*> This subroutine may explicitly set H(i,j) = 0 for i > j and
*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (IHI)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (IHI)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
*> and WI(ILO:IHI). If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
*> the eigenvalues are stored in the same order as on the
*> diagonal of the Schur form returned in H, with
*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
*> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*> WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
*> If WANTZ is .FALSE., then Z is not referenced.
*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*> (The output value of Z when INFO > 0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if WANTZ is .TRUE.
*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension LWORK
*> On exit, if LWORK = -1, WORK(1) returns an estimate of
*> the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N)
*> is sufficient, but LWORK typically as large as 6*N may
*> be required for optimal performance. A workspace query
*> to determine the optimal workspace size is recommended.
*>
*> If LWORK = -1, then DLAQR4 does a workspace query.
*> In this case, DLAQR4 checks the input parameters and
*> estimates the optimal workspace size for the given
*> values of N, ILO and IHI. The estimate is returned
*> in WORK(1). No error message related to LWORK is
*> issued by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = i, DLAQR4 failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO > 0 and WANT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO > 0 and WANTT is .TRUE., then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
*> where U is a orthogonal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO > 0 and WANTZ is .TRUE., then on exit
*>
*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
*>
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of WANTT.)
*>
*> If INFO > 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr4
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*> \n
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*> of Matrix Analysis, volume 23, pages 948--973, 2002.
*>
* =====================================================================
SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
* . deflation window after KEXNW iterations. ====
INTEGER KEXNW
PARAMETER ( KEXNW = 5 )
*
* ==== Exceptional shifts: try to cure rare slow convergence
* . with ad-hoc exceptional shifts every KEXSH iterations.
* . ====
INTEGER KEXSH
PARAMETER ( KEXSH = 6 )
*
* ==== The constants WILK1 and WILK2 are used to form the
* . exceptional shifts. ====
DOUBLE PRECISION WILK1, WILK2
PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
$ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
$ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
LOGICAL SORTED
CHARACTER JBCMPZ*2
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Local Arrays ..
DOUBLE PRECISION ZDUM( 1, 1 )
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR2,
$ DLAQR5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
* ..
* .. Executable Statements ..
INFO = 0
*
* ==== Quick return for N = 0: nothing to do. ====
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = ONE
RETURN
END IF
*
IF( N.LE.NTINY ) THEN
*
* ==== Tiny matrices must use DLAHQR. ====
*
LWKOPT = 1
IF( LWORK.NE.-1 )
$ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
ELSE
*
* ==== Use small bulge multi-shift QR with aggressive early
* . deflation on larger-than-tiny matrices. ====
*
* ==== Hope for the best. ====
*
INFO = 0
*
* ==== Set up job flags for ILAENV. ====
*
IF( WANTT ) THEN
JBCMPZ( 1: 1 ) = 'S'
ELSE
JBCMPZ( 1: 1 ) = 'E'
END IF
IF( WANTZ ) THEN
JBCMPZ( 2: 2 ) = 'V'
ELSE
JBCMPZ( 2: 2 ) = 'N'
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
*
* ==== Workspace query call to DLAQR2 ====
*
CALL DLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
$ N, H, LDH, WORK, -1 )
*
* ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
*
LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== DLAHQR/DLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== Nibble crossover point ====
*
NIBBLE = ILAENV( 14, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NIBBLE = MAX( 0, NIBBLE )
*
* ==== Accumulate reflections during ttswp? Use block
* . 2-by-2 structure during matrix-matrix multiply? ====
*
KACC22 = ILAENV( 16, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
KACC22 = MAX( 0, KACC22 )
KACC22 = MIN( 2, KACC22 )
*
* ==== NWMAX = the largest possible deflation window for
* . which there is sufficient workspace. ====
*
NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
NW = NWMAX
*
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
*
NDFL = 1
*
* ==== ITMAX = iteration limit ====
*
ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
*
* ==== Last row and column in the active block ====
*
KBOT = IHI
*
* ==== Main Loop ====
*
DO 80 IT = 1, ITMAX
*
* ==== Done when KBOT falls below ILO ====
*
IF( KBOT.LT.ILO )
$ GO TO 90
*
* ==== Locate active block ====
*
DO 10 K = KBOT, ILO + 1, -1
IF( H( K, K-1 ).EQ.ZERO )
$ GO TO 20
10 CONTINUE
K = ILO
20 CONTINUE
KTOP = K
*
* ==== Select deflation window size:
* . Typical Case:
* . If possible and advisable, nibble the entire
* . active block. If not, use size MIN(NWR,NWMAX)
* . or MIN(NWR+1,NWMAX) depending upon which has
* . the smaller corresponding subdiagonal entry
* . (a heuristic).
* .
* . Exceptional Case:
* . If there have been no deflations in KEXNW or
* . more iterations, then vary the deflation window
* . size. At first, because, larger windows are,
* . in general, more powerful than smaller ones,
* . rapidly increase the window to the maximum possible.
* . Then, gradually reduce the window size. ====
*
NH = KBOT - KTOP + 1
NWUPBD = MIN( NH, NWMAX )
IF( NDFL.LT.KEXNW ) THEN
NW = MIN( NWUPBD, NWR )
ELSE
NW = MIN( NWUPBD, 2*NW )
END IF
IF( NW.LT.NWMAX ) THEN
IF( NW.GE.NH-1 ) THEN
NW = NH
ELSE
KWTOP = KBOT - NW + 1
IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
$ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
END IF
END IF
IF( NDFL.LT.KEXNW ) THEN
NDEC = -1
ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
NDEC = NDEC + 1
IF( NW-NDEC.LT.2 )
$ NDEC = 0
NW = NW - NDEC
END IF
*
* ==== Aggressive early deflation:
* . split workspace under the subdiagonal into
* . - an nw-by-nw work array V in the lower
* . left-hand-corner,
* . - an NW-by-at-least-NW-but-more-is-better
* . (NW-by-NHO) horizontal work array along
* . the bottom edge,
* . - an at-least-NW-but-more-is-better (NHV-by-NW)
* . vertical work array along the left-hand-edge.
* . ====
*
KV = N - NW + 1
KT = NW + 1
NHO = ( N-NW-1 ) - KT + 1
KWV = NW + 2
NVE = ( N-NW ) - KWV + 1
*
* ==== Aggressive early deflation ====
*
CALL DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
$ ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
$ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
$ WORK, LWORK )
*
* ==== Adjust KBOT accounting for new deflations. ====
*
KBOT = KBOT - LD
*
* ==== KS points to the shifts. ====
*
KS = KBOT - LS + 1
*
* ==== Skip an expensive QR sweep if there is a (partly
* . heuristic) reason to expect that many eigenvalues
* . will deflate without it. Here, the QR sweep is
* . skipped if many eigenvalues have just been deflated
* . or if the remaining active block is small.
*
IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
$ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
*
* ==== NS = nominal number of simultaneous shifts.
* . This may be lowered (slightly) if DLAQR2
* . did not provide that many shifts. ====
*
NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
NS = NS - MOD( NS, 2 )
*
* ==== If there have been no deflations
* . in a multiple of KEXSH iterations,
* . then try exceptional shifts.
* . Otherwise use shifts provided by
* . DLAQR2 above or from the eigenvalues
* . of a trailing principal submatrix. ====
*
IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
KS = KBOT - NS + 1
DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
AA = WILK1*SS + H( I, I )
BB = SS
CC = WILK2*SS
DD = AA
CALL DLANV2( AA, BB, CC, DD, WR( I-1 ),
$ WI( I-1 ),
$ WR( I ), WI( I ), CS, SN )
30 CONTINUE
IF( KS.EQ.KTOP ) THEN
WR( KS+1 ) = H( KS+1, KS+1 )
WI( KS+1 ) = ZERO
WR( KS ) = WR( KS+1 )
WI( KS ) = WI( KS+1 )
END IF
ELSE
*
* ==== Got NS/2 or fewer shifts? Use DLAHQR
* . on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
IF( KBOT-KS+1.LE.NS / 2 ) THEN
KS = KBOT - NS + 1
KT = N - NS + 1
CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
$ H( KT, 1 ), LDH )
CALL DLAHQR( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, WR( KS ), WI( KS ),
$ 1, 1, ZDUM, 1, INF )
KS = KS + INF
*
* ==== In case of a rare QR failure use
* . eigenvalues of the trailing 2-by-2
* . principal submatrix. ====
*
IF( KS.GE.KBOT ) THEN
AA = H( KBOT-1, KBOT-1 )
CC = H( KBOT, KBOT-1 )
BB = H( KBOT-1, KBOT )
DD = H( KBOT, KBOT )
CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
$ WI( KBOT-1 ), WR( KBOT ),
$ WI( KBOT ), CS, SN )
KS = KBOT - 1
END IF
END IF
*
IF( KBOT-KS+1.GT.NS ) THEN
*
* ==== Sort the shifts (Helps a little)
* . Bubble sort keeps complex conjugate
* . pairs together. ====
*
SORTED = .false.
DO 50 K = KBOT, KS + 1, -1
IF( SORTED )
$ GO TO 60
SORTED = .true.
DO 40 I = KS, K - 1
IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
$ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
SORTED = .false.
*
SWAP = WR( I )
WR( I ) = WR( I+1 )
WR( I+1 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I+1 )
WI( I+1 ) = SWAP
END IF
40 CONTINUE
50 CONTINUE
60 CONTINUE
END IF
*
* ==== Shuffle shifts into pairs of real shifts
* . and pairs of complex conjugate shifts
* . assuming complex conjugate shifts are
* . already adjacent to one another. (Yes,
* . they are.) ====
*
DO 70 I = KBOT, KS + 2, -2
IF( WI( I ).NE.-WI( I-1 ) ) THEN
*
SWAP = WR( I )
WR( I ) = WR( I-1 )
WR( I-1 ) = WR( I-2 )
WR( I-2 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I-1 )
WI( I-1 ) = WI( I-2 )
WI( I-2 ) = SWAP
END IF
70 CONTINUE
END IF
*
* ==== If there are only two shifts and both are
* . real, then use only one. ====
*
IF( KBOT-KS+1.EQ.2 ) THEN
IF( WI( KBOT ).EQ.ZERO ) THEN
IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
$ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
WR( KBOT-1 ) = WR( KBOT )
ELSE
WR( KBOT ) = WR( KBOT-1 )
END IF
END IF
END IF
*
* ==== Use up to NS of the the smallest magnitude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====
*
NS = MIN( NS, KBOT-KS+1 )
NS = NS - MOD( NS, 2 )
KS = KBOT - NS + 1
*
* ==== Small-bulge multi-shift QR sweep:
* . split workspace under the subdiagonal into
* . - a KDU-by-KDU work array U in the lower
* . left-hand-corner,
* . - a KDU-by-at-least-KDU-but-more-is-better
* . (KDU-by-NHo) horizontal work array WH along
* . the bottom edge,
* . - and an at-least-KDU-but-more-is-better-by-KDU
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
KWV = KDU + 4
NVE = N - KDU - KWV + 1
*
* ==== Small-bulge multi-shift QR sweep ====
*
CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
$ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
$ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
$ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
END IF
*
* ==== Note progress (or the lack of it). ====
*
IF( LD.GT.0 ) THEN
NDFL = 1
ELSE
NDFL = NDFL + 1
END IF
*
* ==== End of main loop ====
80 CONTINUE
*
* ==== Iteration limit exceeded. Set INFO to show where
* . the problem occurred and exit. ====
*
INFO = KBOT
90 CONTINUE
END IF
*
* ==== Return the optimal value of LWORK. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR4 ====
*
END
*> \brief \b DLAQR5 performs a single small-bulge multi-shift QR sweep.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
* SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
* LDU, NV, WV, LDWV, NH, WH, LDWH )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
* $ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
* $ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR5, called by DLAQR0, performs a
*> single small-bulge multi-shift QR sweep.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> WANTT = .true. if the quasi-triangular Schur factor
*> is being computed. WANTT is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> WANTZ = .true. if the orthogonal Schur factor is being
*> computed. WANTZ is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] KACC22
*> \verbatim
*> KACC22 is INTEGER with value 0, 1, or 2.
*> Specifies the computation mode of far-from-diagonal
*> orthogonal updates.
*> = 0: DLAQR5 does not accumulate reflections and does not
*> use matrix-matrix multiply to update far-from-diagonal
*> matrix entries.
*> = 1: DLAQR5 accumulates reflections and uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries.
*> = 2: Same as KACC22 = 1. This option used to enable exploiting
*> the 2-by-2 structure during matrix multiplications, but
*> this is no longer supported.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> N is the order of the Hessenberg matrix H upon which this
*> subroutine operates.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is INTEGER
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is INTEGER
*> These are the first and last rows and columns of an
*> isolated diagonal block upon which the QR sweep is to be
*> applied. It is assumed without a check that
*> either KTOP = 1 or H(KTOP,KTOP-1) = 0
*> and
*> either KBOT = N or H(KBOT+1,KBOT) = 0.
*> \endverbatim
*>
*> \param[in] NSHFTS
*> \verbatim
*> NSHFTS is INTEGER
*> NSHFTS gives the number of simultaneous shifts. NSHFTS
*> must be positive and even.
*> \endverbatim
*>
*> \param[in,out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array, dimension (NSHFTS)
*> \endverbatim
*>
*> \param[in,out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array, dimension (NSHFTS)
*> SR contains the real parts and SI contains the imaginary
*> parts of the NSHFTS shifts of origin that define the
*> multi-shift QR sweep. On output SR and SI may be
*> reordered.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On input H contains a Hessenberg matrix. On output a
*> multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
*> to the isolated diagonal block in rows and columns KTOP
*> through KBOT.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> LDH is the leading dimension of H just as declared in the
*> calling procedure. LDH >= MAX(1,N).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,IHIZ)
*> If WANTZ = .TRUE., then the QR Sweep orthogonal
*> similarity transformation is accumulated into
*> Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*> If WANTZ = .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> LDA is the leading dimension of Z just as declared in
*> the calling procedure. LDZ >= N.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2)
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> LDV is the leading dimension of V as declared in the
*> calling procedure. LDV >= 3.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> LDU is the leading dimension of U just as declared in the
*> in the calling subroutine. LDU >= 2*NSHFTS.
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> NV is the number of rows in WV agailable for workspace.
*> NV >= 1.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is INTEGER
*> LDWV is the leading dimension of WV as declared in the
*> in the calling subroutine. LDWV >= NV.
*> \endverbatim
*
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> NH is the number of columns in array WH available for
*> workspace. NH >= 1.
*> \endverbatim
*>
*> \param[out] WH
*> \verbatim
*> WH is DOUBLE PRECISION array, dimension (LDWH,NH)
*> \endverbatim
*>
*> \param[in] LDWH
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH >= 2*NSHFTS.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr5
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
*> Lars Karlsson, Daniel Kressner, and Bruno Lang
*>
*> Thijs Steel, Department of Computer science,
*> KU Leuven, Belgium
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*>
*> Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed
*> chains of bulges in multishift QR algorithms.
*> ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
*>
* =====================================================================
SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
$ SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
$ LDU, NV, WV, LDWV, NH, WH, LDWH )
IMPLICIT NONE
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
$ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
$ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, BETA, H11, H12, H21, H22, REFSUM,
$ SAFMAX, SAFMIN, SCL, SMLNUM, SWAP, T1, T2,
$ T3, TST1, TST2, ULP
INTEGER I, I2, I4, INCOL, J, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KRCOL,
$ M, M22, MBOT, MTOP, NBMPS, NDCOL,
$ NS, NU
LOGICAL ACCUM, BMP22
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
*
INTRINSIC ABS, DBLE, MAX, MIN, MOD
* ..
* .. Local Arrays ..
DOUBLE PRECISION VT( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLAQR1, DLARFG, DLASET,
$ DTRMM
* ..
* .. Executable Statements ..
*
* ==== If there are no shifts, then there is nothing to do. ====
*
IF( NSHFTS.LT.2 )
$ RETURN
*
* ==== If the active block is empty or 1-by-1, then there
* . is nothing to do. ====
*
IF( KTOP.GE.KBOT )
$ RETURN
*
* ==== Shuffle shifts into pairs of real shifts and pairs
* . of complex conjugate shifts assuming complex
* . conjugate shifts are already adjacent to one
* . another. ====
*
DO 10 I = 1, NSHFTS - 2, 2
IF( SI( I ).NE.-SI( I+1 ) ) THEN
*
SWAP = SR( I )
SR( I ) = SR( I+1 )
SR( I+1 ) = SR( I+2 )
SR( I+2 ) = SWAP
*
SWAP = SI( I )
SI( I ) = SI( I+1 )
SI( I+1 ) = SI( I+2 )
SI( I+2 ) = SWAP
END IF
10 CONTINUE
*
* ==== NSHFTS is supposed to be even, but if it is odd,
* . then simply reduce it by one. The shuffle above
* . ensures that the dropped shift is real and that
* . the remaining shifts are paired. ====
*
NS = NSHFTS - MOD( NSHFTS, 2 )
*
* ==== Machine constants for deflation ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Use accumulated reflections to update far-from-diagonal
* . entries ? ====
*
ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
* ==== clear trash ====
*
IF( KTOP+2.LE.KBOT )
$ H( KTOP+2, KTOP ) = ZERO
*
* ==== NBMPS = number of 2-shift bulges in the chain ====
*
NBMPS = NS / 2
*
* ==== KDU = width of slab ====
*
KDU = 4*NBMPS
*
* ==== Create and chase chains of NBMPS bulges ====
*
DO 180 INCOL = KTOP - 2*NBMPS + 1, KBOT - 2, 2*NBMPS
*
* JTOP = Index from which updates from the right start.
*
IF( ACCUM ) THEN
JTOP = MAX( KTOP, INCOL )
ELSE IF( WANTT ) THEN
JTOP = 1
ELSE
JTOP = KTOP
END IF
*
NDCOL = INCOL + KDU
IF( ACCUM )
$ CALL DLASET( 'ALL', KDU, KDU, ZERO, ONE, U, LDU )
*
* ==== Near-the-diagonal bulge chase. The following loop
* . performs the near-the-diagonal part of a small bulge
* . multi-shift QR sweep. Each 4*NBMPS column diagonal
* . chunk extends from column INCOL to column NDCOL
* . (including both column INCOL and column NDCOL). The
* . following loop chases a 2*NBMPS+1 column long chain of
* . NBMPS bulges 2*NBMPS columns to the right. (INCOL
* . may be less than KTOP and and NDCOL may be greater than
* . KBOT indicating phantom columns from which to chase
* . bulges before they are actually introduced or to which
* . to chase bulges beyond column KBOT.) ====
*
DO 145 KRCOL = INCOL, MIN( INCOL+2*NBMPS-1, KBOT-2 )
*
* ==== Bulges number MTOP to MBOT are active double implicit
* . shift bulges. There may or may not also be small
* . 2-by-2 bulge, if there is room. The inactive bulges
* . (if any) must wait until the active bulges have moved
* . down the diagonal to make room. The phantom matrix
* . paradigm described above helps keep track. ====
*
MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 )
M22 = MBOT + 1
BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+2*( M22-1 ) ).EQ.
$ ( KBOT-2 )
*
* ==== Generate reflections to chase the chain right
* . one column. (The minimum value of K is KTOP-1.) ====
*
IF ( BMP22 ) THEN
*
* ==== Special case: 2-by-2 reflection at bottom treated
* . separately ====
*
K = KRCOL + 2*( M22-1 )
IF( K.EQ.KTOP-1 ) THEN
CALL DLAQR1( 2, H( K+1, K+1 ), LDH, SR( 2*M22-1 ),
$ SI( 2*M22-1 ), SR( 2*M22 ), SI( 2*M22 ),
$ V( 1, M22 ) )
BETA = V( 1, M22 )
CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
END IF
*
* ==== Perform update from right within
* . computational window. ====
*
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 30 J = JTOP, MIN( KBOT, K+3 )
REFSUM = H( J, K+1 ) + V( 2, M22 )*H( J, K+2 )
H( J, K+1 ) = H( J, K+1 ) - REFSUM*T1
H( J, K+2 ) = H( J, K+2 ) - REFSUM*T2
30 CONTINUE
*
* ==== Perform update from left within
* . computational window. ====
*
IF( ACCUM ) THEN
JBOT = MIN( NDCOL, KBOT )
ELSE IF( WANTT ) THEN
JBOT = N
ELSE
JBOT = KBOT
END IF
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 40 J = K+1, JBOT
REFSUM = H( K+1, J ) + V( 2, M22 )*H( K+2, J )
H( K+1, J ) = H( K+1, J ) - REFSUM*T1
H( K+2, J ) = H( K+2, J ) - REFSUM*T2
40 CONTINUE
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
* . criterion and the Ahues & Tisseur (LAWN 122, 1997)
* . criteria both be satisfied. The latter improves
* . accuracy in some examples. Falling back on an
* . alternate convergence criterion when TST1 or TST2
* . is zero (as done here) is traditional but probably
* . unnecessary. ====
*
IF( K.GE.KTOP ) THEN
IF( H( K+1, K ).NE.ZERO ) THEN
TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
IF( TST1.EQ.ZERO ) THEN
IF( K.GE.KTOP+1 )
$ TST1 = TST1 + ABS( H( K, K-1 ) )
IF( K.GE.KTOP+2 )
$ TST1 = TST1 + ABS( H( K, K-2 ) )
IF( K.GE.KTOP+3 )
$ TST1 = TST1 + ABS( H( K, K-3 ) )
IF( K.LE.KBOT-2 )
$ TST1 = TST1 + ABS( H( K+2, K+1 ) )
IF( K.LE.KBOT-3 )
$ TST1 = TST1 + ABS( H( K+3, K+1 ) )
IF( K.LE.KBOT-4 )
$ TST1 = TST1 + ABS( H( K+4, K+1 ) )
END IF
IF( ABS( H( K+1, K ) )
$ .LE.MAX( SMLNUM, ULP*TST1 ) ) THEN
H12 = MAX( ABS( H( K+1, K ) ),
$ ABS( H( K, K+1 ) ) )
H21 = MIN( ABS( H( K+1, K ) ),
$ ABS( H( K, K+1 ) ) )
H11 = MAX( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
H22 = MIN( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
SCL = H11 + H12
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) ) THEN
H( K+1, K ) = ZERO
END IF
END IF
END IF
END IF
*
* ==== Accumulate orthogonal transformations. ====
*
IF( ACCUM ) THEN
KMS = K - INCOL
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 50 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = U( J, KMS+1 ) + V( 2, M22 )*U( J, KMS+2 )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM*T1
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*T2
50 CONTINUE
ELSE IF( WANTZ ) THEN
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 60 J = ILOZ, IHIZ
REFSUM = Z( J, K+1 )+V( 2, M22 )*Z( J, K+2 )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM*T1
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*T2
60 CONTINUE
END IF
END IF
*
* ==== Normal case: Chain of 3-by-3 reflections ====
*
DO 80 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
IF( K.EQ.KTOP-1 ) THEN
CALL DLAQR1( 3, H( KTOP, KTOP ), LDH, SR( 2*M-1 ),
$ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
$ V( 1, M ) )
ALPHA = V( 1, M )
CALL DLARFG( 3, ALPHA, V( 2, M ), 1, V( 1, M ) )
ELSE
*
* ==== Perform delayed transformation of row below
* . Mth bulge. Exploit fact that first two elements
* . of row are actually zero. ====
*
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
REFSUM = V( 3, M )*H( K+3, K+2 )
H( K+3, K ) = -REFSUM*T1
H( K+3, K+1 ) = -REFSUM*T2
H( K+3, K+2 ) = H( K+3, K+2 ) - REFSUM*T3
*
* ==== Calculate reflection to move
* . Mth bulge one step. ====
*
BETA = H( K+1, K )
V( 2, M ) = H( K+2, K )
V( 3, M ) = H( K+3, K )
CALL DLARFG( 3, BETA, V( 2, M ), 1, V( 1, M ) )
*
* ==== A Bulge may collapse because of vigilant
* . deflation or destructive underflow. In the
* . underflow case, try the two-small-subdiagonals
* . trick to try to reinflate the bulge. ====
*
IF( H( K+3, K ).NE.ZERO .OR. H( K+3, K+1 ).NE.
$ ZERO .OR. H( K+3, K+2 ).EQ.ZERO ) THEN
*
* ==== Typical case: not collapsed (yet). ====
*
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
ELSE
*
* ==== Atypical case: collapsed. Attempt to
* . reintroduce ignoring H(K+1,K) and H(K+2,K).
* . If the fill resulting from the new
* . reflector is too large, then abandon it.
* . Otherwise, use the new one. ====
*
CALL DLAQR1( 3, H( K+1, K+1 ), LDH, SR( 2*M-1 ),
$ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
$ VT )
ALPHA = VT( 1 )
CALL DLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) )
T1 = VT( 1 )
T2 = T1*VT( 2 )
T3 = T1*VT( 3 )
REFSUM = H( K+1, K ) + VT( 2 )*H( K+2, K )
*
IF( ABS( H( K+2, K )-REFSUM*T2 )+
$ ABS( REFSUM*T3 ).GT.ULP*
$ ( ABS( H( K, K ) )+ABS( H( K+1,
$ K+1 ) )+ABS( H( K+2, K+2 ) ) ) ) THEN
*
* ==== Starting a new bulge here would
* . create non-negligible fill. Use
* . the old one with trepidation. ====
*
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
ELSE
*
* ==== Starting a new bulge here would
* . create only negligible fill.
* . Replace the old reflector with
* . the new one. ====
*
H( K+1, K ) = H( K+1, K ) - REFSUM*T1
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
V( 1, M ) = VT( 1 )
V( 2, M ) = VT( 2 )
V( 3, M ) = VT( 3 )
END IF
END IF
END IF
*
* ==== Apply reflection from the right and
* . the first column of update from the left.
* . These updates are required for the vigilant
* . deflation check. We still delay most of the
* . updates from the left for efficiency. ====
*
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 70 J = JTOP, MIN( KBOT, K+3 )
REFSUM = H( J, K+1 ) + V( 2, M )*H( J, K+2 )
$ + V( 3, M )*H( J, K+3 )
H( J, K+1 ) = H( J, K+1 ) - REFSUM*T1
H( J, K+2 ) = H( J, K+2 ) - REFSUM*T2
H( J, K+3 ) = H( J, K+3 ) - REFSUM*T3
70 CONTINUE
*
* ==== Perform update from left for subsequent
* . column. ====
*
REFSUM = H( K+1, K+1 ) + V( 2, M )*H( K+2, K+1 )
$ + V( 3, M )*H( K+3, K+1 )
H( K+1, K+1 ) = H( K+1, K+1 ) - REFSUM*T1
H( K+2, K+1 ) = H( K+2, K+1 ) - REFSUM*T2
H( K+3, K+1 ) = H( K+3, K+1 ) - REFSUM*T3
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
* . criterion and the Ahues & Tisseur (LAWN 122, 1997)
* . criteria both be satisfied. The latter improves
* . accuracy in some examples. Falling back on an
* . alternate convergence criterion when TST1 or TST2
* . is zero (as done here) is traditional but probably
* . unnecessary. ====
*
IF( K.LT.KTOP)
$ CYCLE
IF( H( K+1, K ).NE.ZERO ) THEN
TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
IF( TST1.EQ.ZERO ) THEN
IF( K.GE.KTOP+1 )
$ TST1 = TST1 + ABS( H( K, K-1 ) )
IF( K.GE.KTOP+2 )
$ TST1 = TST1 + ABS( H( K, K-2 ) )
IF( K.GE.KTOP+3 )
$ TST1 = TST1 + ABS( H( K, K-3 ) )
IF( K.LE.KBOT-2 )
$ TST1 = TST1 + ABS( H( K+2, K+1 ) )
IF( K.LE.KBOT-3 )
$ TST1 = TST1 + ABS( H( K+3, K+1 ) )
IF( K.LE.KBOT-4 )
$ TST1 = TST1 + ABS( H( K+4, K+1 ) )
END IF
IF( ABS( H( K+1, K ) ).LE.MAX( SMLNUM, ULP*TST1 ) )
$ THEN
H12 = MAX( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
H21 = MIN( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
H11 = MAX( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
H22 = MIN( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
SCL = H11 + H12
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) ) THEN
H( K+1, K ) = ZERO
END IF
END IF
END IF
80 CONTINUE
*
* ==== Multiply H by reflections from the left ====
*
IF( ACCUM ) THEN
JBOT = MIN( NDCOL, KBOT )
ELSE IF( WANTT ) THEN
JBOT = N
ELSE
JBOT = KBOT
END IF
*
DO 100 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 90 J = MAX( KTOP, KRCOL + 2*M ), JBOT
REFSUM = H( K+1, J ) + V( 2, M )*H( K+2, J )
$ + V( 3, M )*H( K+3, J )
H( K+1, J ) = H( K+1, J ) - REFSUM*T1
H( K+2, J ) = H( K+2, J ) - REFSUM*T2
H( K+3, J ) = H( K+3, J ) - REFSUM*T3
90 CONTINUE
100 CONTINUE
*
* ==== Accumulate orthogonal transformations. ====
*
IF( ACCUM ) THEN
*
* ==== Accumulate U. (If needed, update Z later
* . with an efficient matrix-matrix
* . multiply.) ====
*
DO 120 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
KMS = K - INCOL
I2 = MAX( 1, KTOP-INCOL )
I2 = MAX( I2, KMS-(KRCOL-INCOL)+1 )
I4 = MIN( KDU, KRCOL + 2*( MBOT-1 ) - INCOL + 5 )
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 110 J = I2, I4
REFSUM = U( J, KMS+1 ) + V( 2, M )*U( J, KMS+2 )
$ + V( 3, M )*U( J, KMS+3 )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM*T1
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*T2
U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*T3
110 CONTINUE
120 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
*
DO 140 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 130 J = ILOZ, IHIZ
REFSUM = Z( J, K+1 ) + V( 2, M )*Z( J, K+2 )
$ + V( 3, M )*Z( J, K+3 )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM*T1
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*T2
Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*T3
130 CONTINUE
140 CONTINUE
END IF
*
* ==== End of near-the-diagonal bulge chase. ====
*
145 CONTINUE
*
* ==== Use U (if accumulated) to update far-from-diagonal
* . entries in H. If required, use U to update Z as
* . well. ====
*
IF( ACCUM ) THEN
IF( WANTT ) THEN
JTOP = 1
JBOT = N
ELSE
JTOP = KTOP
JBOT = KBOT
END IF
K1 = MAX( 1, KTOP-INCOL )
NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
* ==== Horizontal Multiply ====
*
DO 150 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
CALL DGEMM( 'C', 'N', NU, JLEN, NU, ONE, U( K1, K1 ),
$ LDU, H( INCOL+K1, JCOL ), LDH, ZERO, WH,
$ LDWH )
CALL DLACPY( 'ALL', NU, JLEN, WH, LDWH,
$ H( INCOL+K1, JCOL ), LDH )
150 CONTINUE
*
* ==== Vertical multiply ====
*
DO 160 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
JLEN = MIN( NV, MAX( KTOP, INCOL )-JROW )
CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ H( JROW, INCOL+K1 ), LDH, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ H( JROW, INCOL+K1 ), LDH )
160 CONTINUE
*
* ==== Z multiply (also vertical) ====
*
IF( WANTZ ) THEN
DO 170 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ Z( JROW, INCOL+K1 ), LDZ, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ Z( JROW, INCOL+K1 ), LDZ )
170 CONTINUE
END IF
END IF
180 CONTINUE
*
* ==== End of DLAQR5 ====
*
END
*> \brief \b DLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQSB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqsb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqsb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqsb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, UPLO
* INTEGER KD, LDAB, N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQSB equilibrates a symmetric band matrix A using the scaling
*> factors in the vector S.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of super-diagonals of the matrix A if UPLO = 'U',
*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**T*U or A = L*L**T of the band
*> matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A.
*> \endverbatim
*>
*> \param[in] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> Ratio of the smallest S(i) to the largest S(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies whether or not equilibration was done.
*> = 'N': No equilibration.
*> = 'Y': Equilibration was done, i.e., A has been replaced by
*> diag(S) * A * diag(S).
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> THRESH is a threshold value used to decide if scaling should be done
*> based on the ratio of the scaling factors. If SCOND < THRESH,
*> scaling is done.
*>
*> LARGE and SMALL are threshold values used to decide if scaling should
*> be done based on the absolute size of the largest matrix element.
*> If AMAX > LARGE or AMAX < SMALL, scaling is done.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqhb
*
* =====================================================================
SUBROUTINE DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
$ EQUED )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, UPLO
INTEGER KD, LDAB, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, THRESH
PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION CJ, LARGE, SMALL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
EQUED = 'N'
RETURN
END IF
*
* Initialize LARGE and SMALL.
*
SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
LARGE = ONE / SMALL
*
IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
*
* No equilibration
*
EQUED = 'N'
ELSE
*
* Replace A by diag(S) * A * diag(S).
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Upper triangle of A is stored in band format.
*
DO 20 J = 1, N
CJ = S( J )
DO 10 I = MAX( 1, J-KD ), J
AB( KD+1+I-J, J ) = CJ*S( I )*AB( KD+1+I-J, J )
10 CONTINUE
20 CONTINUE
ELSE
*
* Lower triangle of A is stored.
*
DO 40 J = 1, N
CJ = S( J )
DO 30 I = J, MIN( N, J+KD )
AB( 1+I-J, J ) = CJ*S( I )*AB( 1+I-J, J )
30 CONTINUE
40 CONTINUE
END IF
EQUED = 'Y'
END IF
*
RETURN
*
* End of DLAQSB
*
END
*> \brief \b DLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQSP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqsp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqsp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqsp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, UPLO
* INTEGER N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQSP equilibrates a symmetric matrix A using the scaling factors
*> in the vector S.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, the equilibrated matrix: diag(S) * A * diag(S), in
*> the same storage format as A.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A.
*> \endverbatim
*>
*> \param[in] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> Ratio of the smallest S(i) to the largest S(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies whether or not equilibration was done.
*> = 'N': No equilibration.
*> = 'Y': Equilibration was done, i.e., A has been replaced by
*> diag(S) * A * diag(S).
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> THRESH is a threshold value used to decide if scaling should be done
*> based on the ratio of the scaling factors. If SCOND < THRESH,
*> scaling is done.
*>
*> LARGE and SMALL are threshold values used to decide if scaling should
*> be done based on the absolute size of the largest matrix element.
*> If AMAX > LARGE or AMAX < SMALL, scaling is done.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqhp
*
* =====================================================================
SUBROUTINE DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, UPLO
INTEGER N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, THRESH
PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, JC
DOUBLE PRECISION CJ, LARGE, SMALL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
EQUED = 'N'
RETURN
END IF
*
* Initialize LARGE and SMALL.
*
SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
LARGE = ONE / SMALL
*
IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
*
* No equilibration
*
EQUED = 'N'
ELSE
*
* Replace A by diag(S) * A * diag(S).
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Upper triangle of A is stored.
*
JC = 1
DO 20 J = 1, N
CJ = S( J )
DO 10 I = 1, J
AP( JC+I-1 ) = CJ*S( I )*AP( JC+I-1 )
10 CONTINUE
JC = JC + J
20 CONTINUE
ELSE
*
* Lower triangle of A is stored.
*
JC = 1
DO 40 J = 1, N
CJ = S( J )
DO 30 I = J, N
AP( JC+I-J ) = CJ*S( I )*AP( JC+I-J )
30 CONTINUE
JC = JC + N - J + 1
40 CONTINUE
END IF
EQUED = 'Y'
END IF
*
RETURN
*
* End of DLAQSP
*
END
*> \brief \b DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQSY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqsy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqsy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqsy.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, UPLO
* INTEGER LDA, N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQSY equilibrates a symmetric matrix A using the scaling factors
*> in the vector S.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if EQUED = 'Y', the equilibrated matrix:
*> diag(S) * A * diag(S).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(N,1).
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A.
*> \endverbatim
*>
*> \param[in] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> Ratio of the smallest S(i) to the largest S(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies whether or not equilibration was done.
*> = 'N': No equilibration.
*> = 'Y': Equilibration was done, i.e., A has been replaced by
*> diag(S) * A * diag(S).
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> THRESH is a threshold value used to decide if scaling should be done
*> based on the ratio of the scaling factors. If SCOND < THRESH,
*> scaling is done.
*>
*> LARGE and SMALL are threshold values used to decide if scaling should
*> be done based on the absolute size of the largest matrix element.
*> If AMAX > LARGE or AMAX < SMALL, scaling is done.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqhe
*
* =====================================================================
SUBROUTINE DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, UPLO
INTEGER LDA, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, THRESH
PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION CJ, LARGE, SMALL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
EQUED = 'N'
RETURN
END IF
*
* Initialize LARGE and SMALL.
*
SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
LARGE = ONE / SMALL
*
IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
*
* No equilibration
*
EQUED = 'N'
ELSE
*
* Replace A by diag(S) * A * diag(S).
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Upper triangle of A is stored.
*
DO 20 J = 1, N
CJ = S( J )
DO 10 I = 1, J
A( I, J ) = CJ*S( I )*A( I, J )
10 CONTINUE
20 CONTINUE
ELSE
*
* Lower triangle of A is stored.
*
DO 40 J = 1, N
CJ = S( J )
DO 30 I = J, N
A( I, J ) = CJ*S( I )*A( I, J )
30 CONTINUE
40 CONTINUE
END IF
EQUED = 'Y'
END IF
*
RETURN
*
* End of DLAQSY
*
END
*> \brief \b DLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK,
* INFO )
*
* .. Scalar Arguments ..
* LOGICAL LREAL, LTRAN
* INTEGER INFO, LDT, N
* DOUBLE PRECISION SCALE, W
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( * ), T( LDT, * ), WORK( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQTR solves the real quasi-triangular system
*>
*> op(T)*p = scale*c, if LREAL = .TRUE.
*>
*> or the complex quasi-triangular systems
*>
*> op(T + iB)*(p+iq) = scale*(c+id), if LREAL = .FALSE.
*>
*> in real arithmetic, where T is upper quasi-triangular.
*> If LREAL = .FALSE., then the first diagonal block of T must be
*> 1 by 1, B is the specially structured matrix
*>
*> B = [ b(1) b(2) ... b(n) ]
*> [ w ]
*> [ w ]
*> [ . ]
*> [ w ]
*>
*> op(A) = A or A**T, A**T denotes the transpose of
*> matrix A.
*>
*> On input, X = [ c ]. On output, X = [ p ].
*> [ d ] [ q ]
*>
*> This subroutine is designed for the condition number estimation
*> in routine DTRSNA.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] LTRAN
*> \verbatim
*> LTRAN is LOGICAL
*> On entry, LTRAN specifies the option of conjugate transpose:
*> = .FALSE., op(T+i*B) = T+i*B,
*> = .TRUE., op(T+i*B) = (T+i*B)**T.
*> \endverbatim
*>
*> \param[in] LREAL
*> \verbatim
*> LREAL is LOGICAL
*> On entry, LREAL specifies the input matrix structure:
*> = .FALSE., the input is complex
*> = .TRUE., the input is real
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of T+i*B. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> On entry, T contains a matrix in Schur canonical form.
*> If LREAL = .FALSE., then the first diagonal block of T mu
*> be 1 by 1.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the matrix T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (N)
*> On entry, B contains the elements to form the matrix
*> B as described above.
*> If LREAL = .TRUE., B is not referenced.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION
*> On entry, W is the diagonal element of the matrix B.
*> If LREAL = .TRUE., W is not referenced.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> On exit, SCALE is the scale factor.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (2*N)
*> On entry, X contains the right hand side of the system.
*> On exit, X is overwritten by the solution.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, INFO is set to
*> 0: successful exit.
*> 1: the some diagonal 1 by 1 block has been perturbed by
*> a small number SMIN to keep nonsingularity.
*> 2: the some diagonal 2 by 2 block has been perturbed by
*> a small number in DLALN2 to keep nonsingularity.
*> NOTE: In the interests of speed, this routine does not
*> check the inputs for errors.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqtr
*
* =====================================================================
SUBROUTINE DLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X,
$ WORK,
$ INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL LREAL, LTRAN
INTEGER INFO, LDT, N
DOUBLE PRECISION SCALE, W
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( * ), T( LDT, * ), WORK( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER I, IERR, J, J1, J2, JNEXT, K, N1, N2
DOUBLE PRECISION BIGNUM, EPS, REC, SCALOC, SI, SMIN, SMINW,
$ SMLNUM, SR, TJJ, TMP, XJ, XMAX, XNORM, Z
* ..
* .. Local Arrays ..
DOUBLE PRECISION D( 2, 2 ), V( 2, 2 )
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DDOT, DLAMCH, DLANGE
EXTERNAL IDAMAX, DASUM, DDOT, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLADIV, DLALN2, DSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Do not test the input parameters for errors
*
NOTRAN = .NOT.LTRAN
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
BIGNUM = ONE / SMLNUM
*
XNORM = DLANGE( 'M', N, N, T, LDT, D )
IF( .NOT.LREAL )
$ XNORM = MAX( XNORM, ABS( W ), DLANGE( 'M', N, 1, B, N, D ) )
SMIN = MAX( SMLNUM, EPS*XNORM )
*
* Compute 1-norm of each column of strictly upper triangular
* part of T to control overflow in triangular solver.
*
WORK( 1 ) = ZERO
DO 10 J = 2, N
WORK( J ) = DASUM( J-1, T( 1, J ), 1 )
10 CONTINUE
*
IF( .NOT.LREAL ) THEN
DO 20 I = 2, N
WORK( I ) = WORK( I ) + ABS( B( I ) )
20 CONTINUE
END IF
*
N2 = 2*N
N1 = N
IF( .NOT.LREAL )
$ N1 = N2
K = IDAMAX( N1, X, 1 )
XMAX = ABS( X( K ) )
SCALE = ONE
*
IF( XMAX.GT.BIGNUM ) THEN
SCALE = BIGNUM / XMAX
CALL DSCAL( N1, SCALE, X, 1 )
XMAX = BIGNUM
END IF
*
IF( LREAL ) THEN
*
IF( NOTRAN ) THEN
*
* Solve T*p = scale*c
*
JNEXT = N
DO 30 J = N, 1, -1
IF( J.GT.JNEXT )
$ GO TO 30
J1 = J
J2 = J
JNEXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNEXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* Meet 1 by 1 diagonal block
*
* Scale to avoid overflow when computing
* x(j) = b(j)/T(j,j)
*
XJ = ABS( X( J1 ) )
TJJ = ABS( T( J1, J1 ) )
TMP = T( J1, J1 )
IF( TJJ.LT.SMIN ) THEN
TMP = SMIN
TJJ = SMIN
INFO = 1
END IF
*
IF( XJ.EQ.ZERO )
$ GO TO 30
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.BIGNUM*TJJ ) THEN
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J1 ) = X( J1 ) / TMP
XJ = ABS( X( J1 ) )
*
* Scale x if necessary to avoid overflow when adding a
* multiple of column j1 of T.
*
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
END IF
END IF
IF( J1.GT.1 ) THEN
CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
$ 1 )
K = IDAMAX( J1-1, X, 1 )
XMAX = ABS( X( K ) )
END IF
*
ELSE
*
* Meet 2 by 2 diagonal block
*
* Call 2 by 2 linear system solve, to take
* care of possible overflow by scaling factor.
*
D( 1, 1 ) = X( J1 )
D( 2, 1 ) = X( J2 )
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, T( J1, J1 ),
$ LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2,
$ SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 2
*
IF( SCALOC.NE.ONE ) THEN
CALL DSCAL( N, SCALOC, X, 1 )
SCALE = SCALE*SCALOC
END IF
X( J1 ) = V( 1, 1 )
X( J2 ) = V( 2, 1 )
*
* Scale V(1,1) (= X(J1)) and/or V(2,1) (=X(J2))
* to avoid overflow in updating right-hand side.
*
XJ = MAX( ABS( V( 1, 1 ) ), ABS( V( 2, 1 ) ) )
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
$ ( BIGNUM-XMAX )*REC ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
END IF
END IF
*
* Update right-hand side
*
IF( J1.GT.1 ) THEN
CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
$ 1 )
CALL DAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X,
$ 1 )
K = IDAMAX( J1-1, X, 1 )
XMAX = ABS( X( K ) )
END IF
*
END IF
*
30 CONTINUE
*
ELSE
*
* Solve T**T*p = scale*c
*
JNEXT = 1
DO 40 J = 1, N
IF( J.LT.JNEXT )
$ GO TO 40
J1 = J
J2 = J
JNEXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNEXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1 by 1 diagonal block
*
* Scale if necessary to avoid overflow in forming the
* right-hand side element by inner product.
*
XJ = ABS( X( J1 ) )
IF( XMAX.GT.ONE ) THEN
REC = ONE / XMAX
IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
X( J1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
$ 1 )
*
XJ = ABS( X( J1 ) )
TJJ = ABS( T( J1, J1 ) )
TMP = T( J1, J1 )
IF( TJJ.LT.SMIN ) THEN
TMP = SMIN
TJJ = SMIN
INFO = 1
END IF
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.BIGNUM*TJJ ) THEN
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J1 ) = X( J1 ) / TMP
XMAX = MAX( XMAX, ABS( X( J1 ) ) )
*
ELSE
*
* 2 by 2 diagonal block
*
* Scale if necessary to avoid overflow in forming the
* right-hand side elements by inner product.
*
XJ = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ) )
IF( XMAX.GT.ONE ) THEN
REC = ONE / XMAX
IF( MAX( WORK( J2 ), WORK( J1 ) ).GT.( BIGNUM-XJ )*
$ REC ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
D( 1, 1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
$ 1 )
D( 2, 1 ) = X( J2 ) - DDOT( J1-1, T( 1, J2 ), 1, X,
$ 1 )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J1, J1 ),
$ LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2,
$ SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 2
*
IF( SCALOC.NE.ONE ) THEN
CALL DSCAL( N, SCALOC, X, 1 )
SCALE = SCALE*SCALOC
END IF
X( J1 ) = V( 1, 1 )
X( J2 ) = V( 2, 1 )
XMAX = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ), XMAX )
*
END IF
40 CONTINUE
END IF
*
ELSE
*
SMINW = MAX( EPS*ABS( W ), SMIN )
IF( NOTRAN ) THEN
*
* Solve (T + iB)*(p+iq) = c+id
*
JNEXT = N
DO 70 J = N, 1, -1
IF( J.GT.JNEXT )
$ GO TO 70
J1 = J
J2 = J
JNEXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNEXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1 by 1 diagonal block
*
* Scale if necessary to avoid overflow in division
*
Z = W
IF( J1.EQ.1 )
$ Z = B( 1 )
XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) )
TJJ = ABS( T( J1, J1 ) ) + ABS( Z )
TMP = T( J1, J1 )
IF( TJJ.LT.SMINW ) THEN
TMP = SMINW
TJJ = SMINW
INFO = 1
END IF
*
IF( XJ.EQ.ZERO )
$ GO TO 70
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.BIGNUM*TJJ ) THEN
REC = ONE / XJ
CALL DSCAL( N2, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
CALL DLADIV( X( J1 ), X( N+J1 ), TMP, Z, SR, SI )
X( J1 ) = SR
X( N+J1 ) = SI
XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) )
*
* Scale x if necessary to avoid overflow when adding a
* multiple of column j1 of T.
*
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN
CALL DSCAL( N2, REC, X, 1 )
SCALE = SCALE*REC
END IF
END IF
*
IF( J1.GT.1 ) THEN
CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
$ 1 )
CALL DAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1,
$ X( N+1 ), 1 )
*
X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 )
X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 )
*
XMAX = ZERO
DO 50 K = 1, J1 - 1
XMAX = MAX( XMAX, ABS( X( K ) )+
$ ABS( X( K+N ) ) )
50 CONTINUE
END IF
*
ELSE
*
* Meet 2 by 2 diagonal block
*
D( 1, 1 ) = X( J1 )
D( 2, 1 ) = X( J2 )
D( 1, 2 ) = X( N+J1 )
D( 2, 2 ) = X( N+J2 )
CALL DLALN2( .FALSE., 2, 2, SMINW, ONE, T( J1,
$ J1 ),
$ LDT, ONE, ONE, D, 2, ZERO, -W, V, 2,
$ SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 2
*
IF( SCALOC.NE.ONE ) THEN
CALL DSCAL( 2*N, SCALOC, X, 1 )
SCALE = SCALOC*SCALE
END IF
X( J1 ) = V( 1, 1 )
X( J2 ) = V( 2, 1 )
X( N+J1 ) = V( 1, 2 )
X( N+J2 ) = V( 2, 2 )
*
* Scale X(J1), .... to avoid overflow in
* updating right hand side.
*
XJ = MAX( ABS( V( 1, 1 ) )+ABS( V( 1, 2 ) ),
$ ABS( V( 2, 1 ) )+ABS( V( 2, 2 ) ) )
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
$ ( BIGNUM-XMAX )*REC ) THEN
CALL DSCAL( N2, REC, X, 1 )
SCALE = SCALE*REC
END IF
END IF
*
* Update the right-hand side.
*
IF( J1.GT.1 ) THEN
CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
$ 1 )
CALL DAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X,
$ 1 )
*
CALL DAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1,
$ X( N+1 ), 1 )
CALL DAXPY( J1-1, -X( N+J2 ), T( 1, J2 ), 1,
$ X( N+1 ), 1 )
*
X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 ) +
$ B( J2 )*X( N+J2 )
X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 ) -
$ B( J2 )*X( J2 )
*
XMAX = ZERO
DO 60 K = 1, J1 - 1
XMAX = MAX( ABS( X( K ) )+ABS( X( K+N ) ),
$ XMAX )
60 CONTINUE
END IF
*
END IF
70 CONTINUE
*
ELSE
*
* Solve (T + iB)**T*(p+iq) = c+id
*
JNEXT = 1
DO 80 J = 1, N
IF( J.LT.JNEXT )
$ GO TO 80
J1 = J
J2 = J
JNEXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNEXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1 by 1 diagonal block
*
* Scale if necessary to avoid overflow in forming the
* right-hand side element by inner product.
*
XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) )
IF( XMAX.GT.ONE ) THEN
REC = ONE / XMAX
IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN
CALL DSCAL( N2, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
X( J1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
$ 1 )
X( N+J1 ) = X( N+J1 ) - DDOT( J1-1, T( 1, J1 ), 1,
$ X( N+1 ), 1 )
IF( J1.GT.1 ) THEN
X( J1 ) = X( J1 ) - B( J1 )*X( N+1 )
X( N+J1 ) = X( N+J1 ) + B( J1 )*X( 1 )
END IF
XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) )
*
Z = W
IF( J1.EQ.1 )
$ Z = B( 1 )
*
* Scale if necessary to avoid overflow in
* complex division
*
TJJ = ABS( T( J1, J1 ) ) + ABS( Z )
TMP = T( J1, J1 )
IF( TJJ.LT.SMINW ) THEN
TMP = SMINW
TJJ = SMINW
INFO = 1
END IF
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.BIGNUM*TJJ ) THEN
REC = ONE / XJ
CALL DSCAL( N2, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
CALL DLADIV( X( J1 ), X( N+J1 ), TMP, -Z, SR, SI )
X( J1 ) = SR
X( J1+N ) = SI
XMAX = MAX( ABS( X( J1 ) )+ABS( X( J1+N ) ), XMAX )
*
ELSE
*
* 2 by 2 diagonal block
*
* Scale if necessary to avoid overflow in forming the
* right-hand side element by inner product.
*
XJ = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ),
$ ABS( X( J2 ) )+ABS( X( N+J2 ) ) )
IF( XMAX.GT.ONE ) THEN
REC = ONE / XMAX
IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
$ ( BIGNUM-XJ ) / XMAX ) THEN
CALL DSCAL( N2, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
D( 1, 1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
$ 1 )
D( 2, 1 ) = X( J2 ) - DDOT( J1-1, T( 1, J2 ), 1, X,
$ 1 )
D( 1, 2 ) = X( N+J1 ) - DDOT( J1-1, T( 1, J1 ), 1,
$ X( N+1 ), 1 )
D( 2, 2 ) = X( N+J2 ) - DDOT( J1-1, T( 1, J2 ), 1,
$ X( N+1 ), 1 )
D( 1, 1 ) = D( 1, 1 ) - B( J1 )*X( N+1 )
D( 2, 1 ) = D( 2, 1 ) - B( J2 )*X( N+1 )
D( 1, 2 ) = D( 1, 2 ) + B( J1 )*X( 1 )
D( 2, 2 ) = D( 2, 2 ) + B( J2 )*X( 1 )
*
CALL DLALN2( .TRUE., 2, 2, SMINW, ONE, T( J1, J1 ),
$ LDT, ONE, ONE, D, 2, ZERO, W, V, 2,
$ SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 2
*
IF( SCALOC.NE.ONE ) THEN
CALL DSCAL( N2, SCALOC, X, 1 )
SCALE = SCALOC*SCALE
END IF
X( J1 ) = V( 1, 1 )
X( J2 ) = V( 2, 1 )
X( N+J1 ) = V( 1, 2 )
X( N+J2 ) = V( 2, 2 )
XMAX = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ),
$ ABS( X( J2 ) )+ABS( X( N+J2 ) ), XMAX )
*
END IF
*
80 CONTINUE
*
END IF
*
END IF
*
RETURN
*
* End of DLAQTR
*
END
*> \brief \b DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAR1V + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlar1v.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlar1v.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlar1v.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD,
* PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
* R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
*
* .. Scalar Arguments ..
* LOGICAL WANTNC
* INTEGER B1, BN, N, NEGCNT, R
* DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
* $ RQCORR, ZTZ
* ..
* .. Array Arguments ..
* INTEGER ISUPPZ( * )
* DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ),
* $ WORK( * )
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAR1V computes the (scaled) r-th column of the inverse of
*> the sumbmatrix in rows B1 through BN of the tridiagonal matrix
*> L D L**T - sigma I. When sigma is close to an eigenvalue, the
*> computed vector is an accurate eigenvector. Usually, r corresponds
*> to the index where the eigenvector is largest in magnitude.
*> The following steps accomplish this computation :
*> (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T,
*> (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
*> (c) Computation of the diagonal elements of the inverse of
*> L D L**T - sigma I by combining the above transforms, and choosing
*> r as the index where the diagonal of the inverse is (one of the)
*> largest in magnitude.
*> (d) Computation of the (scaled) r-th column of the inverse using the
*> twisted factorization obtained by combining the top part of the
*> the stationary and the bottom part of the progressive transform.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix L D L**T.
*> \endverbatim
*>
*> \param[in] B1
*> \verbatim
*> B1 is INTEGER
*> First index of the submatrix of L D L**T.
*> \endverbatim
*>
*> \param[in] BN
*> \verbatim
*> BN is INTEGER
*> Last index of the submatrix of L D L**T.
*> \endverbatim
*>
*> \param[in] LAMBDA
*> \verbatim
*> LAMBDA is DOUBLE PRECISION
*> The shift. In order to compute an accurate eigenvector,
*> LAMBDA should be a good approximation to an eigenvalue
*> of L D L**T.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the unit bidiagonal matrix
*> L, in elements 1 to N-1.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LD
*> \verbatim
*> LD is DOUBLE PRECISION array, dimension (N-1)
*> The n-1 elements L(i)*D(i).
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*> LLD is DOUBLE PRECISION array, dimension (N-1)
*> The n-1 elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] GAPTOL
*> \verbatim
*> GAPTOL is DOUBLE PRECISION
*> Tolerance that indicates when eigenvector entries are negligible
*> w.r.t. their contribution to the residual.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> On input, all entries of Z must be set to 0.
*> On output, Z contains the (scaled) r-th column of the
*> inverse. The scaling is such that Z(R) equals 1.
*> \endverbatim
*>
*> \param[in] WANTNC
*> \verbatim
*> WANTNC is LOGICAL
*> Specifies whether NEGCNT has to be computed.
*> \endverbatim
*>
*> \param[out] NEGCNT
*> \verbatim
*> NEGCNT is INTEGER
*> If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
*> in the matrix factorization L D L**T, and NEGCNT = -1 otherwise.
*> \endverbatim
*>
*> \param[out] ZTZ
*> \verbatim
*> ZTZ is DOUBLE PRECISION
*> The square of the 2-norm of Z.
*> \endverbatim
*>
*> \param[out] MINGMA
*> \verbatim
*> MINGMA is DOUBLE PRECISION
*> The reciprocal of the largest (in magnitude) diagonal
*> element of the inverse of L D L**T - sigma I.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*> R is INTEGER
*> The twist index for the twisted factorization used to
*> compute Z.
*> On input, 0 <= R <= N. If R is input as 0, R is set to
*> the index where (L D L**T - sigma I)^{-1} is largest
*> in magnitude. If 1 <= R <= N, R is unchanged.
*> On output, R contains the twist index used to compute Z.
*> Ideally, R designates the position of the maximum entry in the
*> eigenvector.
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*> ISUPPZ is INTEGER array, dimension (2)
*> The support of the vector in Z, i.e., the vector Z is
*> nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
*> \endverbatim
*>
*> \param[out] NRMINV
*> \verbatim
*> NRMINV is DOUBLE PRECISION
*> NRMINV = 1/SQRT( ZTZ )
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is DOUBLE PRECISION
*> The residual of the FP vector.
*> RESID = ABS( MINGMA )/SQRT( ZTZ )
*> \endverbatim
*>
*> \param[out] RQCORR
*> \verbatim
*> RQCORR is DOUBLE PRECISION
*> The Rayleigh Quotient correction to LAMBDA.
*> RQCORR = MINGMA*TMP
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lar1v
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD,
$ PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
$ R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL WANTNC
INTEGER B1, BN, N, NEGCNT, R
DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
$ RQCORR, ZTZ
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * )
DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ),
$ WORK( * )
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL SAWNAN1, SAWNAN2
INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
$ R2
DOUBLE PRECISION DMINUS, DPLUS, EPS, S, TMP
* ..
* .. External Functions ..
LOGICAL DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL DISNAN, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
EPS = DLAMCH( 'Precision' )
IF( R.EQ.0 ) THEN
R1 = B1
R2 = BN
ELSE
R1 = R
R2 = R
END IF
* Storage for LPLUS
INDLPL = 0
* Storage for UMINUS
INDUMN = N
INDS = 2*N + 1
INDP = 3*N + 1
IF( B1.EQ.1 ) THEN
WORK( INDS ) = ZERO
ELSE
WORK( INDS+B1-1 ) = LLD( B1-1 )
END IF
*
* Compute the stationary transform (using the differential form)
* until the index R2.
*
SAWNAN1 = .FALSE.
NEG1 = 0
S = WORK( INDS+B1-1 ) - LAMBDA
DO 50 I = B1, R1 - 1
DPLUS = D( I ) + S
WORK( INDLPL+I ) = LD( I ) / DPLUS
IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
S = WORK( INDS+I ) - LAMBDA
50 CONTINUE
SAWNAN1 = DISNAN( S )
IF( SAWNAN1 ) GOTO 60
DO 51 I = R1, R2 - 1
DPLUS = D( I ) + S
WORK( INDLPL+I ) = LD( I ) / DPLUS
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
S = WORK( INDS+I ) - LAMBDA
51 CONTINUE
SAWNAN1 = DISNAN( S )
*
60 CONTINUE
IF( SAWNAN1 ) THEN
* Runs a slower version of the above loop if a NaN is detected
NEG1 = 0
S = WORK( INDS+B1-1 ) - LAMBDA
DO 70 I = B1, R1 - 1
DPLUS = D( I ) + S
IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN
WORK( INDLPL+I ) = LD( I ) / DPLUS
IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
IF( WORK( INDLPL+I ).EQ.ZERO )
$ WORK( INDS+I ) = LLD( I )
S = WORK( INDS+I ) - LAMBDA
70 CONTINUE
DO 71 I = R1, R2 - 1
DPLUS = D( I ) + S
IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN
WORK( INDLPL+I ) = LD( I ) / DPLUS
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
IF( WORK( INDLPL+I ).EQ.ZERO )
$ WORK( INDS+I ) = LLD( I )
S = WORK( INDS+I ) - LAMBDA
71 CONTINUE
END IF
*
* Compute the progressive transform (using the differential form)
* until the index R1
*
SAWNAN2 = .FALSE.
NEG2 = 0
WORK( INDP+BN-1 ) = D( BN ) - LAMBDA
DO 80 I = BN - 1, R1, -1
DMINUS = LLD( I ) + WORK( INDP+I )
TMP = D( I ) / DMINUS
IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1
WORK( INDUMN+I ) = L( I )*TMP
WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
80 CONTINUE
TMP = WORK( INDP+R1-1 )
SAWNAN2 = DISNAN( TMP )
IF( SAWNAN2 ) THEN
* Runs a slower version of the above loop if a NaN is detected
NEG2 = 0
DO 100 I = BN-1, R1, -1
DMINUS = LLD( I ) + WORK( INDP+I )
IF(ABS(DMINUS).LT.PIVMIN) DMINUS = -PIVMIN
TMP = D( I ) / DMINUS
IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1
WORK( INDUMN+I ) = L( I )*TMP
WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
IF( TMP.EQ.ZERO )
$ WORK( INDP+I-1 ) = D( I ) - LAMBDA
100 CONTINUE
END IF
*
* Find the index (from R1 to R2) of the largest (in magnitude)
* diagonal element of the inverse
*
MINGMA = WORK( INDS+R1-1 ) + WORK( INDP+R1-1 )
IF( MINGMA.LT.ZERO ) NEG1 = NEG1 + 1
IF( WANTNC ) THEN
NEGCNT = NEG1 + NEG2
ELSE
NEGCNT = -1
ENDIF
IF( ABS(MINGMA).EQ.ZERO )
$ MINGMA = EPS*WORK( INDS+R1-1 )
R = R1
DO 110 I = R1, R2 - 1
TMP = WORK( INDS+I ) + WORK( INDP+I )
IF( TMP.EQ.ZERO )
$ TMP = EPS*WORK( INDS+I )
IF( ABS( TMP ).LE.ABS( MINGMA ) ) THEN
MINGMA = TMP
R = I + 1
END IF
110 CONTINUE
*
* Compute the FP vector: solve N^T v = e_r
*
ISUPPZ( 1 ) = B1
ISUPPZ( 2 ) = BN
Z( R ) = ONE
ZTZ = ONE
*
* Compute the FP vector upwards from R
*
IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN
DO 210 I = R-1, B1, -1
Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I ) = ZERO
ISUPPZ( 1 ) = I + 1
GOTO 220
ENDIF
ZTZ = ZTZ + Z( I )*Z( I )
210 CONTINUE
220 CONTINUE
ELSE
* Run slower loop if NaN occurred.
DO 230 I = R - 1, B1, -1
IF( Z( I+1 ).EQ.ZERO ) THEN
Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 )
ELSE
Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
END IF
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I ) = ZERO
ISUPPZ( 1 ) = I + 1
GO TO 240
END IF
ZTZ = ZTZ + Z( I )*Z( I )
230 CONTINUE
240 CONTINUE
ENDIF
* Compute the FP vector downwards from R in blocks of size BLKSIZ
IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN
DO 250 I = R, BN-1
Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I+1 ) = ZERO
ISUPPZ( 2 ) = I
GO TO 260
END IF
ZTZ = ZTZ + Z( I+1 )*Z( I+1 )
250 CONTINUE
260 CONTINUE
ELSE
* Run slower loop if NaN occurred.
DO 270 I = R, BN - 1
IF( Z( I ).EQ.ZERO ) THEN
Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 )
ELSE
Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
END IF
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I+1 ) = ZERO
ISUPPZ( 2 ) = I
GO TO 280
END IF
ZTZ = ZTZ + Z( I+1 )*Z( I+1 )
270 CONTINUE
280 CONTINUE
END IF
*
* Compute quantities for convergence test
*
TMP = ONE / ZTZ
NRMINV = SQRT( TMP )
RESID = ABS( MINGMA )*NRMINV
RQCORR = MINGMA*TMP
*
*
RETURN
*
* End of DLAR1V
*
END
*> \brief \b DLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAR2V + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlar2v.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlar2v.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlar2v.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAR2V( N, X, Y, Z, INCX, C, S, INCC )
*
* .. Scalar Arguments ..
* INTEGER INCC, INCX, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( * ), S( * ), X( * ), Y( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAR2V applies a vector of real plane rotations from both sides to
*> a sequence of 2-by-2 real symmetric matrices, defined by the elements
*> of the vectors x, y and z. For i = 1,2,...,n
*>
*> ( x(i) z(i) ) := ( c(i) s(i) ) ( x(i) z(i) ) ( c(i) -s(i) )
*> ( z(i) y(i) ) ( -s(i) c(i) ) ( z(i) y(i) ) ( s(i) c(i) )
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of plane rotations to be applied.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCX)
*> The vector x.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCX)
*> The vector y.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCX)
*> The vector z.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X, Y and Z. INCX > 0.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*> The cosines of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*> The sines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCC
*> \verbatim
*> INCC is INTEGER
*> The increment between elements of C and S. INCC > 0.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lar2v
*
* =====================================================================
SUBROUTINE DLAR2V( N, X, Y, Z, INCX, C, S, INCC )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCC, INCX, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( * ), S( * ), X( * ), Y( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IC, IX
DOUBLE PRECISION CI, SI, T1, T2, T3, T4, T5, T6, XI, YI, ZI
* ..
* .. Executable Statements ..
*
IX = 1
IC = 1
DO 10 I = 1, N
XI = X( IX )
YI = Y( IX )
ZI = Z( IX )
CI = C( IC )
SI = S( IC )
T1 = SI*ZI
T2 = CI*ZI
T3 = T2 - SI*XI
T4 = T2 + SI*YI
T5 = CI*XI + T1
T6 = CI*YI - T1
X( IX ) = CI*T5 + SI*T4
Y( IX ) = CI*T6 - SI*T3
Z( IX ) = CI*T4 - SI*T5
IX = IX + INCX
IC = IC + INCC
10 CONTINUE
*
* End of DLAR2V
*
RETURN
END
*> \brief \b DLARF applies an elementary reflector to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER INCV, LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARF applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*> or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*> The vector v in the representation of H. V is not used if
*> TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*> INCV is INTEGER
*> The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larf
*
* =====================================================================
SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL APPLYLEFT
INTEGER I, LASTV, LASTC
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILADLR, ILADLC
EXTERNAL LSAME, ILADLR, ILADLC
* ..
* .. Executable Statements ..
*
APPLYLEFT = LSAME( SIDE, 'L' )
LASTV = 0
LASTC = 0
IF( TAU.NE.ZERO ) THEN
! Set up variables for scanning V. LASTV begins pointing to the end
! of V.
IF( APPLYLEFT ) THEN
LASTV = M
ELSE
LASTV = N
END IF
IF( INCV.GT.0 ) THEN
I = 1 + (LASTV-1) * INCV
ELSE
I = 1
END IF
! Look for the last non-zero row in V.
DO WHILE( LASTV.GT.0 .AND. V( I ).EQ.ZERO )
LASTV = LASTV - 1
I = I - INCV
END DO
IF( APPLYLEFT ) THEN
! Scan for the last non-zero column in C(1:lastv,:).
LASTC = ILADLC(LASTV, N, C, LDC)
ELSE
! Scan for the last non-zero row in C(:,1:lastv).
LASTC = ILADLR(M, LASTV, C, LDC)
END IF
END IF
! Note that lastc.eq.0 renders the BLAS operations null; no special
! case is needed at this level.
IF( APPLYLEFT ) THEN
*
* Form H * C
*
IF( LASTV.GT.0 ) THEN
*
* w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
CALL DGEMV( 'Transpose', LASTV, LASTC, ONE, C, LDC, V,
$ INCV,
$ ZERO, WORK, 1 )
*
* C(1:lastv,1:lastc) := C(...) - v(1:lastv,1) * w(1:lastc,1)**T
*
CALL DGER( LASTV, LASTC, -TAU, V, INCV, WORK, 1, C, LDC )
END IF
ELSE
*
* Form C * H
*
IF( LASTV.GT.0 ) THEN
*
* w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
CALL DGEMV( 'No transpose', LASTC, LASTV, ONE, C, LDC,
$ V, INCV, ZERO, WORK, 1 )
*
* C(1:lastc,1:lastv) := C(...) - w(1:lastc,1) * v(1:lastv,1)**T
*
CALL DGER( LASTC, LASTV, -TAU, WORK, 1, V, INCV, C, LDC )
END IF
END IF
RETURN
*
* End of DLARF
*
END
*> \brief \b DLARF1F applies an elementary reflector to a general rectangular
* matrix assuming v(1) = 1.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF1F + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf1f.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarf1f.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarf1f.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARF1F( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER INCV, LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARF1F applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*> or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*> The vector v in the representation of H. V is not used if
*> TAU = 0. V(1) is not referenced or modified.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*> INCV is INTEGER
*> The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> \endverbatim
*
* To take advantage of the fact that v(1) = 1, we do the following
* v = [ 1 v_2 ]**T
* If SIDE='L'
* |-----|
* | C_1 |
* C =| C_2 |
* |-----|
* C_1\in\mathbb{R}^{1\times n}, C_2\in\mathbb{R}^{m-1\times n}
* So we compute:
* C = HC = (I - \tau vv**T)C
* = C - \tau vv**T C
* w = C**T v = [ C_1**T C_2**T ] [ 1 v_2 ]**T
* = C_1**T + C_2**T v ( DGEMM then DAXPY )
* C = C - \tau vv**T C
* = C - \tau vw**T
* Giving us C_1 = C_1 - \tau w**T ( DAXPY )
* and
* C_2 = C_2 - \tau v_2w**T ( DGER )
* If SIDE='R'
*
* C = [ C_1 C_2 ]
* C_1\in\mathbb{R}^{m\times 1}, C_2\in\mathbb{R}^{m\times n-1}
* So we compute:
* C = CH = C(I - \tau vv**T)
* = C - \tau Cvv**T
*
* w = Cv = [ C_1 C_2 ] [ 1 v_2 ]**T
* = C_1 + C_2v_2 ( DGEMM then DAXPY )
* C = C - \tau Cvv**T
* = C - \tau wv**T
* Giving us C_1 = C_1 - \tau w ( DAXPY )
* and
* C_2 = C_2 - \tau wv_2**T ( DGER )
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larf
*
* =====================================================================
SUBROUTINE DLARF1F( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL APPLYLEFT
INTEGER I, LASTV, LASTC
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER, DAXPY, DSCAL
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILADLR, ILADLC
EXTERNAL LSAME, ILADLR, ILADLC
* ..
* .. Executable Statements ..
*
APPLYLEFT = LSAME( SIDE, 'L' )
LASTV = 1
LASTC = 0
IF( TAU.NE.ZERO ) THEN
! Set up variables for scanning V. LASTV begins pointing to the end
! of V.
IF( APPLYLEFT ) THEN
LASTV = M
ELSE
LASTV = N
END IF
IF( INCV.GT.0 ) THEN
I = 1 + (LASTV-1) * INCV
ELSE
I = 1
END IF
! Look for the last non-zero row in V.
! Since we are assuming that V(1) = 1, and it is not stored, so we
! shouldn't access it.
DO WHILE( LASTV.GT.1 .AND. V( I ).EQ.ZERO )
LASTV = LASTV - 1
I = I - INCV
END DO
IF( APPLYLEFT ) THEN
! Scan for the last non-zero column in C(1:lastv,:).
LASTC = ILADLC(LASTV, N, C, LDC)
ELSE
! Scan for the last non-zero row in C(:,1:lastv).
LASTC = ILADLR(M, LASTV, C, LDC)
END IF
END IF
IF( LASTC.EQ.0 ) THEN
RETURN
END IF
IF( APPLYLEFT ) THEN
*
* Form H * C
*
! Check if lastv = 1. This means v = 1, So we just need to compute
! C := HC = (1-\tau)C.
IF( LASTV.EQ.1 ) THEN
*
* C(1,1:lastc) := ( 1 - tau ) * C(1,1:lastc)
*
CALL DSCAL(LASTC, ONE - TAU, C, LDC)
ELSE
*
* w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
! w(1:lastc,1) := C(2:lastv,1:lastc)**T * v(2:lastv,1)
CALL DGEMV( 'Transpose', LASTV-1, LASTC, ONE, C(1+1,1),
$ LDC, V(1+INCV), INCV, ZERO, WORK, 1)
! w(1:lastc,1) += C(1,1:lastc)**T * v(1,1) = C(1,1:lastc)**T
CALL DAXPY(LASTC, ONE, C, LDC, WORK, 1)
*
* C(1:lastv,1:lastc) := C(...) - tau * v(1:lastv,1) * w(1:lastc,1)**T
*
! C(1, 1:lastc) := C(...) - tau * v(1,1) * w(1:lastc,1)**T
! = C(...) - tau * w(1:lastc,1)**T
CALL DAXPY(LASTC, -TAU, WORK, 1, C, LDC)
! C(2:lastv,1:lastc) := C(...) - tau * v(2:lastv,1)*w(1:lastc,1)**T
CALL DGER(LASTV-1, LASTC, -TAU, V(1+INCV), INCV, WORK, 1,
$ C(1+1,1), LDC)
END IF
ELSE
*
* Form C * H
*
! Check if n = 1. This means v = 1, so we just need to compute
! C := CH = C(1-\tau).
IF( LASTV.EQ.1 ) THEN
*
* C(1:lastc,1) := ( 1 - tau ) * C(1:lastc,1)
*
CALL DSCAL(LASTC, ONE - TAU, C, 1)
ELSE
*
* w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
! w(1:lastc,1) := C(1:lastc,2:lastv) * v(2:lastv,1)
CALL DGEMV( 'No transpose', LASTC, LASTV-1, ONE,
$ C(1,1+1), LDC, V(1+INCV), INCV, ZERO, WORK, 1 )
! w(1:lastc,1) += C(1:lastc,1) v(1,1) = C(1:lastc,1)
CALL DAXPY(LASTC, ONE, C, 1, WORK, 1)
*
* C(1:lastc,1:lastv) := C(...) - tau * w(1:lastc,1) * v(1:lastv,1)**T
*
! C(1:lastc,1) := C(...) - tau * w(1:lastc,1) * v(1,1)**T
! = C(...) - tau * w(1:lastc,1)
CALL DAXPY(LASTC, -TAU, WORK, 1, C, 1)
! C(1:lastc,2:lastv) := C(...) - tau * w(1:lastc,1) * v(2:lastv)**T
CALL DGER( LASTC, LASTV-1, -TAU, WORK, 1, V(1+INCV),
$ INCV, C(1,1+1), LDC )
END IF
END IF
RETURN
*
* End of DLARF1F
*
END
*> \brief \b DLARF1L applies an elementary reflector to a general rectangular
* matrix assuming v(lastv) = 1 where lastv is the last non-zero
* element
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF1L + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf1l.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarf1l.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarf1l.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARF1L( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER INCV, LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARF1L applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*> or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*> The vector v in the representation of H. V is not used if
*> TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*> INCV is INTEGER
*> The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larf
*
* =====================================================================
SUBROUTINE DLARF1L( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL APPLYLEFT
INTEGER I, FIRSTV, LASTV, LASTC
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILADLR, ILADLC
EXTERNAL LSAME, ILADLR, ILADLC
* ..
* .. Executable Statements ..
*
APPLYLEFT = LSAME( SIDE, 'L' )
FIRSTV = 1
LASTC = 0
IF( TAU.NE.ZERO ) THEN
! Set up variables for scanning V. LASTV begins pointing to the end
! of V.
IF( APPLYLEFT ) THEN
LASTV = M
ELSE
LASTV = N
END IF
I = 1
! Look for the last non-zero row in V.
DO WHILE( LASTV.GT.FIRSTV .AND. V( I ).EQ.ZERO )
FIRSTV = FIRSTV + 1
I = I + INCV
END DO
IF( APPLYLEFT ) THEN
! Scan for the last non-zero column in C(1:lastv,:).
LASTC = ILADLC(LASTV, N, C, LDC)
ELSE
! Scan for the last non-zero row in C(:,1:lastv).
LASTC = ILADLR(M, LASTV, C, LDC)
END IF
END IF
IF( LASTC.EQ.0 ) THEN
RETURN
END IF
IF( APPLYLEFT ) THEN
*
* Form H * C
*
IF( LASTV.GT.0 ) THEN
! Check if m = 1. This means v = 1, So we just need to compute
! C := HC = (1-\tau)C.
IF( LASTV.EQ.FIRSTV ) THEN
CALL DSCAL(LASTC, ONE - TAU, C( FIRSTV, 1), LDC)
ELSE
*
* w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
! w(1:lastc,1) := C(1:lastv-1,1:lastc)**T * v(1:lastv-1,1)
CALL DGEMV( 'Transpose', LASTV-FIRSTV, LASTC, ONE,
$ C(FIRSTV,1), LDC, V(I), INCV, ZERO,
$ WORK, 1)
! w(1:lastc,1) += C(lastv,1:lastc)**T * v(lastv,1) = C(lastv,1:lastc)**T
CALL DAXPY(LASTC, ONE, C(LASTV,1), LDC, WORK, 1)
*
* C(1:lastv,1:lastc) := C(...) - tau * v(1:lastv,1) * w(1:lastc,1)**T
*
! C(lastv, 1:lastc) := C(...) - tau * v(lastv,1) * w(1:lastc,1)**T
! = C(...) - tau * w(1:lastc,1)**T
CALL DAXPY(LASTC, -TAU, WORK, 1, C(LASTV,1), LDC)
! C(1:lastv-1,1:lastc) := C(...) - tau * v(1:lastv-1,1)*w(1:lastc,1)**T
CALL DGER(LASTV-FIRSTV, LASTC, -TAU, V(I), INCV,
$ WORK, 1, C(FIRSTV,1), LDC)
END IF
END IF
ELSE
*
* Form C * H
*
IF( LASTV.GT.0 ) THEN
! Check if n = 1. This means v = 1, so we just need to compute
! C := CH = C(1-\tau).
IF( LASTV.EQ.FIRSTV ) THEN
CALL DSCAL(LASTC, ONE - TAU, C, 1)
ELSE
*
* w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
! w(1:lastc,1) := C(1:lastc,1:lastv-1) * v(1:lastv-1,1)
CALL DGEMV( 'No transpose', LASTC, LASTV-FIRSTV,
$ ONE, C(1,FIRSTV), LDC, V(I), INCV, ZERO, WORK, 1 )
! w(1:lastc,1) += C(1:lastc,lastv) * v(lastv,1) = C(1:lastc,lastv)
CALL DAXPY(LASTC, ONE, C(1,LASTV), 1, WORK, 1)
*
* C(1:lastc,1:lastv) := C(...) - tau * w(1:lastc,1) * v(1:lastv,1)**T
*
! C(1:lastc,lastv) := C(...) - tau * w(1:lastc,1) * v(lastv,1)**T
! = C(...) - tau * w(1:lastc,1)
CALL DAXPY(LASTC, -TAU, WORK, 1, C(1,LASTV), 1)
! C(1:lastc,1:lastv-1) := C(...) - tau * w(1:lastc,1) * v(1:lastv-1)**T
CALL DGER( LASTC, LASTV-FIRSTV, -TAU, WORK, 1, V(I),
$ INCV, C(1,FIRSTV), LDC )
END IF
END IF
END IF
RETURN
*
* End of DLARF1L
*
END
*> \brief \b DLARFB applies a block reflector or its transpose to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
* T, LDT, C, LDC, WORK, LDWORK )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, SIDE, STOREV, TRANS
* INTEGER K, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
* $ WORK( LDWORK, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFB applies a real block reflector H or its transpose H**T to a
*> real m by n matrix C, from either the left or the right.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply H or H**T from the Left
*> = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply H (No transpose)
*> = 'T': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Indicates how H is formed from a product of elementary
*> reflectors
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Indicates how the vectors which define the elementary
*> reflectors are stored:
*> = 'C': Columnwise
*> = 'R': Rowwise
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the matrix T (= the number of elementary
*> reflectors whose product defines the block reflector).
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,M) if STOREV = 'R' and SIDE = 'L'
*> (LDV,N) if STOREV = 'R' and SIDE = 'R'
*> The matrix V. See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
*> if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
*> if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The triangular k by k matrix T in the representation of the
*> block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LDWORK,K)
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*> LDWORK is INTEGER
*> The leading dimension of the array WORK.
*> If SIDE = 'L', LDWORK >= max(1,N);
*> if SIDE = 'R', LDWORK >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfb
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The triangular part of V (including its diagonal) is not
*> referenced.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
*> ( v1 1 ) ( 1 v2 v2 v2 )
*> ( v1 v2 1 ) ( 1 v3 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> V = ( v1 v2 v3 ) V = ( v1 v1 1 )
*> ( v1 v2 v3 ) ( v2 v2 v2 1 )
*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
*> ( 1 v3 )
*> ( 1 )
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V,
$ LDV,
$ T, LDT, C, LDC, WORK, LDWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, SIDE, STOREV, TRANS
INTEGER K, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
$ WORK( LDWORK, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
CHARACTER TRANST
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DTRMM
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( LSAME( TRANS, 'N' ) ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
IF( LSAME( STOREV, 'C' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 ) (first K rows)
* ( V2 )
* where V1 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK)
*
* W := C1**T
*
DO 10 J = 1, K
CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
10 CONTINUE
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ N,
$ K, ONE, V, LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C2**T * V2
*
CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K,
$ ONE, C( K+1, 1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N,
$ K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W**T
*
IF( M.GT.K ) THEN
*
* C2 := C2 - V2 * W**T
*
CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K,
$ -ONE, V( K+1, 1 ), LDV, WORK, LDWORK, ONE,
$ C( K+1, 1 ), LDC )
END IF
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N,
$ K,
$ ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W**T
*
DO 30 J = 1, K
DO 20 I = 1, N
C( J, I ) = C( J, I ) - WORK( I, J )
20 CONTINUE
30 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C1
*
DO 40 J = 1, K
CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
40 CONTINUE
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ M,
$ K, ONE, V, LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C2 * V2
*
CALL DGEMM( 'No transpose', 'No transpose', M, K,
$ N-K,
$ ONE, C( 1, K+1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V**T
*
IF( N.GT.K ) THEN
*
* C2 := C2 - W * V2**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE,
$ C( 1, K+1 ), LDC )
END IF
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M,
$ K,
$ ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 60 J = 1, K
DO 50 I = 1, M
C( I, J ) = C( I, J ) - WORK( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
ELSE
*
* Let V = ( V1 )
* ( V2 ) (last K rows)
* where V2 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK)
*
* W := C2**T
*
DO 70 J = 1, K
CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ),
$ 1 )
70 CONTINUE
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ N,
$ K, ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1**T * V1
*
CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N,
$ K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W**T
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1 * W**T
*
CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K,
$ -ONE, V, LDV, WORK, LDWORK, ONE, C, LDC )
END IF
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N,
$ K,
$ ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W**T
*
DO 90 J = 1, K
DO 80 I = 1, N
C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
80 CONTINUE
90 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C2
*
DO 100 J = 1, K
CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
100 CONTINUE
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ M,
$ K, ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1
*
CALL DGEMM( 'No transpose', 'No transpose', M, K,
$ N-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V**T
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
END IF
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M,
$ K,
$ ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W
*
DO 120 J = 1, K
DO 110 I = 1, M
C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
110 CONTINUE
120 CONTINUE
END IF
END IF
*
ELSE IF( LSAME( STOREV, 'R' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 V2 ) (V1: first K columns)
* where V1 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V**T = (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
* W := C1**T
*
DO 130 J = 1, K
CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
130 CONTINUE
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N,
$ K,
$ ONE, V, LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C2**T * V2**T
*
CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K,
$ ONE,
$ C( K+1, 1 ), LDC, V( 1, K+1 ), LDV, ONE,
$ WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N,
$ K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V**T * W**T
*
IF( M.GT.K ) THEN
*
* C2 := C2 - V2**T * W**T
*
CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K,
$ -ONE,
$ V( 1, K+1 ), LDV, WORK, LDWORK, ONE,
$ C( K+1, 1 ), LDC )
END IF
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ N,
$ K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W**T
*
DO 150 J = 1, K
DO 140 I = 1, N
C( J, I ) = C( J, I ) - WORK( I, J )
140 CONTINUE
150 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
* W := C * V**T = (C1*V1**T + C2*V2**T) (stored in WORK)
*
* W := C1
*
DO 160 J = 1, K
CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
160 CONTINUE
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M,
$ K,
$ ONE, V, LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C2 * V2**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K,
$ ONE, C( 1, K+1 ), LDC, V( 1, K+1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( N.GT.K ) THEN
*
* C2 := C2 - W * V2
*
CALL DGEMM( 'No transpose', 'No transpose', M, N-K,
$ K,
$ -ONE, WORK, LDWORK, V( 1, K+1 ), LDV, ONE,
$ C( 1, K+1 ), LDC )
END IF
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ M,
$ K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 180 J = 1, K
DO 170 I = 1, M
C( I, J ) = C( I, J ) - WORK( I, J )
170 CONTINUE
180 CONTINUE
*
END IF
*
ELSE
*
* Let V = ( V1 V2 ) (V2: last K columns)
* where V2 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V**T = (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
* W := C2**T
*
DO 190 J = 1, K
CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ),
$ 1 )
190 CONTINUE
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N,
$ K,
$ ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1**T * V1**T
*
CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K,
$ ONE,
$ C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N,
$ K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V**T * W**T
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1**T * W**T
*
CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K,
$ -ONE,
$ V, LDV, WORK, LDWORK, ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ N,
$ K, ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W**T
*
DO 210 J = 1, K
DO 200 I = 1, N
C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
200 CONTINUE
210 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H' where C = ( C1 C2 )
*
* W := C * V**T = (C1*V1**T + C2*V2**T) (stored in WORK)
*
* W := C2
*
DO 220 J = 1, K
CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
220 CONTINUE
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M,
$ K,
$ ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1
*
CALL DGEMM( 'No transpose', 'No transpose', M, N-K,
$ K,
$ -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ M,
$ K, ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 240 J = 1, K
DO 230 I = 1, M
C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
230 CONTINUE
240 CONTINUE
*
END IF
*
END IF
END IF
*
RETURN
*
* End of DLARFB
*
END
*> \brief \b DLARFG generates an elementary reflector (Householder matrix).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfg.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
* .. Scalar Arguments ..
* INTEGER INCX, N
* DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFG generates a real elementary reflector H of order n, such
*> that
*>
*> H * ( alpha ) = ( beta ), H**T * H = I.
*> ( x ) ( 0 )
*>
*> where alpha and beta are scalars, and x is an (n-1)-element real
*> vector. H is represented in the form
*>
*> H = I - tau * ( 1 ) * ( 1 v**T ) ,
*> ( v )
*>
*> where tau is a real scalar and v is a real (n-1)-element
*> vector.
*>
*> If the elements of x are all zero, then tau = 0 and H is taken to be
*> the unit matrix.
*>
*> Otherwise 1 <= tau <= 2.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> On entry, the value alpha.
*> On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension
*> (1+(N-2)*abs(INCX))
*> On entry, the vector x.
*> On exit, it is overwritten with the vector v.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfg
*
* =====================================================================
SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX, N
DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J, KNT
DOUBLE PRECISION BETA, RSAFMN, SAFMIN, XNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
EXTERNAL DLAMCH, DLAPY2, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN
* ..
* .. External Subroutines ..
EXTERNAL DSCAL
* ..
* .. Executable Statements ..
*
IF( N.LE.1 ) THEN
TAU = ZERO
RETURN
END IF
*
XNORM = DNRM2( N-1, X, INCX )
*
IF( XNORM.EQ.ZERO ) THEN
*
* H = I
*
TAU = ZERO
ELSE
*
* general case
*
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
KNT = 0
IF( ABS( BETA ).LT.SAFMIN ) THEN
*
* XNORM, BETA may be inaccurate; scale X and recompute them
*
RSAFMN = ONE / SAFMIN
10 CONTINUE
KNT = KNT + 1
CALL DSCAL( N-1, RSAFMN, X, INCX )
BETA = BETA*RSAFMN
ALPHA = ALPHA*RSAFMN
IF( (ABS( BETA ).LT.SAFMIN) .AND. (KNT .LT. 20) )
$ GO TO 10
*
* New BETA is at most 1, at least SAFMIN
*
XNORM = DNRM2( N-1, X, INCX )
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
END IF
TAU = ( BETA-ALPHA ) / BETA
CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
*
* If ALPHA is subnormal, it may lose relative accuracy
*
DO 20 J = 1, KNT
BETA = BETA*SAFMIN
20 CONTINUE
ALPHA = BETA
END IF
*
RETURN
*
* End of DLARFG
*
END
*> \brief \b DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFGP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfgp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfgp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfgp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
*
* .. Scalar Arguments ..
* INTEGER INCX, N
* DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFGP generates a real elementary reflector H of order n, such
*> that
*>
*> H * ( alpha ) = ( beta ), H**T * H = I.
*> ( x ) ( 0 )
*>
*> where alpha and beta are scalars, beta is non-negative, and x is
*> an (n-1)-element real vector. H is represented in the form
*>
*> H = I - tau * ( 1 ) * ( 1 v**T ) ,
*> ( v )
*>
*> where tau is a real scalar and v is a real (n-1)-element
*> vector.
*>
*> If the elements of x are all zero, then tau = 0 and H is taken to be
*> the unit matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> On entry, the value alpha.
*> On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension
*> (1+(N-2)*abs(INCX))
*> On entry, the vector x.
*> On exit, it is overwritten with the vector v.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfgp
*
* =====================================================================
SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX, N
DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION TWO, ONE, ZERO
PARAMETER ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J, KNT
DOUBLE PRECISION BETA, BIGNUM, EPS, SAVEALPHA, SMLNUM, XNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
EXTERNAL DLAMCH, DLAPY2, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN
* ..
* .. External Subroutines ..
EXTERNAL DSCAL
* ..
* .. Executable Statements ..
*
IF( N.LE.0 ) THEN
TAU = ZERO
RETURN
END IF
*
EPS = DLAMCH( 'Precision' )
XNORM = DNRM2( N-1, X, INCX )
*
IF( XNORM.LE.EPS*ABS(ALPHA) ) THEN
*
* H = [+/-1, 0; I], sign chosen so ALPHA >= 0.
*
IF( ALPHA.GE.ZERO ) THEN
* When TAU.eq.ZERO, the vector is special-cased to be
* all zeros in the application routines. We do not need
* to clear it.
TAU = ZERO
ELSE
* However, the application routines rely on explicit
* zero checks when TAU.ne.ZERO, and we must clear X.
TAU = TWO
DO J = 1, N-1
X( 1 + (J-1)*INCX ) = 0
END DO
ALPHA = -ALPHA
END IF
ELSE
*
* general case
*
BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'E' )
KNT = 0
IF( ABS( BETA ).LT.SMLNUM ) THEN
*
* XNORM, BETA may be inaccurate; scale X and recompute them
*
BIGNUM = ONE / SMLNUM
10 CONTINUE
KNT = KNT + 1
CALL DSCAL( N-1, BIGNUM, X, INCX )
BETA = BETA*BIGNUM
ALPHA = ALPHA*BIGNUM
IF( (ABS( BETA ).LT.SMLNUM) .AND. (KNT .LT. 20) )
$ GO TO 10
*
* New BETA is at most 1, at least SMLNUM
*
XNORM = DNRM2( N-1, X, INCX )
BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
END IF
SAVEALPHA = ALPHA
ALPHA = ALPHA + BETA
IF( BETA.LT.ZERO ) THEN
BETA = -BETA
TAU = -ALPHA / BETA
ELSE
ALPHA = XNORM * (XNORM/ALPHA)
TAU = ALPHA / BETA
ALPHA = -ALPHA
END IF
*
IF ( ABS(TAU).LE.SMLNUM ) THEN
*
* In the case where the computed TAU ends up being a denormalized number,
* it loses relative accuracy. This is a BIG problem. Solution: flush TAU
* to ZERO. This explains the next IF statement.
*
* (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
* (Thanks Pat. Thanks MathWorks.)
*
IF( SAVEALPHA.GE.ZERO ) THEN
TAU = ZERO
ELSE
TAU = TWO
DO J = 1, N-1
X( 1 + (J-1)*INCX ) = 0
END DO
BETA = -SAVEALPHA
END IF
*
ELSE
*
* This is the general case.
*
CALL DSCAL( N-1, ONE / ALPHA, X, INCX )
*
END IF
*
* If BETA is subnormal, it may lose relative accuracy
*
DO 20 J = 1, KNT
BETA = BETA*SMLNUM
20 CONTINUE
ALPHA = BETA
END IF
*
RETURN
*
* End of DLARFGP
*
END
*> \brief \b DLARFT forms the triangular factor T of a block reflector H = I - vtvH
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarft.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarft.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarft.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
* INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFT forms the triangular factor T of a real block reflector H
*> of order n, which is defined as a product of k elementary reflectors.
*>
*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*>
*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*>
*> If STOREV = 'C', the vector which defines the elementary reflector
*> H(i) is stored in the i-th column of the array V, and
*>
*> H = I - V * T * V**T
*>
*> If STOREV = 'R', the vector which defines the elementary reflector
*> H(i) is stored in the i-th row of the array V, and
*>
*> H = I - V**T * T * V
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies the order in which the elementary reflectors are
*> multiplied to form the block reflector:
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Specifies how the vectors which define the elementary
*> reflectors are stored (see also Further Details):
*> = 'C': columnwise
*> = 'R': rowwise
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the block reflector H. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the triangular factor T (= the number of
*> elementary reflectors). K >= 1.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,N) if STOREV = 'R'
*> The matrix V. See further details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The k by k triangular factor T of the block reflector.
*> If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
*> lower triangular. The rest of the array is not used.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larft
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The elements equal to 1 are not stored.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
*> ( v1 1 ) ( 1 v2 v2 v2 )
*> ( v1 v2 1 ) ( 1 v3 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> V = ( v1 v2 v3 ) V = ( v1 v1 1 )
*> ( v1 v2 v3 ) ( v2 v2 v2 1 )
*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
*> ( 1 v3 )
*> ( 1 )
*> \endverbatim
*>
* =====================================================================
RECURSIVE SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV,
$ TAU, T, LDT )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments
*
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
*
DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* .. Parameters ..
*
DOUBLE PRECISION ONE, NEG_ONE, ZERO
PARAMETER(ONE=1.0D+0, ZERO = 0.0D+0, NEG_ONE=-1.0D+0)
*
* .. Local Scalars ..
*
INTEGER I,J,L
LOGICAL QR,LQ,QL,DIRF,COLV
*
* .. External Subroutines ..
*
EXTERNAL DTRMM,DGEMM,DLACPY
*
* .. External Functions..
*
LOGICAL LSAME
EXTERNAL LSAME
*
* The general scheme used is inspired by the approach inside DGEQRT3
* which was (at the time of writing this code):
* Based on the algorithm of Elmroth and Gustavson,
* IBM J. Res. Develop. Vol 44 No. 4 July 2000.
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF(N.EQ.0.OR.K.EQ.0) THEN
RETURN
END IF
*
* Base case
*
IF(N.EQ.1.OR.K.EQ.1) THEN
T(1,1) = TAU(1)
RETURN
END IF
*
* Beginning of executable statements
*
L = K / 2
*
* Determine what kind of Q we need to compute
* We assume that if the user doesn't provide 'F' for DIRECT,
* then they meant to provide 'B' and if they don't provide
* 'C' for STOREV, then they meant to provide 'R'
*
DIRF = LSAME(DIRECT,'F')
COLV = LSAME(STOREV,'C')
*
* QR happens when we have forward direction in column storage
*
QR = DIRF.AND.COLV
*
* LQ happens when we have forward direction in row storage
*
LQ = DIRF.AND.(.NOT.COLV)
*
* QL happens when we have backward direction in column storage
*
QL = (.NOT.DIRF).AND.COLV
*
* The last case is RQ. Due to how we structured this, if the
* above 3 are false, then RQ must be true, so we never store
* this
* RQ happens when we have backward direction in row storage
* RQ = (.NOT.DIRF).AND.(.NOT.COLV)
*
IF(QR) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} 0 |
* |V_{2,1} V_{2,2}|
* |V_{3,1} V_{3,2}|
* |---------------|
*
* V_{1,1}\in\R^{l,l} unit lower triangular
* V_{2,1}\in\R^{k-l,l} rectangular
* V_{3,1}\in\R^{n-k,l} rectangular
*
* V_{2,2}\in\R^{k-l,k-l} unit lower triangular
* V_{3,2}\in\R^{n-k,k-l} rectangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{l, l} upper triangular
* T_{2,2}\in\R^{k-l, k-l} upper triangular
* T_{1,2}\in\R^{l, k-l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2')
* = I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2'
*
* Define T_{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)
*
* Compute T_{1,2}
* T_{1,2} = V_{2,1}'
*
DO J = 1, L
DO I = 1, K-L
T(J, L+I) = V(L+I, J)
END DO
END DO
*
* T_{1,2} = T_{1,2}*V_{2,2}
*
CALL DTRMM('Right', 'Lower', 'No transpose', 'Unit', L,
$ K-L, ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)
*
* T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE,
$ V(K+1, 1), LDV, V(K+1, L+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1'*V_2
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL DTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL DTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)
ELSE IF(LQ) THEN
*
* Break V apart into 6 components
*
* V = |----------------------|
* |V_{1,1} V_{1,2} V{1,3}|
* |0 V_{2,2} V{2,3}|
* |----------------------|
*
* V_{1,1}\in\R^{l,l} unit upper triangular
* V_{1,2}\in\R^{l,k-l} rectangular
* V_{1,3}\in\R^{l,n-k} rectangular
*
* V_{2,2}\in\R^{k-l,k-l} unit upper triangular
* V_{2,3}\in\R^{k-l,n-k} rectangular
*
* Where l = floor(k/2)
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{l, l} upper triangular
* T_{2,2}\in\R^{k-l, k-l} upper triangular
* T_{1,2}\in\R^{l, k-l} rectangular
*
* Then, consider the product:
*
* (I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2)
* = I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2
*
* Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)
*
* Compute T_{1,2}
* T_{1,2} = V_{1,2}
*
CALL DLACPY('All', L, K-L, V(1, L+1), LDV, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*V_{2,2}'
*
CALL DTRMM('Right', 'Upper', 'Transpose', 'Unit', L, K-L,
$ ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)
*
* T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE,
$ V(1, K+1), LDV, V(L+1, K+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1*V_2'
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL DTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL DTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)
ELSE IF(QL) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} V_{1,2}|
* |V_{2,1} V_{2,2}|
* |0 V_{3,2}|
* |---------------|
*
* V_{1,1}\in\R^{n-k,k-l} rectangular
* V_{2,1}\in\R^{k-l,k-l} unit upper triangular
*
* V_{1,2}\in\R^{n-k,l} rectangular
* V_{2,2}\in\R^{k-l,l} rectangular
* V_{3,2}\in\R^{l,l} unit upper triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\R^{l, l} non-unit lower triangular
* T_{2,1}\in\R^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1')
* = I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1'
*
* Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V(1, K-L+1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}'
*
DO J = 1, K-L
DO I = 1, L
T(K-L+I, J) = V(N-K+J, K-L+I)
END DO
END DO
*
* T_{2,1} = T_{2,1}*V_{2,1}
*
CALL DTRMM('Right', 'Upper', 'No transpose', 'Unit', L,
$ K-L, ONE, V(N-K+1, 1), LDV, T(K-L+1, 1), LDT)
*
* T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE,
$ V(1, K-L+1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)
*
* At this point, we have that T_{2,1} = V_2'*V_1
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL DTRMM('Left', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)
*
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL DTRMM('Right', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
ELSE
*
* Else means RQ case
*
* Break V apart into 6 components
*
* V = |-----------------------|
* |V_{1,1} V_{1,2} 0 |
* |V_{2,1} V_{2,2} V_{2,3}|
* |-----------------------|
*
* V_{1,1}\in\R^{k-l,n-k} rectangular
* V_{1,2}\in\R^{k-l,k-l} unit lower triangular
*
* V_{2,1}\in\R^{l,n-k} rectangular
* V_{2,2}\in\R^{l,k-l} rectangular
* V_{2,3}\in\R^{l,l} unit lower triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\R^{l, l} non-unit lower triangular
* T_{2,1}\in\R^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1)
* = I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1
*
* Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V(K-L+1, 1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}
*
CALL DLACPY('All', L, K-L, V(K-L+1, N-K+1), LDV,
$ T(K-L+1, 1), LDT)
*
* T_{2,1} = T_{2,1}*V_{1,2}'
*
CALL DTRMM('Right', 'Lower', 'Transpose', 'Unit', L, K-L,
$ ONE, V(1, N-K+1), LDV, T(K-L+1, 1), LDT)
*
* T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE,
$ V(K-L+1, 1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)
*
* At this point, we have that T_{2,1} = V_2*V_1'
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL DTRMM('Left', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)
*
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL DTRMM('Right', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
END IF
END SUBROUTINE
*> \brief \b DLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFX applies a real elementary reflector H to a real m by n
*> matrix C, from either the left or the right. H is represented in the
*> form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix
*>
*> This version uses inline code if H has order < 11.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (M) if SIDE = 'L'
*> or (N) if SIDE = 'R'
*> The vector v in the representation of H.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= (1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> WORK is not referenced if H has order < 11.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfx
*
* =====================================================================
SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J
DOUBLE PRECISION SUM, T1, T10, T2, T3, T4, T5, T6, T7, T8, T9,
$ V1, V10, V2, V3, V4, V5, V6, V7, V8, V9
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF
* ..
* .. Executable Statements ..
*
IF( TAU.EQ.ZERO )
$ RETURN
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C, where H has order m.
*
GO TO ( 10, 30, 50, 70, 90, 110, 130, 150,
$ 170, 190 )M
*
* Code for general M
*
CALL DLARF( SIDE, M, N, V, 1, TAU, C, LDC, WORK )
GO TO 410
10 CONTINUE
*
* Special code for 1 x 1 Householder
*
T1 = ONE - TAU*V( 1 )*V( 1 )
DO 20 J = 1, N
C( 1, J ) = T1*C( 1, J )
20 CONTINUE
GO TO 410
30 CONTINUE
*
* Special code for 2 x 2 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
DO 40 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
40 CONTINUE
GO TO 410
50 CONTINUE
*
* Special code for 3 x 3 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
DO 60 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
60 CONTINUE
GO TO 410
70 CONTINUE
*
* Special code for 4 x 4 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
DO 80 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
80 CONTINUE
GO TO 410
90 CONTINUE
*
* Special code for 5 x 5 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
DO 100 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
100 CONTINUE
GO TO 410
110 CONTINUE
*
* Special code for 6 x 6 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
DO 120 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
120 CONTINUE
GO TO 410
130 CONTINUE
*
* Special code for 7 x 7 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
DO 140 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
140 CONTINUE
GO TO 410
150 CONTINUE
*
* Special code for 8 x 8 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
DO 160 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J ) + V8*C( 8, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
C( 8, J ) = C( 8, J ) - SUM*T8
160 CONTINUE
GO TO 410
170 CONTINUE
*
* Special code for 9 x 9 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
DO 180 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
C( 8, J ) = C( 8, J ) - SUM*T8
C( 9, J ) = C( 9, J ) - SUM*T9
180 CONTINUE
GO TO 410
190 CONTINUE
*
* Special code for 10 x 10 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
V10 = V( 10 )
T10 = TAU*V10
DO 200 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J ) +
$ V10*C( 10, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
C( 8, J ) = C( 8, J ) - SUM*T8
C( 9, J ) = C( 9, J ) - SUM*T9
C( 10, J ) = C( 10, J ) - SUM*T10
200 CONTINUE
GO TO 410
ELSE
*
* Form C * H, where H has order n.
*
GO TO ( 210, 230, 250, 270, 290, 310, 330, 350,
$ 370, 390 )N
*
* Code for general N
*
CALL DLARF( SIDE, M, N, V, 1, TAU, C, LDC, WORK )
GO TO 410
210 CONTINUE
*
* Special code for 1 x 1 Householder
*
T1 = ONE - TAU*V( 1 )*V( 1 )
DO 220 J = 1, M
C( J, 1 ) = T1*C( J, 1 )
220 CONTINUE
GO TO 410
230 CONTINUE
*
* Special code for 2 x 2 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
DO 240 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
240 CONTINUE
GO TO 410
250 CONTINUE
*
* Special code for 3 x 3 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
DO 260 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
260 CONTINUE
GO TO 410
270 CONTINUE
*
* Special code for 4 x 4 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
DO 280 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
280 CONTINUE
GO TO 410
290 CONTINUE
*
* Special code for 5 x 5 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
DO 300 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
300 CONTINUE
GO TO 410
310 CONTINUE
*
* Special code for 6 x 6 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
DO 320 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
320 CONTINUE
GO TO 410
330 CONTINUE
*
* Special code for 7 x 7 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
DO 340 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
340 CONTINUE
GO TO 410
350 CONTINUE
*
* Special code for 8 x 8 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
DO 360 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 ) + V8*C( J, 8 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
C( J, 8 ) = C( J, 8 ) - SUM*T8
360 CONTINUE
GO TO 410
370 CONTINUE
*
* Special code for 9 x 9 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
DO 380 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
C( J, 8 ) = C( J, 8 ) - SUM*T8
C( J, 9 ) = C( J, 9 ) - SUM*T9
380 CONTINUE
GO TO 410
390 CONTINUE
*
* Special code for 10 x 10 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
V10 = V( 10 )
T10 = TAU*V10
DO 400 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 ) +
$ V10*C( J, 10 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
C( J, 8 ) = C( J, 8 ) - SUM*T8
C( J, 9 ) = C( J, 9 ) - SUM*T9
C( J, 10 ) = C( J, 10 ) - SUM*T10
400 CONTINUE
GO TO 410
END IF
410 CONTINUE
RETURN
*
* End of DLARFX
*
END
*> \brief \b DLARGV generates a vector of plane rotations with real cosines and real sines.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlargv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlargv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlargv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARGV( N, X, INCX, Y, INCY, C, INCC )
*
* .. Scalar Arguments ..
* INTEGER INCC, INCX, INCY, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( * ), X( * ), Y( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARGV generates a vector of real plane rotations, determined by
*> elements of the real vectors x and y. For i = 1,2,...,n
*>
*> ( c(i) s(i) ) ( x(i) ) = ( a(i) )
*> ( -s(i) c(i) ) ( y(i) ) = ( 0 )
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of plane rotations to be generated.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCX)
*> On entry, the vector x.
*> On exit, x(i) is overwritten by a(i), for i = 1,...,n.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCY)
*> On entry, the vector y.
*> On exit, the sines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> The increment between elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*> The cosines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCC
*> \verbatim
*> INCC is INTEGER
*> The increment between elements of C. INCC > 0.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup largv
*
* =====================================================================
SUBROUTINE DLARGV( N, X, INCX, Y, INCY, C, INCC )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCC, INCX, INCY, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( * ), X( * ), Y( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IC, IX, IY
DOUBLE PRECISION F, G, T, TT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IX = 1
IY = 1
IC = 1
DO 10 I = 1, N
F = X( IX )
G = Y( IY )
IF( G.EQ.ZERO ) THEN
C( IC ) = ONE
ELSE IF( F.EQ.ZERO ) THEN
C( IC ) = ZERO
Y( IY ) = ONE
X( IX ) = G
ELSE IF( ABS( F ).GT.ABS( G ) ) THEN
T = G / F
TT = SQRT( ONE+T*T )
C( IC ) = ONE / TT
Y( IY ) = T*C( IC )
X( IX ) = F*TT
ELSE
T = F / G
TT = SQRT( ONE+T*T )
Y( IY ) = ONE / TT
C( IC ) = T*Y( IY )
X( IX ) = G*TT
END IF
IC = IC + INCC
IY = IY + INCY
IX = IX + INCX
10 CONTINUE
RETURN
*
* End of DLARGV
*
END
*> \brief \b DLARNV returns a vector of random numbers from a uniform or normal distribution.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARNV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarnv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarnv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarnv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARNV( IDIST, ISEED, N, X )
*
* .. Scalar Arguments ..
* INTEGER IDIST, N
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* DOUBLE PRECISION X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARNV returns a vector of n random real numbers from a uniform or
*> normal distribution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] IDIST
*> \verbatim
*> IDIST is INTEGER
*> Specifies the distribution of the random numbers:
*> = 1: uniform (0,1)
*> = 2: uniform (-1,1)
*> = 3: normal (0,1)
*> \endverbatim
*>
*> \param[in,out] ISEED
*> \verbatim
*> ISEED is INTEGER array, dimension (4)
*> On entry, the seed of the random number generator; the array
*> elements must be between 0 and 4095, and ISEED(4) must be
*> odd.
*> On exit, the seed is updated.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of random numbers to be generated.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> The generated random numbers.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larnv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> This routine calls the auxiliary routine DLARUV to generate random
*> real numbers from a uniform (0,1) distribution, in batches of up to
*> 128 using vectorisable code. The Box-Muller method is used to
*> transform numbers from a uniform to a normal distribution.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARNV( IDIST, ISEED, N, X )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IDIST, N
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
DOUBLE PRECISION X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, TWO
PARAMETER ( ONE = 1.0D+0, TWO = 2.0D+0 )
INTEGER LV
PARAMETER ( LV = 128 )
DOUBLE PRECISION TWOPI
PARAMETER ( TWOPI = 6.28318530717958647692528676655900576839D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IL, IL2, IV
* ..
* .. Local Arrays ..
DOUBLE PRECISION U( LV )
* ..
* .. Intrinsic Functions ..
INTRINSIC COS, LOG, MIN, SQRT
* ..
* .. External Subroutines ..
EXTERNAL DLARUV
* ..
* .. Executable Statements ..
*
DO 40 IV = 1, N, LV / 2
IL = MIN( LV / 2, N-IV+1 )
IF( IDIST.EQ.3 ) THEN
IL2 = 2*IL
ELSE
IL2 = IL
END IF
*
* Call DLARUV to generate IL2 numbers from a uniform (0,1)
* distribution (IL2 <= LV)
*
CALL DLARUV( ISEED, IL2, U )
*
IF( IDIST.EQ.1 ) THEN
*
* Copy generated numbers
*
DO 10 I = 1, IL
X( IV+I-1 ) = U( I )
10 CONTINUE
ELSE IF( IDIST.EQ.2 ) THEN
*
* Convert generated numbers to uniform (-1,1) distribution
*
DO 20 I = 1, IL
X( IV+I-1 ) = TWO*U( I ) - ONE
20 CONTINUE
ELSE IF( IDIST.EQ.3 ) THEN
*
* Convert generated numbers to normal (0,1) distribution
*
DO 30 I = 1, IL
X( IV+I-1 ) = SQRT( -TWO*LOG( U( 2*I-1 ) ) )*
$ COS( TWOPI*U( 2*I ) )
30 CONTINUE
END IF
40 CONTINUE
RETURN
*
* End of DLARNV
*
END
*> \brief \b DLARRA computes the splitting points with the specified threshold.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarra.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarra.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarra.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRA( N, D, E, E2, SPLTOL, TNRM,
* NSPLIT, ISPLIT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N, NSPLIT
* DOUBLE PRECISION SPLTOL, TNRM
* ..
* .. Array Arguments ..
* INTEGER ISPLIT( * )
* DOUBLE PRECISION D( * ), E( * ), E2( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Compute the splitting points with threshold SPLTOL.
*> DLARRA sets any "small" off-diagonal elements to zero.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the N diagonal elements of the tridiagonal
*> matrix T.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, the first (N-1) entries contain the subdiagonal
*> elements of the tridiagonal matrix T; E(N) need not be set.
*> On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT,
*> are set to zero, the other entries of E are untouched.
*> \endverbatim
*>
*> \param[in,out] E2
*> \verbatim
*> E2 is DOUBLE PRECISION array, dimension (N)
*> On entry, the first (N-1) entries contain the SQUARES of the
*> subdiagonal elements of the tridiagonal matrix T;
*> E2(N) need not be set.
*> On exit, the entries E2( ISPLIT( I ) ),
*> 1 <= I <= NSPLIT, have been set to zero
*> \endverbatim
*>
*> \param[in] SPLTOL
*> \verbatim
*> SPLTOL is DOUBLE PRECISION
*> The threshold for splitting. Two criteria can be used:
*> SPLTOL<0 : criterion based on absolute off-diagonal value
*> SPLTOL>0 : criterion that preserves relative accuracy
*> \endverbatim
*>
*> \param[in] TNRM
*> \verbatim
*> TNRM is DOUBLE PRECISION
*> The norm of the matrix.
*> \endverbatim
*>
*> \param[out] NSPLIT
*> \verbatim
*> NSPLIT is INTEGER
*> The number of blocks T splits into. 1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[out] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into blocks.
*> The first block consists of rows/columns 1 to ISPLIT(1),
*> the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*> etc., and the NSPLIT-th consists of rows/columns
*> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larra
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRA( N, D, E, E2, SPLTOL, TNRM,
$ NSPLIT, ISPLIT, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N, NSPLIT
DOUBLE PRECISION SPLTOL, TNRM
* ..
* .. Array Arguments ..
INTEGER ISPLIT( * )
DOUBLE PRECISION D( * ), E( * ), E2( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION EABS, TMP1
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
INFO = 0
NSPLIT = 1
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
* Compute splitting points
IF(SPLTOL.LT.ZERO) THEN
* Criterion based on absolute off-diagonal value
TMP1 = ABS(SPLTOL)* TNRM
DO 9 I = 1, N-1
EABS = ABS( E(I) )
IF( EABS .LE. TMP1) THEN
E(I) = ZERO
E2(I) = ZERO
ISPLIT( NSPLIT ) = I
NSPLIT = NSPLIT + 1
END IF
9 CONTINUE
ELSE
* Criterion that guarantees relative accuracy
DO 10 I = 1, N-1
EABS = ABS( E(I) )
IF( EABS .LE. SPLTOL * SQRT(ABS(D(I)))*SQRT(ABS(D(I+1))) )
$ THEN
E(I) = ZERO
E2(I) = ZERO
ISPLIT( NSPLIT ) = I
NSPLIT = NSPLIT + 1
END IF
10 CONTINUE
ENDIF
ISPLIT( NSPLIT ) = N
RETURN
*
* End of DLARRA
*
END
*> \brief \b DLARRB provides limited bisection to locate eigenvalues for more accuracy.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
* RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
* PIVMIN, SPDIAM, TWIST, INFO )
*
* .. Scalar Arguments ..
* INTEGER IFIRST, ILAST, INFO, N, OFFSET, TWIST
* DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPDIAM
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), LLD( * ), W( * ),
* $ WERR( * ), WGAP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given the relatively robust representation(RRR) L D L^T, DLARRB
*> does "limited" bisection to refine the eigenvalues of L D L^T,
*> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
*> guesses for these eigenvalues are input in W, the corresponding estimate
*> of the error in these guesses and their gaps are input in WERR
*> and WGAP, respectively. During bisection, intervals
*> [left, right] are maintained by storing their mid-points and
*> semi-widths in the arrays W and WERR respectively.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*> LLD is DOUBLE PRECISION array, dimension (N-1)
*> The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] IFIRST
*> \verbatim
*> IFIRST is INTEGER
*> The index of the first eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] ILAST
*> \verbatim
*> ILAST is INTEGER
*> The index of the last eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*> RTOL1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*> RTOL2 is DOUBLE PRECISION
*> Tolerance for the convergence of the bisection intervals.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> where GAP is the (estimated) distance to the nearest
*> eigenvalue.
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*> OFFSET is INTEGER
*> Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
*> through ILAST-OFFSET elements of these arrays are to be used.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
*> estimates of the eigenvalues of L D L^T indexed IFIRST through
*> ILAST.
*> On output, these estimates are refined.
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*> WGAP is DOUBLE PRECISION array, dimension (N-1)
*> On input, the (estimated) gaps between consecutive
*> eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between
*> eigenvalues I and I+1. Note that if IFIRST = ILAST
*> then WGAP(IFIRST-OFFSET) must be set to ZERO.
*> On output, these gaps are refined.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION array, dimension (N)
*> On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
*> the errors in the estimates of the corresponding elements in W.
*> On output, these errors are refined.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*> SPDIAM is DOUBLE PRECISION
*> The spectral diameter of the matrix.
*> \endverbatim
*>
*> \param[in] TWIST
*> \verbatim
*> TWIST is INTEGER
*> The twist index for the twisted factorization that is used
*> for the negcount.
*> TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T
*> TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T
*> TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> Error flag.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrb
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
$ RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
$ PIVMIN, SPDIAM, TWIST, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IFIRST, ILAST, INFO, N, OFFSET, TWIST
DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPDIAM
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), LLD( * ), W( * ),
$ WERR( * ), WGAP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, TWO, HALF
PARAMETER ( ZERO = 0.0D0, TWO = 2.0D0,
$ HALF = 0.5D0 )
INTEGER MAXITR
* ..
* .. Local Scalars ..
INTEGER I, I1, II, IP, ITER, K, NEGCNT, NEXT, NINT,
$ OLNINT, PREV, R
DOUBLE PRECISION BACK, CVRGD, GAP, LEFT, LGAP, MID, MNWDTH,
$ RGAP, RIGHT, TMP, WIDTH
* ..
* .. External Functions ..
INTEGER DLANEG
EXTERNAL DLANEG
*
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
MNWDTH = TWO * PIVMIN
*
R = TWIST
IF((R.LT.1).OR.(R.GT.N)) R = N
*
* Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
* The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
* Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
* for an unconverged interval is set to the index of the next unconverged
* interval, and is -1 or 0 for a converged interval. Thus a linked
* list of unconverged intervals is set up.
*
I1 = IFIRST
* The number of unconverged intervals
NINT = 0
* The last unconverged interval found
PREV = 0
RGAP = WGAP( I1-OFFSET )
DO 75 I = I1, ILAST
K = 2*I
II = I - OFFSET
LEFT = W( II ) - WERR( II )
RIGHT = W( II ) + WERR( II )
LGAP = RGAP
RGAP = WGAP( II )
GAP = MIN( LGAP, RGAP )
* Make sure that [LEFT,RIGHT] contains the desired eigenvalue
* Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT
*
* Do while( NEGCNT(LEFT).GT.I-1 )
*
BACK = WERR( II )
20 CONTINUE
NEGCNT = DLANEG( N, D, LLD, LEFT, PIVMIN, R )
IF( NEGCNT.GT.I-1 ) THEN
LEFT = LEFT - BACK
BACK = TWO*BACK
GO TO 20
END IF
*
* Do while( NEGCNT(RIGHT).LT.I )
* Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT
*
BACK = WERR( II )
50 CONTINUE
NEGCNT = DLANEG( N, D, LLD, RIGHT, PIVMIN, R )
IF( NEGCNT.LT.I ) THEN
RIGHT = RIGHT + BACK
BACK = TWO*BACK
GO TO 50
END IF
WIDTH = HALF*ABS( LEFT - RIGHT )
TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
IF( WIDTH.LE.CVRGD .OR. WIDTH.LE.MNWDTH ) THEN
* This interval has already converged and does not need refinement.
* (Note that the gaps might change through refining the
* eigenvalues, however, they can only get bigger.)
* Remove it from the list.
IWORK( K-1 ) = -1
* Make sure that I1 always points to the first unconverged interval
IF((I.EQ.I1).AND.(I.LT.ILAST)) I1 = I + 1
IF((PREV.GE.I1).AND.(I.LE.ILAST)) IWORK( 2*PREV-1 ) = I + 1
ELSE
* unconverged interval found
PREV = I
NINT = NINT + 1
IWORK( K-1 ) = I + 1
IWORK( K ) = NEGCNT
END IF
WORK( K-1 ) = LEFT
WORK( K ) = RIGHT
75 CONTINUE
*
* Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
* and while (ITER.LT.MAXITR)
*
ITER = 0
80 CONTINUE
PREV = I1 - 1
I = I1
OLNINT = NINT
DO 100 IP = 1, OLNINT
K = 2*I
II = I - OFFSET
RGAP = WGAP( II )
LGAP = RGAP
IF(II.GT.1) LGAP = WGAP( II-1 )
GAP = MIN( LGAP, RGAP )
NEXT = IWORK( K-1 )
LEFT = WORK( K-1 )
RIGHT = WORK( K )
MID = HALF*( LEFT + RIGHT )
* semiwidth of interval
WIDTH = RIGHT - MID
TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
IF( ( WIDTH.LE.CVRGD ) .OR. ( WIDTH.LE.MNWDTH ).OR.
$ ( ITER.EQ.MAXITR ) )THEN
* reduce number of unconverged intervals
NINT = NINT - 1
* Mark interval as converged.
IWORK( K-1 ) = 0
IF( I1.EQ.I ) THEN
I1 = NEXT
ELSE
* Prev holds the last unconverged interval previously examined
IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
END IF
I = NEXT
GO TO 100
END IF
PREV = I
*
* Perform one bisection step
*
NEGCNT = DLANEG( N, D, LLD, MID, PIVMIN, R )
IF( NEGCNT.LE.I-1 ) THEN
WORK( K-1 ) = MID
ELSE
WORK( K ) = MID
END IF
I = NEXT
100 CONTINUE
ITER = ITER + 1
* do another loop if there are still unconverged intervals
* However, in the last iteration, all intervals are accepted
* since this is the best we can do.
IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
*
*
* At this point, all the intervals have converged
DO 110 I = IFIRST, ILAST
K = 2*I
II = I - OFFSET
* All intervals marked by '0' have been refined.
IF( IWORK( K-1 ).EQ.0 ) THEN
W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
WERR( II ) = WORK( K ) - W( II )
END IF
110 CONTINUE
*
DO 111 I = IFIRST+1, ILAST
K = 2*I
II = I - OFFSET
WGAP( II-1 ) = MAX( ZERO,
$ W(II) - WERR (II) - W( II-1 ) - WERR( II-1 ))
111 CONTINUE
RETURN
*
* End of DLARRB
*
END
*> \brief \b DLARRC computes the number of eigenvalues of the symmetric tridiagonal matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRC( JOBT, N, VL, VU, D, E, PIVMIN,
* EIGCNT, LCNT, RCNT, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBT
* INTEGER EIGCNT, INFO, LCNT, N, RCNT
* DOUBLE PRECISION PIVMIN, VL, VU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Find the number of eigenvalues of the symmetric tridiagonal matrix T
*> that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T
*> if JOBT = 'L'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBT
*> \verbatim
*> JOBT is CHARACTER*1
*> = 'T': Compute Sturm count for matrix T.
*> = 'L': Compute Sturm count for matrix L D L^T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> The lower bound for the eigenvalues.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> The upper bound for the eigenvalues.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> JOBT = 'T': The N diagonal elements of the tridiagonal matrix T.
*> JOBT = 'L': The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> JOBT = 'T': The N-1 offdiagonal elements of the matrix T.
*> JOBT = 'L': The N-1 offdiagonal elements of the matrix L.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[out] EIGCNT
*> \verbatim
*> EIGCNT is INTEGER
*> The number of eigenvalues of the symmetric tridiagonal matrix T
*> that are in the interval (VL,VU]
*> \endverbatim
*>
*> \param[out] LCNT
*> \verbatim
*> LCNT is INTEGER
*> \endverbatim
*>
*> \param[out] RCNT
*> \verbatim
*> RCNT is INTEGER
*> The left and right negcounts of the interval.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrc
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRC( JOBT, N, VL, VU, D, E, PIVMIN,
$ EIGCNT, LCNT, RCNT, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBT
INTEGER EIGCNT, INFO, LCNT, N, RCNT
DOUBLE PRECISION PIVMIN, VL, VU
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I
LOGICAL MATT
DOUBLE PRECISION LPIVOT, RPIVOT, SL, SU, TMP, TMP2
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
INFO = 0
LCNT = 0
RCNT = 0
EIGCNT = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
MATT = LSAME( JOBT, 'T' )
IF (MATT) THEN
* Sturm sequence count on T
LPIVOT = D( 1 ) - VL
RPIVOT = D( 1 ) - VU
IF( LPIVOT.LE.ZERO ) THEN
LCNT = LCNT + 1
ENDIF
IF( RPIVOT.LE.ZERO ) THEN
RCNT = RCNT + 1
ENDIF
DO 10 I = 1, N-1
TMP = E(I)**2
LPIVOT = ( D( I+1 )-VL ) - TMP/LPIVOT
RPIVOT = ( D( I+1 )-VU ) - TMP/RPIVOT
IF( LPIVOT.LE.ZERO ) THEN
LCNT = LCNT + 1
ENDIF
IF( RPIVOT.LE.ZERO ) THEN
RCNT = RCNT + 1
ENDIF
10 CONTINUE
ELSE
* Sturm sequence count on L D L^T
SL = -VL
SU = -VU
DO 20 I = 1, N - 1
LPIVOT = D( I ) + SL
RPIVOT = D( I ) + SU
IF( LPIVOT.LE.ZERO ) THEN
LCNT = LCNT + 1
ENDIF
IF( RPIVOT.LE.ZERO ) THEN
RCNT = RCNT + 1
ENDIF
TMP = E(I) * D(I) * E(I)
*
TMP2 = TMP / LPIVOT
IF( TMP2.EQ.ZERO ) THEN
SL = TMP - VL
ELSE
SL = SL*TMP2 - VL
END IF
*
TMP2 = TMP / RPIVOT
IF( TMP2.EQ.ZERO ) THEN
SU = TMP - VU
ELSE
SU = SU*TMP2 - VU
END IF
20 CONTINUE
LPIVOT = D( N ) + SL
RPIVOT = D( N ) + SU
IF( LPIVOT.LE.ZERO ) THEN
LCNT = LCNT + 1
ENDIF
IF( RPIVOT.LE.ZERO ) THEN
RCNT = RCNT + 1
ENDIF
ENDIF
EIGCNT = RCNT - LCNT
RETURN
*
* End of DLARRC
*
END
*> \brief \b DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS,
* RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
* M, W, WERR, WL, WU, IBLOCK, INDEXW,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER ORDER, RANGE
* INTEGER IL, INFO, IU, M, N, NSPLIT
* DOUBLE PRECISION PIVMIN, RELTOL, VL, VU, WL, WU
* ..
* .. Array Arguments ..
* INTEGER IBLOCK( * ), INDEXW( * ),
* $ ISPLIT( * ), IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), E2( * ),
* $ GERS( * ), W( * ), WERR( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARRD computes the eigenvalues of a symmetric tridiagonal
*> matrix T to suitable accuracy. This is an auxiliary code to be
*> called from DSTEMR.
*> The user may ask for all eigenvalues, all eigenvalues
*> in the half-open interval (VL, VU], or the IL-th through IU-th
*> eigenvalues.
*>
*> To avoid overflow, the matrix must be scaled so that its
*> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
*> accuracy, it should not be much smaller than that.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': ("All") all eigenvalues will be found.
*> = 'V': ("Value") all eigenvalues in the half-open interval
*> (VL, VU] will be found.
*> = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
*> entire matrix) will be found.
*> \endverbatim
*>
*> \param[in] ORDER
*> \verbatim
*> ORDER is CHARACTER*1
*> = 'B': ("By Block") the eigenvalues will be grouped by
*> split-off block (see IBLOCK, ISPLIT) and
*> ordered from smallest to largest within
*> the block.
*> = 'E': ("Entire matrix")
*> the eigenvalues for the entire matrix
*> will be ordered from smallest to
*> largest.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the tridiagonal matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. Eigenvalues less than or equal
*> to VL, or greater than VU, will not be returned. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. Eigenvalues less than or equal
*> to VL, or greater than VU, will not be returned. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] GERS
*> \verbatim
*> GERS is DOUBLE PRECISION array, dimension (2*N)
*> The N Gerschgorin intervals (the i-th Gerschgorin interval
*> is (GERS(2*i-1), GERS(2*i)).
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*> RELTOL is DOUBLE PRECISION
*> The minimum relative width of an interval. When an interval
*> is narrower than RELTOL times the larger (in
*> magnitude) endpoint, then it is considered to be
*> sufficiently small, i.e., converged. Note: this should
*> always be at least radix*machine epsilon.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*> E2 is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot allowed in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] NSPLIT
*> \verbatim
*> NSPLIT is INTEGER
*> The number of diagonal blocks in the matrix T.
*> 1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into submatrices.
*> The first submatrix consists of rows/columns 1 to ISPLIT(1),
*> the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*> etc., and the NSPLIT-th consists of rows/columns
*> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*> (Only the first NSPLIT elements will actually be used, but
*> since the user cannot know a priori what value NSPLIT will
*> have, N words must be reserved for ISPLIT.)
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The actual number of eigenvalues found. 0 <= M <= N.
*> (See also the description of INFO=2,3.)
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> On exit, the first M elements of W will contain the
*> eigenvalue approximations. DLARRD computes an interval
*> I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
*> approximation is given as the interval midpoint
*> W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
*> WERR(j) = abs( a_j - b_j)/2
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION array, dimension (N)
*> The error bound on the corresponding eigenvalue approximation
*> in W.
*> \endverbatim
*>
*> \param[out] WL
*> \verbatim
*> WL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] WU
*> \verbatim
*> WU is DOUBLE PRECISION
*> The interval (WL, WU] contains all the wanted eigenvalues.
*> If RANGE='V', then WL=VL and WU=VU.
*> If RANGE='A', then WL and WU are the global Gerschgorin bounds
*> on the spectrum.
*> If RANGE='I', then WL and WU are computed by DLAEBZ from the
*> index range specified.
*> \endverbatim
*>
*> \param[out] IBLOCK
*> \verbatim
*> IBLOCK is INTEGER array, dimension (N)
*> At each row/column j where E(j) is zero or small, the
*> matrix T is considered to split into a block diagonal
*> matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
*> block (from 1 to the number of blocks) the eigenvalue W(i)
*> belongs. (DLARRD may use the remaining N-M elements as
*> workspace.)
*> \endverbatim
*>
*> \param[out] INDEXW
*> \verbatim
*> INDEXW is INTEGER array, dimension (N)
*> The indices of the eigenvalues within each block (submatrix);
*> for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
*> i-th eigenvalue W(i) is the j-th eigenvalue in block k.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: some or all of the eigenvalues failed to converge or
*> were not computed:
*> =1 or 3: Bisection failed to converge for some
*> eigenvalues; these eigenvalues are flagged by a
*> negative block number. The effect is that the
*> eigenvalues may not be as accurate as the
*> absolute and relative tolerances. This is
*> generally caused by unexpectedly inaccurate
*> arithmetic.
*> =2 or 3: RANGE='I' only: Not all of the eigenvalues
*> IL:IU were found.
*> Effect: M < IU+1-IL
*> Cause: non-monotonic arithmetic, causing the
*> Sturm sequence to be non-monotonic.
*> Cure: recalculate, using RANGE='A', and pick
*> out eigenvalues IL:IU. In some cases,
*> increasing the PARAMETER "FUDGE" may
*> make things work.
*> = 4: RANGE='I', and the Gershgorin interval
*> initially used was too small. No eigenvalues
*> were computed.
*> Probable cause: your machine has sloppy
*> floating-point arithmetic.
*> Cure: Increase the PARAMETER "FUDGE",
*> recompile, and try again.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> FUDGE DOUBLE PRECISION, default = 2
*> A "fudge factor" to widen the Gershgorin intervals. Ideally,
*> a value of 1 should work, but on machines with sloppy
*> arithmetic, this needs to be larger. The default for
*> publicly released versions should be large enough to handle
*> the worst machine around. Note that this has no effect
*> on accuracy of the solution.
*> \endverbatim
*>
*> \par Contributors:
* ==================
*>
*> W. Kahan, University of California, Berkeley, USA \n
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA \n
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrd
*
* =====================================================================
SUBROUTINE DLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS,
$ RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
$ M, W, WERR, WL, WU, IBLOCK, INDEXW,
$ WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER ORDER, RANGE
INTEGER IL, INFO, IU, M, N, NSPLIT
DOUBLE PRECISION PIVMIN, RELTOL, VL, VU, WL, WU
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), INDEXW( * ),
$ ISPLIT( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), E2( * ),
$ GERS( * ), W( * ), WERR( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, HALF, FUDGE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, HALF = ONE/TWO,
$ FUDGE = TWO )
INTEGER ALLRNG, VALRNG, INDRNG
PARAMETER ( ALLRNG = 1, VALRNG = 2, INDRNG = 3 )
* ..
* .. Local Scalars ..
LOGICAL NCNVRG, TOOFEW
INTEGER I, IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO,
$ IM, IN, IOFF, IOUT, IRANGE, ITMAX, ITMP1,
$ ITMP2, IW, IWOFF, J, JBLK, JDISC, JE, JEE, NB,
$ NWL, NWU
DOUBLE PRECISION ATOLI, EPS, GL, GU, RTOLI, TMP1, TMP2,
$ TNORM, UFLOW, WKILL, WLU, WUL
* ..
* .. Local Arrays ..
INTEGER IDUMMA( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, ILAENV, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLAEBZ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
M = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
* Decode RANGE
*
IF( LSAME( RANGE, 'A' ) ) THEN
IRANGE = ALLRNG
ELSE IF( LSAME( RANGE, 'V' ) ) THEN
IRANGE = VALRNG
ELSE IF( LSAME( RANGE, 'I' ) ) THEN
IRANGE = INDRNG
ELSE
IRANGE = 0
END IF
*
* Check for Errors
*
IF( IRANGE.LE.0 ) THEN
INFO = -1
ELSE IF( .NOT.(LSAME(ORDER,'B').OR.LSAME(ORDER,'E')) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( IRANGE.EQ.VALRNG ) THEN
IF( VL.GE.VU )
$ INFO = -5
ELSE IF( IRANGE.EQ.INDRNG .AND.
$ ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN
INFO = -6
ELSE IF( IRANGE.EQ.INDRNG .AND.
$ ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN
INFO = -7
END IF
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
* Initialize error flags
NCNVRG = .FALSE.
TOOFEW = .FALSE.
* Simplification:
IF( IRANGE.EQ.INDRNG .AND. IL.EQ.1 .AND. IU.EQ.N ) IRANGE = 1
* Get machine constants
EPS = DLAMCH( 'P' )
UFLOW = DLAMCH( 'U' )
* Special Case when N=1
* Treat case of 1x1 matrix for quick return
IF( N.EQ.1 ) THEN
IF( (IRANGE.EQ.ALLRNG).OR.
$ ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR.
$ ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN
M = 1
W(1) = D(1)
* The computation error of the eigenvalue is zero
WERR(1) = ZERO
IBLOCK( 1 ) = 1
INDEXW( 1 ) = 1
ENDIF
RETURN
END IF
* NB is the minimum vector length for vector bisection, or 0
* if only scalar is to be done.
NB = ILAENV( 1, 'DSTEBZ', ' ', N, -1, -1, -1 )
IF( NB.LE.1 ) NB = 0
* Find global spectral radius
GL = D(1)
GU = D(1)
DO 5 I = 1,N
GL = MIN( GL, GERS( 2*I - 1))
GU = MAX( GU, GERS(2*I) )
5 CONTINUE
* Compute global Gerschgorin bounds and spectral diameter
TNORM = MAX( ABS( GL ), ABS( GU ) )
GL = GL - FUDGE*TNORM*EPS*N - FUDGE*TWO*PIVMIN
GU = GU + FUDGE*TNORM*EPS*N + FUDGE*TWO*PIVMIN
* [JAN/28/2009] remove the line below since SPDIAM variable not use
* SPDIAM = GU - GL
* Input arguments for DLAEBZ:
* The relative tolerance. An interval (a,b] lies within
* "relative tolerance" if b-a < RELTOL*max(|a|,|b|),
RTOLI = RELTOL
* Set the absolute tolerance for interval convergence to zero to force
* interval convergence based on relative size of the interval.
* This is dangerous because intervals might not converge when RELTOL is
* small. But at least a very small number should be selected so that for
* strongly graded matrices, the code can get relatively accurate
* eigenvalues.
ATOLI = FUDGE*TWO*UFLOW + FUDGE*TWO*PIVMIN
IF( IRANGE.EQ.INDRNG ) THEN
* RANGE='I': Compute an interval containing eigenvalues
* IL through IU. The initial interval [GL,GU] from the global
* Gerschgorin bounds GL and GU is refined by DLAEBZ.
ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
WORK( N+1 ) = GL
WORK( N+2 ) = GL
WORK( N+3 ) = GU
WORK( N+4 ) = GU
WORK( N+5 ) = GL
WORK( N+6 ) = GU
IWORK( 1 ) = -1
IWORK( 2 ) = -1
IWORK( 3 ) = N + 1
IWORK( 4 ) = N + 1
IWORK( 5 ) = IL - 1
IWORK( 6 ) = IU
*
CALL DLAEBZ( 3, ITMAX, N, 2, 2, NB, ATOLI, RTOLI, PIVMIN,
$ D, E, E2, IWORK( 5 ), WORK( N+1 ), WORK( N+5 ), IOUT,
$ IWORK, W, IBLOCK, IINFO )
IF( IINFO .NE. 0 ) THEN
INFO = IINFO
RETURN
END IF
* On exit, output intervals may not be ordered by ascending negcount
IF( IWORK( 6 ).EQ.IU ) THEN
WL = WORK( N+1 )
WLU = WORK( N+3 )
NWL = IWORK( 1 )
WU = WORK( N+4 )
WUL = WORK( N+2 )
NWU = IWORK( 4 )
ELSE
WL = WORK( N+2 )
WLU = WORK( N+4 )
NWL = IWORK( 2 )
WU = WORK( N+3 )
WUL = WORK( N+1 )
NWU = IWORK( 3 )
END IF
* On exit, the interval [WL, WLU] contains a value with negcount NWL,
* and [WUL, WU] contains a value with negcount NWU.
IF( NWL.LT.0 .OR. NWL.GE.N .OR. NWU.LT.1 .OR. NWU.GT.N ) THEN
INFO = 4
RETURN
END IF
ELSEIF( IRANGE.EQ.VALRNG ) THEN
WL = VL
WU = VU
ELSEIF( IRANGE.EQ.ALLRNG ) THEN
WL = GL
WU = GU
ENDIF
* Find Eigenvalues -- Loop Over blocks and recompute NWL and NWU.
* NWL accumulates the number of eigenvalues .le. WL,
* NWU accumulates the number of eigenvalues .le. WU
M = 0
IEND = 0
INFO = 0
NWL = 0
NWU = 0
*
DO 70 JBLK = 1, NSPLIT
IOFF = IEND
IBEGIN = IOFF + 1
IEND = ISPLIT( JBLK )
IN = IEND - IOFF
*
IF( IN.EQ.1 ) THEN
* 1x1 block
IF( WL.GE.D( IBEGIN )-PIVMIN )
$ NWL = NWL + 1
IF( WU.GE.D( IBEGIN )-PIVMIN )
$ NWU = NWU + 1
IF( IRANGE.EQ.ALLRNG .OR.
$ ( WL.LT.D( IBEGIN )-PIVMIN
$ .AND. WU.GE. D( IBEGIN )-PIVMIN ) ) THEN
M = M + 1
W( M ) = D( IBEGIN )
WERR(M) = ZERO
* The gap for a single block doesn't matter for the later
* algorithm and is assigned an arbitrary large value
IBLOCK( M ) = JBLK
INDEXW( M ) = 1
END IF
* Disabled 2x2 case because of a failure on the following matrix
* RANGE = 'I', IL = IU = 4
* Original Tridiagonal, d = [
* -0.150102010615740E+00
* -0.849897989384260E+00
* -0.128208148052635E-15
* 0.128257718286320E-15
* ];
* e = [
* -0.357171383266986E+00
* -0.180411241501588E-15
* -0.175152352710251E-15
* ];
*
* ELSE IF( IN.EQ.2 ) THEN
** 2x2 block
* DISC = SQRT( (HALF*(D(IBEGIN)-D(IEND)))**2 + E(IBEGIN)**2 )
* TMP1 = HALF*(D(IBEGIN)+D(IEND))
* L1 = TMP1 - DISC
* IF( WL.GE. L1-PIVMIN )
* $ NWL = NWL + 1
* IF( WU.GE. L1-PIVMIN )
* $ NWU = NWU + 1
* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L1-PIVMIN .AND. WU.GE.
* $ L1-PIVMIN ) ) THEN
* M = M + 1
* W( M ) = L1
** The uncertainty of eigenvalues of a 2x2 matrix is very small
* WERR( M ) = EPS * ABS( W( M ) ) * TWO
* IBLOCK( M ) = JBLK
* INDEXW( M ) = 1
* ENDIF
* L2 = TMP1 + DISC
* IF( WL.GE. L2-PIVMIN )
* $ NWL = NWL + 1
* IF( WU.GE. L2-PIVMIN )
* $ NWU = NWU + 1
* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L2-PIVMIN .AND. WU.GE.
* $ L2-PIVMIN ) ) THEN
* M = M + 1
* W( M ) = L2
** The uncertainty of eigenvalues of a 2x2 matrix is very small
* WERR( M ) = EPS * ABS( W( M ) ) * TWO
* IBLOCK( M ) = JBLK
* INDEXW( M ) = 2
* ENDIF
ELSE
* General Case - block of size IN >= 2
* Compute local Gerschgorin interval and use it as the initial
* interval for DLAEBZ
GU = D( IBEGIN )
GL = D( IBEGIN )
TMP1 = ZERO
DO 40 J = IBEGIN, IEND
GL = MIN( GL, GERS( 2*J - 1))
GU = MAX( GU, GERS(2*J) )
40 CONTINUE
* [JAN/28/2009]
* change SPDIAM by TNORM in lines 2 and 3 thereafter
* line 1: remove computation of SPDIAM (not useful anymore)
* SPDIAM = GU - GL
* GL = GL - FUDGE*SPDIAM*EPS*IN - FUDGE*PIVMIN
* GU = GU + FUDGE*SPDIAM*EPS*IN + FUDGE*PIVMIN
GL = GL - FUDGE*TNORM*EPS*IN - FUDGE*PIVMIN
GU = GU + FUDGE*TNORM*EPS*IN + FUDGE*PIVMIN
*
IF( IRANGE.GT.1 ) THEN
IF( GU.LT.WL ) THEN
* the local block contains none of the wanted eigenvalues
NWL = NWL + IN
NWU = NWU + IN
GO TO 70
END IF
* refine search interval if possible, only range (WL,WU] matters
GL = MAX( GL, WL )
GU = MIN( GU, WU )
IF( GL.GE.GU )
$ GO TO 70
END IF
* Find negcount of initial interval boundaries GL and GU
WORK( N+1 ) = GL
WORK( N+IN+1 ) = GU
CALL DLAEBZ( 1, 0, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
$ D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
$ IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IM,
$ IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
IF( IINFO .NE. 0 ) THEN
INFO = IINFO
RETURN
END IF
*
NWL = NWL + IWORK( 1 )
NWU = NWU + IWORK( IN+1 )
IWOFF = M - IWORK( 1 )
* Compute Eigenvalues
ITMAX = INT( ( LOG( GU-GL+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
CALL DLAEBZ( 2, ITMAX, IN, IN, 1, NB, ATOLI, RTOLI,
$ PIVMIN,
$ D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
$ IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IOUT,
$ IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
IF( IINFO .NE. 0 ) THEN
INFO = IINFO
RETURN
END IF
*
* Copy eigenvalues into W and IBLOCK
* Use -JBLK for block number for unconverged eigenvalues.
* Loop over the number of output intervals from DLAEBZ
DO 60 J = 1, IOUT
* eigenvalue approximation is middle point of interval
TMP1 = HALF*( WORK( J+N )+WORK( J+IN+N ) )
* semi length of error interval
TMP2 = HALF*ABS( WORK( J+N )-WORK( J+IN+N ) )
IF( J.GT.IOUT-IINFO ) THEN
* Flag non-convergence.
NCNVRG = .TRUE.
IB = -JBLK
ELSE
IB = JBLK
END IF
DO 50 JE = IWORK( J ) + 1 + IWOFF,
$ IWORK( J+IN ) + IWOFF
W( JE ) = TMP1
WERR( JE ) = TMP2
INDEXW( JE ) = JE - IWOFF
IBLOCK( JE ) = IB
50 CONTINUE
60 CONTINUE
*
M = M + IM
END IF
70 CONTINUE
* If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
* If NWL+1 < IL or NWU > IU, discard extra eigenvalues.
IF( IRANGE.EQ.INDRNG ) THEN
IDISCL = IL - 1 - NWL
IDISCU = NWU - IU
*
IF( IDISCL.GT.0 ) THEN
IM = 0
DO 80 JE = 1, M
* Remove some of the smallest eigenvalues from the left so that
* at the end IDISCL =0. Move all eigenvalues up to the left.
IF( W( JE ).LE.WLU .AND. IDISCL.GT.0 ) THEN
IDISCL = IDISCL - 1
ELSE
IM = IM + 1
W( IM ) = W( JE )
WERR( IM ) = WERR( JE )
INDEXW( IM ) = INDEXW( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
80 CONTINUE
M = IM
END IF
IF( IDISCU.GT.0 ) THEN
* Remove some of the largest eigenvalues from the right so that
* at the end IDISCU =0. Move all eigenvalues up to the left.
IM=M+1
DO 81 JE = M, 1, -1
IF( W( JE ).GE.WUL .AND. IDISCU.GT.0 ) THEN
IDISCU = IDISCU - 1
ELSE
IM = IM - 1
W( IM ) = W( JE )
WERR( IM ) = WERR( JE )
INDEXW( IM ) = INDEXW( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
81 CONTINUE
JEE = 0
DO 82 JE = IM, M
JEE = JEE + 1
W( JEE ) = W( JE )
WERR( JEE ) = WERR( JE )
INDEXW( JEE ) = INDEXW( JE )
IBLOCK( JEE ) = IBLOCK( JE )
82 CONTINUE
M = M-IM+1
END IF
IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
* Code to deal with effects of bad arithmetic. (If N(w) is
* monotone non-decreasing, this should never happen.)
* Some low eigenvalues to be discarded are not in (WL,WLU],
* or high eigenvalues to be discarded are not in (WUL,WU]
* so just kill off the smallest IDISCL/largest IDISCU
* eigenvalues, by marking the corresponding IBLOCK = 0
IF( IDISCL.GT.0 ) THEN
WKILL = WU
DO 100 JDISC = 1, IDISCL
IW = 0
DO 90 JE = 1, M
IF( IBLOCK( JE ).NE.0 .AND.
$ ( W( JE ).LT.WKILL .OR. IW.EQ.0 ) ) THEN
IW = JE
WKILL = W( JE )
END IF
90 CONTINUE
IBLOCK( IW ) = 0
100 CONTINUE
END IF
IF( IDISCU.GT.0 ) THEN
WKILL = WL
DO 120 JDISC = 1, IDISCU
IW = 0
DO 110 JE = 1, M
IF( IBLOCK( JE ).NE.0 .AND.
$ ( W( JE ).GE.WKILL .OR. IW.EQ.0 ) ) THEN
IW = JE
WKILL = W( JE )
END IF
110 CONTINUE
IBLOCK( IW ) = 0
120 CONTINUE
END IF
* Now erase all eigenvalues with IBLOCK set to zero
IM = 0
DO 130 JE = 1, M
IF( IBLOCK( JE ).NE.0 ) THEN
IM = IM + 1
W( IM ) = W( JE )
WERR( IM ) = WERR( JE )
INDEXW( IM ) = INDEXW( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
130 CONTINUE
M = IM
END IF
IF( IDISCL.LT.0 .OR. IDISCU.LT.0 ) THEN
TOOFEW = .TRUE.
END IF
END IF
*
IF(( IRANGE.EQ.ALLRNG .AND. M.NE.N ).OR.
$ ( IRANGE.EQ.INDRNG .AND. M.NE.IU-IL+1 ) ) THEN
TOOFEW = .TRUE.
END IF
* If ORDER='B', do nothing the eigenvalues are already sorted by
* block.
* If ORDER='E', sort the eigenvalues from smallest to largest
IF( LSAME(ORDER,'E') .AND. NSPLIT.GT.1 ) THEN
DO 150 JE = 1, M - 1
IE = 0
TMP1 = W( JE )
DO 140 J = JE + 1, M
IF( W( J ).LT.TMP1 ) THEN
IE = J
TMP1 = W( J )
END IF
140 CONTINUE
IF( IE.NE.0 ) THEN
TMP2 = WERR( IE )
ITMP1 = IBLOCK( IE )
ITMP2 = INDEXW( IE )
W( IE ) = W( JE )
WERR( IE ) = WERR( JE )
IBLOCK( IE ) = IBLOCK( JE )
INDEXW( IE ) = INDEXW( JE )
W( JE ) = TMP1
WERR( JE ) = TMP2
IBLOCK( JE ) = ITMP1
INDEXW( JE ) = ITMP2
END IF
150 CONTINUE
END IF
*
INFO = 0
IF( NCNVRG )
$ INFO = INFO + 1
IF( TOOFEW )
$ INFO = INFO + 2
RETURN
*
* End of DLARRD
*
END
*> \brief \b DLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarre.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarre.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarre.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2,
* RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M,
* W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER RANGE
* INTEGER IL, INFO, IU, M, N, NSPLIT
* DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ),
* $ INDEXW( * )
* DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ),
* $ W( * ),WERR( * ), WGAP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> To find the desired eigenvalues of a given real symmetric
*> tridiagonal matrix T, DLARRE sets any "small" off-diagonal
*> elements to zero, and for each unreduced block T_i, it finds
*> (a) a suitable shift at one end of the block's spectrum,
*> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
*> (c) eigenvalues of each L_i D_i L_i^T.
*> The representations and eigenvalues found are then used by
*> DSTEMR to compute the eigenvectors of T.
*> The accuracy varies depending on whether bisection is used to
*> find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
*> compute all and then discard any unwanted one.
*> As an added benefit, DLARRE also outputs the n
*> Gerschgorin intervals for the matrices L_i D_i L_i^T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': ("All") all eigenvalues will be found.
*> = 'V': ("Value") all eigenvalues in the half-open interval
*> (VL, VU] will be found.
*> = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
*> entire matrix) will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound for the eigenvalues.
*> Eigenvalues less than or equal to VL, or greater than VU,
*> will not be returned. VL < VU.
*> If RANGE='I' or ='A', DLARRE computes bounds on the desired
*> part of the spectrum.
*> \endverbatim
*>
*> \param[in,out] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound for the eigenvalues.
*> Eigenvalues less than or equal to VL, or greater than VU,
*> will not be returned. VL < VU.
*> If RANGE='I' or ='A', DLARRE computes bounds on the desired
*> part of the spectrum.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the N diagonal elements of the tridiagonal
*> matrix T.
*> On exit, the N diagonal elements of the diagonal
*> matrices D_i.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, the first (N-1) entries contain the subdiagonal
*> elements of the tridiagonal matrix T; E(N) need not be set.
*> On exit, E contains the subdiagonal elements of the unit
*> bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
*> 1 <= I <= NSPLIT, contain the base points sigma_i on output.
*> \endverbatim
*>
*> \param[in,out] E2
*> \verbatim
*> E2 is DOUBLE PRECISION array, dimension (N)
*> On entry, the first (N-1) entries contain the SQUARES of the
*> subdiagonal elements of the tridiagonal matrix T;
*> E2(N) need not be set.
*> On exit, the entries E2( ISPLIT( I ) ),
*> 1 <= I <= NSPLIT, have been set to zero
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*> RTOL1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*> RTOL2 is DOUBLE PRECISION
*> Parameters for bisection.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> \endverbatim
*>
*> \param[in] SPLTOL
*> \verbatim
*> SPLTOL is DOUBLE PRECISION
*> The threshold for splitting.
*> \endverbatim
*>
*> \param[out] NSPLIT
*> \verbatim
*> NSPLIT is INTEGER
*> The number of blocks T splits into. 1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[out] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into blocks.
*> The first block consists of rows/columns 1 to ISPLIT(1),
*> the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*> etc., and the NSPLIT-th consists of rows/columns
*> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues (of all L_i D_i L_i^T)
*> found.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the eigenvalues. The
*> eigenvalues of each of the blocks, L_i D_i L_i^T, are
*> sorted in ascending order ( DLARRE may use the
*> remaining N-M elements as workspace).
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION array, dimension (N)
*> The error bound on the corresponding eigenvalue in W.
*> \endverbatim
*>
*> \param[out] WGAP
*> \verbatim
*> WGAP is DOUBLE PRECISION array, dimension (N)
*> The separation from the right neighbor eigenvalue in W.
*> The gap is only with respect to the eigenvalues of the same block
*> as each block has its own representation tree.
*> Exception: at the right end of a block we store the left gap
*> \endverbatim
*>
*> \param[out] IBLOCK
*> \verbatim
*> IBLOCK is INTEGER array, dimension (N)
*> The indices of the blocks (submatrices) associated with the
*> corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
*> W(i) belongs to the first block from the top, =2 if W(i)
*> belongs to the second block, etc.
*> \endverbatim
*>
*> \param[out] INDEXW
*> \verbatim
*> INDEXW is INTEGER array, dimension (N)
*> The indices of the eigenvalues within each block (submatrix);
*> for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
*> i-th eigenvalue W(i) is the 10-th eigenvalue in block 2
*> \endverbatim
*>
*> \param[out] GERS
*> \verbatim
*> GERS is DOUBLE PRECISION array, dimension (2*N)
*> The N Gerschgorin intervals (the i-th Gerschgorin interval
*> is (GERS(2*i-1), GERS(2*i)).
*> \endverbatim
*>
*> \param[out] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (6*N)
*> Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> Workspace.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: A problem occurred in DLARRE.
*> < 0: One of the called subroutines signaled an internal problem.
*> Needs inspection of the corresponding parameter IINFO
*> for further information.
*>
*> =-1: Problem in DLARRD.
*> = 2: No base representation could be found in MAXTRY iterations.
*> Increasing MAXTRY and recompilation might be a remedy.
*> =-3: Problem in DLARRB when computing the refined root
*> representation for DLASQ2.
*> =-4: Problem in DLARRB when preforming bisection on the
*> desired part of the spectrum.
*> =-5: Problem in DLASQ2.
*> =-6: Problem in DLASQ2.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larre
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The base representations are required to suffer very little
*> element growth and consequently define all their eigenvalues to
*> high relative accuracy.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA \n
*>
* =====================================================================
SUBROUTINE DLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2,
$ RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M,
$ W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN,
$ WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER RANGE
INTEGER IL, INFO, IU, M, N, NSPLIT
DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ),
$ INDEXW( * )
DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ),
$ W( * ),WERR( * ), WGAP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION FAC, FOUR, FOURTH, FUDGE, HALF, HNDRD,
$ MAXGROWTH, ONE, PERT, TWO, ZERO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, FOUR=4.0D0,
$ HNDRD = 100.0D0,
$ PERT = 8.0D0,
$ HALF = ONE/TWO, FOURTH = ONE/FOUR, FAC= HALF,
$ MAXGROWTH = 64.0D0, FUDGE = 2.0D0 )
INTEGER MAXTRY, ALLRNG, INDRNG, VALRNG
PARAMETER ( MAXTRY = 6, ALLRNG = 1, INDRNG = 2,
$ VALRNG = 3 )
* ..
* .. Local Scalars ..
LOGICAL FORCEB, NOREP, USEDQD
INTEGER CNT, CNT1, CNT2, I, IBEGIN, IDUM, IEND, IINFO,
$ IN, INDL, INDU, IRANGE, J, JBLK, MB, MM,
$ WBEGIN, WEND
DOUBLE PRECISION AVGAP, BSRTOL, CLWDTH, DMAX, DPIVOT, EABS,
$ EMAX, EOLD, EPS, GL, GU, ISLEFT, ISRGHT, RTL,
$ RTOL, S1, S2, SAFMIN, SGNDEF, SIGMA, SPDIAM,
$ TAU, TMP, TMP1
* ..
* .. Local Arrays ..
INTEGER ISEED( 4 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, LSAME
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLARNV, DLARRA, DLARRB, DLARRC,
$ DLARRD,
$ DLASQ2, DLARRK
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NSPLIT = 0
M = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
* Decode RANGE
*
IF( LSAME( RANGE, 'A' ) ) THEN
IRANGE = ALLRNG
ELSE IF( LSAME( RANGE, 'V' ) ) THEN
IRANGE = VALRNG
ELSE IF( LSAME( RANGE, 'I' ) ) THEN
IRANGE = INDRNG
END IF
* Get machine constants
SAFMIN = DLAMCH( 'S' )
EPS = DLAMCH( 'P' )
* Set parameters
RTL = SQRT(EPS)
BSRTOL = SQRT(EPS)
* Treat case of 1x1 matrix for quick return
IF( N.EQ.1 ) THEN
IF( (IRANGE.EQ.ALLRNG).OR.
$ ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR.
$ ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN
M = 1
W(1) = D(1)
* The computation error of the eigenvalue is zero
WERR(1) = ZERO
WGAP(1) = ZERO
IBLOCK( 1 ) = 1
INDEXW( 1 ) = 1
GERS(1) = D( 1 )
GERS(2) = D( 1 )
ENDIF
* store the shift for the initial RRR, which is zero in this case
E(1) = ZERO
RETURN
END IF
* General case: tridiagonal matrix of order > 1
*
* Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter.
* Compute maximum off-diagonal entry and pivmin.
GL = D(1)
GU = D(1)
EOLD = ZERO
EMAX = ZERO
E(N) = ZERO
DO 5 I = 1,N
WERR(I) = ZERO
WGAP(I) = ZERO
EABS = ABS( E(I) )
IF( EABS .GE. EMAX ) THEN
EMAX = EABS
END IF
TMP1 = EABS + EOLD
GERS( 2*I-1) = D(I) - TMP1
GL = MIN( GL, GERS( 2*I - 1))
GERS( 2*I ) = D(I) + TMP1
GU = MAX( GU, GERS(2*I) )
EOLD = EABS
5 CONTINUE
* The minimum pivot allowed in the Sturm sequence for T
PIVMIN = SAFMIN * MAX( ONE, EMAX**2 )
* Compute spectral diameter. The Gerschgorin bounds give an
* estimate that is wrong by at most a factor of SQRT(2)
SPDIAM = GU - GL
* Compute splitting points
CALL DLARRA( N, D, E, E2, SPLTOL, SPDIAM,
$ NSPLIT, ISPLIT, IINFO )
* Can force use of bisection instead of faster DQDS.
* Option left in the code for future multisection work.
FORCEB = .FALSE.
* Initialize USEDQD, DQDS should be used for ALLRNG unless someone
* explicitly wants bisection.
USEDQD = (( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB))
IF( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ) THEN
* Set interval [VL,VU] that contains all eigenvalues
VL = GL
VU = GU
ELSE
* We call DLARRD to find crude approximations to the eigenvalues
* in the desired range. In case IRANGE = INDRNG, we also obtain the
* interval (VL,VU] that contains all the wanted eigenvalues.
* An interval [LEFT,RIGHT] has converged if
* RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT))
* DLARRD needs a WORK of size 4*N, IWORK of size 3*N
CALL DLARRD( RANGE, 'B', N, VL, VU, IL, IU, GERS,
$ BSRTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
$ MM, W, WERR, VL, VU, IBLOCK, INDEXW,
$ WORK, IWORK, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = -1
RETURN
ENDIF
* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0
DO 14 I = MM+1,N
W( I ) = ZERO
WERR( I ) = ZERO
IBLOCK( I ) = 0
INDEXW( I ) = 0
14 CONTINUE
END IF
***
* Loop over unreduced blocks
IBEGIN = 1
WBEGIN = 1
DO 170 JBLK = 1, NSPLIT
IEND = ISPLIT( JBLK )
IN = IEND - IBEGIN + 1
* 1 X 1 block
IF( IN.EQ.1 ) THEN
IF( (IRANGE.EQ.ALLRNG).OR.( (IRANGE.EQ.VALRNG).AND.
$ ( D( IBEGIN ).GT.VL ).AND.( D( IBEGIN ).LE.VU ) )
$ .OR. ( (IRANGE.EQ.INDRNG).AND.(IBLOCK(WBEGIN).EQ.JBLK))
$ ) THEN
M = M + 1
W( M ) = D( IBEGIN )
WERR(M) = ZERO
* The gap for a single block doesn't matter for the later
* algorithm and is assigned an arbitrary large value
WGAP(M) = ZERO
IBLOCK( M ) = JBLK
INDEXW( M ) = 1
WBEGIN = WBEGIN + 1
ENDIF
* E( IEND ) holds the shift for the initial RRR
E( IEND ) = ZERO
IBEGIN = IEND + 1
GO TO 170
END IF
*
* Blocks of size larger than 1x1
*
* E( IEND ) will hold the shift for the initial RRR, for now set it =0
E( IEND ) = ZERO
*
* Find local outer bounds GL,GU for the block
GL = D(IBEGIN)
GU = D(IBEGIN)
DO 15 I = IBEGIN , IEND
GL = MIN( GERS( 2*I-1 ), GL )
GU = MAX( GERS( 2*I ), GU )
15 CONTINUE
SPDIAM = GU - GL
IF(.NOT. ((IRANGE.EQ.ALLRNG).AND.(.NOT.FORCEB)) ) THEN
* Count the number of eigenvalues in the current block.
MB = 0
DO 20 I = WBEGIN,MM
IF( IBLOCK(I).EQ.JBLK ) THEN
MB = MB+1
ELSE
GOTO 21
ENDIF
20 CONTINUE
21 CONTINUE
IF( MB.EQ.0) THEN
* No eigenvalue in the current block lies in the desired range
* E( IEND ) holds the shift for the initial RRR
E( IEND ) = ZERO
IBEGIN = IEND + 1
GO TO 170
ELSE
* Decide whether dqds or bisection is more efficient
USEDQD = ( (MB .GT. FAC*IN) .AND. (.NOT.FORCEB) )
WEND = WBEGIN + MB - 1
* Calculate gaps for the current block
* In later stages, when representations for individual
* eigenvalues are different, we use SIGMA = E( IEND ).
SIGMA = ZERO
DO 30 I = WBEGIN, WEND - 1
WGAP( I ) = MAX( ZERO,
$ W(I+1)-WERR(I+1) - (W(I)+WERR(I)) )
30 CONTINUE
WGAP( WEND ) = MAX( ZERO,
$ VU - SIGMA - (W( WEND )+WERR( WEND )))
* Find local index of the first and last desired evalue.
INDL = INDEXW(WBEGIN)
INDU = INDEXW( WEND )
ENDIF
ENDIF
IF(( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ).OR.USEDQD) THEN
* Case of DQDS
* Find approximations to the extremal eigenvalues of the block
CALL DLARRK( IN, 1, GL, GU, D(IBEGIN),
$ E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = -1
RETURN
ENDIF
ISLEFT = MAX(GL, TMP - TMP1
$ - HNDRD * EPS* ABS(TMP - TMP1))
CALL DLARRK( IN, IN, GL, GU, D(IBEGIN),
$ E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = -1
RETURN
ENDIF
ISRGHT = MIN(GU, TMP + TMP1
$ + HNDRD * EPS * ABS(TMP + TMP1))
* Improve the estimate of the spectral diameter
SPDIAM = ISRGHT - ISLEFT
ELSE
* Case of bisection
* Find approximations to the wanted extremal eigenvalues
ISLEFT = MAX(GL, W(WBEGIN) - WERR(WBEGIN)
$ - HNDRD * EPS*ABS(W(WBEGIN)- WERR(WBEGIN) ))
ISRGHT = MIN(GU,W(WEND) + WERR(WEND)
$ + HNDRD * EPS * ABS(W(WEND)+ WERR(WEND)))
ENDIF
* Decide whether the base representation for the current block
* L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I
* should be on the left or the right end of the current block.
* The strategy is to shift to the end which is "more populated"
* Furthermore, decide whether to use DQDS for the computation of
* the eigenvalue approximations at the end of DLARRE or bisection.
* dqds is chosen if all eigenvalues are desired or the number of
* eigenvalues to be computed is large compared to the blocksize.
IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
* If all the eigenvalues have to be computed, we use dqd
USEDQD = .TRUE.
* INDL is the local index of the first eigenvalue to compute
INDL = 1
INDU = IN
* MB = number of eigenvalues to compute
MB = IN
WEND = WBEGIN + MB - 1
* Define 1/4 and 3/4 points of the spectrum
S1 = ISLEFT + FOURTH * SPDIAM
S2 = ISRGHT - FOURTH * SPDIAM
ELSE
* DLARRD has computed IBLOCK and INDEXW for each eigenvalue
* approximation.
* choose sigma
IF( USEDQD ) THEN
S1 = ISLEFT + FOURTH * SPDIAM
S2 = ISRGHT - FOURTH * SPDIAM
ELSE
TMP = MIN(ISRGHT,VU) - MAX(ISLEFT,VL)
S1 = MAX(ISLEFT,VL) + FOURTH * TMP
S2 = MIN(ISRGHT,VU) - FOURTH * TMP
ENDIF
ENDIF
* Compute the negcount at the 1/4 and 3/4 points
IF(MB.GT.1) THEN
CALL DLARRC( 'T', IN, S1, S2, D(IBEGIN),
$ E(IBEGIN), PIVMIN, CNT, CNT1, CNT2, IINFO)
ENDIF
IF(MB.EQ.1) THEN
SIGMA = GL
SGNDEF = ONE
ELSEIF( CNT1 - INDL .GE. INDU - CNT2 ) THEN
IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
SIGMA = MAX(ISLEFT,GL)
ELSEIF( USEDQD ) THEN
* use Gerschgorin bound as shift to get pos def matrix
* for dqds
SIGMA = ISLEFT
ELSE
* use approximation of the first desired eigenvalue of the
* block as shift
SIGMA = MAX(ISLEFT,VL)
ENDIF
SGNDEF = ONE
ELSE
IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
SIGMA = MIN(ISRGHT,GU)
ELSEIF( USEDQD ) THEN
* use Gerschgorin bound as shift to get neg def matrix
* for dqds
SIGMA = ISRGHT
ELSE
* use approximation of the first desired eigenvalue of the
* block as shift
SIGMA = MIN(ISRGHT,VU)
ENDIF
SGNDEF = -ONE
ENDIF
* An initial SIGMA has been chosen that will be used for computing
* T - SIGMA I = L D L^T
* Define the increment TAU of the shift in case the initial shift
* needs to be refined to obtain a factorization with not too much
* element growth.
IF( USEDQD ) THEN
* The initial SIGMA was to the outer end of the spectrum
* the matrix is definite and we need not retreat.
TAU = SPDIAM*EPS*N + TWO*PIVMIN
TAU = MAX( TAU,TWO*EPS*ABS(SIGMA) )
ELSE
IF(MB.GT.1) THEN
CLWDTH = W(WEND) + WERR(WEND) - W(WBEGIN) - WERR(WBEGIN)
AVGAP = ABS(CLWDTH / DBLE(WEND-WBEGIN))
IF( SGNDEF.EQ.ONE ) THEN
TAU = HALF*MAX(WGAP(WBEGIN),AVGAP)
TAU = MAX(TAU,WERR(WBEGIN))
ELSE
TAU = HALF*MAX(WGAP(WEND-1),AVGAP)
TAU = MAX(TAU,WERR(WEND))
ENDIF
ELSE
TAU = WERR(WBEGIN)
ENDIF
ENDIF
*
DO 80 IDUM = 1, MAXTRY
* Compute L D L^T factorization of tridiagonal matrix T - sigma I.
* Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of
* pivots in WORK(2*IN+1:3*IN)
DPIVOT = D( IBEGIN ) - SIGMA
WORK( 1 ) = DPIVOT
DMAX = ABS( WORK(1) )
J = IBEGIN
DO 70 I = 1, IN - 1
WORK( 2*IN+I ) = ONE / WORK( I )
TMP = E( J )*WORK( 2*IN+I )
WORK( IN+I ) = TMP
DPIVOT = ( D( J+1 )-SIGMA ) - TMP*E( J )
WORK( I+1 ) = DPIVOT
DMAX = MAX( DMAX, ABS(DPIVOT) )
J = J + 1
70 CONTINUE
* check for element growth
IF( DMAX .GT. MAXGROWTH*SPDIAM ) THEN
NOREP = .TRUE.
ELSE
NOREP = .FALSE.
ENDIF
IF( USEDQD .AND. .NOT.NOREP ) THEN
* Ensure the definiteness of the representation
* All entries of D (of L D L^T) must have the same sign
DO 71 I = 1, IN
TMP = SGNDEF*WORK( I )
IF( TMP.LT.ZERO ) NOREP = .TRUE.
71 CONTINUE
ENDIF
IF(NOREP) THEN
* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin
* shift which makes the matrix definite. So we should end up
* here really only in the case of IRANGE = VALRNG or INDRNG.
IF( IDUM.EQ.MAXTRY-1 ) THEN
IF( SGNDEF.EQ.ONE ) THEN
* The fudged Gerschgorin shift should succeed
SIGMA =
$ GL - FUDGE*SPDIAM*EPS*N - FUDGE*TWO*PIVMIN
ELSE
SIGMA =
$ GU + FUDGE*SPDIAM*EPS*N + FUDGE*TWO*PIVMIN
END IF
ELSE
SIGMA = SIGMA - SGNDEF * TAU
TAU = TWO * TAU
END IF
ELSE
* an initial RRR is found
GO TO 83
END IF
80 CONTINUE
* if the program reaches this point, no base representation could be
* found in MAXTRY iterations.
INFO = 2
RETURN
83 CONTINUE
* At this point, we have found an initial base representation
* T - SIGMA I = L D L^T with not too much element growth.
* Store the shift.
E( IEND ) = SIGMA
* Store D and L.
CALL DCOPY( IN, WORK, 1, D( IBEGIN ), 1 )
CALL DCOPY( IN-1, WORK( IN+1 ), 1, E( IBEGIN ), 1 )
IF(MB.GT.1 ) THEN
*
* Perturb each entry of the base representation by a small
* (but random) relative amount to overcome difficulties with
* glued matrices.
*
DO 122 I = 1, 4
ISEED( I ) = 1
122 CONTINUE
CALL DLARNV(2, ISEED, 2*IN-1, WORK(1))
DO 125 I = 1,IN-1
D(IBEGIN+I-1) = D(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(I))
E(IBEGIN+I-1) = E(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(IN+I))
125 CONTINUE
D(IEND) = D(IEND)*(ONE+EPS*FOUR*WORK(IN))
*
ENDIF
*
* Don't update the Gerschgorin intervals because keeping track
* of the updates would be too much work in DLARRV.
* We update W instead and use it to locate the proper Gerschgorin
* intervals.
* Compute the required eigenvalues of L D L' by bisection or dqds
IF ( .NOT.USEDQD ) THEN
* If DLARRD has been used, shift the eigenvalue approximations
* according to their representation. This is necessary for
* a uniform DLARRV since dqds computes eigenvalues of the
* shifted representation. In DLARRV, W will always hold the
* UNshifted eigenvalue approximation.
DO 134 J=WBEGIN,WEND
W(J) = W(J) - SIGMA
WERR(J) = WERR(J) + ABS(W(J)) * EPS
134 CONTINUE
* call DLARRB to reduce eigenvalue error of the approximations
* from DLARRD
DO 135 I = IBEGIN, IEND-1
WORK( I ) = D( I ) * E( I )**2
135 CONTINUE
* use bisection to find EV from INDL to INDU
CALL DLARRB(IN, D(IBEGIN), WORK(IBEGIN),
$ INDL, INDU, RTOL1, RTOL2, INDL-1,
$ W(WBEGIN), WGAP(WBEGIN), WERR(WBEGIN),
$ WORK( 2*N+1 ), IWORK, PIVMIN, SPDIAM,
$ IN, IINFO )
IF( IINFO .NE. 0 ) THEN
INFO = -4
RETURN
END IF
* DLARRB computes all gaps correctly except for the last one
* Record distance to VU/GU
WGAP( WEND ) = MAX( ZERO,
$ ( VU-SIGMA ) - ( W( WEND ) + WERR( WEND ) ) )
DO 138 I = INDL, INDU
M = M + 1
IBLOCK(M) = JBLK
INDEXW(M) = I
138 CONTINUE
ELSE
* Call dqds to get all eigs (and then possibly delete unwanted
* eigenvalues).
* Note that dqds finds the eigenvalues of the L D L^T representation
* of T to high relative accuracy. High relative accuracy
* might be lost when the shift of the RRR is subtracted to obtain
* the eigenvalues of T. However, T is not guaranteed to define its
* eigenvalues to high relative accuracy anyway.
* Set RTOL to the order of the tolerance used in DLASQ2
* This is an ESTIMATED error, the worst case bound is 4*N*EPS
* which is usually too large and requires unnecessary work to be
* done by bisection when computing the eigenvectors
RTOL = LOG(DBLE(IN)) * FOUR * EPS
J = IBEGIN
DO 140 I = 1, IN - 1
WORK( 2*I-1 ) = ABS( D( J ) )
WORK( 2*I ) = E( J )*E( J )*WORK( 2*I-1 )
J = J + 1
140 CONTINUE
WORK( 2*IN-1 ) = ABS( D( IEND ) )
WORK( 2*IN ) = ZERO
CALL DLASQ2( IN, WORK, IINFO )
IF( IINFO .NE. 0 ) THEN
* If IINFO = -5 then an index is part of a tight cluster
* and should be changed. The index is in IWORK(1) and the
* gap is in WORK(N+1)
INFO = -5
RETURN
ELSE
* Test that all eigenvalues are positive as expected
DO 149 I = 1, IN
IF( WORK( I ).LT.ZERO ) THEN
INFO = -6
RETURN
ENDIF
149 CONTINUE
END IF
IF( SGNDEF.GT.ZERO ) THEN
DO 150 I = INDL, INDU
M = M + 1
W( M ) = WORK( IN-I+1 )
IBLOCK( M ) = JBLK
INDEXW( M ) = I
150 CONTINUE
ELSE
DO 160 I = INDL, INDU
M = M + 1
W( M ) = -WORK( I )
IBLOCK( M ) = JBLK
INDEXW( M ) = I
160 CONTINUE
END IF
DO 165 I = M - MB + 1, M
* the value of RTOL below should be the tolerance in DLASQ2
WERR( I ) = RTOL * ABS( W(I) )
165 CONTINUE
DO 166 I = M - MB + 1, M - 1
* compute the right gap between the intervals
WGAP( I ) = MAX( ZERO,
$ W(I+1)-WERR(I+1) - (W(I)+WERR(I)) )
166 CONTINUE
WGAP( M ) = MAX( ZERO,
$ ( VU-SIGMA ) - ( W( M ) + WERR( M ) ) )
END IF
* proceed with next block
IBEGIN = IEND + 1
WBEGIN = WEND + 1
170 CONTINUE
*
RETURN
*
* End of DLARRE
*
END
*> \brief \b DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
* W, WGAP, WERR,
* SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
* DPLUS, LPLUS, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER CLSTRT, CLEND, INFO, N
* DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ),
* $ LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given the initial representation L D L^T and its cluster of close
*> eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
*> W( CLEND ), DLARRF finds a new relatively robust representation
*> L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
*> eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix (subblock, if the matrix split).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is DOUBLE PRECISION array, dimension (N-1)
*> The (N-1) subdiagonal elements of the unit bidiagonal
*> matrix L.
*> \endverbatim
*>
*> \param[in] LD
*> \verbatim
*> LD is DOUBLE PRECISION array, dimension (N-1)
*> The (N-1) elements L(i)*D(i).
*> \endverbatim
*>
*> \param[in] CLSTRT
*> \verbatim
*> CLSTRT is INTEGER
*> The index of the first eigenvalue in the cluster.
*> \endverbatim
*>
*> \param[in] CLEND
*> \verbatim
*> CLEND is INTEGER
*> The index of the last eigenvalue in the cluster.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension
*> dimension is >= (CLEND-CLSTRT+1)
*> The eigenvalue APPROXIMATIONS of L D L^T in ascending order.
*> W( CLSTRT ) through W( CLEND ) form the cluster of relatively
*> close eigenalues.
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*> WGAP is DOUBLE PRECISION array, dimension
*> dimension is >= (CLEND-CLSTRT+1)
*> The separation from the right neighbor eigenvalue in W.
*> \endverbatim
*>
*> \param[in] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION array, dimension
*> dimension is >= (CLEND-CLSTRT+1)
*> WERR contain the semiwidth of the uncertainty
*> interval of the corresponding eigenvalue APPROXIMATION in W
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*> SPDIAM is DOUBLE PRECISION
*> estimate of the spectral diameter obtained from the
*> Gerschgorin intervals
*> \endverbatim
*>
*> \param[in] CLGAPL
*> \verbatim
*> CLGAPL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CLGAPR
*> \verbatim
*> CLGAPR is DOUBLE PRECISION
*> absolute gap on each end of the cluster.
*> Set by the calling routine to protect against shifts too close
*> to eigenvalues outside the cluster.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot allowed in the Sturm sequence.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> The shift used to form L(+) D(+) L(+)^T.
*> \endverbatim
*>
*> \param[out] DPLUS
*> \verbatim
*> DPLUS is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the diagonal matrix D(+).
*> \endverbatim
*>
*> \param[out] LPLUS
*> \verbatim
*> LPLUS is DOUBLE PRECISION array, dimension (N-1)
*> The first (N-1) elements of LPLUS contain the subdiagonal
*> elements of the unit bidiagonal matrix L(+).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> Signals processing OK (=0) or failure (=1)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrf
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
$ W, WGAP, WERR,
$ SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
$ DPLUS, LPLUS, WORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER CLSTRT, CLEND, INFO, N
DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ),
$ LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION FOUR, MAXGROWTH1, MAXGROWTH2, ONE, QUART, TWO
PARAMETER ( ONE = 1.0D0, TWO = 2.0D0, FOUR = 4.0D0,
$ QUART = 0.25D0,
$ MAXGROWTH1 = 8.D0,
$ MAXGROWTH2 = 8.D0 )
* ..
* .. Local Scalars ..
LOGICAL DORRR1, FORCER, NOFAIL, SAWNAN1, SAWNAN2, TRYRRR1
INTEGER I, INDX, KTRY, KTRYMAX, SLEFT, SRIGHT, SHIFT
PARAMETER ( KTRYMAX = 1, SLEFT = 1, SRIGHT = 2 )
DOUBLE PRECISION AVGAP, BESTSHIFT, CLWDTH, EPS, FACT, FAIL,
$ FAIL2, GROWTHBOUND, LDELTA, LDMAX, LSIGMA,
$ MAX1, MAX2, MINGAP, OLDP, PROD, RDELTA, RDMAX,
$ RRR1, RRR2, RSIGMA, S, SMLGROWTH, TMP, ZNM2
* ..
* .. External Functions ..
LOGICAL DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL DISNAN, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
FACT = DBLE(2**KTRYMAX)
EPS = DLAMCH( 'Precision' )
SHIFT = 0
FORCER = .FALSE.
* Note that we cannot guarantee that for any of the shifts tried,
* the factorization has a small or even moderate element growth.
* There could be Ritz values at both ends of the cluster and despite
* backing off, there are examples where all factorizations tried
* (in IEEE mode, allowing zero pivots & infinities) have INFINITE
* element growth.
* For this reason, we should use PIVMIN in this subroutine so that at
* least the L D L^T factorization exists. It can be checked afterwards
* whether the element growth caused bad residuals/orthogonality.
* Decide whether the code should accept the best among all
* representations despite large element growth or signal INFO=1
* Setting NOFAIL to .FALSE. for quick fix for bug 113
NOFAIL = .FALSE.
*
* Compute the average gap length of the cluster
CLWDTH = ABS(W(CLEND)-W(CLSTRT)) + WERR(CLEND) + WERR(CLSTRT)
AVGAP = CLWDTH / DBLE(CLEND-CLSTRT)
MINGAP = MIN(CLGAPL, CLGAPR)
* Initial values for shifts to both ends of cluster
LSIGMA = MIN(W( CLSTRT ),W( CLEND )) - WERR( CLSTRT )
RSIGMA = MAX(W( CLSTRT ),W( CLEND )) + WERR( CLEND )
* Use a small fudge to make sure that we really shift to the outside
LSIGMA = LSIGMA - ABS(LSIGMA)* FOUR * EPS
RSIGMA = RSIGMA + ABS(RSIGMA)* FOUR * EPS
* Compute upper bounds for how much to back off the initial shifts
LDMAX = QUART * MINGAP + TWO * PIVMIN
RDMAX = QUART * MINGAP + TWO * PIVMIN
LDELTA = MAX(AVGAP,WGAP( CLSTRT ))/FACT
RDELTA = MAX(AVGAP,WGAP( CLEND-1 ))/FACT
*
* Initialize the record of the best representation found
*
S = DLAMCH( 'S' )
SMLGROWTH = ONE / S
FAIL = DBLE(N-1)*MINGAP/(SPDIAM*EPS)
FAIL2 = DBLE(N-1)*MINGAP/(SPDIAM*SQRT(EPS))
BESTSHIFT = LSIGMA
*
* while (KTRY <= KTRYMAX)
KTRY = 0
GROWTHBOUND = MAXGROWTH1*SPDIAM
5 CONTINUE
SAWNAN1 = .FALSE.
SAWNAN2 = .FALSE.
* Ensure that we do not back off too much of the initial shifts
LDELTA = MIN(LDMAX,LDELTA)
RDELTA = MIN(RDMAX,RDELTA)
* Compute the element growth when shifting to both ends of the cluster
* accept the shift if there is no element growth at one of the two ends
* Left end
S = -LSIGMA
DPLUS( 1 ) = D( 1 ) + S
IF(ABS(DPLUS(1)).LT.PIVMIN) THEN
DPLUS(1) = -PIVMIN
* Need to set SAWNAN1 because refined RRR test should not be used
* in this case
SAWNAN1 = .TRUE.
ENDIF
MAX1 = ABS( DPLUS( 1 ) )
DO 6 I = 1, N - 1
LPLUS( I ) = LD( I ) / DPLUS( I )
S = S*LPLUS( I )*L( I ) - LSIGMA
DPLUS( I+1 ) = D( I+1 ) + S
IF(ABS(DPLUS(I+1)).LT.PIVMIN) THEN
DPLUS(I+1) = -PIVMIN
* Need to set SAWNAN1 because refined RRR test should not be used
* in this case
SAWNAN1 = .TRUE.
ENDIF
MAX1 = MAX( MAX1,ABS(DPLUS(I+1)) )
6 CONTINUE
SAWNAN1 = SAWNAN1 .OR. DISNAN( MAX1 )
IF( FORCER .OR.
$ (MAX1.LE.GROWTHBOUND .AND. .NOT.SAWNAN1 ) ) THEN
SIGMA = LSIGMA
SHIFT = SLEFT
GOTO 100
ENDIF
* Right end
S = -RSIGMA
WORK( 1 ) = D( 1 ) + S
IF(ABS(WORK(1)).LT.PIVMIN) THEN
WORK(1) = -PIVMIN
* Need to set SAWNAN2 because refined RRR test should not be used
* in this case
SAWNAN2 = .TRUE.
ENDIF
MAX2 = ABS( WORK( 1 ) )
DO 7 I = 1, N - 1
WORK( N+I ) = LD( I ) / WORK( I )
S = S*WORK( N+I )*L( I ) - RSIGMA
WORK( I+1 ) = D( I+1 ) + S
IF(ABS(WORK(I+1)).LT.PIVMIN) THEN
WORK(I+1) = -PIVMIN
* Need to set SAWNAN2 because refined RRR test should not be used
* in this case
SAWNAN2 = .TRUE.
ENDIF
MAX2 = MAX( MAX2,ABS(WORK(I+1)) )
7 CONTINUE
SAWNAN2 = SAWNAN2 .OR. DISNAN( MAX2 )
IF( FORCER .OR.
$ (MAX2.LE.GROWTHBOUND .AND. .NOT.SAWNAN2 ) ) THEN
SIGMA = RSIGMA
SHIFT = SRIGHT
GOTO 100
ENDIF
* If we are at this point, both shifts led to too much element growth
* Record the better of the two shifts (provided it didn't lead to NaN)
IF(SAWNAN1.AND.SAWNAN2) THEN
* both MAX1 and MAX2 are NaN
GOTO 50
ELSE
IF( .NOT.SAWNAN1 ) THEN
INDX = 1
IF(MAX1.LE.SMLGROWTH) THEN
SMLGROWTH = MAX1
BESTSHIFT = LSIGMA
ENDIF
ENDIF
IF( .NOT.SAWNAN2 ) THEN
IF(SAWNAN1 .OR. MAX2.LE.MAX1) INDX = 2
IF(MAX2.LE.SMLGROWTH) THEN
SMLGROWTH = MAX2
BESTSHIFT = RSIGMA
ENDIF
ENDIF
ENDIF
* If we are here, both the left and the right shift led to
* element growth. If the element growth is moderate, then
* we may still accept the representation, if it passes a
* refined test for RRR. This test supposes that no NaN occurred.
* Moreover, we use the refined RRR test only for isolated clusters.
IF((CLWDTH.LT.MINGAP/DBLE(128)) .AND.
$ (MIN(MAX1,MAX2).LT.FAIL2)
$ .AND.(.NOT.SAWNAN1).AND.(.NOT.SAWNAN2)) THEN
DORRR1 = .TRUE.
ELSE
DORRR1 = .FALSE.
ENDIF
TRYRRR1 = .TRUE.
IF( TRYRRR1 .AND. DORRR1 ) THEN
IF(INDX.EQ.1) THEN
TMP = ABS( DPLUS( N ) )
ZNM2 = ONE
PROD = ONE
OLDP = ONE
DO 15 I = N-1, 1, -1
IF( PROD .LE. EPS ) THEN
PROD =
$ ((DPLUS(I+1)*WORK(N+I+1))/(DPLUS(I)*WORK(N+I)))*OLDP
ELSE
PROD = PROD*ABS(WORK(N+I))
END IF
OLDP = PROD
ZNM2 = ZNM2 + PROD**2
TMP = MAX( TMP, ABS( DPLUS( I ) * PROD ))
15 CONTINUE
RRR1 = TMP/( SPDIAM * SQRT( ZNM2 ) )
IF (RRR1.LE.MAXGROWTH2) THEN
SIGMA = LSIGMA
SHIFT = SLEFT
GOTO 100
ENDIF
ELSE IF(INDX.EQ.2) THEN
TMP = ABS( WORK( N ) )
ZNM2 = ONE
PROD = ONE
OLDP = ONE
DO 16 I = N-1, 1, -1
IF( PROD .LE. EPS ) THEN
PROD = ((WORK(I+1)*LPLUS(I+1))/(WORK(I)*LPLUS(I)))*OLDP
ELSE
PROD = PROD*ABS(LPLUS(I))
END IF
OLDP = PROD
ZNM2 = ZNM2 + PROD**2
TMP = MAX( TMP, ABS( WORK( I ) * PROD ))
16 CONTINUE
RRR2 = TMP/( SPDIAM * SQRT( ZNM2 ) )
IF (RRR2.LE.MAXGROWTH2) THEN
SIGMA = RSIGMA
SHIFT = SRIGHT
GOTO 100
ENDIF
END IF
ENDIF
50 CONTINUE
IF (KTRY.LT.KTRYMAX) THEN
* If we are here, both shifts failed also the RRR test.
* Back off to the outside
LSIGMA = MAX( LSIGMA - LDELTA,
$ LSIGMA - LDMAX)
RSIGMA = MIN( RSIGMA + RDELTA,
$ RSIGMA + RDMAX )
LDELTA = TWO * LDELTA
RDELTA = TWO * RDELTA
KTRY = KTRY + 1
GOTO 5
ELSE
* None of the representations investigated satisfied our
* criteria. Take the best one we found.
IF((SMLGROWTH.LT.FAIL).OR.NOFAIL) THEN
LSIGMA = BESTSHIFT
RSIGMA = BESTSHIFT
FORCER = .TRUE.
GOTO 5
ELSE
INFO = 1
RETURN
ENDIF
END IF
100 CONTINUE
IF (SHIFT.EQ.SLEFT) THEN
ELSEIF (SHIFT.EQ.SRIGHT) THEN
* store new L and D back into DPLUS, LPLUS
CALL DCOPY( N, WORK, 1, DPLUS, 1 )
CALL DCOPY( N-1, WORK(N+1), 1, LPLUS, 1 )
ENDIF
RETURN
*
* End of DLARRF
*
END
*> \brief \b DLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRJ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrj.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrj.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrj.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
* RTOL, OFFSET, W, WERR, WORK, IWORK,
* PIVMIN, SPDIAM, INFO )
*
* .. Scalar Arguments ..
* INTEGER IFIRST, ILAST, INFO, N, OFFSET
* DOUBLE PRECISION PIVMIN, RTOL, SPDIAM
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E2( * ), W( * ),
* $ WERR( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given the initial eigenvalue approximations of T, DLARRJ
*> does bisection to refine the eigenvalues of T,
*> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
*> guesses for these eigenvalues are input in W, the corresponding estimate
*> of the error in these guesses in WERR. During bisection, intervals
*> [left, right] are maintained by storing their mid-points and
*> semi-widths in the arrays W and WERR respectively.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*> E2 is DOUBLE PRECISION array, dimension (N-1)
*> The Squares of the (N-1) subdiagonal elements of T.
*> \endverbatim
*>
*> \param[in] IFIRST
*> \verbatim
*> IFIRST is INTEGER
*> The index of the first eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] ILAST
*> \verbatim
*> ILAST is INTEGER
*> The index of the last eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] RTOL
*> \verbatim
*> RTOL is DOUBLE PRECISION
*> Tolerance for the convergence of the bisection intervals.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT < RTOL*MAX(|LEFT|,|RIGHT|).
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*> OFFSET is INTEGER
*> Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
*> through ILAST-OFFSET elements of these arrays are to be used.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
*> estimates of the eigenvalues of L D L^T indexed IFIRST through
*> ILAST.
*> On output, these estimates are refined.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION array, dimension (N)
*> On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
*> the errors in the estimates of the corresponding elements in W.
*> On output, these errors are refined.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*> SPDIAM is DOUBLE PRECISION
*> The spectral diameter of T.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> Error flag.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrj
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
$ RTOL, OFFSET, W, WERR, WORK, IWORK,
$ PIVMIN, SPDIAM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IFIRST, ILAST, INFO, N, OFFSET
DOUBLE PRECISION PIVMIN, RTOL, SPDIAM
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E2( * ), W( * ),
$ WERR( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, HALF
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ HALF = 0.5D0 )
INTEGER MAXITR
* ..
* .. Local Scalars ..
INTEGER CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
$ OLNINT, P, PREV, SAVI1
DOUBLE PRECISION DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
*
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
*
* Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
* The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
* Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
* for an unconverged interval is set to the index of the next unconverged
* interval, and is -1 or 0 for a converged interval. Thus a linked
* list of unconverged intervals is set up.
*
I1 = IFIRST
I2 = ILAST
* The number of unconverged intervals
NINT = 0
* The last unconverged interval found
PREV = 0
DO 75 I = I1, I2
K = 2*I
II = I - OFFSET
LEFT = W( II ) - WERR( II )
MID = W(II)
RIGHT = W( II ) + WERR( II )
WIDTH = RIGHT - MID
TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
* The following test prevents the test of converged intervals
IF( WIDTH.LT.RTOL*TMP ) THEN
* This interval has already converged and does not need refinement.
* (Note that the gaps might change through refining the
* eigenvalues, however, they can only get bigger.)
* Remove it from the list.
IWORK( K-1 ) = -1
* Make sure that I1 always points to the first unconverged interval
IF((I.EQ.I1).AND.(I.LT.I2)) I1 = I + 1
IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
ELSE
* unconverged interval found
PREV = I
* Make sure that [LEFT,RIGHT] contains the desired eigenvalue
*
* Do while( CNT(LEFT).GT.I-1 )
*
FAC = ONE
20 CONTINUE
CNT = 0
S = LEFT
DPLUS = D( 1 ) - S
IF( DPLUS.LT.ZERO ) CNT = CNT + 1
DO 30 J = 2, N
DPLUS = D( J ) - S - E2( J-1 )/DPLUS
IF( DPLUS.LT.ZERO ) CNT = CNT + 1
30 CONTINUE
IF( CNT.GT.I-1 ) THEN
LEFT = LEFT - WERR( II )*FAC
FAC = TWO*FAC
GO TO 20
END IF
*
* Do while( CNT(RIGHT).LT.I )
*
FAC = ONE
50 CONTINUE
CNT = 0
S = RIGHT
DPLUS = D( 1 ) - S
IF( DPLUS.LT.ZERO ) CNT = CNT + 1
DO 60 J = 2, N
DPLUS = D( J ) - S - E2( J-1 )/DPLUS
IF( DPLUS.LT.ZERO ) CNT = CNT + 1
60 CONTINUE
IF( CNT.LT.I ) THEN
RIGHT = RIGHT + WERR( II )*FAC
FAC = TWO*FAC
GO TO 50
END IF
NINT = NINT + 1
IWORK( K-1 ) = I + 1
IWORK( K ) = CNT
END IF
WORK( K-1 ) = LEFT
WORK( K ) = RIGHT
75 CONTINUE
SAVI1 = I1
*
* Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
* and while (ITER.LT.MAXITR)
*
ITER = 0
80 CONTINUE
PREV = I1 - 1
I = I1
OLNINT = NINT
DO 100 P = 1, OLNINT
K = 2*I
II = I - OFFSET
NEXT = IWORK( K-1 )
LEFT = WORK( K-1 )
RIGHT = WORK( K )
MID = HALF*( LEFT + RIGHT )
* semiwidth of interval
WIDTH = RIGHT - MID
TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
IF( ( WIDTH.LT.RTOL*TMP ) .OR.
$ (ITER.EQ.MAXITR) )THEN
* reduce number of unconverged intervals
NINT = NINT - 1
* Mark interval as converged.
IWORK( K-1 ) = 0
IF( I1.EQ.I ) THEN
I1 = NEXT
ELSE
* Prev holds the last unconverged interval previously examined
IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
END IF
I = NEXT
GO TO 100
END IF
PREV = I
*
* Perform one bisection step
*
CNT = 0
S = MID
DPLUS = D( 1 ) - S
IF( DPLUS.LT.ZERO ) CNT = CNT + 1
DO 90 J = 2, N
DPLUS = D( J ) - S - E2( J-1 )/DPLUS
IF( DPLUS.LT.ZERO ) CNT = CNT + 1
90 CONTINUE
IF( CNT.LE.I-1 ) THEN
WORK( K-1 ) = MID
ELSE
WORK( K ) = MID
END IF
I = NEXT
100 CONTINUE
ITER = ITER + 1
* do another loop if there are still unconverged intervals
* However, in the last iteration, all intervals are accepted
* since this is the best we can do.
IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
*
*
* At this point, all the intervals have converged
DO 110 I = SAVI1, ILAST
K = 2*I
II = I - OFFSET
* All intervals marked by '0' have been refined.
IF( IWORK( K-1 ).EQ.0 ) THEN
W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
WERR( II ) = WORK( K ) - W( II )
END IF
110 CONTINUE
*
RETURN
*
* End of DLARRJ
*
END
*> \brief \b DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrk.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrk.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrk.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRK( N, IW, GL, GU,
* D, E2, PIVMIN, RELTOL, W, WERR, INFO)
*
* .. Scalar Arguments ..
* INTEGER INFO, IW, N
* DOUBLE PRECISION PIVMIN, RELTOL, GL, GU, W, WERR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E2( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARRK computes one eigenvalue of a symmetric tridiagonal
*> matrix T to suitable accuracy. This is an auxiliary code to be
*> called from DSTEMR.
*>
*> To avoid overflow, the matrix must be scaled so that its
*> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
*> accuracy, it should not be much smaller than that.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the tridiagonal matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] IW
*> \verbatim
*> IW is INTEGER
*> The index of the eigenvalues to be returned.
*> \endverbatim
*>
*> \param[in] GL
*> \verbatim
*> GL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] GU
*> \verbatim
*> GU is DOUBLE PRECISION
*> An upper and a lower bound on the eigenvalue.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*> E2 is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot allowed in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*> RELTOL is DOUBLE PRECISION
*> The minimum relative width of an interval. When an interval
*> is narrower than RELTOL times the larger (in
*> magnitude) endpoint, then it is considered to be
*> sufficiently small, i.e., converged. Note: this should
*> always be at least radix*machine epsilon.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION
*> The error bound on the corresponding eigenvalue approximation
*> in W.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: Eigenvalue converged
*> = -1: Eigenvalue did NOT converge
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> FUDGE DOUBLE PRECISION, default = 2
*> A "fudge factor" to widen the Gershgorin intervals.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrk
*
* =====================================================================
SUBROUTINE DLARRK( N, IW, GL, GU,
$ D, E2, PIVMIN, RELTOL, W, WERR, INFO)
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, IW, N
DOUBLE PRECISION PIVMIN, RELTOL, GL, GU, W, WERR
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E2( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION FUDGE, HALF, TWO, ZERO
PARAMETER ( HALF = 0.5D0, TWO = 2.0D0,
$ FUDGE = TWO, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, IT, ITMAX, NEGCNT
DOUBLE PRECISION ATOLI, EPS, LEFT, MID, RIGHT, RTOLI, TMP1,
$ TMP2, TNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
INFO = 0
RETURN
END IF
*
* Get machine constants
EPS = DLAMCH( 'P' )
TNORM = MAX( ABS( GL ), ABS( GU ) )
RTOLI = RELTOL
ATOLI = FUDGE*TWO*PIVMIN
ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
INFO = -1
LEFT = GL - FUDGE*TNORM*EPS*N - FUDGE*TWO*PIVMIN
RIGHT = GU + FUDGE*TNORM*EPS*N + FUDGE*TWO*PIVMIN
IT = 0
10 CONTINUE
*
* Check if interval converged or maximum number of iterations reached
*
TMP1 = ABS( RIGHT - LEFT )
TMP2 = MAX( ABS(RIGHT), ABS(LEFT) )
IF( TMP1.LT.MAX( ATOLI, PIVMIN, RTOLI*TMP2 ) ) THEN
INFO = 0
GOTO 30
ENDIF
IF(IT.GT.ITMAX)
$ GOTO 30
*
* Count number of negative pivots for mid-point
*
IT = IT + 1
MID = HALF * (LEFT + RIGHT)
NEGCNT = 0
TMP1 = D( 1 ) - MID
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
IF( TMP1.LE.ZERO )
$ NEGCNT = NEGCNT + 1
*
DO 20 I = 2, N
TMP1 = D( I ) - E2( I-1 ) / TMP1 - MID
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
IF( TMP1.LE.ZERO )
$ NEGCNT = NEGCNT + 1
20 CONTINUE
IF(NEGCNT.GE.IW) THEN
RIGHT = MID
ELSE
LEFT = MID
ENDIF
GOTO 10
30 CONTINUE
*
* Converged or maximum number of iterations reached
*
W = HALF * (LEFT + RIGHT)
WERR = HALF * ABS( RIGHT - LEFT )
RETURN
*
* End of DLARRK
*
END
*> \brief \b DLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRR( N, D, E, INFO )
*
* .. Scalar Arguments ..
* INTEGER N, INFO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Perform tests to decide whether the symmetric tridiagonal matrix T
*> warrants expensive computations which guarantee high relative accuracy
*> in the eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, the first (N-1) entries contain the subdiagonal
*> elements of the tridiagonal matrix T; E(N) is set to ZERO.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> INFO = 0(default) : the matrix warrants computations preserving
*> relative accuracy.
*> INFO = 1 : the matrix warrants computations guaranteeing
*> only absolute accuracy.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrr
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRR( N, D, E, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER N, INFO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, RELCOND
PARAMETER ( ZERO = 0.0D0,
$ RELCOND = 0.999D0 )
* ..
* .. Local Scalars ..
INTEGER I
LOGICAL YESREL
DOUBLE PRECISION EPS, SAFMIN, SMLNUM, RMIN, TMP, TMP2,
$ OFFDIG, OFFDIG2
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
INFO = 0
RETURN
END IF
*
* As a default, do NOT go for relative-accuracy preserving computations.
INFO = 1
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
RMIN = SQRT( SMLNUM )
* Tests for relative accuracy
*
* Test for scaled diagonal dominance
* Scale the diagonal entries to one and check whether the sum of the
* off-diagonals is less than one
*
* The sdd relative error bounds have a 1/(1- 2*x) factor in them,
* x = max(OFFDIG + OFFDIG2), so when x is close to 1/2, no relative
* accuracy is promised. In the notation of the code fragment below,
* 1/(1 - (OFFDIG + OFFDIG2)) is the condition number.
* We don't think it is worth going into "sdd mode" unless the relative
* condition number is reasonable, not 1/macheps.
* The threshold should be compatible with other thresholds used in the
* code. We set OFFDIG + OFFDIG2 <= .999 =: RELCOND, it corresponds
* to losing at most 3 decimal digits: 1 / (1 - (OFFDIG + OFFDIG2)) <= 1000
* instead of the current OFFDIG + OFFDIG2 < 1
*
YESREL = .TRUE.
OFFDIG = ZERO
TMP = SQRT(ABS(D(1)))
IF (TMP.LT.RMIN) YESREL = .FALSE.
IF(.NOT.YESREL) GOTO 11
DO 10 I = 2, N
TMP2 = SQRT(ABS(D(I)))
IF (TMP2.LT.RMIN) YESREL = .FALSE.
IF(.NOT.YESREL) GOTO 11
OFFDIG2 = ABS(E(I-1))/(TMP*TMP2)
IF(OFFDIG+OFFDIG2.GE.RELCOND) YESREL = .FALSE.
IF(.NOT.YESREL) GOTO 11
TMP = TMP2
OFFDIG = OFFDIG2
10 CONTINUE
11 CONTINUE
IF( YESREL ) THEN
INFO = 0
RETURN
ELSE
ENDIF
*
*
* *** MORE TO BE IMPLEMENTED ***
*
*
* Test if the lower bidiagonal matrix L from T = L D L^T
* (zero shift facto) is well conditioned
*
*
* Test if the upper bidiagonal matrix U from T = U D U^T
* (zero shift facto) is well conditioned.
* In this case, the matrix needs to be flipped and, at the end
* of the eigenvector computation, the flip needs to be applied
* to the computed eigenvectors (and the support)
*
*
RETURN
*
* End of DLARRR
*
END
*> \brief \b DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRV( N, VL, VU, D, L, PIVMIN,
* ISPLIT, M, DOL, DOU, MINRGP,
* RTOL1, RTOL2, W, WERR, WGAP,
* IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER DOL, DOU, INFO, LDZ, M, N
* DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
* $ ISUPPZ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
* $ WGAP( * ), WORK( * )
* DOUBLE PRECISION Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARRV computes the eigenvectors of the tridiagonal matrix
*> T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
*> The input eigenvalues should have been computed by DLARRE.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> Lower bound of the interval that contains the desired
*> eigenvalues. VL < VU. Needed to compute gaps on the left or right
*> end of the extremal eigenvalues in the desired RANGE.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> Upper bound of the interval that contains the desired
*> eigenvalues. VL < VU.
*> Note: VU is currently not used by this implementation of DLARRV, VU is
*> passed to DLARRV because it could be used compute gaps on the right end
*> of the extremal eigenvalues. However, with not much initial accuracy in
*> LAMBDA and VU, the formula can lead to an overestimation of the right gap
*> and thus to inadequately early RQI 'convergence'. This is currently
*> prevented this by forcing a small right gap. And so it turns out that VU
*> is currently not used by this implementation of DLARRV.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the N diagonal elements of the diagonal matrix D.
*> On exit, D may be overwritten.
*> \endverbatim
*>
*> \param[in,out] L
*> \verbatim
*> L is DOUBLE PRECISION array, dimension (N)
*> On entry, the (N-1) subdiagonal elements of the unit
*> bidiagonal matrix L are in elements 1 to N-1 of L
*> (if the matrix is not split.) At the end of each block
*> is stored the corresponding shift as given by DLARRE.
*> On exit, L is overwritten.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot allowed in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into blocks.
*> The first block consists of rows/columns 1 to
*> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
*> through ISPLIT( 2 ), etc.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The total number of input eigenvalues. 0 <= M <= N.
*> \endverbatim
*>
*> \param[in] DOL
*> \verbatim
*> DOL is INTEGER
*> \endverbatim
*>
*> \param[in] DOU
*> \verbatim
*> DOU is INTEGER
*> If the user wants to compute only selected eigenvectors from all
*> the eigenvalues supplied, he can specify an index range DOL:DOU.
*> Or else the setting DOL=1, DOU=M should be applied.
*> Note that DOL and DOU refer to the order in which the eigenvalues
*> are stored in W.
*> If the user wants to compute only selected eigenpairs, then
*> the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
*> computed eigenvectors. All other columns of Z are set to zero.
*> \endverbatim
*>
*> \param[in] MINRGP
*> \verbatim
*> MINRGP is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*> RTOL1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*> RTOL2 is DOUBLE PRECISION
*> Parameters for bisection.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements of W contain the APPROXIMATE eigenvalues for
*> which eigenvectors are to be computed. The eigenvalues
*> should be grouped by split-off block and ordered from
*> smallest to largest within the block ( The output array
*> W from DLARRE is expected here ). Furthermore, they are with
*> respect to the shift of the corresponding root representation
*> for their block. On exit, W holds the eigenvalues of the
*> UNshifted matrix.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the semiwidth of the uncertainty
*> interval of the corresponding eigenvalue in W
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*> WGAP is DOUBLE PRECISION array, dimension (N)
*> The separation from the right neighbor eigenvalue in W.
*> \endverbatim
*>
*> \param[in] IBLOCK
*> \verbatim
*> IBLOCK is INTEGER array, dimension (N)
*> The indices of the blocks (submatrices) associated with the
*> corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
*> W(i) belongs to the first block from the top, =2 if W(i)
*> belongs to the second block, etc.
*> \endverbatim
*>
*> \param[in] INDEXW
*> \verbatim
*> INDEXW is INTEGER array, dimension (N)
*> The indices of the eigenvalues within each block (submatrix);
*> for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
*> i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.
*> \endverbatim
*>
*> \param[in] GERS
*> \verbatim
*> GERS is DOUBLE PRECISION array, dimension (2*N)
*> The N Gerschgorin intervals (the i-th Gerschgorin interval
*> is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
*> be computed from the original UNshifted matrix.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*> If INFO = 0, the first M columns of Z contain the
*> orthonormal eigenvectors of the matrix T
*> corresponding to the input eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*> The support of the eigenvectors in Z, i.e., the indices
*> indicating the nonzero elements in Z. The I-th eigenvector
*> is nonzero only in elements ISUPPZ( 2*I-1 ) through
*> ISUPPZ( 2*I ).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (12*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*>
*> > 0: A problem occurred in DLARRV.
*> < 0: One of the called subroutines signaled an internal problem.
*> Needs inspection of the corresponding parameter IINFO
*> for further information.
*>
*> =-1: Problem in DLARRB when refining a child's eigenvalues.
*> =-2: Problem in DLARRF when computing the RRR of a child.
*> When a child is inside a tight cluster, it can be difficult
*> to find an RRR. A partial remedy from the user's point of
*> view is to make the parameter MINRGP smaller and recompile.
*> However, as the orthogonality of the computed vectors is
*> proportional to 1/MINRGP, the user should be aware that
*> he might be trading in precision when he decreases MINRGP.
*> =-3: Problem in DLARRB when refining a single eigenvalue
*> after the Rayleigh correction was rejected.
*> = 5: The Rayleigh Quotient Iteration failed to converge to
*> full accuracy in MAXITR steps.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrv
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRV( N, VL, VU, D, L, PIVMIN,
$ ISPLIT, M, DOL, DOU, MINRGP,
$ RTOL1, RTOL2, W, WERR, WGAP,
$ IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
$ WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER DOL, DOU, INFO, LDZ, M, N
DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
$ ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
$ WGAP( * ), WORK( * )
DOUBLE PRECISION Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXITR
PARAMETER ( MAXITR = 10 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, HALF
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, THREE = 3.0D0,
$ FOUR = 4.0D0, HALF = 0.5D0)
* ..
* .. Local Scalars ..
LOGICAL ESKIP, NEEDBS, STP2II, TRYRQC, USEDBS, USEDRQ
INTEGER DONE, I, IBEGIN, IDONE, IEND, II, IINDC1,
$ IINDC2, IINDR, IINDWK, IINFO, IM, IN, INDEIG,
$ INDLD, INDLLD, INDWRK, ISUPMN, ISUPMX, ITER,
$ ITMP1, J, JBLK, K, MINIWSIZE, MINWSIZE, NCLUS,
$ NDEPTH, NEGCNT, NEWCLS, NEWFST, NEWFTT, NEWLST,
$ NEWSIZ, OFFSET, OLDCLS, OLDFST, OLDIEN, OLDLST,
$ OLDNCL, P, PARITY, Q, WBEGIN, WEND, WINDEX,
$ WINDMN, WINDPL, ZFROM, ZTO, ZUSEDL, ZUSEDU,
$ ZUSEDW
DOUBLE PRECISION BSTRES, BSTW, EPS, FUDGE, GAP, GAPTOL, GL, GU,
$ LAMBDA, LEFT, LGAP, MINGMA, NRMINV, RESID,
$ RGAP, RIGHT, RQCORR, RQTOL, SAVGAP, SGNDEF,
$ SIGMA, SPDIAM, SSIGMA, TAU, TMP, TOL, ZTZ
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAR1V, DLARRB, DLARRF,
$ DLASET,
$ DSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
* ..
* .. Executable Statements ..
* ..
INFO = 0
*
* Quick return if possible
*
IF( (N.LE.0).OR.(M.LE.0) ) THEN
RETURN
END IF
*
* The first N entries of WORK are reserved for the eigenvalues
INDLD = N+1
INDLLD= 2*N+1
INDWRK= 3*N+1
MINWSIZE = 12 * N
DO 5 I= 1,MINWSIZE
WORK( I ) = ZERO
5 CONTINUE
* IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
* factorization used to compute the FP vector
IINDR = 0
* IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
* layer and the one above.
IINDC1 = N
IINDC2 = 2*N
IINDWK = 3*N + 1
MINIWSIZE = 7 * N
DO 10 I= 1,MINIWSIZE
IWORK( I ) = 0
10 CONTINUE
ZUSEDL = 1
IF(DOL.GT.1) THEN
* Set lower bound for use of Z
ZUSEDL = DOL-1
ENDIF
ZUSEDU = M
IF(DOU.LT.M) THEN
* Set lower bound for use of Z
ZUSEDU = DOU+1
ENDIF
* The width of the part of Z that is used
ZUSEDW = ZUSEDU - ZUSEDL + 1
CALL DLASET( 'Full', N, ZUSEDW, ZERO, ZERO,
$ Z(1,ZUSEDL), LDZ )
EPS = DLAMCH( 'Precision' )
RQTOL = TWO * EPS
*
* Set expert flags for standard code.
TRYRQC = .TRUE.
IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
ELSE
* Only selected eigenpairs are computed. Since the other evalues
* are not refined by RQ iteration, bisection has to compute to full
* accuracy.
RTOL1 = FOUR * EPS
RTOL2 = FOUR * EPS
ENDIF
* The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
* desired eigenvalues. The support of the nonzero eigenvector
* entries is contained in the interval IBEGIN:IEND.
* Remark that if k eigenpairs are desired, then the eigenvectors
* are stored in k contiguous columns of Z.
* DONE is the number of eigenvectors already computed
DONE = 0
IBEGIN = 1
WBEGIN = 1
DO 170 JBLK = 1, IBLOCK( M )
IEND = ISPLIT( JBLK )
SIGMA = L( IEND )
* Find the eigenvectors of the submatrix indexed IBEGIN
* through IEND.
WEND = WBEGIN - 1
15 CONTINUE
IF( WEND.LT.M ) THEN
IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN
WEND = WEND + 1
GO TO 15
END IF
END IF
IF( WEND.LT.WBEGIN ) THEN
IBEGIN = IEND + 1
GO TO 170
ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN
IBEGIN = IEND + 1
WBEGIN = WEND + 1
GO TO 170
END IF
* Find local spectral diameter of the block
GL = GERS( 2*IBEGIN-1 )
GU = GERS( 2*IBEGIN )
DO 20 I = IBEGIN+1 , IEND
GL = MIN( GERS( 2*I-1 ), GL )
GU = MAX( GERS( 2*I ), GU )
20 CONTINUE
SPDIAM = GU - GL
* OLDIEN is the last index of the previous block
OLDIEN = IBEGIN - 1
* Calculate the size of the current block
IN = IEND - IBEGIN + 1
* The number of eigenvalues in the current block
IM = WEND - WBEGIN + 1
* This is for a 1x1 block
IF( IBEGIN.EQ.IEND ) THEN
DONE = DONE+1
Z( IBEGIN, WBEGIN ) = ONE
ISUPPZ( 2*WBEGIN-1 ) = IBEGIN
ISUPPZ( 2*WBEGIN ) = IBEGIN
W( WBEGIN ) = W( WBEGIN ) + SIGMA
WORK( WBEGIN ) = W( WBEGIN )
IBEGIN = IEND + 1
WBEGIN = WBEGIN + 1
GO TO 170
END IF
* The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
* Note that these can be approximations, in this case, the corresp.
* entries of WERR give the size of the uncertainty interval.
* The eigenvalue approximations will be refined when necessary as
* high relative accuracy is required for the computation of the
* corresponding eigenvectors.
CALL DCOPY( IM, W( WBEGIN ), 1,
$ WORK( WBEGIN ), 1 )
* We store in W the eigenvalue approximations w.r.t. the original
* matrix T.
DO 30 I=1,IM
W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA
30 CONTINUE
* NDEPTH is the current depth of the representation tree
NDEPTH = 0
* PARITY is either 1 or 0
PARITY = 1
* NCLUS is the number of clusters for the next level of the
* representation tree, we start with NCLUS = 1 for the root
NCLUS = 1
IWORK( IINDC1+1 ) = 1
IWORK( IINDC1+2 ) = IM
* IDONE is the number of eigenvectors already computed in the current
* block
IDONE = 0
* loop while( IDONE.LT.IM )
* generate the representation tree for the current block and
* compute the eigenvectors
40 CONTINUE
IF( IDONE.LT.IM ) THEN
* This is a crude protection against infinitely deep trees
IF( NDEPTH.GT.M ) THEN
INFO = -2
RETURN
ENDIF
* breadth first processing of the current level of the representation
* tree: OLDNCL = number of clusters on current level
OLDNCL = NCLUS
* reset NCLUS to count the number of child clusters
NCLUS = 0
*
PARITY = 1 - PARITY
IF( PARITY.EQ.0 ) THEN
OLDCLS = IINDC1
NEWCLS = IINDC2
ELSE
OLDCLS = IINDC2
NEWCLS = IINDC1
END IF
* Process the clusters on the current level
DO 150 I = 1, OLDNCL
J = OLDCLS + 2*I
* OLDFST, OLDLST = first, last index of current cluster.
* cluster indices start with 1 and are relative
* to WBEGIN when accessing W, WGAP, WERR, Z
OLDFST = IWORK( J-1 )
OLDLST = IWORK( J )
IF( NDEPTH.GT.0 ) THEN
* Retrieve relatively robust representation (RRR) of cluster
* that has been computed at the previous level
* The RRR is stored in Z and overwritten once the eigenvectors
* have been computed or when the cluster is refined
IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
* Get representation from location of the leftmost evalue
* of the cluster
J = WBEGIN + OLDFST - 1
ELSE
IF(WBEGIN+OLDFST-1.LT.DOL) THEN
* Get representation from the left end of Z array
J = DOL - 1
ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN
* Get representation from the right end of Z array
J = DOU
ELSE
J = WBEGIN + OLDFST - 1
ENDIF
ENDIF
CALL DCOPY( IN, Z( IBEGIN, J ), 1, D( IBEGIN ), 1 )
CALL DCOPY( IN-1, Z( IBEGIN, J+1 ), 1, L( IBEGIN ),
$ 1 )
SIGMA = Z( IEND, J+1 )
* Set the corresponding entries in Z to zero
CALL DLASET( 'Full', IN, 2, ZERO, ZERO,
$ Z( IBEGIN, J), LDZ )
END IF
* Compute DL and DLL of current RRR
DO 50 J = IBEGIN, IEND-1
TMP = D( J )*L( J )
WORK( INDLD-1+J ) = TMP
WORK( INDLLD-1+J ) = TMP*L( J )
50 CONTINUE
IF( NDEPTH.GT.0 ) THEN
* P and Q are index of the first and last eigenvalue to compute
* within the current block
P = INDEXW( WBEGIN-1+OLDFST )
Q = INDEXW( WBEGIN-1+OLDLST )
* Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET
* through the Q-OFFSET elements of these arrays are to be used.
* OFFSET = P-OLDFST
OFFSET = INDEXW( WBEGIN ) - 1
* perform limited bisection (if necessary) to get approximate
* eigenvalues to the precision needed.
CALL DLARRB( IN, D( IBEGIN ),
$ WORK(INDLLD+IBEGIN-1),
$ P, Q, RTOL1, RTOL2, OFFSET,
$ WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN),
$ WORK( INDWRK ), IWORK( IINDWK ),
$ PIVMIN, SPDIAM, IN, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = -1
RETURN
ENDIF
* We also recompute the extremal gaps. W holds all eigenvalues
* of the unshifted matrix and must be used for computation
* of WGAP, the entries of WORK might stem from RRRs with
* different shifts. The gaps from WBEGIN-1+OLDFST to
* WBEGIN-1+OLDLST are correctly computed in DLARRB.
* However, we only allow the gaps to become greater since
* this is what should happen when we decrease WERR
IF( OLDFST.GT.1) THEN
WGAP( WBEGIN+OLDFST-2 ) =
$ MAX(WGAP(WBEGIN+OLDFST-2),
$ W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1)
$ - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) )
ENDIF
IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN
WGAP( WBEGIN+OLDLST-1 ) =
$ MAX(WGAP(WBEGIN+OLDLST-1),
$ W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST)
$ - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) )
ENDIF
* Each time the eigenvalues in WORK get refined, we store
* the newly found approximation with all shifts applied in W
DO 53 J=OLDFST,OLDLST
W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA
53 CONTINUE
END IF
* Process the current node.
NEWFST = OLDFST
DO 140 J = OLDFST, OLDLST
IF( J.EQ.OLDLST ) THEN
* we are at the right end of the cluster, this is also the
* boundary of the child cluster
NEWLST = J
ELSE IF ( WGAP( WBEGIN + J -1).GE.
$ MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN
* the right relative gap is big enough, the child cluster
* (NEWFST,..,NEWLST) is well separated from the following
NEWLST = J
ELSE
* inside a child cluster, the relative gap is not
* big enough.
GOTO 140
END IF
* Compute size of child cluster found
NEWSIZ = NEWLST - NEWFST + 1
* NEWFTT is the place in Z where the new RRR or the computed
* eigenvector is to be stored
IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
* Store representation at location of the leftmost evalue
* of the cluster
NEWFTT = WBEGIN + NEWFST - 1
ELSE
IF(WBEGIN+NEWFST-1.LT.DOL) THEN
* Store representation at the left end of Z array
NEWFTT = DOL - 1
ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN
* Store representation at the right end of Z array
NEWFTT = DOU
ELSE
NEWFTT = WBEGIN + NEWFST - 1
ENDIF
ENDIF
IF( NEWSIZ.GT.1) THEN
*
* Current child is not a singleton but a cluster.
* Compute and store new representation of child.
*
*
* Compute left and right cluster gap.
*
* LGAP and RGAP are not computed from WORK because
* the eigenvalue approximations may stem from RRRs
* different shifts. However, W hold all eigenvalues
* of the unshifted matrix. Still, the entries in WGAP
* have to be computed from WORK since the entries
* in W might be of the same order so that gaps are not
* exhibited correctly for very close eigenvalues.
IF( NEWFST.EQ.1 ) THEN
LGAP = MAX( ZERO,
$ W(WBEGIN)-WERR(WBEGIN) - VL )
ELSE
LGAP = WGAP( WBEGIN+NEWFST-2 )
ENDIF
RGAP = WGAP( WBEGIN+NEWLST-1 )
*
* Compute left- and rightmost eigenvalue of child
* to high precision in order to shift as close
* as possible and obtain as large relative gaps
* as possible
*
DO 55 K =1,2
IF(K.EQ.1) THEN
P = INDEXW( WBEGIN-1+NEWFST )
ELSE
P = INDEXW( WBEGIN-1+NEWLST )
ENDIF
OFFSET = INDEXW( WBEGIN ) - 1
CALL DLARRB( IN, D(IBEGIN),
$ WORK( INDLLD+IBEGIN-1 ),P,P,
$ RQTOL, RQTOL, OFFSET,
$ WORK(WBEGIN),WGAP(WBEGIN),
$ WERR(WBEGIN),WORK( INDWRK ),
$ IWORK( IINDWK ), PIVMIN, SPDIAM,
$ IN, IINFO )
55 CONTINUE
*
IF((WBEGIN+NEWLST-1.LT.DOL).OR.
$ (WBEGIN+NEWFST-1.GT.DOU)) THEN
* if the cluster contains no desired eigenvalues
* skip the computation of that branch of the rep. tree
*
* We could skip before the refinement of the extremal
* eigenvalues of the child, but then the representation
* tree could be different from the one when nothing is
* skipped. For this reason we skip at this place.
IDONE = IDONE + NEWLST - NEWFST + 1
GOTO 139
ENDIF
*
* Compute RRR of child cluster.
* Note that the new RRR is stored in Z
*
* DLARRF needs LWORK = 2*N
CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
$ WORK(INDLD+IBEGIN-1),
$ NEWFST, NEWLST, WORK(WBEGIN),
$ WGAP(WBEGIN), WERR(WBEGIN),
$ SPDIAM, LGAP, RGAP, PIVMIN, TAU,
$ Z(IBEGIN, NEWFTT),Z(IBEGIN, NEWFTT+1),
$ WORK( INDWRK ), IINFO )
IF( IINFO.EQ.0 ) THEN
* a new RRR for the cluster was found by DLARRF
* update shift and store it
SSIGMA = SIGMA + TAU
Z( IEND, NEWFTT+1 ) = SSIGMA
* WORK() are the midpoints and WERR() the semi-width
* Note that the entries in W are unchanged.
DO 116 K = NEWFST, NEWLST
FUDGE =
$ THREE*EPS*ABS(WORK(WBEGIN+K-1))
WORK( WBEGIN + K - 1 ) =
$ WORK( WBEGIN + K - 1) - TAU
FUDGE = FUDGE +
$ FOUR*EPS*ABS(WORK(WBEGIN+K-1))
* Fudge errors
WERR( WBEGIN + K - 1 ) =
$ WERR( WBEGIN + K - 1 ) + FUDGE
* Gaps are not fudged. Provided that WERR is small
* when eigenvalues are close, a zero gap indicates
* that a new representation is needed for resolving
* the cluster. A fudge could lead to a wrong decision
* of judging eigenvalues 'separated' which in
* reality are not. This could have a negative impact
* on the orthogonality of the computed eigenvectors.
116 CONTINUE
NCLUS = NCLUS + 1
K = NEWCLS + 2*NCLUS
IWORK( K-1 ) = NEWFST
IWORK( K ) = NEWLST
ELSE
INFO = -2
RETURN
ENDIF
ELSE
*
* Compute eigenvector of singleton
*
ITER = 0
*
TOL = FOUR * LOG(DBLE(IN)) * EPS
*
K = NEWFST
WINDEX = WBEGIN + K - 1
WINDMN = MAX(WINDEX - 1,1)
WINDPL = MIN(WINDEX + 1,M)
LAMBDA = WORK( WINDEX )
DONE = DONE + 1
* Check if eigenvector computation is to be skipped
IF((WINDEX.LT.DOL).OR.
$ (WINDEX.GT.DOU)) THEN
ESKIP = .TRUE.
GOTO 125
ELSE
ESKIP = .FALSE.
ENDIF
LEFT = WORK( WINDEX ) - WERR( WINDEX )
RIGHT = WORK( WINDEX ) + WERR( WINDEX )
INDEIG = INDEXW( WINDEX )
* Note that since we compute the eigenpairs for a child,
* all eigenvalue approximations are w.r.t the same shift.
* In this case, the entries in WORK should be used for
* computing the gaps since they exhibit even very small
* differences in the eigenvalues, as opposed to the
* entries in W which might "look" the same.
IF( K .EQ. 1) THEN
* In the case RANGE='I' and with not much initial
* accuracy in LAMBDA and VL, the formula
* LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
* can lead to an overestimation of the left gap and
* thus to inadequately early RQI 'convergence'.
* Prevent this by forcing a small left gap.
LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
ELSE
LGAP = WGAP(WINDMN)
ENDIF
IF( K .EQ. IM) THEN
* In the case RANGE='I' and with not much initial
* accuracy in LAMBDA and VU, the formula
* can lead to an overestimation of the right gap and
* thus to inadequately early RQI 'convergence'.
* Prevent this by forcing a small right gap.
RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
ELSE
RGAP = WGAP(WINDEX)
ENDIF
GAP = MIN( LGAP, RGAP )
IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN
* The eigenvector support can become wrong
* because significant entries could be cut off due to a
* large GAPTOL parameter in LAR1V. Prevent this.
GAPTOL = ZERO
ELSE
GAPTOL = GAP * EPS
ENDIF
ISUPMN = IN
ISUPMX = 1
* Update WGAP so that it holds the minimum gap
* to the left or the right. This is crucial in the
* case where bisection is used to ensure that the
* eigenvalue is refined up to the required precision.
* The correct value is restored afterwards.
SAVGAP = WGAP(WINDEX)
WGAP(WINDEX) = GAP
* We want to use the Rayleigh Quotient Correction
* as often as possible since it converges quadratically
* when we are close enough to the desired eigenvalue.
* However, the Rayleigh Quotient can have the wrong sign
* and lead us away from the desired eigenvalue. In this
* case, the best we can do is to use bisection.
USEDBS = .FALSE.
USEDRQ = .FALSE.
* Bisection is initially turned off unless it is forced
NEEDBS = .NOT.TRYRQC
120 CONTINUE
* Check if bisection should be used to refine eigenvalue
IF(NEEDBS) THEN
* Take the bisection as new iterate
USEDBS = .TRUE.
ITMP1 = IWORK( IINDR+WINDEX )
OFFSET = INDEXW( WBEGIN ) - 1
CALL DLARRB( IN, D(IBEGIN),
$ WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG,
$ ZERO, TWO*EPS, OFFSET,
$ WORK(WBEGIN),WGAP(WBEGIN),
$ WERR(WBEGIN),WORK( INDWRK ),
$ IWORK( IINDWK ), PIVMIN, SPDIAM,
$ ITMP1, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = -3
RETURN
ENDIF
LAMBDA = WORK( WINDEX )
* Reset twist index from inaccurate LAMBDA to
* force computation of true MINGMA
IWORK( IINDR+WINDEX ) = 0
ENDIF
* Given LAMBDA, compute the eigenvector.
CALL DLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
$ L( IBEGIN ), WORK(INDLD+IBEGIN-1),
$ WORK(INDLLD+IBEGIN-1),
$ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
$ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
$ IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ),
$ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
IF(ITER .EQ. 0) THEN
BSTRES = RESID
BSTW = LAMBDA
ELSEIF(RESID.LT.BSTRES) THEN
BSTRES = RESID
BSTW = LAMBDA
ENDIF
ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 ))
ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX ))
ITER = ITER + 1
* sin alpha <= |resid|/gap
* Note that both the residual and the gap are
* proportional to the matrix, so ||T|| doesn't play
* a role in the quotient
*
* Convergence test for Rayleigh-Quotient iteration
* (omitted when Bisection has been used)
*
IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
$ RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS)
$ THEN
* We need to check that the RQCORR update doesn't
* move the eigenvalue away from the desired one and
* towards a neighbor. -> protection with bisection
IF(INDEIG.LE.NEGCNT) THEN
* The wanted eigenvalue lies to the left
SGNDEF = -ONE
ELSE
* The wanted eigenvalue lies to the right
SGNDEF = ONE
ENDIF
* We only use the RQCORR if it improves the
* the iterate reasonably.
IF( ( RQCORR*SGNDEF.GE.ZERO )
$ .AND.( LAMBDA + RQCORR.LE. RIGHT)
$ .AND.( LAMBDA + RQCORR.GE. LEFT)
$ ) THEN
USEDRQ = .TRUE.
* Store new midpoint of bisection interval in WORK
IF(SGNDEF.EQ.ONE) THEN
* The current LAMBDA is on the left of the true
* eigenvalue
LEFT = LAMBDA
* We prefer to assume that the error estimate
* is correct. We could make the interval not
* as a bracket but to be modified if the RQCORR
* chooses to. In this case, the RIGHT side should
* be modified as follows:
* RIGHT = MAX(RIGHT, LAMBDA + RQCORR)
ELSE
* The current LAMBDA is on the right of the true
* eigenvalue
RIGHT = LAMBDA
* See comment about assuming the error estimate is
* correct above.
* LEFT = MIN(LEFT, LAMBDA + RQCORR)
ENDIF
WORK( WINDEX ) =
$ HALF * (RIGHT + LEFT)
* Take RQCORR since it has the correct sign and
* improves the iterate reasonably
LAMBDA = LAMBDA + RQCORR
* Update width of error interval
WERR( WINDEX ) =
$ HALF * (RIGHT-LEFT)
ELSE
NEEDBS = .TRUE.
ENDIF
IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN
* The eigenvalue is computed to bisection accuracy
* compute eigenvector and stop
USEDBS = .TRUE.
GOTO 120
ELSEIF( ITER.LT.MAXITR ) THEN
GOTO 120
ELSEIF( ITER.EQ.MAXITR ) THEN
NEEDBS = .TRUE.
GOTO 120
ELSE
INFO = 5
RETURN
END IF
ELSE
STP2II = .FALSE.
IF(USEDRQ .AND. USEDBS .AND.
$ BSTRES.LE.RESID) THEN
LAMBDA = BSTW
STP2II = .TRUE.
ENDIF
IF (STP2II) THEN
* improve error angle by second step
CALL DLAR1V( IN, 1, IN, LAMBDA,
$ D( IBEGIN ), L( IBEGIN ),
$ WORK(INDLD+IBEGIN-1),
$ WORK(INDLLD+IBEGIN-1),
$ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
$ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
$ IWORK( IINDR+WINDEX ),
$ ISUPPZ( 2*WINDEX-1 ),
$ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
ENDIF
WORK( WINDEX ) = LAMBDA
END IF
*
* Compute FP-vector support w.r.t. whole matrix
*
ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN
ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN
ZFROM = ISUPPZ( 2*WINDEX-1 )
ZTO = ISUPPZ( 2*WINDEX )
ISUPMN = ISUPMN + OLDIEN
ISUPMX = ISUPMX + OLDIEN
* Ensure vector is ok if support in the RQI has changed
IF(ISUPMN.LT.ZFROM) THEN
DO 122 II = ISUPMN,ZFROM-1
Z( II, WINDEX ) = ZERO
122 CONTINUE
ENDIF
IF(ISUPMX.GT.ZTO) THEN
DO 123 II = ZTO+1,ISUPMX
Z( II, WINDEX ) = ZERO
123 CONTINUE
ENDIF
CALL DSCAL( ZTO-ZFROM+1, NRMINV,
$ Z( ZFROM, WINDEX ), 1 )
125 CONTINUE
* Update W
W( WINDEX ) = LAMBDA+SIGMA
* Recompute the gaps on the left and right
* But only allow them to become larger and not
* smaller (which can only happen through "bad"
* cancellation and doesn't reflect the theory
* where the initial gaps are underestimated due
* to WERR being too crude.)
IF(.NOT.ESKIP) THEN
IF( K.GT.1) THEN
WGAP( WINDMN ) = MAX( WGAP(WINDMN),
$ W(WINDEX)-WERR(WINDEX)
$ - W(WINDMN)-WERR(WINDMN) )
ENDIF
IF( WINDEX.LT.WEND ) THEN
WGAP( WINDEX ) = MAX( SAVGAP,
$ W( WINDPL )-WERR( WINDPL )
$ - W( WINDEX )-WERR( WINDEX) )
ENDIF
ENDIF
IDONE = IDONE + 1
ENDIF
* here ends the code for the current child
*
139 CONTINUE
* Proceed to any remaining child nodes
NEWFST = J + 1
140 CONTINUE
150 CONTINUE
NDEPTH = NDEPTH + 1
GO TO 40
END IF
IBEGIN = IEND + 1
WBEGIN = WEND + 1
170 CONTINUE
*
RETURN
*
* End of DLARRV
*
END
*> \brief \b DLARSCL2 performs reciprocal diagonal scaling on a matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARSCL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarscl2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarscl2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarscl2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARSCL2 ( M, N, D, X, LDX )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDX
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARSCL2 performs a reciprocal diagonal scaling on a matrix:
*> x <-- inv(D) * x
*> where the diagonal matrix D is stored as a vector.
*>
*> Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
*> standard.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of D and X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of X. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (M)
*> Diagonal matrix D, stored as a vector of length M.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,N)
*> On entry, the matrix X to be scaled by D.
*> On exit, the scaled matrix.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the matrix X. LDX >= M.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larscl2
*
* =====================================================================
SUBROUTINE DLARSCL2 ( M, N, D, X, LDX )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER M, N, LDX
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. Executable Statements ..
*
DO J = 1, N
DO I = 1, M
X( I, J ) = X( I, J ) / D( I )
END DO
END DO
RETURN
END
*> \brief \b DLARTGP generates a plane rotation so that the diagonal is nonnegative.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTGP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartgp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartgp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartgp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARTGP( F, G, CS, SN, R )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CS, F, G, R, SN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARTGP generates a plane rotation so that
*>
*> [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1.
*> [ -SN CS ] [ G ] [ 0 ]
*>
*> This is a slower, more accurate version of the Level 1 BLAS routine DROTG,
*> with the following other differences:
*> F and G are unchanged on return.
*> If G=0, then CS=(+/-)1 and SN=0.
*> If F=0 and (G .ne. 0), then CS=0 and SN=(+/-)1.
*>
*> The sign is chosen so that R >= 0.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The first component of vector to be rotated.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The second component of vector to be rotated.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> The cosine of the rotation.
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> The sine of the rotation.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is DOUBLE PRECISION
*> The nonzero component of the rotated vector.
*>
*> This version has a few statements commented out for thread safety
*> (machine parameters are computed on each entry). 10 feb 03, SJH.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lartgp
*
* =====================================================================
SUBROUTINE DLARTGP( F, G, CS, SN, R )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION CS, F, G, R, SN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
* LOGICAL FIRST
INTEGER COUNT, I
DOUBLE PRECISION EPS, F1, G1, SAFMIN, SAFMN2, SAFMX2, SCALE
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, SIGN, SQRT
* ..
* .. Save statement ..
* SAVE FIRST, SAFMX2, SAFMIN, SAFMN2
* ..
* .. Data statements ..
* DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
* IF( FIRST ) THEN
SAFMIN = DLAMCH( 'S' )
EPS = DLAMCH( 'E' )
SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
$ LOG( DLAMCH( 'B' ) ) / TWO )
SAFMX2 = ONE / SAFMN2
* FIRST = .FALSE.
* END IF
IF( G.EQ.ZERO ) THEN
CS = SIGN( ONE, F )
SN = ZERO
R = ABS( F )
ELSE IF( F.EQ.ZERO ) THEN
CS = ZERO
SN = SIGN( ONE, G )
R = ABS( G )
ELSE
F1 = F
G1 = G
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 ) THEN
COUNT = 0
10 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMN2
G1 = G1*SAFMN2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 .AND. COUNT .LT. 20 )
$ GO TO 10
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 20 I = 1, COUNT
R = R*SAFMX2
20 CONTINUE
ELSE IF( SCALE.LE.SAFMN2 ) THEN
COUNT = 0
30 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMX2
G1 = G1*SAFMX2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.LE.SAFMN2 )
$ GO TO 30
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 40 I = 1, COUNT
R = R*SAFMN2
40 CONTINUE
ELSE
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
END IF
IF( R.LT.ZERO ) THEN
CS = -CS
SN = -SN
R = -R
END IF
END IF
RETURN
*
* End of DLARTGP
*
END
*> \brief \b DLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartgs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartgs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartgs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARTGS( X, Y, SIGMA, CS, SN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CS, SIGMA, SN, X, Y
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARTGS generates a plane rotation designed to introduce a bulge in
*> Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
*> problem. X and Y are the top-row entries, and SIGMA is the shift.
*> The computed CS and SN define a plane rotation satisfying
*>
*> [ CS SN ] . [ X^2 - SIGMA ] = [ R ],
*> [ -SN CS ] [ X * Y ] [ 0 ]
*>
*> with R nonnegative. If X^2 - SIGMA and X * Y are 0, then the
*> rotation is by PI/2.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION
*> The (1,1) entry of an upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*> Y is DOUBLE PRECISION
*> The (1,2) entry of an upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> The shift.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> The cosine of the rotation.
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> The sine of the rotation.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lartgs
*
* =====================================================================
SUBROUTINE DLARTGS( X, Y, SIGMA, CS, SN )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION CS, SIGMA, SN, X, Y
* ..
*
* ===================================================================
*
* .. Parameters ..
DOUBLE PRECISION NEGONE, ONE, ZERO
PARAMETER ( NEGONE = -1.0D0, ONE = 1.0D0, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION R, S, THRESH, W, Z
* ..
* .. External Subroutines ..
EXTERNAL DLARTGP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* .. Executable Statements ..
*
THRESH = DLAMCH('E')
*
* Compute the first column of B**T*B - SIGMA^2*I, up to a scale
* factor.
*
IF( (SIGMA .EQ. ZERO .AND. ABS(X) .LT. THRESH) .OR.
$ (ABS(X) .EQ. SIGMA .AND. Y .EQ. ZERO) ) THEN
Z = ZERO
W = ZERO
ELSE IF( SIGMA .EQ. ZERO ) THEN
IF( X .GE. ZERO ) THEN
Z = X
W = Y
ELSE
Z = -X
W = -Y
END IF
ELSE IF( ABS(X) .LT. THRESH ) THEN
Z = -SIGMA*SIGMA
W = ZERO
ELSE
IF( X .GE. ZERO ) THEN
S = ONE
ELSE
S = NEGONE
END IF
Z = S * (ABS(X)-SIGMA) * (S+SIGMA/X)
W = S * Y
END IF
*
* Generate the rotation.
* CALL DLARTGP( Z, W, CS, SN, R ) might seem more natural;
* reordering the arguments ensures that if Z = 0 then the rotation
* is by PI/2.
*
CALL DLARTGP( W, Z, SN, CS, R )
*
RETURN
*
* End DLARTGS
*
END
*> \brief \b DLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARTV( N, X, INCX, Y, INCY, C, S, INCC )
*
* .. Scalar Arguments ..
* INTEGER INCC, INCX, INCY, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( * ), S( * ), X( * ), Y( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARTV applies a vector of real plane rotations to elements of the
*> real vectors x and y. For i = 1,2,...,n
*>
*> ( x(i) ) := ( c(i) s(i) ) ( x(i) )
*> ( y(i) ) ( -s(i) c(i) ) ( y(i) )
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of plane rotations to be applied.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCX)
*> The vector x.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array,
*> dimension (1+(N-1)*INCY)
*> The vector y.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> The increment between elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*> The cosines of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*> The sines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCC
*> \verbatim
*> INCC is INTEGER
*> The increment between elements of C and S. INCC > 0.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lartv
*
* =====================================================================
SUBROUTINE DLARTV( N, X, INCX, Y, INCY, C, S, INCC )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCC, INCX, INCY, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( * ), S( * ), X( * ), Y( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IC, IX, IY
DOUBLE PRECISION XI, YI
* ..
* .. Executable Statements ..
*
IX = 1
IY = 1
IC = 1
DO 10 I = 1, N
XI = X( IX )
YI = Y( IY )
X( IX ) = C( IC )*XI + S( IC )*YI
Y( IY ) = C( IC )*YI - S( IC )*XI
IX = IX + INCX
IY = IY + INCY
IC = IC + INCC
10 CONTINUE
RETURN
*
* End of DLARTV
*
END
*> \brief \b DLARUV returns a vector of n random real numbers from a uniform distribution.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARUV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaruv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaruv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaruv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARUV( ISEED, N, X )
*
* .. Scalar Arguments ..
* INTEGER N
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* DOUBLE PRECISION X( N )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARUV returns a vector of n random real numbers from a uniform (0,1)
*> distribution (n <= 128).
*>
*> This is an auxiliary routine called by DLARNV and ZLARNV.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] ISEED
*> \verbatim
*> ISEED is INTEGER array, dimension (4)
*> On entry, the seed of the random number generator; the array
*> elements must be between 0 and 4095, and ISEED(4) must be
*> odd.
*> On exit, the seed is updated.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of random numbers to be generated. N <= 128.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> The generated random numbers.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laruv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> This routine uses a multiplicative congruential method with modulus
*> 2**48 and multiplier 33952834046453 (see G.S.Fishman,
*> 'Multiplicative congruential random number generators with modulus
*> 2**b: an exhaustive analysis for b = 32 and a partial analysis for
*> b = 48', Math. Comp. 189, pp 331-344, 1990).
*>
*> 48-bit integers are stored in 4 integer array elements with 12 bits
*> per element. Hence the routine is portable across machines with
*> integers of 32 bits or more.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARUV( ISEED, N, X )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER N
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
DOUBLE PRECISION X( N )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
INTEGER LV, IPW2
DOUBLE PRECISION R
PARAMETER ( LV = 128, IPW2 = 4096, R = ONE / IPW2 )
* ..
* .. Local Scalars ..
INTEGER I, I1, I2, I3, I4, IT1, IT2, IT3, IT4, J
* ..
* .. Local Arrays ..
INTEGER MM( LV, 4 )
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MIN, MOD
* ..
* .. Data statements ..
DATA ( MM( 1, J ), J = 1, 4 ) / 494, 322, 2508,
$ 2549 /
DATA ( MM( 2, J ), J = 1, 4 ) / 2637, 789, 3754,
$ 1145 /
DATA ( MM( 3, J ), J = 1, 4 ) / 255, 1440, 1766,
$ 2253 /
DATA ( MM( 4, J ), J = 1, 4 ) / 2008, 752, 3572,
$ 305 /
DATA ( MM( 5, J ), J = 1, 4 ) / 1253, 2859, 2893,
$ 3301 /
DATA ( MM( 6, J ), J = 1, 4 ) / 3344, 123, 307,
$ 1065 /
DATA ( MM( 7, J ), J = 1, 4 ) / 4084, 1848, 1297,
$ 3133 /
DATA ( MM( 8, J ), J = 1, 4 ) / 1739, 643, 3966,
$ 2913 /
DATA ( MM( 9, J ), J = 1, 4 ) / 3143, 2405, 758,
$ 3285 /
DATA ( MM( 10, J ), J = 1, 4 ) / 3468, 2638, 2598,
$ 1241 /
DATA ( MM( 11, J ), J = 1, 4 ) / 688, 2344, 3406,
$ 1197 /
DATA ( MM( 12, J ), J = 1, 4 ) / 1657, 46, 2922,
$ 3729 /
DATA ( MM( 13, J ), J = 1, 4 ) / 1238, 3814, 1038,
$ 2501 /
DATA ( MM( 14, J ), J = 1, 4 ) / 3166, 913, 2934,
$ 1673 /
DATA ( MM( 15, J ), J = 1, 4 ) / 1292, 3649, 2091,
$ 541 /
DATA ( MM( 16, J ), J = 1, 4 ) / 3422, 339, 2451,
$ 2753 /
DATA ( MM( 17, J ), J = 1, 4 ) / 1270, 3808, 1580,
$ 949 /
DATA ( MM( 18, J ), J = 1, 4 ) / 2016, 822, 1958,
$ 2361 /
DATA ( MM( 19, J ), J = 1, 4 ) / 154, 2832, 2055,
$ 1165 /
DATA ( MM( 20, J ), J = 1, 4 ) / 2862, 3078, 1507,
$ 4081 /
DATA ( MM( 21, J ), J = 1, 4 ) / 697, 3633, 1078,
$ 2725 /
DATA ( MM( 22, J ), J = 1, 4 ) / 1706, 2970, 3273,
$ 3305 /
DATA ( MM( 23, J ), J = 1, 4 ) / 491, 637, 17,
$ 3069 /
DATA ( MM( 24, J ), J = 1, 4 ) / 931, 2249, 854,
$ 3617 /
DATA ( MM( 25, J ), J = 1, 4 ) / 1444, 2081, 2916,
$ 3733 /
DATA ( MM( 26, J ), J = 1, 4 ) / 444, 4019, 3971,
$ 409 /
DATA ( MM( 27, J ), J = 1, 4 ) / 3577, 1478, 2889,
$ 2157 /
DATA ( MM( 28, J ), J = 1, 4 ) / 3944, 242, 3831,
$ 1361 /
DATA ( MM( 29, J ), J = 1, 4 ) / 2184, 481, 2621,
$ 3973 /
DATA ( MM( 30, J ), J = 1, 4 ) / 1661, 2075, 1541,
$ 1865 /
DATA ( MM( 31, J ), J = 1, 4 ) / 3482, 4058, 893,
$ 2525 /
DATA ( MM( 32, J ), J = 1, 4 ) / 657, 622, 736,
$ 1409 /
DATA ( MM( 33, J ), J = 1, 4 ) / 3023, 3376, 3992,
$ 3445 /
DATA ( MM( 34, J ), J = 1, 4 ) / 3618, 812, 787,
$ 3577 /
DATA ( MM( 35, J ), J = 1, 4 ) / 1267, 234, 2125,
$ 77 /
DATA ( MM( 36, J ), J = 1, 4 ) / 1828, 641, 2364,
$ 3761 /
DATA ( MM( 37, J ), J = 1, 4 ) / 164, 4005, 2460,
$ 2149 /
DATA ( MM( 38, J ), J = 1, 4 ) / 3798, 1122, 257,
$ 1449 /
DATA ( MM( 39, J ), J = 1, 4 ) / 3087, 3135, 1574,
$ 3005 /
DATA ( MM( 40, J ), J = 1, 4 ) / 2400, 2640, 3912,
$ 225 /
DATA ( MM( 41, J ), J = 1, 4 ) / 2870, 2302, 1216,
$ 85 /
DATA ( MM( 42, J ), J = 1, 4 ) / 3876, 40, 3248,
$ 3673 /
DATA ( MM( 43, J ), J = 1, 4 ) / 1905, 1832, 3401,
$ 3117 /
DATA ( MM( 44, J ), J = 1, 4 ) / 1593, 2247, 2124,
$ 3089 /
DATA ( MM( 45, J ), J = 1, 4 ) / 1797, 2034, 2762,
$ 1349 /
DATA ( MM( 46, J ), J = 1, 4 ) / 1234, 2637, 149,
$ 2057 /
DATA ( MM( 47, J ), J = 1, 4 ) / 3460, 1287, 2245,
$ 413 /
DATA ( MM( 48, J ), J = 1, 4 ) / 328, 1691, 166,
$ 65 /
DATA ( MM( 49, J ), J = 1, 4 ) / 2861, 496, 466,
$ 1845 /
DATA ( MM( 50, J ), J = 1, 4 ) / 1950, 1597, 4018,
$ 697 /
DATA ( MM( 51, J ), J = 1, 4 ) / 617, 2394, 1399,
$ 3085 /
DATA ( MM( 52, J ), J = 1, 4 ) / 2070, 2584, 190,
$ 3441 /
DATA ( MM( 53, J ), J = 1, 4 ) / 3331, 1843, 2879,
$ 1573 /
DATA ( MM( 54, J ), J = 1, 4 ) / 769, 336, 153,
$ 3689 /
DATA ( MM( 55, J ), J = 1, 4 ) / 1558, 1472, 2320,
$ 2941 /
DATA ( MM( 56, J ), J = 1, 4 ) / 2412, 2407, 18,
$ 929 /
DATA ( MM( 57, J ), J = 1, 4 ) / 2800, 433, 712,
$ 533 /
DATA ( MM( 58, J ), J = 1, 4 ) / 189, 2096, 2159,
$ 2841 /
DATA ( MM( 59, J ), J = 1, 4 ) / 287, 1761, 2318,
$ 4077 /
DATA ( MM( 60, J ), J = 1, 4 ) / 2045, 2810, 2091,
$ 721 /
DATA ( MM( 61, J ), J = 1, 4 ) / 1227, 566, 3443,
$ 2821 /
DATA ( MM( 62, J ), J = 1, 4 ) / 2838, 442, 1510,
$ 2249 /
DATA ( MM( 63, J ), J = 1, 4 ) / 209, 41, 449,
$ 2397 /
DATA ( MM( 64, J ), J = 1, 4 ) / 2770, 1238, 1956,
$ 2817 /
DATA ( MM( 65, J ), J = 1, 4 ) / 3654, 1086, 2201,
$ 245 /
DATA ( MM( 66, J ), J = 1, 4 ) / 3993, 603, 3137,
$ 1913 /
DATA ( MM( 67, J ), J = 1, 4 ) / 192, 840, 3399,
$ 1997 /
DATA ( MM( 68, J ), J = 1, 4 ) / 2253, 3168, 1321,
$ 3121 /
DATA ( MM( 69, J ), J = 1, 4 ) / 3491, 1499, 2271,
$ 997 /
DATA ( MM( 70, J ), J = 1, 4 ) / 2889, 1084, 3667,
$ 1833 /
DATA ( MM( 71, J ), J = 1, 4 ) / 2857, 3438, 2703,
$ 2877 /
DATA ( MM( 72, J ), J = 1, 4 ) / 2094, 2408, 629,
$ 1633 /
DATA ( MM( 73, J ), J = 1, 4 ) / 1818, 1589, 2365,
$ 981 /
DATA ( MM( 74, J ), J = 1, 4 ) / 688, 2391, 2431,
$ 2009 /
DATA ( MM( 75, J ), J = 1, 4 ) / 1407, 288, 1113,
$ 941 /
DATA ( MM( 76, J ), J = 1, 4 ) / 634, 26, 3922,
$ 2449 /
DATA ( MM( 77, J ), J = 1, 4 ) / 3231, 512, 2554,
$ 197 /
DATA ( MM( 78, J ), J = 1, 4 ) / 815, 1456, 184,
$ 2441 /
DATA ( MM( 79, J ), J = 1, 4 ) / 3524, 171, 2099,
$ 285 /
DATA ( MM( 80, J ), J = 1, 4 ) / 1914, 1677, 3228,
$ 1473 /
DATA ( MM( 81, J ), J = 1, 4 ) / 516, 2657, 4012,
$ 2741 /
DATA ( MM( 82, J ), J = 1, 4 ) / 164, 2270, 1921,
$ 3129 /
DATA ( MM( 83, J ), J = 1, 4 ) / 303, 2587, 3452,
$ 909 /
DATA ( MM( 84, J ), J = 1, 4 ) / 2144, 2961, 3901,
$ 2801 /
DATA ( MM( 85, J ), J = 1, 4 ) / 3480, 1970, 572,
$ 421 /
DATA ( MM( 86, J ), J = 1, 4 ) / 119, 1817, 3309,
$ 4073 /
DATA ( MM( 87, J ), J = 1, 4 ) / 3357, 676, 3171,
$ 2813 /
DATA ( MM( 88, J ), J = 1, 4 ) / 837, 1410, 817,
$ 2337 /
DATA ( MM( 89, J ), J = 1, 4 ) / 2826, 3723, 3039,
$ 1429 /
DATA ( MM( 90, J ), J = 1, 4 ) / 2332, 2803, 1696,
$ 1177 /
DATA ( MM( 91, J ), J = 1, 4 ) / 2089, 3185, 1256,
$ 1901 /
DATA ( MM( 92, J ), J = 1, 4 ) / 3780, 184, 3715,
$ 81 /
DATA ( MM( 93, J ), J = 1, 4 ) / 1700, 663, 2077,
$ 1669 /
DATA ( MM( 94, J ), J = 1, 4 ) / 3712, 499, 3019,
$ 2633 /
DATA ( MM( 95, J ), J = 1, 4 ) / 150, 3784, 1497,
$ 2269 /
DATA ( MM( 96, J ), J = 1, 4 ) / 2000, 1631, 1101,
$ 129 /
DATA ( MM( 97, J ), J = 1, 4 ) / 3375, 1925, 717,
$ 1141 /
DATA ( MM( 98, J ), J = 1, 4 ) / 1621, 3912, 51,
$ 249 /
DATA ( MM( 99, J ), J = 1, 4 ) / 3090, 1398, 981,
$ 3917 /
DATA ( MM( 100, J ), J = 1, 4 ) / 3765, 1349, 1978,
$ 2481 /
DATA ( MM( 101, J ), J = 1, 4 ) / 1149, 1441, 1813,
$ 3941 /
DATA ( MM( 102, J ), J = 1, 4 ) / 3146, 2224, 3881,
$ 2217 /
DATA ( MM( 103, J ), J = 1, 4 ) / 33, 2411, 76,
$ 2749 /
DATA ( MM( 104, J ), J = 1, 4 ) / 3082, 1907, 3846,
$ 3041 /
DATA ( MM( 105, J ), J = 1, 4 ) / 2741, 3192, 3694,
$ 1877 /
DATA ( MM( 106, J ), J = 1, 4 ) / 359, 2786, 1682,
$ 345 /
DATA ( MM( 107, J ), J = 1, 4 ) / 3316, 382, 124,
$ 2861 /
DATA ( MM( 108, J ), J = 1, 4 ) / 1749, 37, 1660,
$ 1809 /
DATA ( MM( 109, J ), J = 1, 4 ) / 185, 759, 3997,
$ 3141 /
DATA ( MM( 110, J ), J = 1, 4 ) / 2784, 2948, 479,
$ 2825 /
DATA ( MM( 111, J ), J = 1, 4 ) / 2202, 1862, 1141,
$ 157 /
DATA ( MM( 112, J ), J = 1, 4 ) / 2199, 3802, 886,
$ 2881 /
DATA ( MM( 113, J ), J = 1, 4 ) / 1364, 2423, 3514,
$ 3637 /
DATA ( MM( 114, J ), J = 1, 4 ) / 1244, 2051, 1301,
$ 1465 /
DATA ( MM( 115, J ), J = 1, 4 ) / 2020, 2295, 3604,
$ 2829 /
DATA ( MM( 116, J ), J = 1, 4 ) / 3160, 1332, 1888,
$ 2161 /
DATA ( MM( 117, J ), J = 1, 4 ) / 2785, 1832, 1836,
$ 3365 /
DATA ( MM( 118, J ), J = 1, 4 ) / 2772, 2405, 1990,
$ 361 /
DATA ( MM( 119, J ), J = 1, 4 ) / 1217, 3638, 2058,
$ 2685 /
DATA ( MM( 120, J ), J = 1, 4 ) / 1822, 3661, 692,
$ 3745 /
DATA ( MM( 121, J ), J = 1, 4 ) / 1245, 327, 1194,
$ 2325 /
DATA ( MM( 122, J ), J = 1, 4 ) / 2252, 3660, 20,
$ 3609 /
DATA ( MM( 123, J ), J = 1, 4 ) / 3904, 716, 3285,
$ 3821 /
DATA ( MM( 124, J ), J = 1, 4 ) / 2774, 1842, 2046,
$ 3537 /
DATA ( MM( 125, J ), J = 1, 4 ) / 997, 3987, 2107,
$ 517 /
DATA ( MM( 126, J ), J = 1, 4 ) / 2573, 1368, 3508,
$ 3017 /
DATA ( MM( 127, J ), J = 1, 4 ) / 1148, 1848, 3525,
$ 2141 /
DATA ( MM( 128, J ), J = 1, 4 ) / 545, 2366, 3801,
$ 1537 /
* ..
* .. Executable Statements ..
*
* Quick return for N < 1
IF ( N < 1 ) THEN
RETURN
END IF
*
I1 = ISEED( 1 )
I2 = ISEED( 2 )
I3 = ISEED( 3 )
I4 = ISEED( 4 )
*
DO 10 I = 1, MIN( N, LV )
*
20 CONTINUE
*
* Multiply the seed by i-th power of the multiplier modulo 2**48
*
IT4 = I4*MM( I, 4 )
IT3 = IT4 / IPW2
IT4 = IT4 - IPW2*IT3
IT3 = IT3 + I3*MM( I, 4 ) + I4*MM( I, 3 )
IT2 = IT3 / IPW2
IT3 = IT3 - IPW2*IT2
IT2 = IT2 + I2*MM( I, 4 ) + I3*MM( I, 3 ) + I4*MM( I, 2 )
IT1 = IT2 / IPW2
IT2 = IT2 - IPW2*IT1
IT1 = IT1 + I1*MM( I, 4 ) + I2*MM( I, 3 ) + I3*MM( I, 2 ) +
$ I4*MM( I, 1 )
IT1 = MOD( IT1, IPW2 )
*
* Convert 48-bit integer to a real number in the interval (0,1)
*
X( I ) = R*( DBLE( IT1 )+R*( DBLE( IT2 )+R*( DBLE( IT3 )+R*
$ DBLE( IT4 ) ) ) )
*
IF (X( I ).EQ.1.0D0) THEN
* If a real number has n bits of precision, and the first
* n bits of the 48-bit integer above happen to be all 1 (which
* will occur about once every 2**n calls), then X( I ) will
* be rounded to exactly 1.0.
* Since X( I ) is not supposed to return exactly 0.0 or 1.0,
* the statistically correct thing to do in this situation is
* simply to iterate again.
* N.B. the case X( I ) = 0.0 should not be possible.
I1 = I1 + 2
I2 = I2 + 2
I3 = I3 + 2
I4 = I4 + 2
GOTO 20
END IF
*
10 CONTINUE
*
* Return final value of seed
*
ISEED( 1 ) = IT1
ISEED( 2 ) = IT2
ISEED( 3 ) = IT3
ISEED( 4 ) = IT4
RETURN
*
* End of DLARUV
*
END
*> \brief \b DLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARZ( SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER INCV, L, LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARZ applies a real elementary reflector H to a real M-by-N
*> matrix C, from either the left or the right. H is represented in the
*> form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*>
*>
*> H is a product of k elementary reflectors as returned by DTZRZF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of entries of the vector V containing
*> the meaningful part of the Householder vectors.
*> If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (1+(L-1)*abs(INCV))
*> The vector v in the representation of H as returned by
*> DTZRZF. V is not used if TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*> INCV is INTEGER
*> The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larz
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARZ( SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, L, LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGER
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C
*
IF( TAU.NE.ZERO ) THEN
*
* w( 1:n ) = C( 1, 1:n )
*
CALL DCOPY( N, C, LDC, WORK, 1 )
*
* w( 1:n ) = w( 1:n ) + C( m-l+1:m, 1:n )**T * v( 1:l )
*
CALL DGEMV( 'Transpose', L, N, ONE, C( M-L+1, 1 ), LDC,
$ V,
$ INCV, ONE, WORK, 1 )
*
* C( 1, 1:n ) = C( 1, 1:n ) - tau * w( 1:n )
*
CALL DAXPY( N, -TAU, WORK, 1, C, LDC )
*
* C( m-l+1:m, 1:n ) = C( m-l+1:m, 1:n ) - ...
* tau * v( 1:l ) * w( 1:n )**T
*
CALL DGER( L, N, -TAU, V, INCV, WORK, 1, C( M-L+1, 1 ),
$ LDC )
END IF
*
ELSE
*
* Form C * H
*
IF( TAU.NE.ZERO ) THEN
*
* w( 1:m ) = C( 1:m, 1 )
*
CALL DCOPY( M, C, 1, WORK, 1 )
*
* w( 1:m ) = w( 1:m ) + C( 1:m, n-l+1:n, 1:n ) * v( 1:l )
*
CALL DGEMV( 'No transpose', M, L, ONE, C( 1, N-L+1 ),
$ LDC,
$ V, INCV, ONE, WORK, 1 )
*
* C( 1:m, 1 ) = C( 1:m, 1 ) - tau * w( 1:m )
*
CALL DAXPY( M, -TAU, WORK, 1, C, 1 )
*
* C( 1:m, n-l+1:n ) = C( 1:m, n-l+1:n ) - ...
* tau * w( 1:m ) * v( 1:l )**T
*
CALL DGER( M, L, -TAU, WORK, 1, V, INCV, C( 1, N-L+1 ),
$ LDC )
*
END IF
*
END IF
*
RETURN
*
* End of DLARZ
*
END
*> \brief \b DLARZB applies a block reflector or its transpose to a general matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARZB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarzb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarzb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarzb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V,
* LDV, T, LDT, C, LDC, WORK, LDWORK )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, SIDE, STOREV, TRANS
* INTEGER K, L, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
* $ WORK( LDWORK, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARZB applies a real block reflector H or its transpose H**T to
*> a real distributed M-by-N C from the left or the right.
*>
*> Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply H or H**T from the Left
*> = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply H (No transpose)
*> = 'C': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Indicates how H is formed from a product of elementary
*> reflectors
*> = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Indicates how the vectors which define the elementary
*> reflectors are stored:
*> = 'C': Columnwise (not supported yet)
*> = 'R': Rowwise
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the matrix T (= the number of elementary
*> reflectors whose product defines the block reflector).
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of columns of the matrix V containing the
*> meaningful part of the Householder reflectors.
*> If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NV).
*> If STOREV = 'C', NV = K; if STOREV = 'R', NV = L.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The triangular K-by-K matrix T in the representation of the
*> block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LDWORK,K)
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*> LDWORK is INTEGER
*> The leading dimension of the array WORK.
*> If SIDE = 'L', LDWORK >= max(1,N);
*> if SIDE = 'R', LDWORK >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larzb
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V,
$ LDV, T, LDT, C, LDC, WORK, LDWORK )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, SIDE, STOREV, TRANS
INTEGER K, L, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
$ WORK( LDWORK, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
CHARACTER TRANST
INTEGER I, INFO, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DTRMM, XERBLA
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
* Check for currently supported options
*
INFO = 0
IF( .NOT.LSAME( DIRECT, 'B' ) ) THEN
INFO = -3
ELSE IF( .NOT.LSAME( STOREV, 'R' ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLARZB', -INFO )
RETURN
END IF
*
IF( LSAME( TRANS, 'N' ) ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C
*
* W( 1:n, 1:k ) = C( 1:k, 1:n )**T
*
DO 10 J = 1, K
CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
10 CONTINUE
*
* W( 1:n, 1:k ) = W( 1:n, 1:k ) + ...
* C( m-l+1:m, 1:n )**T * V( 1:k, 1:l )**T
*
IF( L.GT.0 )
$ CALL DGEMM( 'Transpose', 'Transpose', N, K, L, ONE,
$ C( M-L+1, 1 ), LDC, V, LDV, ONE, WORK, LDWORK )
*
* W( 1:n, 1:k ) = W( 1:n, 1:k ) * T**T or W( 1:m, 1:k ) * T
*
CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, ONE,
$ T,
$ LDT, WORK, LDWORK )
*
* C( 1:k, 1:n ) = C( 1:k, 1:n ) - W( 1:n, 1:k )**T
*
DO 30 J = 1, N
DO 20 I = 1, K
C( I, J ) = C( I, J ) - WORK( J, I )
20 CONTINUE
30 CONTINUE
*
* C( m-l+1:m, 1:n ) = C( m-l+1:m, 1:n ) - ...
* V( 1:k, 1:l )**T * W( 1:n, 1:k )**T
*
IF( L.GT.0 )
$ CALL DGEMM( 'Transpose', 'Transpose', L, N, K, -ONE, V,
$ LDV,
$ WORK, LDWORK, ONE, C( M-L+1, 1 ), LDC )
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T
*
* W( 1:m, 1:k ) = C( 1:m, 1:k )
*
DO 40 J = 1, K
CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
40 CONTINUE
*
* W( 1:m, 1:k ) = W( 1:m, 1:k ) + ...
* C( 1:m, n-l+1:n ) * V( 1:k, 1:l )**T
*
IF( L.GT.0 )
$ CALL DGEMM( 'No transpose', 'Transpose', M, K, L, ONE,
$ C( 1, N-L+1 ), LDC, V, LDV, ONE, WORK, LDWORK )
*
* W( 1:m, 1:k ) = W( 1:m, 1:k ) * T or W( 1:m, 1:k ) * T**T
*
CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K, ONE,
$ T,
$ LDT, WORK, LDWORK )
*
* C( 1:m, 1:k ) = C( 1:m, 1:k ) - W( 1:m, 1:k )
*
DO 60 J = 1, K
DO 50 I = 1, M
C( I, J ) = C( I, J ) - WORK( I, J )
50 CONTINUE
60 CONTINUE
*
* C( 1:m, n-l+1:n ) = C( 1:m, n-l+1:n ) - ...
* W( 1:m, 1:k ) * V( 1:k, 1:l )
*
IF( L.GT.0 )
$ CALL DGEMM( 'No transpose', 'No transpose', M, L, K,
$ -ONE,
$ WORK, LDWORK, V, LDV, ONE, C( 1, N-L+1 ), LDC )
*
END IF
*
RETURN
*
* End of DLARZB
*
END
*> \brief \b DLARZT forms the triangular factor T of a block reflector H = I - vtvH.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARZT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarzt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarzt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarzt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
* INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARZT forms the triangular factor T of a real block reflector
*> H of order > n, which is defined as a product of k elementary
*> reflectors.
*>
*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*>
*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*>
*> If STOREV = 'C', the vector which defines the elementary reflector
*> H(i) is stored in the i-th column of the array V, and
*>
*> H = I - V * T * V**T
*>
*> If STOREV = 'R', the vector which defines the elementary reflector
*> H(i) is stored in the i-th row of the array V, and
*>
*> H = I - V**T * T * V
*>
*> Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies the order in which the elementary reflectors are
*> multiplied to form the block reflector:
*> = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Specifies how the vectors which define the elementary
*> reflectors are stored (see also Further Details):
*> = 'C': columnwise (not supported yet)
*> = 'R': rowwise
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the block reflector H. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the triangular factor T (= the number of
*> elementary reflectors). K >= 1.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,N) if STOREV = 'R'
*> The matrix V. See further details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The k by k triangular factor T of the block reflector.
*> If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
*> lower triangular. The rest of the array is not used.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larzt
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The elements equal to 1 are not stored; the corresponding
*> array elements are modified but restored on exit. The rest of the
*> array is not used.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> ______V_____
*> ( v1 v2 v3 ) / \
*> ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
*> V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
*> ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
*> ( v1 v2 v3 )
*> . . .
*> . . .
*> 1 . .
*> 1 .
*> 1
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> ______V_____
*> 1 / \
*> . 1 ( 1 . . . . v1 v1 v1 v1 v1 )
*> . . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
*> . . . ( . . 1 . . v3 v3 v3 v3 v3 )
*> . . .
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*> V = ( v1 v2 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DTRMV, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Check for currently supported options
*
INFO = 0
IF( .NOT.LSAME( DIRECT, 'B' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( STOREV, 'R' ) ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLARZT', -INFO )
RETURN
END IF
*
DO 20 I = K, 1, -1
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO 10 J = I, K
T( J, I ) = ZERO
10 CONTINUE
ELSE
*
* general case
*
IF( I.LT.K ) THEN
*
* T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)**T
*
CALL DGEMV( 'No transpose', K-I, N, -TAU( I ),
$ V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO,
$ T( I+1, I ), 1 )
*
* T(i+1:k,i) = T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
$ T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
END IF
T( I, I ) = TAU( I )
END IF
20 CONTINUE
RETURN
*
* End of DLARZT
*
END
*> \brief \b DLAS2 computes singular values of a 2-by-2 triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlas2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlas2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlas2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION F, G, H, SSMAX, SSMIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAS2 computes the singular values of the 2-by-2 matrix
*> [ F G ]
*> [ 0 H ].
*> On return, SSMIN is the smaller singular value and SSMAX is the
*> larger singular value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is DOUBLE PRECISION
*> The smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*> SSMAX is DOUBLE PRECISION
*> The larger singular value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup las2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Barring over/underflow, all output quantities are correct to within
*> a few units in the last place (ulps), even in the absence of a guard
*> digit in addition/subtraction.
*>
*> In IEEE arithmetic, the code works correctly if one matrix element is
*> infinite.
*>
*> Overflow will not occur unless the largest singular value itself
*> overflows, or is within a few ulps of overflow.
*>
*> Underflow is harmless if underflow is gradual. Otherwise, results
*> may correspond to a matrix modified by perturbations of size near
*> the underflow threshold.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION F, G, H, SSMAX, SSMIN
* ..
*
* ====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
FA = ABS( F )
GA = ABS( G )
HA = ABS( H )
FHMN = MIN( FA, HA )
FHMX = MAX( FA, HA )
IF( FHMN.EQ.ZERO ) THEN
SSMIN = ZERO
IF( FHMX.EQ.ZERO ) THEN
SSMAX = GA
ELSE
SSMAX = MAX( FHMX, GA )*SQRT( ONE+
$ ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
END IF
ELSE
IF( GA.LT.FHMX ) THEN
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
AU = ( GA / FHMX )**2
C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
SSMIN = FHMN*C
SSMAX = FHMX / C
ELSE
AU = FHMX / GA
IF( AU.EQ.ZERO ) THEN
*
* Avoid possible harmful underflow if exponent range
* asymmetric (true SSMIN may not underflow even if
* AU underflows)
*
SSMIN = ( FHMN*FHMX ) / GA
SSMAX = GA
ELSE
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
$ SQRT( ONE+( AT*AU )**2 ) )
SSMIN = ( FHMN*C )*AU
SSMIN = SSMIN + SSMIN
SSMAX = GA / ( C+C )
END IF
END IF
END IF
RETURN
*
* End of DLAS2
*
END
*> \brief \b DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASCL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlascl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlascl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlascl.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TYPE
* INTEGER INFO, KL, KU, LDA, M, N
* DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASCL multiplies the M by N real matrix A by the real scalar
*> CTO/CFROM. This is done without over/underflow as long as the final
*> result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
*> A may be full, upper triangular, lower triangular, upper Hessenberg,
*> or banded.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TYPE
*> \verbatim
*> TYPE is CHARACTER*1
*> TYPE indices the storage type of the input matrix.
*> = 'G': A is a full matrix.
*> = 'L': A is a lower triangular matrix.
*> = 'U': A is an upper triangular matrix.
*> = 'H': A is an upper Hessenberg matrix.
*> = 'B': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the lower
*> half stored.
*> = 'Q': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the upper
*> half stored.
*> = 'Z': A is a band matrix with lower bandwidth KL and upper
*> bandwidth KU. See DGBTRF for storage details.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The lower bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The upper bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] CFROM
*> \verbatim
*> CFROM is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CTO
*> \verbatim
*> CTO is DOUBLE PRECISION
*>
*> The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
*> without over/underflow if the final result CTO*A(I,J)/CFROM
*> can be represented without over/underflow. CFROM must be
*> nonzero.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The matrix to be multiplied by CTO/CFROM. See TYPE for the
*> storage type.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
*> TYPE = 'B', LDA >= KL+1;
*> TYPE = 'Q', LDA >= KU+1;
*> TYPE = 'Z', LDA >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> 0 - successful exit
*> <0 - if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lascl
*
* =====================================================================
SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA,
$ INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, KL, KU, LDA, M, N
DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL DONE
INTEGER I, ITYPE, J, K1, K2, K3, K4
DOUBLE PRECISION BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
*
IF( LSAME( TYPE, 'G' ) ) THEN
ITYPE = 0
ELSE IF( LSAME( TYPE, 'L' ) ) THEN
ITYPE = 1
ELSE IF( LSAME( TYPE, 'U' ) ) THEN
ITYPE = 2
ELSE IF( LSAME( TYPE, 'H' ) ) THEN
ITYPE = 3
ELSE IF( LSAME( TYPE, 'B' ) ) THEN
ITYPE = 4
ELSE IF( LSAME( TYPE, 'Q' ) ) THEN
ITYPE = 5
ELSE IF( LSAME( TYPE, 'Z' ) ) THEN
ITYPE = 6
ELSE
ITYPE = -1
END IF
*
IF( ITYPE.EQ.-1 ) THEN
INFO = -1
ELSE IF( CFROM.EQ.ZERO .OR. DISNAN(CFROM) ) THEN
INFO = -4
ELSE IF( DISNAN(CTO) ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR.
$ ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN
INFO = -7
ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN
INFO = -9
ELSE IF( ITYPE.GE.4 ) THEN
IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN
INFO = -2
ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR.
$ ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) )
$ THEN
INFO = -3
ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR.
$ ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR.
$ ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN
INFO = -9
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASCL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 )
$ RETURN
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
CFROMC = CFROM
CTOC = CTO
*
10 CONTINUE
CFROM1 = CFROMC*SMLNUM
IF( CFROM1.EQ.CFROMC ) THEN
! CFROMC is an inf. Multiply by a correctly signed zero for
! finite CTOC, or a NaN if CTOC is infinite.
MUL = CTOC / CFROMC
DONE = .TRUE.
CTO1 = CTOC
ELSE
CTO1 = CTOC / BIGNUM
IF( CTO1.EQ.CTOC ) THEN
! CTOC is either 0 or an inf. In both cases, CTOC itself
! serves as the correct multiplication factor.
MUL = CTOC
DONE = .TRUE.
CFROMC = ONE
ELSE IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN
MUL = SMLNUM
DONE = .FALSE.
CFROMC = CFROM1
ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN
MUL = BIGNUM
DONE = .FALSE.
CTOC = CTO1
ELSE
MUL = CTOC / CFROMC
DONE = .TRUE.
IF (MUL .EQ. ONE)
$ RETURN
END IF
END IF
*
IF( ITYPE.EQ.0 ) THEN
*
* Full matrix
*
DO 30 J = 1, N
DO 20 I = 1, M
A( I, J ) = A( I, J )*MUL
20 CONTINUE
30 CONTINUE
*
ELSE IF( ITYPE.EQ.1 ) THEN
*
* Lower triangular matrix
*
DO 50 J = 1, N
DO 40 I = J, M
A( I, J ) = A( I, J )*MUL
40 CONTINUE
50 CONTINUE
*
ELSE IF( ITYPE.EQ.2 ) THEN
*
* Upper triangular matrix
*
DO 70 J = 1, N
DO 60 I = 1, MIN( J, M )
A( I, J ) = A( I, J )*MUL
60 CONTINUE
70 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* Upper Hessenberg matrix
*
DO 90 J = 1, N
DO 80 I = 1, MIN( J+1, M )
A( I, J ) = A( I, J )*MUL
80 CONTINUE
90 CONTINUE
*
ELSE IF( ITYPE.EQ.4 ) THEN
*
* Lower half of a symmetric band matrix
*
K3 = KL + 1
K4 = N + 1
DO 110 J = 1, N
DO 100 I = 1, MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
100 CONTINUE
110 CONTINUE
*
ELSE IF( ITYPE.EQ.5 ) THEN
*
* Upper half of a symmetric band matrix
*
K1 = KU + 2
K3 = KU + 1
DO 130 J = 1, N
DO 120 I = MAX( K1-J, 1 ), K3
A( I, J ) = A( I, J )*MUL
120 CONTINUE
130 CONTINUE
*
ELSE IF( ITYPE.EQ.6 ) THEN
*
* Band matrix
*
K1 = KL + KU + 2
K2 = KL + 1
K3 = 2*KL + KU + 1
K4 = KL + KU + 1 + M
DO 150 J = 1, N
DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
140 CONTINUE
150 CONTINUE
*
END IF
*
IF( .NOT.DONE )
$ GO TO 10
*
RETURN
*
* End of DLASCL
*
END
*> \brief \b DLASCL2 performs diagonal scaling on a matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASCL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlascl2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlascl2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlascl2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASCL2 ( M, N, D, X, LDX )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDX
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASCL2 performs a diagonal scaling on a matrix:
*> x <-- D * x
*> where the diagonal matrix D is stored as a vector.
*>
*> Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
*> standard.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of D and X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of X. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, length M
*> Diagonal matrix D, stored as a vector of length M.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,N)
*> On entry, the matrix X to be scaled by D.
*> On exit, the scaled matrix.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the matrix X. LDX >= M.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lascl2
*
* =====================================================================
SUBROUTINE DLASCL2 ( M, N, D, X, LDX )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER M, N, LDX
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. Executable Statements ..
*
DO J = 1, N
DO I = 1, M
X( I, J ) = X( I, J ) * D( I )
END DO
END DO
RETURN
END
*> \brief \b DLASD0 computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK,
* WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDU, LDVT, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Using a divide and conquer approach, DLASD0 computes the singular
*> value decomposition (SVD) of a real upper bidiagonal N-by-M
*> matrix B with diagonal D and offdiagonal E, where M = N + SQRE.
*> The algorithm computes orthogonal matrices U and VT such that
*> B = U * S * VT. The singular values S are overwritten on D.
*>
*> A related subroutine, DLASDA, computes only the singular values,
*> and optionally, the singular vectors in compact form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, the row dimension of the upper bidiagonal matrix.
*> This is also the dimension of the main diagonal array D.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> Specifies the column dimension of the bidiagonal matrix.
*> = 0: The bidiagonal matrix has column dimension M = N;
*> = 1: The bidiagonal matrix has column dimension M = N+1;
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry D contains the main diagonal of the bidiagonal
*> matrix.
*> On exit D, if INFO = 0, contains its singular values.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (M-1)
*> Contains the subdiagonal entries of the bidiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU, N)
*> On exit, U contains the left singular vectors,
*> if U passed in as (N, N) Identity.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> On entry, leading dimension of U.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT, M)
*> On exit, VT**T contains the right singular vectors,
*> if VT passed in as (M, M) Identity.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> On entry, leading dimension of VT.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> On entry, maximum size of the subproblems at the
*> bottom of the computation tree.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*M**2+2*M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd0
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ,
$ IWORK,
$ WORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDU, LDVT, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, I1, IC, IDXQ, IDXQC, IM1, INODE, ITEMP, IWK,
$ J, LF, LL, LVL, M, NCC, ND, NDB1, NDIML, NDIMR,
$ NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQREI
DOUBLE PRECISION ALPHA, BETA
* ..
* .. External Subroutines ..
EXTERNAL DLASD1, DLASDQ, DLASDT, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -2
END IF
*
M = N + SQRE
*
IF( LDU.LT.N ) THEN
INFO = -6
ELSE IF( LDVT.LT.M ) THEN
INFO = -8
ELSE IF( SMLSIZ.LT.3 ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD0', -INFO )
RETURN
END IF
*
* If the input matrix is too small, call DLASDQ to find the SVD.
*
IF( N.LE.SMLSIZ ) THEN
CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDVT, U, LDU,
$ U,
$ LDU, WORK, INFO )
RETURN
END IF
*
* Set up the computation tree.
*
INODE = 1
NDIML = INODE + N
NDIMR = NDIML + N
IDXQ = NDIMR + N
IWK = IDXQ + N
CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
$ IWORK( NDIMR ), SMLSIZ )
*
* For the nodes on bottom level of the tree, solve
* their subproblems by DLASDQ.
*
NDB1 = ( ND+1 ) / 2
NCC = 0
DO 30 I = NDB1, ND
*
* IC : center row of each node
* NL : number of rows of left subproblem
* NR : number of rows of right subproblem
* NLF: starting row of the left subproblem
* NRF: starting row of the right subproblem
*
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NLP1 = NL + 1
NR = IWORK( NDIMR+I1 )
NRP1 = NR + 1
NLF = IC - NL
NRF = IC + 1
SQREI = 1
CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
$ E( NLF ),
$ VT( NLF, NLF ), LDVT, U( NLF, NLF ), LDU,
$ U( NLF, NLF ), LDU, WORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
ITEMP = IDXQ + NLF - 2
DO 10 J = 1, NL
IWORK( ITEMP+J ) = J
10 CONTINUE
IF( I.EQ.ND ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
NRP1 = NR + SQREI
CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
$ E( NRF ),
$ VT( NRF, NRF ), LDVT, U( NRF, NRF ), LDU,
$ U( NRF, NRF ), LDU, WORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
ITEMP = IDXQ + IC
DO 20 J = 1, NR
IWORK( ITEMP+J-1 ) = J
20 CONTINUE
30 CONTINUE
*
* Now conquer each subproblem bottom-up.
*
DO 50 LVL = NLVL, 1, -1
*
* Find the first node LF and last node LL on the
* current level LVL.
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 40 I = LF, LL
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
IF( ( SQRE.EQ.0 ) .AND. ( I.EQ.LL ) ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
IDXQC = IDXQ + NLF - 1
ALPHA = D( IC )
BETA = E( IC )
CALL DLASD1( NL, NR, SQREI, D( NLF ), ALPHA, BETA,
$ U( NLF, NLF ), LDU, VT( NLF, NLF ), LDVT,
$ IWORK( IDXQC ), IWORK( IWK ), WORK, INFO )
*
* Report the possible convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
40 CONTINUE
50 CONTINUE
*
RETURN
*
* End of DLASD0
*
END
*> \brief \b DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
* IDXQ, IWORK, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDU, LDVT, NL, NR, SQRE
* DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
* INTEGER IDXQ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
*> where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.
*>
*> A related subroutine DLASD7 handles the case in which the singular
*> values (and the singular vectors in factored form) are desired.
*>
*> DLASD1 computes the SVD as follows:
*>
*> ( D1(in) 0 0 0 )
*> B = U(in) * ( Z1**T a Z2**T b ) * VT(in)
*> ( 0 0 D2(in) 0 )
*>
*> = U(out) * ( D(out) 0) * VT(out)
*>
*> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
*> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
*> elsewhere; and the entry b is empty if SQRE = 0.
*>
*> The left singular vectors of the original matrix are stored in U, and
*> the transpose of the right singular vectors are stored in VT, and the
*> singular values are in D. The algorithm consists of three stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple singular values or when there are zeros in
*> the Z vector. For each such occurrence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine DLASD2.
*>
*> The second stage consists of calculating the updated
*> singular values. This is done by finding the square roots of the
*> roots of the secular equation via the routine DLASD4 (as called
*> by DLASD3). This routine also calculates the singular vectors of
*> the current problem.
*>
*> The final stage consists of computing the updated singular vectors
*> directly using the updated singular values. The singular vectors
*> for the current problem are multiplied with the singular vectors
*> from the overall problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: the lower block is an NR-by-NR square matrix.
*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*> The bidiagonal matrix has row dimension N = NL + NR + 1,
*> and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array,
*> dimension (N = NL+NR+1).
*> On entry D(1:NL,1:NL) contains the singular values of the
*> upper block; and D(NL+2:N) contains the singular values of
*> the lower block. On exit D(1:N) contains the singular values
*> of the modified matrix.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in,out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> Contains the off-diagonal element associated with the added
*> row.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension(LDU,N)
*> On entry U(1:NL, 1:NL) contains the left singular vectors of
*> the upper block; U(NL+2:N, NL+2:N) contains the left singular
*> vectors of the lower block. On exit U contains the left
*> singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension(LDVT,M)
*> where M = N + SQRE.
*> On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
*> vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
*> the right singular vectors of the lower block. On exit
*> VT**T contains the right singular vectors of the
*> bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= max( 1, M ).
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*> IDXQ is INTEGER array, dimension(N)
*> This contains the permutation which will reintegrate the
*> subproblem just solved back into sorted order, i.e.
*> D( IDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension( 4 * N )
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd1
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT,
$ LDVT,
$ IDXQ, IWORK, WORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDU, LDVT, NL, NR, SQRE
DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
INTEGER IDXQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
*
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2,
$ IVT2, IZ, K, LDQ, LDU2, LDVT2, M, N, N1, N2
DOUBLE PRECISION ORGNRM
* ..
* .. External Subroutines ..
EXTERNAL DLAMRG, DLASCL, DLASD2, DLASD3,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( NL.LT.1 ) THEN
INFO = -1
ELSE IF( NR.LT.1 ) THEN
INFO = -2
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD1', -INFO )
RETURN
END IF
*
N = NL + NR + 1
M = N + SQRE
*
* The following values are for bookkeeping purposes only. They are
* integer pointers which indicate the portion of the workspace
* used by a particular array in DLASD2 and DLASD3.
*
LDU2 = N
LDVT2 = M
*
IZ = 1
ISIGMA = IZ + M
IU2 = ISIGMA + N
IVT2 = IU2 + LDU2*N
IQ = IVT2 + LDVT2*M
*
IDX = 1
IDXC = IDX + N
COLTYP = IDXC + N
IDXP = COLTYP + N
*
* Scale.
*
ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
D( NL+1 ) = ZERO
DO 10 I = 1, N
IF( ABS( D( I ) ).GT.ORGNRM ) THEN
ORGNRM = ABS( D( I ) )
END IF
10 CONTINUE
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
ALPHA = ALPHA / ORGNRM
BETA = BETA / ORGNRM
*
* Deflate singular values.
*
CALL DLASD2( NL, NR, SQRE, K, D, WORK( IZ ), ALPHA, BETA, U,
$ LDU,
$ VT, LDVT, WORK( ISIGMA ), WORK( IU2 ), LDU2,
$ WORK( IVT2 ), LDVT2, IWORK( IDXP ), IWORK( IDX ),
$ IWORK( IDXC ), IDXQ, IWORK( COLTYP ), INFO )
*
* Solve Secular Equation and update singular vectors.
*
LDQ = K
CALL DLASD3( NL, NR, SQRE, K, D, WORK( IQ ), LDQ,
$ WORK( ISIGMA ),
$ U, LDU, WORK( IU2 ), LDU2, VT, LDVT, WORK( IVT2 ),
$ LDVT2, IWORK( IDXC ), IWORK( COLTYP ), WORK( IZ ),
$ INFO )
*
* Report the convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
*
* Unscale.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
* Prepare the IDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL DLAMRG( N1, N2, D, 1, -1, IDXQ )
*
RETURN
*
* End of DLASD1
*
END
*> \brief \b DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
* LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
* IDXC, IDXQ, COLTYP, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
* DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
* INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
* $ IDXQ( * )
* DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ),
* $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
* $ Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD2 merges the two sets of singular values together into a single
*> sorted set. Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur: when two or more
*> singular values are close together or if there is a tiny entry in the
*> Z vector. For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*>
*> DLASD2 is called from DLASD1.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: the lower block is an NR-by-NR square matrix.
*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*> The bidiagonal matrix has N = NL + NR + 1 rows and
*> M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> Contains the dimension of the non-deflated matrix,
*> This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension(N)
*> On entry D contains the singular values of the two submatrices
*> to be combined. On exit D contains the trailing (N-K) updated
*> singular values (those which were deflated) sorted into
*> increasing order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension(N)
*> On exit Z contains the updating row vector in the secular
*> equation.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> Contains the off-diagonal element associated with the added
*> row.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension(LDU,N)
*> On entry U contains the left singular vectors of two
*> submatrices in the two square blocks with corners at (1,1),
*> (NL, NL), and (NL+2, NL+2), (N,N).
*> On exit U contains the trailing (N-K) updated left singular
*> vectors (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= N.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension(LDVT,M)
*> On entry VT**T contains the right singular vectors of two
*> submatrices in the two square blocks with corners at (1,1),
*> (NL+1, NL+1), and (NL+2, NL+2), (M,M).
*> On exit VT**T contains the trailing (N-K) updated right singular
*> vectors (those which were deflated) in its last N-K columns.
*> In case SQRE =1, the last row of VT spans the right null
*> space.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= M.
*> \endverbatim
*>
*> \param[out] DSIGMA
*> \verbatim
*> DSIGMA is DOUBLE PRECISION array, dimension (N)
*> Contains a copy of the diagonal elements (K-1 singular values
*> and one zero) in the secular equation.
*> \endverbatim
*>
*> \param[out] U2
*> \verbatim
*> U2 is DOUBLE PRECISION array, dimension(LDU2,N)
*> Contains a copy of the first K-1 left singular vectors which
*> will be used by DLASD3 in a matrix multiply (DGEMM) to solve
*> for the new left singular vectors. U2 is arranged into four
*> blocks. The first block contains a column with 1 at NL+1 and
*> zero everywhere else; the second block contains non-zero
*> entries only at and above NL; the third contains non-zero
*> entries only below NL+1; and the fourth is dense.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*> LDU2 is INTEGER
*> The leading dimension of the array U2. LDU2 >= N.
*> \endverbatim
*>
*> \param[out] VT2
*> \verbatim
*> VT2 is DOUBLE PRECISION array, dimension(LDVT2,N)
*> VT2**T contains a copy of the first K right singular vectors
*> which will be used by DLASD3 in a matrix multiply (DGEMM) to
*> solve for the new right singular vectors. VT2 is arranged into
*> three blocks. The first block contains a row that corresponds
*> to the special 0 diagonal element in SIGMA; the second block
*> contains non-zeros only at and before NL +1; the third block
*> contains non-zeros only at and after NL +2.
*> \endverbatim
*>
*> \param[in] LDVT2
*> \verbatim
*> LDVT2 is INTEGER
*> The leading dimension of the array VT2. LDVT2 >= M.
*> \endverbatim
*>
*> \param[out] IDXP
*> \verbatim
*> IDXP is INTEGER array, dimension(N)
*> This will contain the permutation used to place deflated
*> values of D at the end of the array. On output IDXP(2:K)
*> points to the nondeflated D-values and IDXP(K+1:N)
*> points to the deflated singular values.
*> \endverbatim
*>
*> \param[out] IDX
*> \verbatim
*> IDX is INTEGER array, dimension(N)
*> This will contain the permutation used to sort the contents of
*> D into ascending order.
*> \endverbatim
*>
*> \param[out] IDXC
*> \verbatim
*> IDXC is INTEGER array, dimension(N)
*> This will contain the permutation used to arrange the columns
*> of the deflated U matrix into three groups: the first group
*> contains non-zero entries only at and above NL, the second
*> contains non-zero entries only below NL+2, and the third is
*> dense.
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*> IDXQ is INTEGER array, dimension(N)
*> This contains the permutation which separately sorts the two
*> sub-problems in D into ascending order. Note that entries in
*> the first hlaf of this permutation must first be moved one
*> position backward; and entries in the second half
*> must first have NL+1 added to their values.
*> \endverbatim
*>
*> \param[out] COLTYP
*> \verbatim
*> COLTYP is INTEGER array, dimension(N)
*> As workspace, this will contain a label which will indicate
*> which of the following types a column in the U2 matrix or a
*> row in the VT2 matrix is:
*> 1 : non-zero in the upper half only
*> 2 : non-zero in the lower half only
*> 3 : dense
*> 4 : deflated
*>
*> On exit, it is an array of dimension 4, with COLTYP(I) being
*> the dimension of the I-th type columns.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd2
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU,
$ VT,
$ LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
$ IDXC, IDXQ, COLTYP, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
$ IDXQ( * )
DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ),
$ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
$ Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, EIGHT
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ EIGHT = 8.0D+0 )
* ..
* .. Local Arrays ..
INTEGER CTOT( 4 ), PSM( 4 )
* ..
* .. Local Scalars ..
INTEGER CT, I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M,
$ N, NLP1, NLP2
DOUBLE PRECISION C, EPS, HLFTOL, S, TAU, TOL, Z1
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAMRG, DLASET, DROT,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( NL.LT.1 ) THEN
INFO = -1
ELSE IF( NR.LT.1 ) THEN
INFO = -2
ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
INFO = -3
END IF
*
N = NL + NR + 1
M = N + SQRE
*
IF( LDU.LT.N ) THEN
INFO = -10
ELSE IF( LDVT.LT.M ) THEN
INFO = -12
ELSE IF( LDU2.LT.N ) THEN
INFO = -15
ELSE IF( LDVT2.LT.M ) THEN
INFO = -17
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD2', -INFO )
RETURN
END IF
*
NLP1 = NL + 1
NLP2 = NL + 2
*
* Generate the first part of the vector Z; and move the singular
* values in the first part of D one position backward.
*
Z1 = ALPHA*VT( NLP1, NLP1 )
Z( 1 ) = Z1
DO 10 I = NL, 1, -1
Z( I+1 ) = ALPHA*VT( I, NLP1 )
D( I+1 ) = D( I )
IDXQ( I+1 ) = IDXQ( I ) + 1
10 CONTINUE
*
* Generate the second part of the vector Z.
*
DO 20 I = NLP2, M
Z( I ) = BETA*VT( I, NLP2 )
20 CONTINUE
*
* Initialize some reference arrays.
*
DO 30 I = 2, NLP1
COLTYP( I ) = 1
30 CONTINUE
DO 40 I = NLP2, N
COLTYP( I ) = 2
40 CONTINUE
*
* Sort the singular values into increasing order
*
DO 50 I = NLP2, N
IDXQ( I ) = IDXQ( I ) + NLP1
50 CONTINUE
*
* DSIGMA, IDXC, IDXC, and the first column of U2
* are used as storage space.
*
DO 60 I = 2, N
DSIGMA( I ) = D( IDXQ( I ) )
U2( I, 1 ) = Z( IDXQ( I ) )
IDXC( I ) = COLTYP( IDXQ( I ) )
60 CONTINUE
*
CALL DLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) )
*
DO 70 I = 2, N
IDXI = 1 + IDX( I )
D( I ) = DSIGMA( IDXI )
Z( I ) = U2( IDXI, 1 )
COLTYP( I ) = IDXC( IDXI )
70 CONTINUE
*
* Calculate the allowable deflation tolerance
*
EPS = DLAMCH( 'Epsilon' )
TOL = MAX( ABS( ALPHA ), ABS( BETA ) )
TOL = EIGHT*EPS*MAX( ABS( D( N ) ), TOL )
*
* There are 2 kinds of deflation -- first a value in the z-vector
* is small, second two (or more) singular values are very close
* together (their difference is small).
*
* If the value in the z-vector is small, we simply permute the
* array so that the corresponding singular value is moved to the
* end.
*
* If two values in the D-vector are close, we perform a two-sided
* rotation designed to make one of the corresponding z-vector
* entries zero, and then permute the array so that the deflated
* singular value is moved to the end.
*
* If there are multiple singular values then the problem deflates.
* Here the number of equal singular values are found. As each equal
* singular value is found, an elementary reflector is computed to
* rotate the corresponding singular subspace so that the
* corresponding components of Z are zero in this new basis.
*
K = 1
K2 = N + 1
DO 80 J = 2, N
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
COLTYP( J ) = 4
IF( J.EQ.N )
$ GO TO 120
ELSE
JPREV = J
GO TO 90
END IF
80 CONTINUE
90 CONTINUE
J = JPREV
100 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 110
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
COLTYP( J ) = 4
ELSE
*
* Check if singular values are close enough to allow deflation.
*
IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN
*
* Deflation is possible.
*
S = Z( JPREV )
C = Z( J )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = DLAPY2( C, S )
C = C / TAU
S = -S / TAU
Z( J ) = TAU
Z( JPREV ) = ZERO
*
* Apply back the Givens rotation to the left and right
* singular vector matrices.
*
IDXJP = IDXQ( IDX( JPREV )+1 )
IDXJ = IDXQ( IDX( J )+1 )
IF( IDXJP.LE.NLP1 ) THEN
IDXJP = IDXJP - 1
END IF
IF( IDXJ.LE.NLP1 ) THEN
IDXJ = IDXJ - 1
END IF
CALL DROT( N, U( 1, IDXJP ), 1, U( 1, IDXJ ), 1, C, S )
CALL DROT( M, VT( IDXJP, 1 ), LDVT, VT( IDXJ, 1 ), LDVT,
$ C,
$ S )
IF( COLTYP( J ).NE.COLTYP( JPREV ) ) THEN
COLTYP( J ) = 3
END IF
COLTYP( JPREV ) = 4
K2 = K2 - 1
IDXP( K2 ) = JPREV
JPREV = J
ELSE
K = K + 1
U2( K, 1 ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
JPREV = J
END IF
END IF
GO TO 100
110 CONTINUE
*
* Record the last singular value.
*
K = K + 1
U2( K, 1 ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
*
120 CONTINUE
*
* Count up the total number of the various types of columns, then
* form a permutation which positions the four column types into
* four groups of uniform structure (although one or more of these
* groups may be empty).
*
DO 130 J = 1, 4
CTOT( J ) = 0
130 CONTINUE
DO 140 J = 2, N
CT = COLTYP( J )
CTOT( CT ) = CTOT( CT ) + 1
140 CONTINUE
*
* PSM(*) = Position in SubMatrix (of types 1 through 4)
*
PSM( 1 ) = 2
PSM( 2 ) = 2 + CTOT( 1 )
PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
*
* Fill out the IDXC array so that the permutation which it induces
* will place all type-1 columns first, all type-2 columns next,
* then all type-3's, and finally all type-4's, starting from the
* second column. This applies similarly to the rows of VT.
*
DO 150 J = 2, N
JP = IDXP( J )
CT = COLTYP( JP )
IDXC( PSM( CT ) ) = J
PSM( CT ) = PSM( CT ) + 1
150 CONTINUE
*
* Sort the singular values and corresponding singular vectors into
* DSIGMA, U2, and VT2 respectively. The singular values/vectors
* which were not deflated go into the first K slots of DSIGMA, U2,
* and VT2 respectively, while those which were deflated go into the
* last N - K slots, except that the first column/row will be treated
* separately.
*
DO 160 J = 2, N
JP = IDXP( J )
DSIGMA( J ) = D( JP )
IDXJ = IDXQ( IDX( IDXP( IDXC( J ) ) )+1 )
IF( IDXJ.LE.NLP1 ) THEN
IDXJ = IDXJ - 1
END IF
CALL DCOPY( N, U( 1, IDXJ ), 1, U2( 1, J ), 1 )
CALL DCOPY( M, VT( IDXJ, 1 ), LDVT, VT2( J, 1 ), LDVT2 )
160 CONTINUE
*
* Determine DSIGMA(1), DSIGMA(2) and Z(1)
*
DSIGMA( 1 ) = ZERO
HLFTOL = TOL / TWO
IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL )
$ DSIGMA( 2 ) = HLFTOL
IF( M.GT.N ) THEN
Z( 1 ) = DLAPY2( Z1, Z( M ) )
IF( Z( 1 ).LE.TOL ) THEN
C = ONE
S = ZERO
Z( 1 ) = TOL
ELSE
C = Z1 / Z( 1 )
S = Z( M ) / Z( 1 )
END IF
ELSE
IF( ABS( Z1 ).LE.TOL ) THEN
Z( 1 ) = TOL
ELSE
Z( 1 ) = Z1
END IF
END IF
*
* Move the rest of the updating row to Z.
*
CALL DCOPY( K-1, U2( 2, 1 ), 1, Z( 2 ), 1 )
*
* Determine the first column of U2, the first row of VT2 and the
* last row of VT.
*
CALL DLASET( 'A', N, 1, ZERO, ZERO, U2, LDU2 )
U2( NLP1, 1 ) = ONE
IF( M.GT.N ) THEN
DO 170 I = 1, NLP1
VT( M, I ) = -S*VT( NLP1, I )
VT2( 1, I ) = C*VT( NLP1, I )
170 CONTINUE
DO 180 I = NLP2, M
VT2( 1, I ) = S*VT( M, I )
VT( M, I ) = C*VT( M, I )
180 CONTINUE
ELSE
CALL DCOPY( M, VT( NLP1, 1 ), LDVT, VT2( 1, 1 ), LDVT2 )
END IF
IF( M.GT.N ) THEN
CALL DCOPY( M, VT( M, 1 ), LDVT, VT2( M, 1 ), LDVT2 )
END IF
*
* The deflated singular values and their corresponding vectors go
* into the back of D, U, and V respectively.
*
IF( N.GT.K ) THEN
CALL DCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 )
CALL DLACPY( 'A', N, N-K, U2( 1, K+1 ), LDU2, U( 1, K+1 ),
$ LDU )
CALL DLACPY( 'A', N-K, M, VT2( K+1, 1 ), LDVT2, VT( K+1,
$ 1 ),
$ LDVT )
END IF
*
* Copy CTOT into COLTYP for referencing in DLASD3.
*
DO 190 J = 1, 4
COLTYP( J ) = CTOT( J )
190 CONTINUE
*
RETURN
*
* End of DLASD2
*
END
*> \brief \b DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
* LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
* $ SQRE
* ..
* .. Array Arguments ..
* INTEGER CTOT( * ), IDXC( * )
* DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
* $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
* $ Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD3 finds all the square roots of the roots of the secular
*> equation, as defined by the values in D and Z. It makes the
*> appropriate calls to DLASD4 and then updates the singular
*> vectors by matrix multiplication.
*>
*> DLASD3 is called from DLASD1.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: the lower block is an NR-by-NR square matrix.
*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*> The bidiagonal matrix has N = NL + NR + 1 rows and
*> M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The size of the secular equation, 1 =< K = < N.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension(K)
*> On exit the square roots of the roots of the secular equation,
*> in ascending order.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,K)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= K.
*> \endverbatim
*>
*> \param[in] DSIGMA
*> \verbatim
*> DSIGMA is DOUBLE PRECISION array, dimension(K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU, N)
*> The last N - K columns of this matrix contain the deflated
*> left singular vectors.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= N.
*> \endverbatim
*>
*> \param[in] U2
*> \verbatim
*> U2 is DOUBLE PRECISION array, dimension (LDU2, N)
*> The first K columns of this matrix contain the non-deflated
*> left singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*> LDU2 is INTEGER
*> The leading dimension of the array U2. LDU2 >= N.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT, M)
*> The last M - K columns of VT**T contain the deflated
*> right singular vectors.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= N.
*> \endverbatim
*>
*> \param[in,out] VT2
*> \verbatim
*> VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
*> The first K columns of VT2**T contain the non-deflated
*> right singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDVT2
*> \verbatim
*> LDVT2 is INTEGER
*> The leading dimension of the array VT2. LDVT2 >= N.
*> \endverbatim
*>
*> \param[in] IDXC
*> \verbatim
*> IDXC is INTEGER array, dimension ( N )
*> The permutation used to arrange the columns of U (and rows of
*> VT) into three groups: the first group contains non-zero
*> entries only at and above (or before) NL +1; the second
*> contains non-zero entries only at and below (or after) NL+2;
*> and the third is dense. The first column of U and the row of
*> VT are treated separately, however.
*>
*> The rows of the singular vectors found by DLASD4
*> must be likewise permuted before the matrix multiplies can
*> take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*> CTOT is INTEGER array, dimension ( 4 )
*> A count of the total number of the various types of columns
*> in U (or rows in VT), as described in IDXC. The fourth column
*> type is any column which has been deflated.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating row vector.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd3
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU,
$ U2,
$ LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
$ INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
$ SQRE
* ..
* .. Array Arguments ..
INTEGER CTOT( * ), IDXC( * )
DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
$ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
$ Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, NEGONE
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0,
$ NEGONE = -1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
DOUBLE PRECISION RHO, TEMP
* ..
* .. External Functions ..
DOUBLE PRECISION DNRM2
EXTERNAL DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLASCL, DLASD4,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( NL.LT.1 ) THEN
INFO = -1
ELSE IF( NR.LT.1 ) THEN
INFO = -2
ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
INFO = -3
END IF
*
N = NL + NR + 1
M = N + SQRE
NLP1 = NL + 1
NLP2 = NL + 2
*
IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN
INFO = -4
ELSE IF( LDQ.LT.K ) THEN
INFO = -7
ELSE IF( LDU.LT.N ) THEN
INFO = -10
ELSE IF( LDU2.LT.N ) THEN
INFO = -12
ELSE IF( LDVT.LT.M ) THEN
INFO = -14
ELSE IF( LDVT2.LT.M ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD3', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.1 ) THEN
D( 1 ) = ABS( Z( 1 ) )
CALL DCOPY( M, VT2( 1, 1 ), LDVT2, VT( 1, 1 ), LDVT )
IF( Z( 1 ).GT.ZERO ) THEN
CALL DCOPY( N, U2( 1, 1 ), 1, U( 1, 1 ), 1 )
ELSE
DO 10 I = 1, N
U( I, 1 ) = -U2( I, 1 )
10 CONTINUE
END IF
RETURN
END IF
*
* Keep a copy of Z.
*
CALL DCOPY( K, Z, 1, Q, 1 )
*
* Normalize Z.
*
RHO = DNRM2( K, Z, 1 )
CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
RHO = RHO*RHO
*
* Find the new singular values.
*
DO 30 J = 1, K
CALL DLASD4( K, J, DSIGMA, Z, U( 1, J ), RHO, D( J ),
$ VT( 1, J ), INFO )
*
* If the zero finder fails, report the convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
30 CONTINUE
*
* Compute updated Z.
*
DO 60 I = 1, K
Z( I ) = U( I, K )*VT( I, K )
DO 40 J = 1, I - 1
Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
$ ( DSIGMA( I )-DSIGMA( J ) ) /
$ ( DSIGMA( I )+DSIGMA( J ) ) )
40 CONTINUE
DO 50 J = I, K - 1
Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
$ ( DSIGMA( I )-DSIGMA( J+1 ) ) /
$ ( DSIGMA( I )+DSIGMA( J+1 ) ) )
50 CONTINUE
Z( I ) = SIGN( SQRT( ABS( Z( I ) ) ), Q( I, 1 ) )
60 CONTINUE
*
* Compute left singular vectors of the modified diagonal matrix,
* and store related information for the right singular vectors.
*
DO 90 I = 1, K
VT( 1, I ) = Z( 1 ) / U( 1, I ) / VT( 1, I )
U( 1, I ) = NEGONE
DO 70 J = 2, K
VT( J, I ) = Z( J ) / U( J, I ) / VT( J, I )
U( J, I ) = DSIGMA( J )*VT( J, I )
70 CONTINUE
TEMP = DNRM2( K, U( 1, I ), 1 )
Q( 1, I ) = U( 1, I ) / TEMP
DO 80 J = 2, K
JC = IDXC( J )
Q( J, I ) = U( JC, I ) / TEMP
80 CONTINUE
90 CONTINUE
*
* Update the left singular vector matrix.
*
IF( K.EQ.2 ) THEN
CALL DGEMM( 'N', 'N', N, K, K, ONE, U2, LDU2, Q, LDQ, ZERO,
$ U,
$ LDU )
GO TO 100
END IF
IF( CTOT( 1 ).GT.0 ) THEN
CALL DGEMM( 'N', 'N', NL, K, CTOT( 1 ), ONE, U2( 1, 2 ),
$ LDU2,
$ Q( 2, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
IF( CTOT( 3 ).GT.0 ) THEN
KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1,
$ KTEMP ),
$ LDU2, Q( KTEMP, 1 ), LDQ, ONE, U( 1, 1 ), LDU )
END IF
ELSE IF( CTOT( 3 ).GT.0 ) THEN
KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
$ LDU2, Q( KTEMP, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
ELSE
CALL DLACPY( 'F', NL, K, U2, LDU2, U, LDU )
END IF
CALL DCOPY( K, Q( 1, 1 ), LDQ, U( NLP1, 1 ), LDU )
KTEMP = 2 + CTOT( 1 )
CTEMP = CTOT( 2 ) + CTOT( 3 )
CALL DGEMM( 'N', 'N', NR, K, CTEMP, ONE, U2( NLP2, KTEMP ),
$ LDU2,
$ Q( KTEMP, 1 ), LDQ, ZERO, U( NLP2, 1 ), LDU )
*
* Generate the right singular vectors.
*
100 CONTINUE
DO 120 I = 1, K
TEMP = DNRM2( K, VT( 1, I ), 1 )
Q( I, 1 ) = VT( 1, I ) / TEMP
DO 110 J = 2, K
JC = IDXC( J )
Q( I, J ) = VT( JC, I ) / TEMP
110 CONTINUE
120 CONTINUE
*
* Update the right singular vector matrix.
*
IF( K.EQ.2 ) THEN
CALL DGEMM( 'N', 'N', K, M, K, ONE, Q, LDQ, VT2, LDVT2,
$ ZERO,
$ VT, LDVT )
RETURN
END IF
KTEMP = 1 + CTOT( 1 )
CALL DGEMM( 'N', 'N', K, NLP1, KTEMP, ONE, Q( 1, 1 ), LDQ,
$ VT2( 1, 1 ), LDVT2, ZERO, VT( 1, 1 ), LDVT )
KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
IF( KTEMP.LE.LDVT2 )
$ CALL DGEMM( 'N', 'N', K, NLP1, CTOT( 3 ), ONE, Q( 1,
$ KTEMP ),
$ LDQ, VT2( KTEMP, 1 ), LDVT2, ONE, VT( 1, 1 ),
$ LDVT )
*
KTEMP = CTOT( 1 ) + 1
NRP1 = NR + SQRE
IF( KTEMP.GT.1 ) THEN
DO 130 I = 1, K
Q( I, KTEMP ) = Q( I, 1 )
130 CONTINUE
DO 140 I = NLP2, M
VT2( KTEMP, I ) = VT2( 1, I )
140 CONTINUE
END IF
CTEMP = 1 + CTOT( 2 ) + CTOT( 3 )
CALL DGEMM( 'N', 'N', K, NRP1, CTEMP, ONE, Q( 1, KTEMP ), LDQ,
$ VT2( KTEMP, NLP2 ), LDVT2, ZERO, VT( 1, NLP2 ), LDVT )
*
RETURN
*
* End of DLASD3
*
END
*> \brief \b DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd4.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER I, INFO, N
* DOUBLE PRECISION RHO, SIGMA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the square root of the I-th updated
*> eigenvalue of a positive symmetric rank-one modification to
*> a positive diagonal matrix whose entries are given as the squares
*> of the corresponding entries in the array d, and that
*>
*> 0 <= D(i) < D(j) for i < j
*>
*> and that RHO > 0. This is arranged by the calling routine, and is
*> no loss in generality. The rank-one modified system is thus
*>
*> diag( D ) * diag( D ) + RHO * Z * Z_transpose.
*>
*> where we assume the Euclidean norm of Z is 1.
*>
*> The method consists of approximating the rational functions in the
*> secular equation by simpler interpolating rational functions.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of all arrays.
*> \endverbatim
*>
*> \param[in] I
*> \verbatim
*> I is INTEGER
*> The index of the eigenvalue to be computed. 1 <= I <= N.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension ( N )
*> The original eigenvalues. It is assumed that they are in
*> order, 0 <= D(I) < D(J) for I < J.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( N )
*> The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension ( N )
*> If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th
*> component. If N = 1, then DELTA(1) = 1. The vector DELTA
*> contains the information necessary to construct the
*> (singular) eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> The computed sigma_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension ( N )
*> If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th
*> component. If N = 1, then WORK( 1 ) = 1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = 1, the updating process failed.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> Logical variable ORGATI (origin-at-i?) is used for distinguishing
*> whether D(i) or D(i+1) is treated as the origin.
*>
*> ORGATI = .true. origin at i
*> ORGATI = .false. origin at i+1
*>
*> Logical variable SWTCH3 (switch-for-3-poles?) is for noting
*> if we are working with THREE poles!
*>
*> MAXIT is the maximum number of iterations allowed for each
*> eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd4
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I, INFO, N
DOUBLE PRECISION RHO, SIGMA
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 400 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ THREE = 3.0D+0, FOUR = 4.0D+0, EIGHT = 8.0D+0,
$ TEN = 10.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ORGATI, SWTCH, SWTCH3, GEOMAVG
INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER
DOUBLE PRECISION A, B, C, DELSQ, DELSQ2, SQ2, DPHI, DPSI, DTIIM,
$ DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS,
$ ERRETM, ETA, PHI, PREW, PSI, RHOINV, SGLB,
$ SGUB, TAU, TAU2, TEMP, TEMP1, TEMP2, W
* ..
* .. Local Arrays ..
DOUBLE PRECISION DD( 3 ), ZZ( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DLAED6, DLASD5
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Since this routine is called in an inner loop, we do no argument
* checking.
*
* Quick return for N=1 and 2.
*
INFO = 0
IF( N.EQ.1 ) THEN
*
* Presumably, I=1 upon entry
*
SIGMA = SQRT( D( 1 )*D( 1 )+RHO*Z( 1 )*Z( 1 ) )
DELTA( 1 ) = ONE
WORK( 1 ) = ONE
RETURN
END IF
IF( N.EQ.2 ) THEN
CALL DLASD5( I, D, Z, DELTA, RHO, SIGMA, WORK )
RETURN
END IF
*
* Compute machine epsilon
*
EPS = DLAMCH( 'Epsilon' )
RHOINV = ONE / RHO
TAU2= ZERO
*
* The case I = N
*
IF( I.EQ.N ) THEN
*
* Initialize some basic variables
*
II = N - 1
NITER = 1
*
* Calculate initial guess
*
TEMP = RHO / TWO
*
* If ||Z||_2 is not one, then TEMP should be set to
* RHO * ||Z||_2^2 / TWO
*
TEMP1 = TEMP / ( D( N )+SQRT( D( N )*D( N )+TEMP ) )
DO 10 J = 1, N
WORK( J ) = D( J ) + D( N ) + TEMP1
DELTA( J ) = ( D( J )-D( N ) ) - TEMP1
10 CONTINUE
*
PSI = ZERO
DO 20 J = 1, N - 2
PSI = PSI + Z( J )*Z( J ) / ( DELTA( J )*WORK( J ) )
20 CONTINUE
*
C = RHOINV + PSI
W = C + Z( II )*Z( II ) / ( DELTA( II )*WORK( II ) ) +
$ Z( N )*Z( N ) / ( DELTA( N )*WORK( N ) )
*
IF( W.LE.ZERO ) THEN
TEMP1 = SQRT( D( N )*D( N )+RHO )
TEMP = Z( N-1 )*Z( N-1 ) / ( ( D( N-1 )+TEMP1 )*
$ ( D( N )-D( N-1 )+RHO / ( D( N )+TEMP1 ) ) ) +
$ Z( N )*Z( N ) / RHO
*
* The following TAU2 is to approximate
* SIGMA_n^2 - D( N )*D( N )
*
IF( C.LE.TEMP ) THEN
TAU = RHO
ELSE
DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DELSQ
IF( A.LT.ZERO ) THEN
TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
END IF
*
* It can be proved that
* D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU2 <= D(N)^2+RHO
*
ELSE
DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DELSQ
*
* The following TAU2 is to approximate
* SIGMA_n^2 - D( N )*D( N )
*
IF( A.LT.ZERO ) THEN
TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
*
* It can be proved that
* D(N)^2 < D(N)^2+TAU2 < SIGMA(N)^2 < D(N)^2+RHO/2
*
END IF
*
* The following TAU is to approximate SIGMA_n - D( N )
*
* TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
*
SIGMA = D( N ) + TAU
DO 30 J = 1, N
DELTA( J ) = ( D( J )-D( N ) ) - TAU
WORK( J ) = D( J ) + D( N ) + TAU
30 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 40 J = 1, II
TEMP = Z( J ) / ( DELTA( J )*WORK( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
40 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / ( DELTA( N )*WORK( N ) )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
GO TO 240
END IF
*
* Calculate the new step
*
NITER = NITER + 1
DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
DTNSQ = WORK( N )*DELTA( N )
C = W - DTNSQ1*DPSI - DTNSQ*DPHI
A = ( DTNSQ+DTNSQ1 )*W - DTNSQ*DTNSQ1*( DPSI+DPHI )
B = DTNSQ*DTNSQ1*W
IF( C.LT.ZERO )
$ C = ABS( C )
IF( C.EQ.ZERO ) THEN
ETA = RHO - SIGMA*SIGMA
ELSE IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = ETA - DTNSQ
IF( TEMP.GT.RHO )
$ ETA = RHO + DTNSQ
*
ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
TAU = TAU + ETA
SIGMA = SIGMA + ETA
*
DO 50 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
WORK( J ) = WORK( J ) + ETA
50 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 60 J = 1, II
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
60 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TAU2 = WORK( N )*DELTA( N )
TEMP = Z( N ) / TAU2
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Main loop to update the values of the array DELTA
*
ITER = NITER + 1
*
DO 90 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
GO TO 240
END IF
*
* Calculate the new step
*
DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
DTNSQ = WORK( N )*DELTA( N )
C = W - DTNSQ1*DPSI - DTNSQ*DPHI
A = ( DTNSQ+DTNSQ1 )*W - DTNSQ1*DTNSQ*( DPSI+DPHI )
B = DTNSQ1*DTNSQ*W
IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = ETA - DTNSQ
IF( TEMP.LE.ZERO )
$ ETA = ETA / TWO
*
ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
TAU = TAU + ETA
SIGMA = SIGMA + ETA
*
DO 70 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
WORK( J ) = WORK( J ) + ETA
70 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 80 J = 1, II
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
80 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TAU2 = WORK( N )*DELTA( N )
TEMP = Z( N ) / TAU2
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
90 CONTINUE
*
* Return with INFO = 1, NITER = MAXIT and not converged
*
INFO = 1
GO TO 240
*
* End for the case I = N
*
ELSE
*
* The case for I < N
*
NITER = 1
IP1 = I + 1
*
* Calculate initial guess
*
DELSQ = ( D( IP1 )-D( I ) )*( D( IP1 )+D( I ) )
DELSQ2 = DELSQ / TWO
SQ2=SQRT( ( D( I )*D( I )+D( IP1 )*D( IP1 ) ) / TWO )
TEMP = DELSQ2 / ( D( I )+SQ2 )
DO 100 J = 1, N
WORK( J ) = D( J ) + D( I ) + TEMP
DELTA( J ) = ( D( J )-D( I ) ) - TEMP
100 CONTINUE
*
PSI = ZERO
DO 110 J = 1, I - 1
PSI = PSI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
110 CONTINUE
*
PHI = ZERO
DO 120 J = N, I + 2, -1
PHI = PHI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
120 CONTINUE
C = RHOINV + PSI + PHI
W = C + Z( I )*Z( I ) / ( WORK( I )*DELTA( I ) ) +
$ Z( IP1 )*Z( IP1 ) / ( WORK( IP1 )*DELTA( IP1 ) )
*
GEOMAVG = .FALSE.
IF( W.GT.ZERO ) THEN
*
* d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2
*
* We choose d(i) as origin.
*
ORGATI = .TRUE.
II = I
SGLB = ZERO
SGUB = DELSQ2 / ( D( I )+SQ2 )
A = C*DELSQ + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
B = Z( I )*Z( I )*DELSQ
IF( A.GT.ZERO ) THEN
TAU2 = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
ELSE
TAU2 = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
END IF
*
* TAU2 now is an estimation of SIGMA^2 - D( I )^2. The
* following, however, is the corresponding estimation of
* SIGMA - D( I ).
*
TAU = TAU2 / ( D( I )+SQRT( D( I )*D( I )+TAU2 ) )
TEMP = SQRT(EPS)
IF( (D(I).LE.TEMP*D(IP1)).AND.(ABS(Z(I)).LE.TEMP)
$ .AND.(D(I).GT.ZERO) ) THEN
TAU = MIN( TEN*D(I), SGUB )
GEOMAVG = .TRUE.
END IF
ELSE
*
* (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2
*
* We choose d(i+1) as origin.
*
ORGATI = .FALSE.
II = IP1
SGLB = -DELSQ2 / ( D( II )+SQ2 )
SGUB = ZERO
A = C*DELSQ - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
B = Z( IP1 )*Z( IP1 )*DELSQ
IF( A.LT.ZERO ) THEN
TAU2 = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
ELSE
TAU2 = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
END IF
*
* TAU2 now is an estimation of SIGMA^2 - D( IP1 )^2. The
* following, however, is the corresponding estimation of
* SIGMA - D( IP1 ).
*
TAU = TAU2 / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+
$ TAU2 ) ) )
END IF
*
SIGMA = D( II ) + TAU
DO 130 J = 1, N
WORK( J ) = D( J ) + D( II ) + TAU
DELTA( J ) = ( D( J )-D( II ) ) - TAU
130 CONTINUE
IIM1 = II - 1
IIP1 = II + 1
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 150 J = 1, IIM1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
150 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 160 J = N, IIP1, -1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
160 CONTINUE
*
W = RHOINV + PHI + PSI
*
* W is the value of the secular function with
* its ii-th element removed.
*
SWTCH3 = .FALSE.
IF( ORGATI ) THEN
IF( W.LT.ZERO )
$ SWTCH3 = .TRUE.
ELSE
IF( W.GT.ZERO )
$ SWTCH3 = .TRUE.
END IF
IF( II.EQ.1 .OR. II.EQ.N )
$ SWTCH3 = .FALSE.
*
TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = W + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
$ + THREE*ABS( TEMP )
* $ + ABS( TAU2 )*DW
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
GO TO 240
END IF
*
IF( W.LE.ZERO ) THEN
SGLB = MAX( SGLB, TAU )
ELSE
SGUB = MIN( SGUB, TAU )
END IF
*
* Calculate the new step
*
NITER = NITER + 1
IF( .NOT.SWTCH3 ) THEN
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI )
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
DTIIM = WORK( IIM1 )*DELTA( IIM1 )
DTIIP = WORK( IIP1 )*DELTA( IIP1 )
TEMP = RHOINV + PSI + PHI
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DTIIM
TEMP1 = TEMP1*TEMP1
C = ( TEMP - DTIIP*( DPSI+DPHI ) ) -
$ ( D( IIM1 )-D( IIP1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
IF( DPSI.LT.TEMP1 ) THEN
ZZ( 3 ) = DTIIP*DTIIP*DPHI
ELSE
ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
END IF
ELSE
TEMP1 = Z( IIP1 ) / DTIIP
TEMP1 = TEMP1*TEMP1
C = ( TEMP - DTIIM*( DPSI+DPHI ) ) -
$ ( D( IIP1 )-D( IIM1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
IF( DPHI.LT.TEMP1 ) THEN
ZZ( 1 ) = DTIIM*DTIIM*DPSI
ELSE
ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
END IF
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
ZZ( 2 ) = Z( II )*Z( II )
DD( 1 ) = DTIIM
DD( 2 ) = DELTA( II )*WORK( II )
DD( 3 ) = DTIIP
CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
IF( INFO.NE.0 ) THEN
*
* If INFO is not 0, i.e., DLAED6 failed, switch back
* to 2 pole interpolation.
*
SWTCH3 = .FALSE.
INFO = 0
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI)
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
END IF
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
*
ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
TEMP = TAU + ETA
IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( SGUB-TAU ) / TWO
ELSE
ETA = ( SGLB-TAU ) / TWO
END IF
IF( GEOMAVG ) THEN
IF( W .LT. ZERO ) THEN
IF( TAU .GT. ZERO ) THEN
ETA = SQRT(SGUB*TAU)-TAU
END IF
ELSE
IF( SGLB .GT. ZERO ) THEN
ETA = SQRT(SGLB*TAU)-TAU
END IF
END IF
END IF
END IF
*
PREW = W
*
TAU = TAU + ETA
SIGMA = SIGMA + ETA
*
DO 170 J = 1, N
WORK( J ) = WORK( J ) + ETA
DELTA( J ) = DELTA( J ) - ETA
170 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 180 J = 1, IIM1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
180 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 190 J = N, IIP1, -1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
190 CONTINUE
*
TAU2 = WORK( II )*DELTA( II )
TEMP = Z( II ) / TAU2
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = RHOINV + PHI + PSI + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
$ + THREE*ABS( TEMP )
* $ + ABS( TAU2 )*DW
*
SWTCH = .FALSE.
IF( ORGATI ) THEN
IF( -W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
ELSE
IF( W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
END IF
*
* Main loop to update the values of the array DELTA and WORK
*
ITER = NITER + 1
*
DO 230 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
* $ .OR. (SGUB-SGLB).LE.EIGHT*ABS(SGUB+SGLB) ) THEN
GO TO 240
END IF
*
IF( W.LE.ZERO ) THEN
SGLB = MAX( SGLB, TAU )
ELSE
SGUB = MIN( SGUB, TAU )
END IF
*
* Calculate the new step
*
IF( .NOT.SWTCH3 ) THEN
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
END IF
ELSE
TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
IF( ORGATI ) THEN
DPSI = DPSI + TEMP*TEMP
ELSE
DPHI = DPHI + TEMP*TEMP
END IF
C = W - DTISQ*DPSI - DTIPSQ*DPHI
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
$ ( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) +
$ DTISQ*DTISQ*( DPSI+DPHI )
END IF
ELSE
A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
DTIIM = WORK( IIM1 )*DELTA( IIM1 )
DTIIP = WORK( IIP1 )*DELTA( IIP1 )
TEMP = RHOINV + PSI + PHI
IF( SWTCH ) THEN
C = TEMP - DTIIM*DPSI - DTIIP*DPHI
ZZ( 1 ) = DTIIM*DTIIM*DPSI
ZZ( 3 ) = DTIIP*DTIIP*DPHI
ELSE
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DTIIM
TEMP1 = TEMP1*TEMP1
TEMP2 = ( D( IIM1 )-D( IIP1 ) )*
$ ( D( IIM1 )+D( IIP1 ) )*TEMP1
C = TEMP - DTIIP*( DPSI+DPHI ) - TEMP2
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
IF( DPSI.LT.TEMP1 ) THEN
ZZ( 3 ) = DTIIP*DTIIP*DPHI
ELSE
ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
END IF
ELSE
TEMP1 = Z( IIP1 ) / DTIIP
TEMP1 = TEMP1*TEMP1
TEMP2 = ( D( IIP1 )-D( IIM1 ) )*
$ ( D( IIM1 )+D( IIP1 ) )*TEMP1
C = TEMP - DTIIM*( DPSI+DPHI ) - TEMP2
IF( DPHI.LT.TEMP1 ) THEN
ZZ( 1 ) = DTIIM*DTIIM*DPSI
ELSE
ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
END IF
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
END IF
DD( 1 ) = DTIIM
DD( 2 ) = DELTA( II )*WORK( II )
DD( 3 ) = DTIIP
CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
IF( INFO.NE.0 ) THEN
*
* If INFO is not 0, i.e., DLAED6 failed, switch
* back to two pole interpolation
*
SWTCH3 = .FALSE.
INFO = 0
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I )/DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 )/DTIPSQ )**2
END IF
ELSE
TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
IF( ORGATI ) THEN
DPSI = DPSI + TEMP*TEMP
ELSE
DPHI = DPHI + TEMP*TEMP
END IF
C = W - DTISQ*DPSI - DTIPSQ*DPHI
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
$ ( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) +
$ DTISQ*DTISQ*( DPSI+DPHI )
END IF
ELSE
A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
END IF
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
*
ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
TEMP=TAU+ETA
IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( SGUB-TAU ) / TWO
ELSE
ETA = ( SGLB-TAU ) / TWO
END IF
IF( GEOMAVG ) THEN
IF( W .LT. ZERO ) THEN
IF( TAU .GT. ZERO ) THEN
ETA = SQRT(SGUB*TAU)-TAU
END IF
ELSE
IF( SGLB .GT. ZERO ) THEN
ETA = SQRT(SGLB*TAU)-TAU
END IF
END IF
END IF
END IF
*
PREW = W
*
TAU = TAU + ETA
SIGMA = SIGMA + ETA
*
DO 200 J = 1, N
WORK( J ) = WORK( J ) + ETA
DELTA( J ) = DELTA( J ) - ETA
200 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 210 J = 1, IIM1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
210 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 220 J = N, IIP1, -1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
220 CONTINUE
*
TAU2 = WORK( II )*DELTA( II )
TEMP = Z( II ) / TAU2
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = RHOINV + PHI + PSI + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
$ + THREE*ABS( TEMP )
* $ + ABS( TAU2 )*DW
*
IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN )
$ SWTCH = .NOT.SWTCH
*
230 CONTINUE
*
* Return with INFO = 1, NITER = MAXIT and not converged
*
INFO = 1
*
END IF
*
240 CONTINUE
RETURN
*
* End of DLASD4
*
END
*> \brief \b DLASD5 computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
*
* .. Scalar Arguments ..
* INTEGER I
* DOUBLE PRECISION DSIGMA, RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the square root of the I-th eigenvalue
*> of a positive symmetric rank-one modification of a 2-by-2 diagonal
*> matrix
*>
*> diag( D ) * diag( D ) + RHO * Z * transpose(Z) .
*>
*> The diagonal entries in the array D are assumed to satisfy
*>
*> 0 <= D(i) < D(j) for i < j .
*>
*> We also assume RHO > 0 and that the Euclidean norm of the vector
*> Z is one.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I
*> \verbatim
*> I is INTEGER
*> The index of the eigenvalue to be computed. I = 1 or I = 2.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension ( 2 )
*> The original eigenvalues. We assume 0 <= D(1) < D(2).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 2 )
*> The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension ( 2 )
*> Contains (D(j) - sigma_I) in its j-th component.
*> The vector DELTA contains the information necessary
*> to construct the eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DSIGMA
*> \verbatim
*> DSIGMA is DOUBLE PRECISION
*> The computed sigma_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension ( 2 )
*> WORK contains (D(j) + sigma_I) in its j-th component.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd5
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I
DOUBLE PRECISION DSIGMA, RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ THREE = 3.0D+0, FOUR = 4.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION B, C, DEL, DELSQ, TAU, W
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
DEL = D( 2 ) - D( 1 )
DELSQ = DEL*( D( 2 )+D( 1 ) )
IF( I.EQ.1 ) THEN
W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )-
$ Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL
IF( W.GT.ZERO ) THEN
B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 1 )*Z( 1 )*DELSQ
*
* B > ZERO, always
*
* The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
*
TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
*
* The following TAU is DSIGMA - D( 1 )
*
TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) )
DSIGMA = D( 1 ) + TAU
DELTA( 1 ) = -TAU
DELTA( 2 ) = DEL - TAU
WORK( 1 ) = TWO*D( 1 ) + TAU
WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 )
* DELTA( 1 ) = -Z( 1 ) / TAU
* DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
ELSE
B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DELSQ
*
* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
*
IF( B.GT.ZERO ) THEN
TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
ELSE
TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
END IF
*
* The following TAU is DSIGMA - D( 2 )
*
TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) )
DSIGMA = D( 2 ) + TAU
DELTA( 1 ) = -( DEL+TAU )
DELTA( 2 ) = -TAU
WORK( 1 ) = D( 1 ) + TAU + D( 2 )
WORK( 2 ) = TWO*D( 2 ) + TAU
* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
* DELTA( 2 ) = -Z( 2 ) / TAU
END IF
* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
* DELTA( 1 ) = DELTA( 1 ) / TEMP
* DELTA( 2 ) = DELTA( 2 ) / TEMP
ELSE
*
* Now I=2
*
B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DELSQ
*
* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
*
IF( B.GT.ZERO ) THEN
TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
ELSE
TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
END IF
*
* The following TAU is DSIGMA - D( 2 )
*
TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) )
DSIGMA = D( 2 ) + TAU
DELTA( 1 ) = -( DEL+TAU )
DELTA( 2 ) = -TAU
WORK( 1 ) = D( 1 ) + TAU + D( 2 )
WORK( 2 ) = TWO*D( 2 ) + TAU
* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
* DELTA( 2 ) = -Z( 2 ) / TAU
* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
* DELTA( 1 ) = DELTA( 1 ) / TEMP
* DELTA( 2 ) = DELTA( 2 ) / TEMP
END IF
RETURN
*
* End of DLASD5
*
END
*> \brief \b DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd6.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
* IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
* LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
* $ NR, SQRE
* DOUBLE PRECISION ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
* $ PERM( * )
* DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ),
* $ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
* $ VF( * ), VL( * ), WORK( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD6 computes the SVD of an updated upper bidiagonal matrix B
*> obtained by merging two smaller ones by appending a row. This
*> routine is used only for the problem which requires all singular
*> values and optionally singular vector matrices in factored form.
*> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
*> A related subroutine, DLASD1, handles the case in which all singular
*> values and singular vectors of the bidiagonal matrix are desired.
*>
*> DLASD6 computes the SVD as follows:
*>
*> ( D1(in) 0 0 0 )
*> B = U(in) * ( Z1**T a Z2**T b ) * VT(in)
*> ( 0 0 D2(in) 0 )
*>
*> = U(out) * ( D(out) 0) * VT(out)
*>
*> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
*> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
*> elsewhere; and the entry b is empty if SQRE = 0.
*>
*> The singular values of B can be computed using D1, D2, the first
*> components of all the right singular vectors of the lower block, and
*> the last components of all the right singular vectors of the upper
*> block. These components are stored and updated in VF and VL,
*> respectively, in DLASD6. Hence U and VT are not explicitly
*> referenced.
*>
*> The singular values are stored in D. The algorithm consists of two
*> stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple singular values or if there is a zero
*> in the Z vector. For each such occurrence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine DLASD7.
*>
*> The second stage consists of calculating the updated
*> singular values. This is done by finding the roots of the
*> secular equation via the routine DLASD4 (as called by DLASD8).
*> This routine also updates VF and VL and computes the distances
*> between the updated singular values and the old singular
*> values.
*>
*> DLASD6 is called from DLASDA.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether singular vectors are to be computed in
*> factored form:
*> = 0: Compute singular values only.
*> = 1: Compute singular vectors in factored form as well.
*> \endverbatim
*>
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: the lower block is an NR-by-NR square matrix.
*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*> The bidiagonal matrix has row dimension N = NL + NR + 1,
*> and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension ( NL+NR+1 ).
*> On entry D(1:NL,1:NL) contains the singular values of the
*> upper block, and D(NL+2:N) contains the singular values
*> of the lower block. On exit D(1:N) contains the singular
*> values of the modified matrix.
*> \endverbatim
*>
*> \param[in,out] VF
*> \verbatim
*> VF is DOUBLE PRECISION array, dimension ( M )
*> On entry, VF(1:NL+1) contains the first components of all
*> right singular vectors of the upper block; and VF(NL+2:M)
*> contains the first components of all right singular vectors
*> of the lower block. On exit, VF contains the first components
*> of all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension ( M )
*> On entry, VL(1:NL+1) contains the last components of all
*> right singular vectors of the upper block; and VL(NL+2:M)
*> contains the last components of all right singular vectors of
*> the lower block. On exit, VL contains the last components of
*> all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in,out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> Contains the off-diagonal element associated with the added
*> row.
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*> IDXQ is INTEGER array, dimension ( N )
*> This contains the permutation which will reintegrate the
*> subproblem just solved back into sorted order, i.e.
*> D( IDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is INTEGER array, dimension ( N )
*> The permutations (from deflation and sorting) to be applied
*> to each block. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> The number of Givens rotations which took place in this
*> subproblem. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER
*> leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*> Each number indicates the C or S value to be used in the
*> corresponding Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*> LDGNUM is INTEGER
*> The leading dimension of GIVNUM and POLES, must be at least N.
*> \endverbatim
*>
*> \param[out] POLES
*> \verbatim
*> POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*> On exit, POLES(1,*) is an array containing the new singular
*> values obtained from solving the secular equation, and
*> POLES(2,*) is an array containing the poles in the secular
*> equation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*> DIFL is DOUBLE PRECISION array, dimension ( N )
*> On exit, DIFL(I) is the distance between I-th updated
*> (undeflated) singular value and the I-th (undeflated) old
*> singular value.
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*> DIFR is DOUBLE PRECISION array,
*> dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
*> dimension ( K ) if ICOMPQ = 0.
*> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
*> defined and will not be referenced.
*>
*> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
*> normalizing factors for the right singular vector matrix.
*>
*> See DLASD8 for details on DIFL and DIFR.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( M )
*> The first elements of this array contain the components
*> of the deflation-adjusted updating row vector.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> Contains the dimension of the non-deflated matrix,
*> This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> C contains garbage if SQRE =0 and the C-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION
*> S contains garbage if SQRE =0 and the S-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension ( 4 * M )
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension ( 3 * N )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd6
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA,
$ BETA,
$ IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
$ LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
$ IWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
$ NR, SQRE
DOUBLE PRECISION ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
$ PERM( * )
DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ),
$ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
$ VF( * ), VL( * ), WORK( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
$ N, N1, N2
DOUBLE PRECISION ORGNRM
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAMRG, DLASCL, DLASD7, DLASD8,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
N = NL + NR + 1
M = N + SQRE
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( NL.LT.1 ) THEN
INFO = -2
ELSE IF( NR.LT.1 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -14
ELSE IF( LDGNUM.LT.N ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD6', -INFO )
RETURN
END IF
*
* The following values are for bookkeeping purposes only. They are
* integer pointers which indicate the portion of the workspace
* used by a particular array in DLASD7 and DLASD8.
*
ISIGMA = 1
IW = ISIGMA + N
IVFW = IW + M
IVLW = IVFW + M
*
IDX = 1
IDXC = IDX + N
IDXP = IDXC + N
*
* Scale.
*
ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
D( NL+1 ) = ZERO
DO 10 I = 1, N
IF( ABS( D( I ) ).GT.ORGNRM ) THEN
ORGNRM = ABS( D( I ) )
END IF
10 CONTINUE
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
ALPHA = ALPHA / ORGNRM
BETA = BETA / ORGNRM
*
* Sort and Deflate singular values.
*
CALL DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
$ WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
$ WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
$ INFO )
*
* Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
*
CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
$ WORK( ISIGMA ), WORK( IW ), INFO )
*
* Report the possible convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
*
* Save the poles if ICOMPQ = 1.
*
IF( ICOMPQ.EQ.1 ) THEN
CALL DCOPY( K, D, 1, POLES( 1, 1 ), 1 )
CALL DCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
END IF
*
* Unscale.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
* Prepare the IDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL DLAMRG( N1, N2, D, 1, -1, IDXQ )
*
RETURN
*
* End of DLASD6
*
END
*> \brief \b DLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD7 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd7.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd7.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd7.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL,
* VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ,
* PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
* C, S, INFO )
*
* .. Scalar Arguments ..
* INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
* $ NR, SQRE
* DOUBLE PRECISION ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ),
* $ IDXQ( * ), PERM( * )
* DOUBLE PRECISION D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ),
* $ VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ),
* $ ZW( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD7 merges the two sets of singular values together into a single
*> sorted set. Then it tries to deflate the size of the problem. There
*> are two ways in which deflation can occur: when two or more singular
*> values are close together or if there is a tiny entry in the Z
*> vector. For each such occurrence the order of the related
*> secular equation problem is reduced by one.
*>
*> DLASD7 is called from DLASD6.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether singular vectors are to be computed
*> in compact form, as follows:
*> = 0: Compute singular values only.
*> = 1: Compute singular vectors of upper
*> bidiagonal matrix in compact form.
*> \endverbatim
*>
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: the lower block is an NR-by-NR square matrix.
*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*> The bidiagonal matrix has
*> N = NL + NR + 1 rows and
*> M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> Contains the dimension of the non-deflated matrix, this is
*> the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension ( N )
*> On entry D contains the singular values of the two submatrices
*> to be combined. On exit D contains the trailing (N-K) updated
*> singular values (those which were deflated) sorted into
*> increasing order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( M )
*> On exit Z contains the updating row vector in the secular
*> equation.
*> \endverbatim
*>
*> \param[out] ZW
*> \verbatim
*> ZW is DOUBLE PRECISION array, dimension ( M )
*> Workspace for Z.
*> \endverbatim
*>
*> \param[in,out] VF
*> \verbatim
*> VF is DOUBLE PRECISION array, dimension ( M )
*> On entry, VF(1:NL+1) contains the first components of all
*> right singular vectors of the upper block; and VF(NL+2:M)
*> contains the first components of all right singular vectors
*> of the lower block. On exit, VF contains the first components
*> of all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[out] VFW
*> \verbatim
*> VFW is DOUBLE PRECISION array, dimension ( M )
*> Workspace for VF.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension ( M )
*> On entry, VL(1:NL+1) contains the last components of all
*> right singular vectors of the upper block; and VL(NL+2:M)
*> contains the last components of all right singular vectors
*> of the lower block. On exit, VL contains the last components
*> of all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[out] VLW
*> \verbatim
*> VLW is DOUBLE PRECISION array, dimension ( M )
*> Workspace for VL.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> Contains the off-diagonal element associated with the added
*> row.
*> \endverbatim
*>
*> \param[out] DSIGMA
*> \verbatim
*> DSIGMA is DOUBLE PRECISION array, dimension ( N )
*> Contains a copy of the diagonal elements (K-1 singular values
*> and one zero) in the secular equation.
*> \endverbatim
*>
*> \param[out] IDX
*> \verbatim
*> IDX is INTEGER array, dimension ( N )
*> This will contain the permutation used to sort the contents of
*> D into ascending order.
*> \endverbatim
*>
*> \param[out] IDXP
*> \verbatim
*> IDXP is INTEGER array, dimension ( N )
*> This will contain the permutation used to place deflated
*> values of D at the end of the array. On output IDXP(2:K)
*> points to the nondeflated D-values and IDXP(K+1:N)
*> points to the deflated singular values.
*> \endverbatim
*>
*> \param[in] IDXQ
*> \verbatim
*> IDXQ is INTEGER array, dimension ( N )
*> This contains the permutation which separately sorts the two
*> sub-problems in D into ascending order. Note that entries in
*> the first half of this permutation must first be moved one
*> position backward; and entries in the second half
*> must first have NL+1 added to their values.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is INTEGER array, dimension ( N )
*> The permutations (from deflation and sorting) to be applied
*> to each singular block. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> The number of Givens rotations which took place in this
*> subproblem. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER
*> The leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*> Each number indicates the C or S value to be used in the
*> corresponding Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*> LDGNUM is INTEGER
*> The leading dimension of GIVNUM, must be at least N.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> C contains garbage if SQRE =0 and the C-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION
*> S contains garbage if SQRE =0 and the S-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd7
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW,
$ VL,
$ VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ,
$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
$ C, S, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
$ NR, SQRE
DOUBLE PRECISION ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ),
$ IDXQ( * ), PERM( * )
DOUBLE PRECISION D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ),
$ VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ),
$ ZW( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, EIGHT
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ EIGHT = 8.0D+0 )
* ..
* .. Local Scalars ..
*
INTEGER I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M, N,
$ NLP1, NLP2
DOUBLE PRECISION EPS, HLFTOL, TAU, TOL, Z1
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAMRG, DROT, XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
N = NL + NR + 1
M = N + SQRE
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( NL.LT.1 ) THEN
INFO = -2
ELSE IF( NR.LT.1 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -22
ELSE IF( LDGNUM.LT.N ) THEN
INFO = -24
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD7', -INFO )
RETURN
END IF
*
NLP1 = NL + 1
NLP2 = NL + 2
IF( ICOMPQ.EQ.1 ) THEN
GIVPTR = 0
END IF
*
* Generate the first part of the vector Z and move the singular
* values in the first part of D one position backward.
*
Z1 = ALPHA*VL( NLP1 )
VL( NLP1 ) = ZERO
TAU = VF( NLP1 )
DO 10 I = NL, 1, -1
Z( I+1 ) = ALPHA*VL( I )
VL( I ) = ZERO
VF( I+1 ) = VF( I )
D( I+1 ) = D( I )
IDXQ( I+1 ) = IDXQ( I ) + 1
10 CONTINUE
VF( 1 ) = TAU
*
* Generate the second part of the vector Z.
*
DO 20 I = NLP2, M
Z( I ) = BETA*VF( I )
VF( I ) = ZERO
20 CONTINUE
*
* Sort the singular values into increasing order
*
DO 30 I = NLP2, N
IDXQ( I ) = IDXQ( I ) + NLP1
30 CONTINUE
*
* DSIGMA, IDXC, IDXC, and ZW are used as storage space.
*
DO 40 I = 2, N
DSIGMA( I ) = D( IDXQ( I ) )
ZW( I ) = Z( IDXQ( I ) )
VFW( I ) = VF( IDXQ( I ) )
VLW( I ) = VL( IDXQ( I ) )
40 CONTINUE
*
CALL DLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) )
*
DO 50 I = 2, N
IDXI = 1 + IDX( I )
D( I ) = DSIGMA( IDXI )
Z( I ) = ZW( IDXI )
VF( I ) = VFW( IDXI )
VL( I ) = VLW( IDXI )
50 CONTINUE
*
* Calculate the allowable deflation tolerance
*
EPS = DLAMCH( 'Epsilon' )
TOL = MAX( ABS( ALPHA ), ABS( BETA ) )
TOL = EIGHT*EIGHT*EPS*MAX( ABS( D( N ) ), TOL )
*
* There are 2 kinds of deflation -- first a value in the z-vector
* is small, second two (or more) singular values are very close
* together (their difference is small).
*
* If the value in the z-vector is small, we simply permute the
* array so that the corresponding singular value is moved to the
* end.
*
* If two values in the D-vector are close, we perform a two-sided
* rotation designed to make one of the corresponding z-vector
* entries zero, and then permute the array so that the deflated
* singular value is moved to the end.
*
* If there are multiple singular values then the problem deflates.
* Here the number of equal singular values are found. As each equal
* singular value is found, an elementary reflector is computed to
* rotate the corresponding singular subspace so that the
* corresponding components of Z are zero in this new basis.
*
K = 1
K2 = N + 1
DO 60 J = 2, N
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
IF( J.EQ.N )
$ GO TO 100
ELSE
JPREV = J
GO TO 70
END IF
60 CONTINUE
70 CONTINUE
J = JPREV
80 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 90
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
ELSE
*
* Check if singular values are close enough to allow deflation.
*
IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN
*
* Deflation is possible.
*
S = Z( JPREV )
C = Z( J )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = DLAPY2( C, S )
Z( J ) = TAU
Z( JPREV ) = ZERO
C = C / TAU
S = -S / TAU
*
* Record the appropriate Givens rotation
*
IF( ICOMPQ.EQ.1 ) THEN
GIVPTR = GIVPTR + 1
IDXJP = IDXQ( IDX( JPREV )+1 )
IDXJ = IDXQ( IDX( J )+1 )
IF( IDXJP.LE.NLP1 ) THEN
IDXJP = IDXJP - 1
END IF
IF( IDXJ.LE.NLP1 ) THEN
IDXJ = IDXJ - 1
END IF
GIVCOL( GIVPTR, 2 ) = IDXJP
GIVCOL( GIVPTR, 1 ) = IDXJ
GIVNUM( GIVPTR, 2 ) = C
GIVNUM( GIVPTR, 1 ) = S
END IF
CALL DROT( 1, VF( JPREV ), 1, VF( J ), 1, C, S )
CALL DROT( 1, VL( JPREV ), 1, VL( J ), 1, C, S )
K2 = K2 - 1
IDXP( K2 ) = JPREV
JPREV = J
ELSE
K = K + 1
ZW( K ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
JPREV = J
END IF
END IF
GO TO 80
90 CONTINUE
*
* Record the last singular value.
*
K = K + 1
ZW( K ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
*
100 CONTINUE
*
* Sort the singular values into DSIGMA. The singular values which
* were not deflated go into the first K slots of DSIGMA, except
* that DSIGMA(1) is treated separately.
*
DO 110 J = 2, N
JP = IDXP( J )
DSIGMA( J ) = D( JP )
VFW( J ) = VF( JP )
VLW( J ) = VL( JP )
110 CONTINUE
IF( ICOMPQ.EQ.1 ) THEN
DO 120 J = 2, N
JP = IDXP( J )
PERM( J ) = IDXQ( IDX( JP )+1 )
IF( PERM( J ).LE.NLP1 ) THEN
PERM( J ) = PERM( J ) - 1
END IF
120 CONTINUE
END IF
*
* The deflated singular values go back into the last N - K slots of
* D.
*
CALL DCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 )
*
* Determine DSIGMA(1), DSIGMA(2), Z(1), VF(1), VL(1), VF(M), and
* VL(M).
*
DSIGMA( 1 ) = ZERO
HLFTOL = TOL / TWO
IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL )
$ DSIGMA( 2 ) = HLFTOL
IF( M.GT.N ) THEN
Z( 1 ) = DLAPY2( Z1, Z( M ) )
IF( Z( 1 ).LE.TOL ) THEN
C = ONE
S = ZERO
Z( 1 ) = TOL
ELSE
C = Z1 / Z( 1 )
S = -Z( M ) / Z( 1 )
END IF
CALL DROT( 1, VF( M ), 1, VF( 1 ), 1, C, S )
CALL DROT( 1, VL( M ), 1, VL( 1 ), 1, C, S )
ELSE
IF( ABS( Z1 ).LE.TOL ) THEN
Z( 1 ) = TOL
ELSE
Z( 1 ) = Z1
END IF
END IF
*
* Restore Z, VF, and VL.
*
CALL DCOPY( K-1, ZW( 2 ), 1, Z( 2 ), 1 )
CALL DCOPY( N-1, VFW( 2 ), 1, VF( 2 ), 1 )
CALL DCOPY( N-1, VLW( 2 ), 1, VL( 2 ), 1 )
*
RETURN
*
* End of DLASD7
*
END
*> \brief \b DLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD8 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd8.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd8.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd8.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
* DSIGMA, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, K, LDDIFR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ),
* $ DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
* $ Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD8 finds the square roots of the roots of the secular equation,
*> as defined by the values in DSIGMA and Z. It makes the appropriate
*> calls to DLASD4, and stores, for each element in D, the distance
*> to its two nearest poles (elements in DSIGMA). It also updates
*> the arrays VF and VL, the first and last components of all the
*> right singular vectors of the original bidiagonal matrix.
*>
*> DLASD8 is called from DLASD6.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether singular vectors are to be computed in
*> factored form in the calling routine:
*> = 0: Compute singular values only.
*> = 1: Compute singular vectors in factored form as well.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved
*> by DLASD4. K >= 1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension ( K )
*> On output, D contains the updated singular values.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( K )
*> On entry, the first K elements of this array contain the
*> components of the deflation-adjusted updating row vector.
*> On exit, Z is updated.
*> \endverbatim
*>
*> \param[in,out] VF
*> \verbatim
*> VF is DOUBLE PRECISION array, dimension ( K )
*> On entry, VF contains information passed through DBEDE8.
*> On exit, VF contains the first K components of the first
*> components of all right singular vectors of the bidiagonal
*> matrix.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension ( K )
*> On entry, VL contains information passed through DBEDE8.
*> On exit, VL contains the first K components of the last
*> components of all right singular vectors of the bidiagonal
*> matrix.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*> DIFL is DOUBLE PRECISION array, dimension ( K )
*> On exit, DIFL(I) = D(I) - DSIGMA(I).
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*> DIFR is DOUBLE PRECISION array,
*> dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
*> dimension ( K ) if ICOMPQ = 0.
*> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
*> defined and will not be referenced.
*>
*> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
*> normalizing factors for the right singular vector matrix.
*> \endverbatim
*>
*> \param[in] LDDIFR
*> \verbatim
*> LDDIFR is INTEGER
*> The leading dimension of DIFR, must be at least K.
*> \endverbatim
*>
*> \param[in] DSIGMA
*> \verbatim
*> DSIGMA is DOUBLE PRECISION array, dimension ( K )
*> On entry, the first K elements of this array contain the old
*> roots of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*K)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd8
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
$ DSIGMA, WORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, K, LDDIFR
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ),
$ DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
$ Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J
DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLASCL, DLASD4, DLASET,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DDOT, DLAMC3, DNRM2
EXTERNAL DDOT, DLAMC3, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( K.LT.1 ) THEN
INFO = -2
ELSE IF( LDDIFR.LT.K ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD8', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.1 ) THEN
D( 1 ) = ABS( Z( 1 ) )
DIFL( 1 ) = D( 1 )
IF( ICOMPQ.EQ.1 ) THEN
DIFL( 2 ) = ONE
DIFR( 1, 2 ) = ONE
END IF
RETURN
END IF
*
* Book keeping.
*
IWK1 = 1
IWK2 = IWK1 + K
IWK3 = IWK2 + K
IWK2I = IWK2 - 1
IWK3I = IWK3 - 1
*
* Normalize Z.
*
RHO = DNRM2( K, Z, 1 )
CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
RHO = RHO*RHO
*
* Initialize WORK(IWK3).
*
CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K )
*
* Compute the updated singular values, the arrays DIFL, DIFR,
* and the updated Z.
*
DO 40 J = 1, K
CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ),
$ WORK( IWK2 ), INFO )
*
* If the root finder fails, report the convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J )
DIFL( J ) = -WORK( J )
DIFR( J, 1 ) = -WORK( J+1 )
DO 20 I = 1, J - 1
WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
$ WORK( IWK2I+I ) / ( DSIGMA( I )-
$ DSIGMA( J ) ) / ( DSIGMA( I )+
$ DSIGMA( J ) )
20 CONTINUE
DO 30 I = J + 1, K
WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
$ WORK( IWK2I+I ) / ( DSIGMA( I )-
$ DSIGMA( J ) ) / ( DSIGMA( I )+
$ DSIGMA( J ) )
30 CONTINUE
40 CONTINUE
*
* Compute updated Z.
*
DO 50 I = 1, K
Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) )
50 CONTINUE
*
* Update VF and VL.
*
DO 80 J = 1, K
DIFLJ = DIFL( J )
DJ = D( J )
DSIGJ = -DSIGMA( J )
IF( J.LT.K ) THEN
DIFRJ = -DIFR( J, 1 )
DSIGJP = -DSIGMA( J+1 )
END IF
WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ )
*
* Use calls to the subroutine DLAMC3 to enforce the parentheses
* (x+y)+z. The goal is to prevent optimizing compilers
* from doing x+(y+z).
*
DO 60 I = 1, J - 1
WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ),
$ DSIGJ )-DIFLJ )
$ / ( DSIGMA( I )+DJ )
60 CONTINUE
DO 70 I = J + 1, K
WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ),
$ DSIGJP )+DIFRJ )
$ / ( DSIGMA( I )+DJ )
70 CONTINUE
TEMP = DNRM2( K, WORK, 1 )
WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP
WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP
IF( ICOMPQ.EQ.1 ) THEN
DIFR( J, 2 ) = TEMP
END IF
80 CONTINUE
*
CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 )
CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 )
*
RETURN
*
* End of DLASD8
*
END
*> \brief \b DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASDA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasda.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasda.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasda.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
* DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
* PERM, GIVNUM, C, S, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
* $ K( * ), PERM( LDGCOL, * )
* DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
* $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
* $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
* $ Z( LDU, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Using a divide and conquer approach, DLASDA computes the singular
*> value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
*> B with diagonal D and offdiagonal E, where M = N + SQRE. The
*> algorithm computes the singular values in the SVD B = U * S * VT.
*> The orthogonal matrices U and VT are optionally computed in
*> compact form.
*>
*> A related subroutine, DLASD0, computes the singular values and
*> the singular vectors in explicit form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether singular vectors are to be computed
*> in compact form, as follows
*> = 0: Compute singular values only.
*> = 1: Compute singular vectors of upper bidiagonal
*> matrix in compact form.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> The maximum size of the subproblems at the bottom of the
*> computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The row dimension of the upper bidiagonal matrix. This is
*> also the dimension of the main diagonal array D.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> Specifies the column dimension of the bidiagonal matrix.
*> = 0: The bidiagonal matrix has column dimension M = N;
*> = 1: The bidiagonal matrix has column dimension M = N + 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension ( N )
*> On entry D contains the main diagonal of the bidiagonal
*> matrix. On exit D, if INFO = 0, contains its singular values.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension ( M-1 )
*> Contains the subdiagonal entries of the bidiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array,
*> dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
*> singular vector matrices of all subproblems at the bottom
*> level.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER, LDU = > N.
*> The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
*> GIVNUM, and Z.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array,
*> dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
*> singular vector matrices of all subproblems at the bottom
*> level.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER array,
*> dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
*> If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
*> secular equation on the computation tree.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*> DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ),
*> where NLVL = floor(log_2 (N/SMLSIZ))).
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*> DIFR is DOUBLE PRECISION array,
*> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
*> dimension ( N ) if ICOMPQ = 0.
*> If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
*> record distances between singular values on the I-th
*> level and singular values on the (I -1)-th level, and
*> DIFR(1:N, 2 * I ) contains the normalizing factors for
*> the right singular vector matrix. See DLASD8 for details.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array,
*> dimension ( LDU, NLVL ) if ICOMPQ = 1 and
*> dimension ( N ) if ICOMPQ = 0.
*> The first K elements of Z(1, I) contain the components of
*> the deflation-adjusted updating row vector for subproblems
*> on the I-th level.
*> \endverbatim
*>
*> \param[out] POLES
*> \verbatim
*> POLES is DOUBLE PRECISION array,
*> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
*> POLES(1, 2*I) contain the new and old singular values
*> involved in the secular equations on the I-th level.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array,
*> dimension ( N ) if ICOMPQ = 1, and not referenced if
*> ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
*> the number of Givens rotations performed on the I-th
*> problem on the computation tree.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array,
*> dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
*> referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
*> GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
*> of Givens rotations performed on the I-th level on the
*> computation tree.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER, LDGCOL = > N.
*> The leading dimension of arrays GIVCOL and PERM.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is INTEGER array,
*> dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
*> permutations done on the I-th level of the computation tree.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array,
*> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not
*> referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
*> GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
*> values of Givens rotations performed on the I-th level on
*> the computation tree.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION array,
*> dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
*> If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
*> C( I ) contains the C-value of a Givens rotation related to
*> the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension ( N ) if
*> ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
*> and the I-th subproblem is not square, on exit, S( I )
*> contains the S-value of a Givens rotation related to
*> the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasda
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT,
$ K,
$ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
$ PERM, GIVNUM, C, S, WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
$ K( * ), PERM( LDGCOL, * )
DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
$ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
$ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
$ Z( LDU, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,
$ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,
$ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,
$ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI
DOUBLE PRECISION ALPHA, BETA
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLASD6, DLASDQ, DLASDT, DLASET,
$ XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( SMLSIZ.LT.3 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( LDU.LT.( N+SQRE ) ) THEN
INFO = -8
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -17
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASDA', -INFO )
RETURN
END IF
*
M = N + SQRE
*
* If the input matrix is too small, call DLASDQ to find the SVD.
*
IF( N.LE.SMLSIZ ) THEN
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U,
$ LDU,
$ U, LDU, WORK, INFO )
ELSE
CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U,
$ LDU,
$ U, LDU, WORK, INFO )
END IF
RETURN
END IF
*
* Book-keeping and set up the computation tree.
*
INODE = 1
NDIML = INODE + N
NDIMR = NDIML + N
IDXQ = NDIMR + N
IWK = IDXQ + N
*
NCC = 0
NRU = 0
*
SMLSZP = SMLSIZ + 1
VF = 1
VL = VF + M
NWORK1 = VL + M
NWORK2 = NWORK1 + SMLSZP*SMLSZP
*
CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
$ IWORK( NDIMR ), SMLSIZ )
*
* for the nodes on bottom level of the tree, solve
* their subproblems by DLASDQ.
*
NDB1 = ( ND+1 ) / 2
DO 30 I = NDB1, ND
*
* IC : center row of each node
* NL : number of rows of left subproblem
* NR : number of rows of right subproblem
* NLF: starting row of the left subproblem
* NRF: starting row of the right subproblem
*
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NLP1 = NL + 1
NR = IWORK( NDIMR+I1 )
NLF = IC - NL
NRF = IC + 1
IDXQI = IDXQ + NLF - 2
VFI = VF + NLF - 1
VLI = VL + NLF - 1
SQREI = 1
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),
$ SMLSZP )
CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ),
$ E( NLF ), WORK( NWORK1 ), SMLSZP,
$ WORK( NWORK2 ), NL, WORK( NWORK2 ), NL,
$ WORK( NWORK2 ), INFO )
ITEMP = NWORK1 + NL*SMLSZP
CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
ELSE
CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )
CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ),
$ LDU )
CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
$ E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU,
$ U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO )
CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 )
END IF
IF( INFO.NE.0 ) THEN
RETURN
END IF
DO 10 J = 1, NL
IWORK( IDXQI+J ) = J
10 CONTINUE
IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN
SQREI = 0
ELSE
SQREI = 1
END IF
IDXQI = IDXQI + NLP1
VFI = VFI + NLP1
VLI = VLI + NLP1
NRP1 = NR + SQREI
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),
$ SMLSZP )
CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ),
$ E( NRF ), WORK( NWORK1 ), SMLSZP,
$ WORK( NWORK2 ), NR, WORK( NWORK2 ), NR,
$ WORK( NWORK2 ), INFO )
ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP
CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
ELSE
CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )
CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ),
$ LDU )
CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
$ E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU,
$ U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO )
CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 )
END IF
IF( INFO.NE.0 ) THEN
RETURN
END IF
DO 20 J = 1, NR
IWORK( IDXQI+J ) = J
20 CONTINUE
30 CONTINUE
*
* Now conquer each subproblem bottom-up.
*
J = 2**NLVL
DO 50 LVL = NLVL, 1, -1
LVL2 = LVL*2 - 1
*
* Find the first node LF and last node LL on
* the current level LVL.
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 40 I = LF, LL
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
NRF = IC + 1
IF( I.EQ.LL ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
VFI = VF + NLF - 1
VLI = VL + NLF - 1
IDXQI = IDXQ + NLF - 1
ALPHA = D( IC )
BETA = E( IC )
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
$ WORK( VFI ), WORK( VLI ), ALPHA, BETA,
$ IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL,
$ LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z,
$ K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ),
$ IWORK( IWK ), INFO )
ELSE
J = J - 1
CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
$ WORK( VFI ), WORK( VLI ), ALPHA, BETA,
$ IWORK( IDXQI ), PERM( NLF, LVL ),
$ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
$ GIVNUM( NLF, LVL2 ), LDU,
$ POLES( NLF, LVL2 ), DIFL( NLF, LVL ),
$ DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ),
$ C( J ), S( J ), WORK( NWORK1 ),
$ IWORK( IWK ), INFO )
END IF
IF( INFO.NE.0 ) THEN
RETURN
END IF
40 CONTINUE
50 CONTINUE
*
RETURN
*
* End of DLASDA
*
END
*> \brief \b DLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASDQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasdq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasdq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasdq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASDQ( UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT,
* U, LDU, C, LDC, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU, SQRE
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASDQ computes the singular value decomposition (SVD) of a real
*> (upper or lower) bidiagonal matrix with diagonal D and offdiagonal
*> E, accumulating the transformations if desired. Letting B denote
*> the input bidiagonal matrix, the algorithm computes orthogonal
*> matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose
*> of P). The singular values S are overwritten on D.
*>
*> The input matrix U is changed to U * Q if desired.
*> The input matrix VT is changed to P**T * VT if desired.
*> The input matrix C is changed to Q**T * C if desired.
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices With
*> Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
*> LAPACK Working Note #3, for a detailed description of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the input bidiagonal matrix
*> is upper or lower bidiagonal, and whether it is square are
*> not.
*> UPLO = 'U' or 'u' B is upper bidiagonal.
*> UPLO = 'L' or 'l' B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: then the input matrix is N-by-N.
*> = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and
*> (N+1)-by-N if UPLU = 'L'.
*>
*> The bidiagonal matrix has
*> N = NL + NR + 1 rows and
*> M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the number of rows and columns
*> in the matrix. N must be at least 0.
*> \endverbatim
*>
*> \param[in] NCVT
*> \verbatim
*> NCVT is INTEGER
*> On entry, NCVT specifies the number of columns of
*> the matrix VT. NCVT must be at least 0.
*> \endverbatim
*>
*> \param[in] NRU
*> \verbatim
*> NRU is INTEGER
*> On entry, NRU specifies the number of rows of
*> the matrix U. NRU must be at least 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*> NCC is INTEGER
*> On entry, NCC specifies the number of columns of
*> the matrix C. NCC must be at least 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D contains the diagonal entries of the
*> bidiagonal matrix whose SVD is desired. On normal exit,
*> D contains the singular values in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array.
*> dimension is (N-1) if SQRE = 0 and N if SQRE = 1.
*> On entry, the entries of E contain the offdiagonal entries
*> of the bidiagonal matrix whose SVD is desired. On normal
*> exit, E will contain 0. If the algorithm does not converge,
*> D and E will contain the diagonal and superdiagonal entries
*> of a bidiagonal matrix orthogonally equivalent to the one
*> given as input.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
*> On entry, contains a matrix which on exit has been
*> premultiplied by P**T, dimension N-by-NCVT if SQRE = 0
*> and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> On entry, LDVT specifies the leading dimension of VT as
*> declared in the calling (sub) program. LDVT must be at
*> least 1. If NCVT is nonzero LDVT must also be at least N.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU, N)
*> On entry, contains a matrix which on exit has been
*> postmultiplied by Q, dimension NRU-by-N if SQRE = 0
*> and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0).
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> On entry, LDU specifies the leading dimension of U as
*> declared in the calling (sub) program. LDU must be at
*> least max( 1, NRU ) .
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC, NCC)
*> On entry, contains an N-by-NCC matrix which on exit
*> has been premultiplied by Q**T dimension N-by-NCC if SQRE = 0
*> and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> On entry, LDC specifies the leading dimension of C as
*> declared in the calling (sub) program. LDC must be at
*> least 1. If NCC is nonzero, LDC must also be at least N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> Workspace. Only referenced if one of NCVT, NRU, or NCC is
*> nonzero, and if N is at least 2.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, a value of 0 indicates a successful exit.
*> If INFO < 0, argument number -INFO is illegal.
*> If INFO > 0, the algorithm did not converge, and INFO
*> specifies how many superdiagonals did not converge.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasdq
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASDQ( UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT,
$ LDVT,
$ U, LDU, C, LDC, WORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU, SQRE
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ROTATE
INTEGER I, ISUB, IUPLO, J, NP1, SQRE1
DOUBLE PRECISION CS, R, SMIN, SN
* ..
* .. External Subroutines ..
EXTERNAL DBDSQR, DLARTG, DLASR, DSWAP,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IUPLO = 0
IF( LSAME( UPLO, 'U' ) )
$ IUPLO = 1
IF( LSAME( UPLO, 'L' ) )
$ IUPLO = 2
IF( IUPLO.EQ.0 ) THEN
INFO = -1
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NCVT.LT.0 ) THEN
INFO = -4
ELSE IF( NRU.LT.0 ) THEN
INFO = -5
ELSE IF( NCC.LT.0 ) THEN
INFO = -6
ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR.
$ ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN
INFO = -10
ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN
INFO = -12
ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR.
$ ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASDQ', -INFO )
RETURN
END IF
IF( N.EQ.0 )
$ RETURN
*
* ROTATE is true if any singular vectors desired, false otherwise
*
ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 )
NP1 = N + 1
SQRE1 = SQRE
*
* If matrix non-square upper bidiagonal, rotate to be lower
* bidiagonal. The rotations are on the right.
*
IF( ( IUPLO.EQ.1 ) .AND. ( SQRE1.EQ.1 ) ) THEN
DO 10 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
IF( ROTATE ) THEN
WORK( I ) = CS
WORK( N+I ) = SN
END IF
10 CONTINUE
CALL DLARTG( D( N ), E( N ), CS, SN, R )
D( N ) = R
E( N ) = ZERO
IF( ROTATE ) THEN
WORK( N ) = CS
WORK( N+N ) = SN
END IF
IUPLO = 2
SQRE1 = 0
*
* Update singular vectors if desired.
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', NP1, NCVT, WORK( 1 ),
$ WORK( NP1 ), VT, LDVT )
END IF
*
* If matrix lower bidiagonal, rotate to be upper bidiagonal
* by applying Givens rotations on the left.
*
IF( IUPLO.EQ.2 ) THEN
DO 20 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
IF( ROTATE ) THEN
WORK( I ) = CS
WORK( N+I ) = SN
END IF
20 CONTINUE
*
* If matrix (N+1)-by-N lower bidiagonal, one additional
* rotation is needed.
*
IF( SQRE1.EQ.1 ) THEN
CALL DLARTG( D( N ), E( N ), CS, SN, R )
D( N ) = R
IF( ROTATE ) THEN
WORK( N ) = CS
WORK( N+N ) = SN
END IF
END IF
*
* Update singular vectors if desired.
*
IF( NRU.GT.0 ) THEN
IF( SQRE1.EQ.0 ) THEN
CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ),
$ WORK( NP1 ), U, LDU )
ELSE
CALL DLASR( 'R', 'V', 'F', NRU, NP1, WORK( 1 ),
$ WORK( NP1 ), U, LDU )
END IF
END IF
IF( NCC.GT.0 ) THEN
IF( SQRE1.EQ.0 ) THEN
CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ),
$ WORK( NP1 ), C, LDC )
ELSE
CALL DLASR( 'L', 'V', 'F', NP1, NCC, WORK( 1 ),
$ WORK( NP1 ), C, LDC )
END IF
END IF
END IF
*
* Call DBDSQR to compute the SVD of the reduced real
* N-by-N upper bidiagonal matrix.
*
CALL DBDSQR( 'U', N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C,
$ LDC, WORK, INFO )
*
* Sort the singular values into ascending order (insertion sort on
* singular values, but only one transposition per singular vector)
*
DO 40 I = 1, N
*
* Scan for smallest D(I).
*
ISUB = I
SMIN = D( I )
DO 30 J = I + 1, N
IF( D( J ).LT.SMIN ) THEN
ISUB = J
SMIN = D( J )
END IF
30 CONTINUE
IF( ISUB.NE.I ) THEN
*
* Swap singular values and vectors.
*
D( ISUB ) = D( I )
D( I ) = SMIN
IF( NCVT.GT.0 )
$ CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( I, 1 ),
$ LDVT )
IF( NRU.GT.0 )
$ CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, I ), 1 )
IF( NCC.GT.0 )
$ CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( I, 1 ), LDC )
END IF
40 CONTINUE
*
RETURN
*
* End of DLASDQ
*
END
*> \brief \b DLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASDT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasdt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasdt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasdt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASDT( N, LVL, ND, INODE, NDIML, NDIMR, MSUB )
*
* .. Scalar Arguments ..
* INTEGER LVL, MSUB, N, ND
* ..
* .. Array Arguments ..
* INTEGER INODE( * ), NDIML( * ), NDIMR( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASDT creates a tree of subproblems for bidiagonal divide and
*> conquer.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, the number of diagonal elements of the
*> bidiagonal matrix.
*> \endverbatim
*>
*> \param[out] LVL
*> \verbatim
*> LVL is INTEGER
*> On exit, the number of levels on the computation tree.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*> ND is INTEGER
*> On exit, the number of nodes on the tree.
*> \endverbatim
*>
*> \param[out] INODE
*> \verbatim
*> INODE is INTEGER array, dimension ( N )
*> On exit, centers of subproblems.
*> \endverbatim
*>
*> \param[out] NDIML
*> \verbatim
*> NDIML is INTEGER array, dimension ( N )
*> On exit, row dimensions of left children.
*> \endverbatim
*>
*> \param[out] NDIMR
*> \verbatim
*> NDIMR is INTEGER array, dimension ( N )
*> On exit, row dimensions of right children.
*> \endverbatim
*>
*> \param[in] MSUB
*> \verbatim
*> MSUB is INTEGER
*> On entry, the maximum row dimension each subproblem at the
*> bottom of the tree can be of.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasdt
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASDT( N, LVL, ND, INODE, NDIML, NDIMR, MSUB )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LVL, MSUB, N, ND
* ..
* .. Array Arguments ..
INTEGER INODE( * ), NDIML( * ), NDIMR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IL, IR, LLST, MAXN, NCRNT, NLVL
DOUBLE PRECISION TEMP
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, INT, LOG, MAX
* ..
* .. Executable Statements ..
*
* Find the number of levels on the tree.
*
MAXN = MAX( 1, N )
TEMP = LOG( DBLE( MAXN ) / DBLE( MSUB+1 ) ) / LOG( TWO )
LVL = INT( TEMP ) + 1
*
I = N / 2
INODE( 1 ) = I + 1
NDIML( 1 ) = I
NDIMR( 1 ) = N - I - 1
IL = 0
IR = 1
LLST = 1
DO 20 NLVL = 1, LVL - 1
*
* Constructing the tree at (NLVL+1)-st level. The number of
* nodes created on this level is LLST * 2.
*
DO 10 I = 0, LLST - 1
IL = IL + 2
IR = IR + 2
NCRNT = LLST + I
NDIML( IL ) = NDIML( NCRNT ) / 2
NDIMR( IL ) = NDIML( NCRNT ) - NDIML( IL ) - 1
INODE( IL ) = INODE( NCRNT ) - NDIMR( IL ) - 1
NDIML( IR ) = NDIMR( NCRNT ) / 2
NDIMR( IR ) = NDIMR( NCRNT ) - NDIML( IR ) - 1
INODE( IR ) = INODE( NCRNT ) + NDIML( IR ) + 1
10 CONTINUE
LLST = LLST*2
20 CONTINUE
ND = LLST*2 - 1
*
RETURN
*
* End of DLASDT
*
END
*> \brief \b DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASET + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaset.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaset.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaset.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, M, N
* DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASET initializes an m-by-n matrix A to BETA on the diagonal and
*> ALPHA on the offdiagonals.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies the part of the matrix A to be set.
*> = 'U': Upper triangular part is set; the strictly lower
*> triangular part of A is not changed.
*> = 'L': Lower triangular part is set; the strictly upper
*> triangular part of A is not changed.
*> Otherwise: All of the matrix A is set.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> The constant to which the offdiagonal elements are to be set.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> The constant to which the diagonal elements are to be set.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On exit, the leading m-by-n submatrix of A is set as follows:
*>
*> if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n,
*> if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n,
*> otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j,
*>
*> and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laset
*
* =====================================================================
SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, M, N
DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Set the strictly upper triangular or trapezoidal part of the
* array to ALPHA.
*
DO 20 J = 2, N
DO 10 I = 1, MIN( J-1, M )
A( I, J ) = ALPHA
10 CONTINUE
20 CONTINUE
*
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
*
* Set the strictly lower triangular or trapezoidal part of the
* array to ALPHA.
*
DO 40 J = 1, MIN( M, N )
DO 30 I = J + 1, M
A( I, J ) = ALPHA
30 CONTINUE
40 CONTINUE
*
ELSE
*
* Set the leading m-by-n submatrix to ALPHA.
*
DO 60 J = 1, N
DO 50 I = 1, M
A( I, J ) = ALPHA
50 CONTINUE
60 CONTINUE
END IF
*
* Set the first min(M,N) diagonal elements to BETA.
*
DO 70 I = 1, MIN( M, N )
A( I, I ) = BETA
70 CONTINUE
*
RETURN
*
* End of DLASET
*
END
*> \brief \b DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ1 computes the singular values of a real N-by-N bidiagonal
*> matrix with diagonal D and off-diagonal E. The singular values
*> are computed to high relative accuracy, in the absence of
*> denormalization, underflow and overflow. The algorithm was first
*> presented in
*>
*> "Accurate singular values and differential qd algorithms" by K. V.
*> Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
*> 1994,
*>
*> and the present implementation is described in "An implementation of
*> the dqds Algorithm (Positive Case)", LAPACK Working Note.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D contains the diagonal elements of the
*> bidiagonal matrix whose SVD is desired. On normal exit,
*> D contains the singular values in decreasing order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, elements E(1:N-1) contain the off-diagonal elements
*> of the bidiagonal matrix whose SVD is desired.
*> On exit, E is overwritten.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm failed
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 100*N
*> iterations (in inner while loop) On exit D and E
*> represent a matrix with the same singular values
*> which the calling subroutine could use to finish the
*> computation, or even feed back into DLASQ1
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq1
*
* =====================================================================
SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, IINFO
DOUBLE PRECISION EPS, SCALE, SAFMIN, SIGMN, SIGMX
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAS2, DLASCL, DLASQ2, DLASRT,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DLASQ1', -INFO )
RETURN
ELSE IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
D( 1 ) = ABS( D( 1 ) )
RETURN
ELSE IF( N.EQ.2 ) THEN
CALL DLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX )
D( 1 ) = SIGMX
D( 2 ) = SIGMN
RETURN
END IF
*
* Estimate the largest singular value.
*
SIGMX = ZERO
DO 10 I = 1, N - 1
D( I ) = ABS( D( I ) )
SIGMX = MAX( SIGMX, ABS( E( I ) ) )
10 CONTINUE
D( N ) = ABS( D( N ) )
*
* Early return if SIGMX is zero (matrix is already diagonal).
*
IF( SIGMX.EQ.ZERO ) THEN
CALL DLASRT( 'D', N, D, IINFO )
RETURN
END IF
*
DO 20 I = 1, N
SIGMX = MAX( SIGMX, D( I ) )
20 CONTINUE
*
* Copy D and E into WORK (in the Z format) and scale (squaring the
* input data makes scaling by a power of the radix pointless).
*
EPS = DLAMCH( 'Precision' )
SAFMIN = DLAMCH( 'Safe minimum' )
SCALE = SQRT( EPS / SAFMIN )
CALL DCOPY( N, D, 1, WORK( 1 ), 2 )
CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )
CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,
$ IINFO )
*
* Compute the q's and e's.
*
DO 30 I = 1, 2*N - 1
WORK( I ) = WORK( I )**2
30 CONTINUE
WORK( 2*N ) = ZERO
*
CALL DLASQ2( N, WORK, INFO )
*
IF( INFO.EQ.0 ) THEN
DO 40 I = 1, N
D( I ) = SQRT( WORK( I ) )
40 CONTINUE
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
ELSE IF( INFO.EQ.2 ) THEN
*
* Maximum number of iterations exceeded. Move data from WORK
* into D and E so the calling subroutine can try to finish
*
DO I = 1, N
D( I ) = SQRT( WORK( 2*I-1 ) )
E( I ) = SQRT( WORK( 2*I ) )
END DO
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, E, N, IINFO )
END IF
*
RETURN
*
* End of DLASQ1
*
END
*> \brief \b DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ2( N, Z, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ2 computes all the eigenvalues of the symmetric positive
*> definite tridiagonal matrix associated with the qd array Z to high
*> relative accuracy are computed to high relative accuracy, in the
*> absence of denormalization, underflow and overflow.
*>
*> To see the relation of Z to the tridiagonal matrix, let L be a
*> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
*> let U be an upper bidiagonal matrix with 1's above and diagonal
*> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
*> symmetric tridiagonal to which it is similar.
*>
*> Note : DLASQ2 defines a logical variable, IEEE, which is true
*> on machines which follow ieee-754 floating-point standard in their
*> handling of infinities and NaNs, and false otherwise. This variable
*> is passed to DLASQ3.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> On entry Z holds the qd array. On exit, entries 1 to N hold
*> the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
*> trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
*> N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
*> holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
*> shifts that failed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if the i-th argument is a scalar and had an illegal
*> value, then INFO = -i, if the i-th argument is an
*> array and the j-entry had an illegal value, then
*> INFO = -(i*100+j)
*> > 0: the algorithm failed
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 100*N
*> iterations (in inner while loop). On exit Z holds
*> a qd array with the same eigenvalues as the given Z.
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Local Variables: I0:N0 defines a current unreduced segment of Z.
*> The shifts are accumulated in SIGMA. Iteration count is in ITER.
*> Ping-pong is controlled by PP (alternates between 0 and 1).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASQ2( N, Z, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CBIAS
PARAMETER ( CBIAS = 1.50D0 )
DOUBLE PRECISION ZERO, HALF, ONE, TWO, FOUR, HUNDRD
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
$ TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
LOGICAL IEEE
INTEGER I0, I1, I4, IINFO, IPN4, ITER, IWHILA, IWHILB,
$ K, KMIN, N0, N1, NBIG, NDIV, NFAIL, PP, SPLT,
$ TTYPE
DOUBLE PRECISION D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX,
$ QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL,
$ TOL2, TRACE, ZMAX, TEMPE, TEMPQ
* ..
* .. External Subroutines ..
EXTERNAL DLASQ3, DLASRT, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments.
* (in case DLASQ2 is not called by DLASQ1)
*
INFO = 0
EPS = DLAMCH( 'Precision' )
SAFMIN = DLAMCH( 'Safe minimum' )
TOL = EPS*HUNDRD
TOL2 = TOL**2
*
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DLASQ2', 1 )
RETURN
ELSE IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
*
* 1-by-1 case.
*
IF( Z( 1 ).LT.ZERO ) THEN
INFO = -201
CALL XERBLA( 'DLASQ2', 2 )
END IF
RETURN
ELSE IF( N.EQ.2 ) THEN
*
* 2-by-2 case.
*
IF( Z( 1 ).LT.ZERO ) THEN
INFO = -201
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( 2 ).LT.ZERO ) THEN
INFO = -202
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( 3 ).LT.ZERO ) THEN
INFO = -203
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN
D = Z( 3 )
Z( 3 ) = Z( 1 )
Z( 1 ) = D
END IF
Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 )
IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN
T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) )
S = Z( 3 )*( Z( 2 ) / T )
IF( S.LE.T ) THEN
S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) )
ELSE
S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
END IF
T = Z( 1 ) + ( S+Z( 2 ) )
Z( 3 ) = Z( 3 )*( Z( 1 ) / T )
Z( 1 ) = T
END IF
Z( 2 ) = Z( 3 )
Z( 6 ) = Z( 2 ) + Z( 1 )
RETURN
END IF
*
* Check for negative data and compute sums of q's and e's.
*
Z( 2*N ) = ZERO
EMIN = Z( 2 )
QMAX = ZERO
ZMAX = ZERO
D = ZERO
E = ZERO
*
DO 10 K = 1, 2*( N-1 ), 2
IF( Z( K ).LT.ZERO ) THEN
INFO = -( 200+K )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( K+1 ).LT.ZERO ) THEN
INFO = -( 200+K+1 )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
END IF
D = D + Z( K )
E = E + Z( K+1 )
QMAX = MAX( QMAX, Z( K ) )
EMIN = MIN( EMIN, Z( K+1 ) )
ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) )
10 CONTINUE
IF( Z( 2*N-1 ).LT.ZERO ) THEN
INFO = -( 200+2*N-1 )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
END IF
D = D + Z( 2*N-1 )
QMAX = MAX( QMAX, Z( 2*N-1 ) )
ZMAX = MAX( QMAX, ZMAX )
*
* Check for diagonality.
*
IF( E.EQ.ZERO ) THEN
DO 20 K = 2, N
Z( K ) = Z( 2*K-1 )
20 CONTINUE
CALL DLASRT( 'D', N, Z, IINFO )
Z( 2*N-1 ) = D
RETURN
END IF
*
TRACE = D + E
*
* Check for zero data.
*
IF( TRACE.EQ.ZERO ) THEN
Z( 2*N-1 ) = ZERO
RETURN
END IF
*
* Check whether the machine is IEEE conformable.
*
IEEE = ( ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 )
*
* Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...).
*
DO 30 K = 2*N, 2, -2
Z( 2*K ) = ZERO
Z( 2*K-1 ) = Z( K )
Z( 2*K-2 ) = ZERO
Z( 2*K-3 ) = Z( K-1 )
30 CONTINUE
*
I0 = 1
N0 = N
*
* Reverse the qd-array, if warranted.
*
IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN
IPN4 = 4*( I0+N0 )
DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( I4-3 )
Z( I4-3 ) = Z( IPN4-I4-3 )
Z( IPN4-I4-3 ) = TEMP
TEMP = Z( I4-1 )
Z( I4-1 ) = Z( IPN4-I4-5 )
Z( IPN4-I4-5 ) = TEMP
40 CONTINUE
END IF
*
* Initial split checking via dqd and Li's test.
*
PP = 0
*
DO 80 K = 1, 2
*
D = Z( 4*N0+PP-3 )
DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4
IF( Z( I4-1 ).LE.TOL2*D ) THEN
Z( I4-1 ) = -ZERO
D = Z( I4-3 )
ELSE
D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) )
END IF
50 CONTINUE
*
* dqd maps Z to ZZ plus Li's test.
*
EMIN = Z( 4*I0+PP+1 )
D = Z( 4*I0+PP-3 )
DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4
Z( I4-2*PP-2 ) = D + Z( I4-1 )
IF( Z( I4-1 ).LE.TOL2*D ) THEN
Z( I4-1 ) = -ZERO
Z( I4-2*PP-2 ) = D
Z( I4-2*PP ) = ZERO
D = Z( I4+1 )
ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND.
$ SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN
TEMP = Z( I4+1 ) / Z( I4-2*PP-2 )
Z( I4-2*PP ) = Z( I4-1 )*TEMP
D = D*TEMP
ELSE
Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) )
D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) )
END IF
EMIN = MIN( EMIN, Z( I4-2*PP ) )
60 CONTINUE
Z( 4*N0-PP-2 ) = D
*
* Now find qmax.
*
QMAX = Z( 4*I0-PP-2 )
DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4
QMAX = MAX( QMAX, Z( I4 ) )
70 CONTINUE
*
* Prepare for the next iteration on K.
*
PP = 1 - PP
80 CONTINUE
*
* Initialise variables to pass to DLASQ3.
*
TTYPE = 0
DMIN1 = ZERO
DMIN2 = ZERO
DN = ZERO
DN1 = ZERO
DN2 = ZERO
G = ZERO
TAU = ZERO
*
ITER = 2
NFAIL = 0
NDIV = 2*( N0-I0 )
*
DO 160 IWHILA = 1, N + 1
IF( N0.LT.1 )
$ GO TO 170
*
* While array unfinished do
*
* E(N0) holds the value of SIGMA when submatrix in I0:N0
* splits from the rest of the array, but is negated.
*
DESIG = ZERO
IF( N0.EQ.N ) THEN
SIGMA = ZERO
ELSE
SIGMA = -Z( 4*N0-1 )
END IF
IF( SIGMA.LT.ZERO ) THEN
INFO = 1
RETURN
END IF
*
* Find last unreduced submatrix's top index I0, find QMAX and
* EMIN. Find Gershgorin-type bound if Q's much greater than E's.
*
EMAX = ZERO
IF( N0.GT.I0 ) THEN
EMIN = ABS( Z( 4*N0-5 ) )
ELSE
EMIN = ZERO
END IF
QMIN = Z( 4*N0-3 )
QMAX = QMIN
DO 90 I4 = 4*N0, 8, -4
IF( Z( I4-5 ).LE.ZERO )
$ GO TO 100
IF( QMIN.GE.FOUR*EMAX ) THEN
QMIN = MIN( QMIN, Z( I4-3 ) )
EMAX = MAX( EMAX, Z( I4-5 ) )
END IF
QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) )
EMIN = MIN( EMIN, Z( I4-5 ) )
90 CONTINUE
I4 = 4
*
100 CONTINUE
I0 = I4 / 4
PP = 0
*
IF( N0-I0.GT.1 ) THEN
DEE = Z( 4*I0-3 )
DEEMIN = DEE
KMIN = I0
DO 110 I4 = 4*I0+1, 4*N0-3, 4
DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) )
IF( DEE.LE.DEEMIN ) THEN
DEEMIN = DEE
KMIN = ( I4+3 )/4
END IF
110 CONTINUE
IF( (KMIN-I0)*2.LT.N0-KMIN .AND.
$ DEEMIN.LE.HALF*Z(4*N0-3) ) THEN
IPN4 = 4*( I0+N0 )
PP = 2
DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( I4-3 )
Z( I4-3 ) = Z( IPN4-I4-3 )
Z( IPN4-I4-3 ) = TEMP
TEMP = Z( I4-2 )
Z( I4-2 ) = Z( IPN4-I4-2 )
Z( IPN4-I4-2 ) = TEMP
TEMP = Z( I4-1 )
Z( I4-1 ) = Z( IPN4-I4-5 )
Z( IPN4-I4-5 ) = TEMP
TEMP = Z( I4 )
Z( I4 ) = Z( IPN4-I4-4 )
Z( IPN4-I4-4 ) = TEMP
120 CONTINUE
END IF
END IF
*
* Put -(initial shift) into DMIN.
*
DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) )
*
* Now I0:N0 is unreduced.
* PP = 0 for ping, PP = 1 for pong.
* PP = 2 indicates that flipping was applied to the Z array and
* and that the tests for deflation upon entry in DLASQ3
* should not be performed.
*
NBIG = 100*( N0-I0+1 )
DO 140 IWHILB = 1, NBIG
IF( I0.GT.N0 )
$ GO TO 150
*
* While submatrix unfinished take a good dqds step.
*
CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX,
$ NFAIL,
$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
$ DN2, G, TAU )
*
PP = 1 - PP
*
* When EMIN is very small check for splits.
*
IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN
IF( Z( 4*N0 ).LE.TOL2*QMAX .OR.
$ Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN
SPLT = I0 - 1
QMAX = Z( 4*I0-3 )
EMIN = Z( 4*I0-1 )
OLDEMN = Z( 4*I0 )
DO 130 I4 = 4*I0, 4*( N0-3 ), 4
IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR.
$ Z( I4-1 ).LE.TOL2*SIGMA ) THEN
Z( I4-1 ) = -SIGMA
SPLT = I4 / 4
QMAX = ZERO
EMIN = Z( I4+3 )
OLDEMN = Z( I4+4 )
ELSE
QMAX = MAX( QMAX, Z( I4+1 ) )
EMIN = MIN( EMIN, Z( I4-1 ) )
OLDEMN = MIN( OLDEMN, Z( I4 ) )
END IF
130 CONTINUE
Z( 4*N0-1 ) = EMIN
Z( 4*N0 ) = OLDEMN
I0 = SPLT + 1
END IF
END IF
*
140 CONTINUE
*
INFO = 2
*
* Maximum number of iterations exceeded, restore the shift
* SIGMA and place the new d's and e's in a qd array.
* This might need to be done for several blocks
*
I1 = I0
N1 = N0
145 CONTINUE
TEMPQ = Z( 4*I0-3 )
Z( 4*I0-3 ) = Z( 4*I0-3 ) + SIGMA
DO K = I0+1, N0
TEMPE = Z( 4*K-5 )
Z( 4*K-5 ) = Z( 4*K-5 ) * (TEMPQ / Z( 4*K-7 ))
TEMPQ = Z( 4*K-3 )
Z( 4*K-3 ) = Z( 4*K-3 ) + SIGMA + TEMPE - Z( 4*K-5 )
END DO
*
* Prepare to do this on the previous block if there is one
*
IF( I1.GT.1 ) THEN
N1 = I1-1
DO WHILE( ( I1.GE.2 ) .AND. ( Z(4*I1-5).GE.ZERO ) )
I1 = I1 - 1
END DO
SIGMA = -Z(4*N1-1)
GO TO 145
END IF
DO K = 1, N
Z( 2*K-1 ) = Z( 4*K-3 )
*
* Only the block 1..N0 is unfinished. The rest of the e's
* must be essentially zero, although sometimes other data
* has been stored in them.
*
IF( K.LT.N0 ) THEN
Z( 2*K ) = Z( 4*K-1 )
ELSE
Z( 2*K ) = 0
END IF
END DO
RETURN
*
* end IWHILB
*
150 CONTINUE
*
160 CONTINUE
*
INFO = 3
RETURN
*
* end IWHILA
*
170 CONTINUE
*
* Move q's to the front.
*
DO 180 K = 2, N
Z( K ) = Z( 4*K-3 )
180 CONTINUE
*
* Sort and compute sum of eigenvalues.
*
CALL DLASRT( 'D', N, Z, IINFO )
*
E = ZERO
DO 190 K = N, 1, -1
E = E + Z( K )
190 CONTINUE
*
* Store trace, sum(eigenvalues) and information on performance.
*
Z( 2*N+1 ) = TRACE
Z( 2*N+2 ) = E
Z( 2*N+3 ) = DBLE( ITER )
Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 )
Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER )
RETURN
*
* End of DLASQ2
*
END
*> \brief \b DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
* ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
* DN2, G, TAU )
*
* .. Scalar Arguments ..
* LOGICAL IEEE
* INTEGER I0, ITER, N0, NDIV, NFAIL, PP
* DOUBLE PRECISION DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
* $ QMAX, SIGMA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
*> In case of failure it changes shifts, and tries again until output
*> is positive.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in,out] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N0 )
*> Z holds the qd array.
*> \endverbatim
*>
*> \param[in,out] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> PP=2 indicates that flipping was applied to the Z array
*> and that the initial tests for deflation should not be
*> performed.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> Sum of shifts used in current segment.
*> \endverbatim
*>
*> \param[in,out] DESIG
*> \verbatim
*> DESIG is DOUBLE PRECISION
*> Lower order part of SIGMA
*> \endverbatim
*>
*> \param[in] QMAX
*> \verbatim
*> QMAX is DOUBLE PRECISION
*> Maximum value of q.
*> \endverbatim
*>
*> \param[in,out] NFAIL
*> \verbatim
*> NFAIL is INTEGER
*> Increment NFAIL by 1 each time the shift was too big.
*> \endverbatim
*>
*> \param[in,out] ITER
*> \verbatim
*> ITER is INTEGER
*> Increment ITER by 1 for each iteration.
*> \endverbatim
*>
*> \param[in,out] NDIV
*> \verbatim
*> NDIV is INTEGER
*> Increment NDIV by 1 for each division.
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*> IEEE is LOGICAL
*> Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).
*> \endverbatim
*>
*> \param[in,out] TTYPE
*> \verbatim
*> TTYPE is INTEGER
*> Shift type.
*> \endverbatim
*>
*> \param[in,out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN1
*> \verbatim
*> DN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN2
*> \verbatim
*> DN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*> G is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*>
*> These are passed as arguments in order to save their values
*> between calls to DLASQ3.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq3
*
* =====================================================================
SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX,
$ NFAIL,
$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
$ DN2, G, TAU )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER I0, ITER, N0, NDIV, NFAIL, PP
DOUBLE PRECISION DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
$ QMAX, SIGMA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CBIAS
PARAMETER ( CBIAS = 1.50D0 )
DOUBLE PRECISION ZERO, QURTR, HALF, ONE, TWO, HUNDRD
PARAMETER ( ZERO = 0.0D0, QURTR = 0.250D0, HALF = 0.5D0,
$ ONE = 1.0D0, TWO = 2.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
INTEGER IPN4, J4, N0IN, NN, TTYPE
DOUBLE PRECISION EPS, S, T, TEMP, TOL, TOL2
* ..
* .. External Subroutines ..
EXTERNAL DLASQ4, DLASQ5, DLASQ6
* ..
* .. External Function ..
DOUBLE PRECISION DLAMCH
LOGICAL DISNAN
EXTERNAL DISNAN, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
N0IN = N0
EPS = DLAMCH( 'Precision' )
TOL = EPS*HUNDRD
TOL2 = TOL**2
*
* Check for deflation.
*
10 CONTINUE
*
IF( N0.LT.I0 )
$ RETURN
IF( N0.EQ.I0 )
$ GO TO 20
NN = 4*N0 + PP
IF( N0.EQ.( I0+1 ) )
$ GO TO 40
*
* Check whether E(N0-1) is negligible, 1 eigenvalue.
*
IF( Z( NN-5 ).GT.TOL2*( SIGMA+Z( NN-3 ) ) .AND.
$ Z( NN-2*PP-4 ).GT.TOL2*Z( NN-7 ) )
$ GO TO 30
*
20 CONTINUE
*
Z( 4*N0-3 ) = Z( 4*N0+PP-3 ) + SIGMA
N0 = N0 - 1
GO TO 10
*
* Check whether E(N0-2) is negligible, 2 eigenvalues.
*
30 CONTINUE
*
IF( Z( NN-9 ).GT.TOL2*SIGMA .AND.
$ Z( NN-2*PP-8 ).GT.TOL2*Z( NN-11 ) )
$ GO TO 50
*
40 CONTINUE
*
IF( Z( NN-3 ).GT.Z( NN-7 ) ) THEN
S = Z( NN-3 )
Z( NN-3 ) = Z( NN-7 )
Z( NN-7 ) = S
END IF
T = HALF*( ( Z( NN-7 )-Z( NN-3 ) )+Z( NN-5 ) )
IF( Z( NN-5 ).GT.Z( NN-3 )*TOL2.AND.T.NE.ZERO ) THEN
S = Z( NN-3 )*( Z( NN-5 ) / T )
IF( S.LE.T ) THEN
S = Z( NN-3 )*( Z( NN-5 ) /
$ ( T*( ONE+SQRT( ONE+S / T ) ) ) )
ELSE
S = Z( NN-3 )*( Z( NN-5 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
END IF
T = Z( NN-7 ) + ( S+Z( NN-5 ) )
Z( NN-3 ) = Z( NN-3 )*( Z( NN-7 ) / T )
Z( NN-7 ) = T
END IF
Z( 4*N0-7 ) = Z( NN-7 ) + SIGMA
Z( 4*N0-3 ) = Z( NN-3 ) + SIGMA
N0 = N0 - 2
GO TO 10
*
50 CONTINUE
IF( PP.EQ.2 )
$ PP = 0
*
* Reverse the qd-array, if warranted.
*
IF( DMIN.LE.ZERO .OR. N0.LT.N0IN ) THEN
IF( CBIAS*Z( 4*I0+PP-3 ).LT.Z( 4*N0+PP-3 ) ) THEN
IPN4 = 4*( I0+N0 )
DO 60 J4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( J4-3 )
Z( J4-3 ) = Z( IPN4-J4-3 )
Z( IPN4-J4-3 ) = TEMP
TEMP = Z( J4-2 )
Z( J4-2 ) = Z( IPN4-J4-2 )
Z( IPN4-J4-2 ) = TEMP
TEMP = Z( J4-1 )
Z( J4-1 ) = Z( IPN4-J4-5 )
Z( IPN4-J4-5 ) = TEMP
TEMP = Z( J4 )
Z( J4 ) = Z( IPN4-J4-4 )
Z( IPN4-J4-4 ) = TEMP
60 CONTINUE
IF( N0-I0.LE.4 ) THEN
Z( 4*N0+PP-1 ) = Z( 4*I0+PP-1 )
Z( 4*N0-PP ) = Z( 4*I0-PP )
END IF
DMIN2 = MIN( DMIN2, Z( 4*N0+PP-1 ) )
Z( 4*N0+PP-1 ) = MIN( Z( 4*N0+PP-1 ), Z( 4*I0+PP-1 ),
$ Z( 4*I0+PP+3 ) )
Z( 4*N0-PP ) = MIN( Z( 4*N0-PP ), Z( 4*I0-PP ),
$ Z( 4*I0-PP+4 ) )
QMAX = MAX( QMAX, Z( 4*I0+PP-3 ), Z( 4*I0+PP+1 ) )
DMIN = -ZERO
END IF
END IF
*
* Choose a shift.
*
CALL DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1,
$ DN2, TAU, TTYPE, G )
*
* Call dqds until DMIN > 0.
*
70 CONTINUE
*
CALL DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, IEEE, EPS )
*
NDIV = NDIV + ( N0-I0+2 )
ITER = ITER + 1
*
* Check status.
*
IF( DMIN.GE.ZERO .AND. DMIN1.GE.ZERO ) THEN
*
* Success.
*
GO TO 90
*
ELSE IF( DMIN.LT.ZERO .AND. DMIN1.GT.ZERO .AND.
$ Z( 4*( N0-1 )-PP ).LT.TOL*( SIGMA+DN1 ) .AND.
$ ABS( DN ).LT.TOL*SIGMA ) THEN
*
* Convergence hidden by negative DN.
*
Z( 4*( N0-1 )-PP+2 ) = ZERO
DMIN = ZERO
GO TO 90
ELSE IF( DMIN.LT.ZERO ) THEN
*
* TAU too big. Select new TAU and try again.
*
NFAIL = NFAIL + 1
IF( TTYPE.LT.-22 ) THEN
*
* Failed twice. Play it safe.
*
TAU = ZERO
ELSE IF( DMIN1.GT.ZERO ) THEN
*
* Late failure. Gives excellent shift.
*
TAU = ( TAU+DMIN )*( ONE-TWO*EPS )
TTYPE = TTYPE - 11
ELSE
*
* Early failure. Divide by 4.
*
TAU = QURTR*TAU
TTYPE = TTYPE - 12
END IF
GO TO 70
ELSE IF( DISNAN( DMIN ) ) THEN
*
* NaN.
*
IF( TAU.EQ.ZERO ) THEN
GO TO 80
ELSE
TAU = ZERO
GO TO 70
END IF
ELSE
*
* Possible underflow. Play it safe.
*
GO TO 80
END IF
*
* Risk of underflow.
*
80 CONTINUE
CALL DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DN1, DN2 )
NDIV = NDIV + ( N0-I0+2 )
ITER = ITER + 1
TAU = ZERO
*
90 CONTINUE
IF( TAU.LT.SIGMA ) THEN
DESIG = DESIG + TAU
T = SIGMA + DESIG
DESIG = DESIG - ( T-SIGMA )
ELSE
T = SIGMA + TAU
DESIG = SIGMA - ( T-TAU ) + DESIG
END IF
SIGMA = T
*
RETURN
*
* End of DLASQ3
*
END
*> \brief \b DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq4.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
* DN1, DN2, TAU, TTYPE, G )
*
* .. Scalar Arguments ..
* INTEGER I0, N0, N0IN, PP, TTYPE
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ4 computes an approximation TAU to the smallest eigenvalue
*> using values of d from the previous transform.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N0 )
*> Z holds the qd array.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] N0IN
*> \verbatim
*> N0IN is INTEGER
*> The value of N0 at start of EIGTEST.
*> \endverbatim
*>
*> \param[in] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[in] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[in] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[in] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N)
*> \endverbatim
*>
*> \param[in] DN1
*> \verbatim
*> DN1 is DOUBLE PRECISION
*> d(N-1)
*> \endverbatim
*>
*> \param[in] DN2
*> \verbatim
*> DN2 is DOUBLE PRECISION
*> d(N-2)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> This is the shift.
*> \endverbatim
*>
*> \param[out] TTYPE
*> \verbatim
*> TTYPE is INTEGER
*> Shift type.
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*> G is DOUBLE PRECISION
*> G is passed as an argument in order to save its value between
*> calls to DLASQ4.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq4
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> CNST1 = 9/16
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, TAU, TTYPE, G )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I0, N0, N0IN, PP, TTYPE
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CNST1, CNST2, CNST3
PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
$ CNST3 = 1.050D0 )
DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0,
$ HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
INTEGER I4, NN, NP
DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* A negative DMIN forces the shift to take that absolute value
* TTYPE records the type of shift.
*
IF( DMIN.LE.ZERO ) THEN
TAU = -DMIN
TTYPE = -1
RETURN
END IF
*
NN = 4*N0 + PP
IF( N0IN.EQ.N0 ) THEN
*
* No eigenvalues deflated.
*
IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
*
B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
A2 = Z( NN-7 ) + Z( NN-5 )
*
* Cases 2 and 3.
*
IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
GAP2 = DMIN2 - A2 - DMIN2*QURTR
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
GAP1 = A2 - DN - ( B2 / GAP2 )*B2
ELSE
GAP1 = A2 - DN - ( B1+B2 )
END IF
IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
TTYPE = -2
ELSE
S = ZERO
IF( DN.GT.B1 )
$ S = DN - B1
IF( A2.GT.( B1+B2 ) )
$ S = MIN( S, A2-( B1+B2 ) )
S = MAX( S, THIRD*DMIN )
TTYPE = -3
END IF
ELSE
*
* Case 4.
*
TTYPE = -4
S = QURTR*DMIN
IF( DMIN.EQ.DN ) THEN
GAM = DN
A2 = ZERO
IF( Z( NN-5 ) .GT. Z( NN-7 ) )
$ RETURN
B2 = Z( NN-5 ) / Z( NN-7 )
NP = NN - 9
ELSE
NP = NN - 2*PP
GAM = DN1
IF( Z( NP-4 ) .GT. Z( NP-2 ) )
$ RETURN
A2 = Z( NP-4 ) / Z( NP-2 )
IF( Z( NN-9 ) .GT. Z( NN-11 ) )
$ RETURN
B2 = Z( NN-9 ) / Z( NN-11 )
NP = NN - 13
END IF
*
* Approximate contribution to norm squared from I < NN-1.
*
A2 = A2 + B2
DO 10 I4 = NP, 4*I0 - 1 + PP, -4
IF( B2.EQ.ZERO )
$ GO TO 20
B1 = B2
IF( Z( I4 ) .GT. Z( I4-2 ) )
$ RETURN
B2 = B2*( Z( I4 ) / Z( I4-2 ) )
A2 = A2 + B2
IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
$ GO TO 20
10 CONTINUE
20 CONTINUE
A2 = CNST3*A2
*
* Rayleigh quotient residual bound.
*
IF( A2.LT.CNST1 )
$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
END IF
ELSE IF( DMIN.EQ.DN2 ) THEN
*
* Case 5.
*
TTYPE = -5
S = QURTR*DMIN
*
* Compute contribution to norm squared from I > NN-2.
*
NP = NN - 2*PP
B1 = Z( NP-2 )
B2 = Z( NP-6 )
GAM = DN2
IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
$ RETURN
A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
*
* Approximate contribution to norm squared from I < NN-2.
*
IF( N0-I0.GT.2 ) THEN
B2 = Z( NN-13 ) / Z( NN-15 )
A2 = A2 + B2
DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
IF( B2.EQ.ZERO )
$ GO TO 40
B1 = B2
IF( Z( I4 ) .GT. Z( I4-2 ) )
$ RETURN
B2 = B2*( Z( I4 ) / Z( I4-2 ) )
A2 = A2 + B2
IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
$ GO TO 40
30 CONTINUE
40 CONTINUE
A2 = CNST3*A2
END IF
*
IF( A2.LT.CNST1 )
$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
ELSE
*
* Case 6, no information to guide us.
*
IF( TTYPE.EQ.-6 ) THEN
G = G + THIRD*( ONE-G )
ELSE IF( TTYPE.EQ.-18 ) THEN
G = QURTR*THIRD
ELSE
G = QURTR
END IF
S = G*DMIN
TTYPE = -6
END IF
*
ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
*
* One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
*
IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
*
* Cases 7 and 8.
*
TTYPE = -7
S = THIRD*DMIN1
IF( Z( NN-5 ).GT.Z( NN-7 ) )
$ RETURN
B1 = Z( NN-5 ) / Z( NN-7 )
B2 = B1
IF( B2.EQ.ZERO )
$ GO TO 60
DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
A2 = B1
IF( Z( I4 ).GT.Z( I4-2 ) )
$ RETURN
B1 = B1*( Z( I4 ) / Z( I4-2 ) )
B2 = B2 + B1
IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
$ GO TO 60
50 CONTINUE
60 CONTINUE
B2 = SQRT( CNST3*B2 )
A2 = DMIN1 / ( ONE+B2**2 )
GAP2 = HALF*DMIN2 - A2
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
ELSE
S = MAX( S, A2*( ONE-CNST2*B2 ) )
TTYPE = -8
END IF
ELSE
*
* Case 9.
*
S = QURTR*DMIN1
IF( DMIN1.EQ.DN1 )
$ S = HALF*DMIN1
TTYPE = -9
END IF
*
ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
*
* Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
*
* Cases 10 and 11.
*
IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
TTYPE = -10
S = THIRD*DMIN2
IF( Z( NN-5 ).GT.Z( NN-7 ) )
$ RETURN
B1 = Z( NN-5 ) / Z( NN-7 )
B2 = B1
IF( B2.EQ.ZERO )
$ GO TO 80
DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
IF( Z( I4 ).GT.Z( I4-2 ) )
$ RETURN
B1 = B1*( Z( I4 ) / Z( I4-2 ) )
B2 = B2 + B1
IF( HUNDRD*B1.LT.B2 )
$ GO TO 80
70 CONTINUE
80 CONTINUE
B2 = SQRT( CNST3*B2 )
A2 = DMIN2 / ( ONE+B2**2 )
GAP2 = Z( NN-7 ) + Z( NN-9 ) -
$ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
ELSE
S = MAX( S, A2*( ONE-CNST2*B2 ) )
END IF
ELSE
S = QURTR*DMIN2
TTYPE = -11
END IF
ELSE IF( N0IN.GT.( N0+2 ) ) THEN
*
* Case 12, more than two eigenvalues deflated. No information.
*
S = ZERO
TTYPE = -12
END IF
*
TAU = S
RETURN
*
* End of DLASQ4
*
END
*> \brief \b DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
* DNM1, DNM2, IEEE, EPS )
*
* .. Scalar Arguments ..
* LOGICAL IEEE
* INTEGER I0, N0, PP
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU, SIGMA, EPS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ5 computes one dqds transform in ping-pong form, one
*> version for IEEE machines another for non IEEE machines.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*> an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> This is the shift.
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> This is the accumulated shift up to this step.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*> DNM1 is DOUBLE PRECISION
*> d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*> DNM2 is DOUBLE PRECISION
*> d(N0-2).
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*> IEEE is LOGICAL
*> Flag for IEEE or non IEEE arithmetic.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*> EPS is DOUBLE PRECISION
*> This is the value of epsilon used.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq5
*
* =====================================================================
SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1,
$ DMIN2,
$ DN, DNM1, DNM2, IEEE, EPS )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER I0, N0, PP
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU,
$ SIGMA, EPS
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameter ..
DOUBLE PRECISION ZERO, HALF
PARAMETER ( ZERO = 0.0D0, HALF = 0.5 )
* ..
* .. Local Scalars ..
INTEGER J4, J4P2
DOUBLE PRECISION D, EMIN, TEMP, DTHRESH
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( ( N0-I0-1 ).LE.0 )
$ RETURN
*
DTHRESH = EPS*(SIGMA+TAU)
IF( TAU.LT.DTHRESH*HALF ) TAU = ZERO
IF( TAU.NE.ZERO ) THEN
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 ) - TAU
DMIN = D
DMIN1 = -Z( J4 )
*
IF( IEEE ) THEN
*
* Code for IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 10 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
TEMP = Z( J4+1 ) / Z( J4-2 )
D = D*TEMP - TAU
DMIN = MIN( DMIN, D )
Z( J4 ) = Z( J4-1 )*TEMP
EMIN = MIN( Z( J4 ), EMIN )
10 CONTINUE
ELSE
DO 20 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
TEMP = Z( J4+2 ) / Z( J4-3 )
D = D*TEMP - TAU
DMIN = MIN( DMIN, D )
Z( J4-1 ) = Z( J4 )*TEMP
EMIN = MIN( Z( J4-1 ), EMIN )
20 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DN )
*
ELSE
*
* Code for non IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 30 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
30 CONTINUE
ELSE
DO 40 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
40 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( DNM2.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( DNM1.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DN )
*
END IF
ELSE
* This is the version that sets d's to zero if they are small enough
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 ) - TAU
DMIN = D
DMIN1 = -Z( J4 )
IF( IEEE ) THEN
*
* Code for IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 50 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
TEMP = Z( J4+1 ) / Z( J4-2 )
D = D*TEMP - TAU
IF( D.LT.DTHRESH ) D = ZERO
DMIN = MIN( DMIN, D )
Z( J4 ) = Z( J4-1 )*TEMP
EMIN = MIN( Z( J4 ), EMIN )
50 CONTINUE
ELSE
DO 60 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
TEMP = Z( J4+2 ) / Z( J4-3 )
D = D*TEMP - TAU
IF( D.LT.DTHRESH ) D = ZERO
DMIN = MIN( DMIN, D )
Z( J4-1 ) = Z( J4 )*TEMP
EMIN = MIN( Z( J4-1 ), EMIN )
60 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DN )
*
ELSE
*
* Code for non IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 70 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
END IF
IF( D.LT.DTHRESH) D = ZERO
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
70 CONTINUE
ELSE
DO 80 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
END IF
IF( D.LT.DTHRESH) D = ZERO
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
80 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( DNM2.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( DNM1.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DN )
*
END IF
END IF
*
Z( J4+2 ) = DN
Z( 4*N0-PP ) = EMIN
RETURN
*
* End of DLASQ5
*
END
*> \brief \b DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq6.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
* DNM1, DNM2 )
*
* .. Scalar Arguments ..
* INTEGER I0, N0, PP
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ6 computes one dqd (shift equal to zero) transform in
*> ping-pong form, with protection against underflow and overflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*> an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*> DNM1 is DOUBLE PRECISION
*> d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*> DNM2 is DOUBLE PRECISION
*> d(N0-2).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq6
*
* =====================================================================
SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
$ DNM1, DNM2 )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I0, N0, PP
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameter ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER J4, J4P2
DOUBLE PRECISION D, EMIN, SAFMIN, TEMP
* ..
* .. External Function ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( ( N0-I0-1 ).LE.0 )
$ RETURN
*
SAFMIN = DLAMCH( 'Safe minimum' )
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 )
DMIN = D
*
IF( PP.EQ.0 ) THEN
DO 10 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
D = Z( J4+1 )
DMIN = D
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4+1 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4+1 ) ) THEN
TEMP = Z( J4+1 ) / Z( J4-2 )
Z( J4 ) = Z( J4-1 )*TEMP
D = D*TEMP
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
10 CONTINUE
ELSE
DO 20 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( Z( J4-3 ).EQ.ZERO ) THEN
Z( J4-1 ) = ZERO
D = Z( J4+2 )
DMIN = D
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4+2 ).LT.Z( J4-3 ) .AND.
$ SAFMIN*Z( J4-3 ).LT.Z( J4+2 ) ) THEN
TEMP = Z( J4+2 ) / Z( J4-3 )
Z( J4-1 ) = Z( J4 )*TEMP
D = D*TEMP
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) )
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
20 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
DNM1 = Z( J4P2+2 )
DMIN = DNM1
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
TEMP = Z( J4P2+2 ) / Z( J4-2 )
Z( J4 ) = Z( J4P2 )*TEMP
DNM1 = DNM2*TEMP
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
DN = Z( J4P2+2 )
DMIN = DN
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
TEMP = Z( J4P2+2 ) / Z( J4-2 )
Z( J4 ) = Z( J4P2 )*TEMP
DN = DNM1*TEMP
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, DN )
*
Z( J4+2 ) = DN
Z( 4*N0-PP ) = EMIN
RETURN
*
* End of DLASQ6
*
END
*> \brief \b DLASR applies a sequence of plane rotations to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, PIVOT, SIDE
* INTEGER LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASR applies a sequence of plane rotations to a real matrix A,
*> from either the left or the right.
*>
*> When SIDE = 'L', the transformation takes the form
*>
*> A := P*A
*>
*> and when SIDE = 'R', the transformation takes the form
*>
*> A := A*P**T
*>
*> where P is an orthogonal matrix consisting of a sequence of z plane
*> rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
*> and P**T is the transpose of P.
*>
*> When DIRECT = 'F' (Forward sequence), then
*>
*> P = P(z-1) * ... * P(2) * P(1)
*>
*> and when DIRECT = 'B' (Backward sequence), then
*>
*> P = P(1) * P(2) * ... * P(z-1)
*>
*> where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
*>
*> R(k) = ( c(k) s(k) )
*> = ( -s(k) c(k) ).
*>
*> When PIVOT = 'V' (Variable pivot), the rotation is performed
*> for the plane (k,k+1), i.e., P(k) has the form
*>
*> P(k) = ( 1 )
*> ( ... )
*> ( 1 )
*> ( c(k) s(k) )
*> ( -s(k) c(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*>
*> where R(k) appears as a rank-2 modification to the identity matrix in
*> rows and columns k and k+1.
*>
*> When PIVOT = 'T' (Top pivot), the rotation is performed for the
*> plane (1,k+1), so P(k) has the form
*>
*> P(k) = ( c(k) s(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*> ( -s(k) c(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*>
*> where R(k) appears in rows and columns 1 and k+1.
*>
*> Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
*> performed for the plane (k,z), giving P(k) the form
*>
*> P(k) = ( 1 )
*> ( ... )
*> ( 1 )
*> ( c(k) s(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*> ( -s(k) c(k) )
*>
*> where R(k) appears in rows and columns k and z. The rotations are
*> performed without ever forming P(k) explicitly.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> Specifies whether the plane rotation matrix P is applied to
*> A on the left or the right.
*> = 'L': Left, compute A := P*A
*> = 'R': Right, compute A:= A*P**T
*> \endverbatim
*>
*> \param[in] PIVOT
*> \verbatim
*> PIVOT is CHARACTER*1
*> Specifies the plane for which P(k) is a plane rotation
*> matrix.
*> = 'V': Variable pivot, the plane (k,k+1)
*> = 'T': Top pivot, the plane (1,k+1)
*> = 'B': Bottom pivot, the plane (k,z)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies whether P is a forward or backward sequence of
*> plane rotations.
*> = 'F': Forward, P = P(z-1)*...*P(2)*P(1)
*> = 'B': Backward, P = P(1)*P(2)*...*P(z-1)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. If m <= 1, an immediate
*> return is effected.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. If n <= 1, an
*> immediate return is effected.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> The cosines c(k) of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> The sines s(k) of the plane rotations. The 2-by-2 plane
*> rotation part of the matrix P(k), R(k), has the form
*> R(k) = ( c(k) s(k) )
*> ( -s(k) c(k) ).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The M-by-N matrix A. On exit, A is overwritten by P*A if
*> SIDE = 'L' or by A*P**T if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasr
*
* =====================================================================
SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, PIVOT, SIDE
INTEGER LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J
DOUBLE PRECISION CTEMP, STEMP, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( .NOT.( LSAME( SIDE, 'L' ) .OR.
$ LSAME( SIDE, 'R' ) ) ) THEN
INFO = 1
ELSE IF( .NOT.( LSAME( PIVOT, 'V' ) .OR. LSAME( PIVOT,
$ 'T' ) .OR. LSAME( PIVOT, 'B' ) ) ) THEN
INFO = 2
ELSE IF( .NOT.( LSAME( DIRECT, 'F' ) .OR.
$ LSAME( DIRECT, 'B' ) ) )
$ THEN
INFO = 3
ELSE IF( M.LT.0 ) THEN
INFO = 4
ELSE IF( N.LT.0 ) THEN
INFO = 5
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = 9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASR ', INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )
$ RETURN
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form P * A
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 20 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 10 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
10 CONTINUE
END IF
20 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 40 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 30 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
30 CONTINUE
END IF
40 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 60 J = 2, M
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 50 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
50 CONTINUE
END IF
60 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 80 J = M, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 70 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
70 CONTINUE
END IF
80 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 100 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 90 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
90 CONTINUE
END IF
100 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 120 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 110 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
110 CONTINUE
END IF
120 CONTINUE
END IF
END IF
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form A * P**T
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 140 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 130 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
130 CONTINUE
END IF
140 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 160 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 150 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
150 CONTINUE
END IF
160 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 180 J = 2, N
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 170 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
170 CONTINUE
END IF
180 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 200 J = N, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 190 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
190 CONTINUE
END IF
200 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 220 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 210 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
210 CONTINUE
END IF
220 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 240 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 230 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
230 CONTINUE
END IF
240 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of DLASR
*
END
*> \brief \b DLASRT sorts numbers in increasing or decreasing order.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASRT( ID, N, D, INFO )
*
* .. Scalar Arguments ..
* CHARACTER ID
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Sort the numbers in D in increasing order (if ID = 'I') or
*> in decreasing order (if ID = 'D' ).
*>
*> Use Quick Sort, reverting to Insertion sort on arrays of
*> size <= 20. Dimension of STACK limits N to about 2**32.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ID
*> \verbatim
*> ID is CHARACTER*1
*> = 'I': sort D in increasing order;
*> = 'D': sort D in decreasing order.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of the array D.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the array to be sorted.
*> On exit, D has been sorted into increasing order
*> (D(1) <= ... <= D(N) ) or into decreasing order
*> (D(1) >= ... >= D(N) ), depending on ID.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasrt
*
* =====================================================================
SUBROUTINE DLASRT( ID, N, D, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER ID
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER SELECT
PARAMETER ( SELECT = 20 )
* ..
* .. Local Scalars ..
INTEGER DIR, ENDD, I, J, START, STKPNT
DOUBLE PRECISION D1, D2, D3, DMNMX, TMP
* ..
* .. Local Arrays ..
INTEGER STACK( 2, 32 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
DIR = -1
IF( LSAME( ID, 'D' ) ) THEN
DIR = 0
ELSE IF( LSAME( ID, 'I' ) ) THEN
DIR = 1
END IF
IF( DIR.EQ.-1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASRT', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
STKPNT = 1
STACK( 1, 1 ) = 1
STACK( 2, 1 ) = N
10 CONTINUE
START = STACK( 1, STKPNT )
ENDD = STACK( 2, STKPNT )
STKPNT = STKPNT - 1
IF( ENDD-START.LE.SELECT .AND. ENDD-START.GT.0 ) THEN
*
* Do Insertion sort on D( START:ENDD )
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
DO 30 I = START + 1, ENDD
DO 20 J = I, START + 1, -1
IF( D( J ).GT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 30
END IF
20 CONTINUE
30 CONTINUE
*
ELSE
*
* Sort into increasing order
*
DO 50 I = START + 1, ENDD
DO 40 J = I, START + 1, -1
IF( D( J ).LT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 50
END IF
40 CONTINUE
50 CONTINUE
*
END IF
*
ELSE IF( ENDD-START.GT.SELECT ) THEN
*
* Partition D( START:ENDD ) and stack parts, largest one first
*
* Choose partition entry as median of 3
*
D1 = D( START )
D2 = D( ENDD )
I = ( START+ENDD ) / 2
D3 = D( I )
IF( D1.LT.D2 ) THEN
IF( D3.LT.D1 ) THEN
DMNMX = D1
ELSE IF( D3.LT.D2 ) THEN
DMNMX = D3
ELSE
DMNMX = D2
END IF
ELSE
IF( D3.LT.D2 ) THEN
DMNMX = D2
ELSE IF( D3.LT.D1 ) THEN
DMNMX = D3
ELSE
DMNMX = D1
END IF
END IF
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
I = START - 1
J = ENDD + 1
60 CONTINUE
70 CONTINUE
J = J - 1
IF( D( J ).LT.DMNMX )
$ GO TO 70
80 CONTINUE
I = I + 1
IF( D( I ).GT.DMNMX )
$ GO TO 80
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 60
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
ELSE
*
* Sort into increasing order
*
I = START - 1
J = ENDD + 1
90 CONTINUE
100 CONTINUE
J = J - 1
IF( D( J ).GT.DMNMX )
$ GO TO 100
110 CONTINUE
I = I + 1
IF( D( I ).LT.DMNMX )
$ GO TO 110
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 90
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
END IF
END IF
IF( STKPNT.GT.0 )
$ GO TO 10
RETURN
*
* End of DLASRT
*
END
*> \brief \b DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASV2 computes the singular value decomposition of a 2-by-2
*> triangular matrix
*> [ F G ]
*> [ 0 H ].
*> On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
*> smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and
*> right singular vectors for abs(SSMAX), giving the decomposition
*>
*> [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ]
*> [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is DOUBLE PRECISION
*> abs(SSMIN) is the smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*> SSMAX is DOUBLE PRECISION
*> abs(SSMAX) is the larger singular value.
*> \endverbatim
*>
*> \param[out] SNL
*> \verbatim
*> SNL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSL
*> \verbatim
*> CSL is DOUBLE PRECISION
*> The vector (CSL, SNL) is a unit left singular vector for the
*> singular value abs(SSMAX).
*> \endverbatim
*>
*> \param[out] SNR
*> \verbatim
*> SNR is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSR
*> \verbatim
*> CSR is DOUBLE PRECISION
*> The vector (CSR, SNR) is a unit right singular vector for the
*> singular value abs(SSMAX).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasv2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Any input parameter may be aliased with any output parameter.
*>
*> Barring over/underflow and assuming a guard digit in subtraction, all
*> output quantities are correct to within a few units in the last
*> place (ulps).
*>
*> In IEEE arithmetic, the code works correctly if one matrix element is
*> infinite.
*>
*> Overflow will not occur unless the largest singular value itself
*> overflows or is within a few ulps of overflow.
*>
*> Underflow is harmless if underflow is gradual. Otherwise, results
*> may correspond to a matrix modified by perturbations of size near
*> the underflow threshold.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION FOUR
PARAMETER ( FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
LOGICAL GASMAL, SWAP
INTEGER PMAX
DOUBLE PRECISION A, CLT, CRT, D, FA, FT, GA, GT, HA, HT, L, M,
$ MM, R, S, SLT, SRT, T, TEMP, TSIGN, TT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Executable Statements ..
*
FT = F
FA = ABS( FT )
HT = H
HA = ABS( H )
*
* PMAX points to the maximum absolute element of matrix
* PMAX = 1 if F largest in absolute values
* PMAX = 2 if G largest in absolute values
* PMAX = 3 if H largest in absolute values
*
PMAX = 1
SWAP = ( HA.GT.FA )
IF( SWAP ) THEN
PMAX = 3
TEMP = FT
FT = HT
HT = TEMP
TEMP = FA
FA = HA
HA = TEMP
*
* Now FA .ge. HA
*
END IF
GT = G
GA = ABS( GT )
IF( GA.EQ.ZERO ) THEN
*
* Diagonal matrix
*
SSMIN = HA
SSMAX = FA
CLT = ONE
CRT = ONE
SLT = ZERO
SRT = ZERO
ELSE
GASMAL = .TRUE.
IF( GA.GT.FA ) THEN
PMAX = 2
IF( ( FA / GA ).LT.DLAMCH( 'EPS' ) ) THEN
*
* Case of very large GA
*
GASMAL = .FALSE.
SSMAX = GA
IF( HA.GT.ONE ) THEN
SSMIN = FA / ( GA / HA )
ELSE
SSMIN = ( FA / GA )*HA
END IF
CLT = ONE
SLT = HT / GT
SRT = ONE
CRT = FT / GT
END IF
END IF
IF( GASMAL ) THEN
*
* Normal case
*
D = FA - HA
IF( D.EQ.FA ) THEN
*
* Copes with infinite F or H
*
L = ONE
ELSE
L = D / FA
END IF
*
* Note that 0 .le. L .le. 1
*
M = GT / FT
*
* Note that abs(M) .le. 1/macheps
*
T = TWO - L
*
* Note that T .ge. 1
*
MM = M*M
TT = T*T
S = SQRT( TT+MM )
*
* Note that 1 .le. S .le. 1 + 1/macheps
*
IF( L.EQ.ZERO ) THEN
R = ABS( M )
ELSE
R = SQRT( L*L+MM )
END IF
*
* Note that 0 .le. R .le. 1 + 1/macheps
*
A = HALF*( S+R )
*
* Note that 1 .le. A .le. 1 + abs(M)
*
SSMIN = HA / A
SSMAX = FA*A
IF( MM.EQ.ZERO ) THEN
*
* Note that M is very tiny
*
IF( L.EQ.ZERO ) THEN
T = SIGN( TWO, FT )*SIGN( ONE, GT )
ELSE
T = GT / SIGN( D, FT ) + M / T
END IF
ELSE
T = ( M / ( S+T )+M / ( R+L ) )*( ONE+A )
END IF
L = SQRT( T*T+FOUR )
CRT = TWO / L
SRT = T / L
CLT = ( CRT+SRT*M ) / A
SLT = ( HT / FT )*SRT / A
END IF
END IF
IF( SWAP ) THEN
CSL = SRT
SNL = CRT
CSR = SLT
SNR = CLT
ELSE
CSL = CLT
SNL = SLT
CSR = CRT
SNR = SRT
END IF
*
* Correct signs of SSMAX and SSMIN
*
IF( PMAX.EQ.1 )
$ TSIGN = SIGN( ONE, CSR )*SIGN( ONE, CSL )*SIGN( ONE, F )
IF( PMAX.EQ.2 )
$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, CSL )*SIGN( ONE, G )
IF( PMAX.EQ.3 )
$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, SNL )*SIGN( ONE, H )
SSMAX = SIGN( SSMAX, TSIGN )
SSMIN = SIGN( SSMIN, TSIGN*SIGN( ONE, F )*SIGN( ONE, H ) )
RETURN
*
* End of DLASV2
*
END
*> \brief \b DLASWP performs a series of row interchanges on a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASWP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaswp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaswp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaswp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASWP( N, A, LDA, K1, K2, IPIV, INCX )
*
* .. Scalar Arguments ..
* INTEGER INCX, K1, K2, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASWP performs a series of row interchanges on the matrix A.
*> One row interchange is initiated for each of rows K1 through K2 of A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the matrix of column dimension N to which the row
*> interchanges will be applied.
*> On exit, the permuted matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> \endverbatim
*>
*> \param[in] K1
*> \verbatim
*> K1 is INTEGER
*> The first element of IPIV for which a row interchange will
*> be done.
*> \endverbatim
*>
*> \param[in] K2
*> \verbatim
*> K2 is INTEGER
*> (K2-K1+1) is the number of elements of IPIV for which a row
*> interchange will be done.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
*> The vector of pivot indices. Only the elements in positions
*> K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
*> IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
*> interchanged.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between successive values of IPIV. If INCX
*> is negative, the pivots are applied in reverse order.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laswp
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified by
*> R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASWP( N, A, LDA, K1, K2, IPIV, INCX )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX, K1, K2, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, I1, I2, INC, IP, IX, IX0, J, K, N32
DOUBLE PRECISION TEMP
* ..
* .. Executable Statements ..
*
* Interchange row I with row IPIV(K1+(I-K1)*abs(INCX)) for each of rows
* K1 through K2.
*
IF( INCX.GT.0 ) THEN
IX0 = K1
I1 = K1
I2 = K2
INC = 1
ELSE IF( INCX.LT.0 ) THEN
IX0 = K1 + ( K1-K2 )*INCX
I1 = K2
I2 = K1
INC = -1
ELSE
RETURN
END IF
*
N32 = ( N / 32 )*32
IF( N32.NE.0 ) THEN
DO 30 J = 1, N32, 32
IX = IX0
DO 20 I = I1, I2, INC
IP = IPIV( IX )
IF( IP.NE.I ) THEN
DO 10 K = J, J + 31
TEMP = A( I, K )
A( I, K ) = A( IP, K )
A( IP, K ) = TEMP
10 CONTINUE
END IF
IX = IX + INCX
20 CONTINUE
30 CONTINUE
END IF
IF( N32.NE.N ) THEN
N32 = N32 + 1
IX = IX0
DO 50 I = I1, I2, INC
IP = IPIV( IX )
IF( IP.NE.I ) THEN
DO 40 K = N32, N
TEMP = A( I, K )
A( I, K ) = A( IP, K )
A( IP, K ) = TEMP
40 CONTINUE
END IF
IX = IX + INCX
50 CONTINUE
END IF
*
RETURN
*
* End of DLASWP
*
END
*> \brief \b DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASY2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasy2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasy2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasy2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
* LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
* .. Scalar Arguments ..
* LOGICAL LTRANL, LTRANR
* INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
* DOUBLE PRECISION SCALE, XNORM
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
*>
*> op(TL)*X + ISGN*X*op(TR) = SCALE*B,
*>
*> where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
*> -1. op(T) = T or T**T, where T**T denotes the transpose of T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] LTRANL
*> \verbatim
*> LTRANL is LOGICAL
*> On entry, LTRANL specifies the op(TL):
*> = .FALSE., op(TL) = TL,
*> = .TRUE., op(TL) = TL**T.
*> \endverbatim
*>
*> \param[in] LTRANR
*> \verbatim
*> LTRANR is LOGICAL
*> On entry, LTRANR specifies the op(TR):
*> = .FALSE., op(TR) = TR,
*> = .TRUE., op(TR) = TR**T.
*> \endverbatim
*>
*> \param[in] ISGN
*> \verbatim
*> ISGN is INTEGER
*> On entry, ISGN specifies the sign of the equation
*> as described before. ISGN may only be 1 or -1.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> On entry, N1 specifies the order of matrix TL.
*> N1 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> On entry, N2 specifies the order of matrix TR.
*> N2 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] TL
*> \verbatim
*> TL is DOUBLE PRECISION array, dimension (LDTL,2)
*> On entry, TL contains an N1 by N1 matrix.
*> \endverbatim
*>
*> \param[in] LDTL
*> \verbatim
*> LDTL is INTEGER
*> The leading dimension of the matrix TL. LDTL >= max(1,N1).
*> \endverbatim
*>
*> \param[in] TR
*> \verbatim
*> TR is DOUBLE PRECISION array, dimension (LDTR,2)
*> On entry, TR contains an N2 by N2 matrix.
*> \endverbatim
*>
*> \param[in] LDTR
*> \verbatim
*> LDTR is INTEGER
*> The leading dimension of the matrix TR. LDTR >= max(1,N2).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,2)
*> On entry, the N1 by N2 matrix B contains the right-hand
*> side of the equation.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the matrix B. LDB >= max(1,N1).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> On exit, SCALE contains the scale factor. SCALE is chosen
*> less than or equal to 1 to prevent the solution overflowing.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,2)
*> On exit, X contains the N1 by N2 solution.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the matrix X. LDX >= max(1,N1).
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*> XNORM is DOUBLE PRECISION
*> On exit, XNORM is the infinity-norm of the solution.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, INFO is set to
*> 0: successful exit.
*> 1: TL and TR have too close eigenvalues, so TL or
*> TR is perturbed to get a nonsingular equation.
*> NOTE: In the interests of speed, this routine does not
*> check the inputs for errors.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasy2
*
* =====================================================================
SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL LTRANL, LTRANR
INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
DOUBLE PRECISION SCALE, XNORM
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TWO, HALF, EIGHT
PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL BSWAP, XSWAP
INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
$ TEMP, U11, U12, U22, XMAX
* ..
* .. Local Arrays ..
LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
$ LOCU22( 4 )
DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Data statements ..
DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
$ LOCU22 / 4, 3, 2, 1 /
DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
* ..
* .. Executable Statements ..
*
* Do not check the input parameters for errors
*
INFO = 0
*
* Quick return if possible
*
IF( N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
SGN = ISGN
*
K = N1 + N1 + N2 - 2
GO TO ( 10, 20, 30, 50 )K
*
* 1 by 1: TL11*X + SGN*X*TR11 = B11
*
10 CONTINUE
TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
BET = ABS( TAU1 )
IF( BET.LE.SMLNUM ) THEN
TAU1 = SMLNUM
BET = SMLNUM
INFO = 1
END IF
*
SCALE = ONE
GAM = ABS( B( 1, 1 ) )
IF( SMLNUM*GAM.GT.BET )
$ SCALE = ONE / GAM
*
X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
XNORM = ABS( X( 1, 1 ) )
RETURN
*
* 1 by 2:
* TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
* [TR21 TR22]
*
20 CONTINUE
*
SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
$ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
IF( LTRANR ) THEN
TMP( 2 ) = SGN*TR( 2, 1 )
TMP( 3 ) = SGN*TR( 1, 2 )
ELSE
TMP( 2 ) = SGN*TR( 1, 2 )
TMP( 3 ) = SGN*TR( 2, 1 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 1, 2 )
GO TO 40
*
* 2 by 1:
* op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
* [TL21 TL22] [X21] [X21] [B21]
*
30 CONTINUE
SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
$ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
IF( LTRANL ) THEN
TMP( 2 ) = TL( 1, 2 )
TMP( 3 ) = TL( 2, 1 )
ELSE
TMP( 2 ) = TL( 2, 1 )
TMP( 3 ) = TL( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
40 CONTINUE
*
* Solve 2 by 2 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
IPIV = IDAMAX( 4, TMP, 1 )
U11 = TMP( IPIV )
IF( ABS( U11 ).LE.SMIN ) THEN
INFO = 1
U11 = SMIN
END IF
U12 = TMP( LOCU12( IPIV ) )
L21 = TMP( LOCL21( IPIV ) ) / U11
U22 = TMP( LOCU22( IPIV ) ) - U12*L21
XSWAP = XSWPIV( IPIV )
BSWAP = BSWPIV( IPIV )
IF( ABS( U22 ).LE.SMIN ) THEN
INFO = 1
U22 = SMIN
END IF
IF( BSWAP ) THEN
TEMP = BTMP( 2 )
BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
BTMP( 1 ) = TEMP
ELSE
BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
END IF
SCALE = ONE
IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
$ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
END IF
X2( 2 ) = BTMP( 2 ) / U22
X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
IF( XSWAP ) THEN
TEMP = X2( 2 )
X2( 2 ) = X2( 1 )
X2( 1 ) = TEMP
END IF
X( 1, 1 ) = X2( 1 )
IF( N1.EQ.1 ) THEN
X( 1, 2 ) = X2( 2 )
XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
ELSE
X( 2, 1 ) = X2( 2 )
XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
END IF
RETURN
*
* 2 by 2:
* op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
* [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
*
* Solve equivalent 4 by 4 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
50 CONTINUE
SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
$ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
$ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
SMIN = MAX( EPS*SMIN, SMLNUM )
BTMP( 1 ) = ZERO
CALL DCOPY( 16, BTMP, 0, T16, 1 )
T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
IF( LTRANL ) THEN
T16( 1, 2 ) = TL( 2, 1 )
T16( 2, 1 ) = TL( 1, 2 )
T16( 3, 4 ) = TL( 2, 1 )
T16( 4, 3 ) = TL( 1, 2 )
ELSE
T16( 1, 2 ) = TL( 1, 2 )
T16( 2, 1 ) = TL( 2, 1 )
T16( 3, 4 ) = TL( 1, 2 )
T16( 4, 3 ) = TL( 2, 1 )
END IF
IF( LTRANR ) THEN
T16( 1, 3 ) = SGN*TR( 1, 2 )
T16( 2, 4 ) = SGN*TR( 1, 2 )
T16( 3, 1 ) = SGN*TR( 2, 1 )
T16( 4, 2 ) = SGN*TR( 2, 1 )
ELSE
T16( 1, 3 ) = SGN*TR( 2, 1 )
T16( 2, 4 ) = SGN*TR( 2, 1 )
T16( 3, 1 ) = SGN*TR( 1, 2 )
T16( 4, 2 ) = SGN*TR( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
BTMP( 3 ) = B( 1, 2 )
BTMP( 4 ) = B( 2, 2 )
*
* Perform elimination
*
DO 100 I = 1, 3
XMAX = ZERO
DO 70 IP = I, 4
DO 60 JP = I, 4
IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
XMAX = ABS( T16( IP, JP ) )
IPSV = IP
JPSV = JP
END IF
60 CONTINUE
70 CONTINUE
IF( IPSV.NE.I ) THEN
CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
TEMP = BTMP( I )
BTMP( I ) = BTMP( IPSV )
BTMP( IPSV ) = TEMP
END IF
IF( JPSV.NE.I )
$ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
JPIV( I ) = JPSV
IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
INFO = 1
T16( I, I ) = SMIN
END IF
DO 90 J = I + 1, 4
T16( J, I ) = T16( J, I ) / T16( I, I )
BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
DO 80 K = I + 1, 4
T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
80 CONTINUE
90 CONTINUE
100 CONTINUE
IF( ABS( T16( 4, 4 ) ).LT.SMIN ) THEN
INFO = 1
T16( 4, 4 ) = SMIN
END IF
SCALE = ONE
IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
BTMP( 3 ) = BTMP( 3 )*SCALE
BTMP( 4 ) = BTMP( 4 )*SCALE
END IF
DO 120 I = 1, 4
K = 5 - I
TEMP = ONE / T16( K, K )
TMP( K ) = BTMP( K )*TEMP
DO 110 J = K + 1, 4
TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
110 CONTINUE
120 CONTINUE
DO 130 I = 1, 3
IF( JPIV( 4-I ).NE.4-I ) THEN
TEMP = TMP( 4-I )
TMP( 4-I ) = TMP( JPIV( 4-I ) )
TMP( JPIV( 4-I ) ) = TEMP
END IF
130 CONTINUE
X( 1, 1 ) = TMP( 1 )
X( 2, 1 ) = TMP( 2 )
X( 1, 2 ) = TMP( 3 )
X( 2, 2 ) = TMP( 4 )
XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
$ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
RETURN
*
* End of DLASY2
*
END
*> \brief \b DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASYF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasyf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasyf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KB, LDA, LDW, N, NB
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), W( LDW, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASYF computes a partial factorization of a real symmetric matrix A
*> using the Bunch-Kaufman diagonal pivoting method. The partial
*> factorization has the form:
*>
*> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
*> ( 0 U22 ) ( 0 D ) ( U12**T U22**T )
*>
*> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'
*> ( L21 I ) ( 0 A22 ) ( 0 I )
*>
*> where the order of D is at most NB. The actual order is returned in
*> the argument KB, and is either NB or NB-1, or N if N <= NB.
*>
*> DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code
*> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
*> A22 (if UPLO = 'L').
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The maximum number of columns of the matrix A that should be
*> factored. NB should be at least 2 to allow for 2-by-2 pivot
*> blocks.
*> \endverbatim
*>
*> \param[out] KB
*> \verbatim
*> KB is INTEGER
*> The number of columns of A that were actually factored.
*> KB is either NB-1 or NB, or N if N <= NB.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit, A contains details of the partial factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D.
*>
*> If UPLO = 'U':
*> Only the last KB elements of IPIV are set.
*>
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
*> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*> is a 2-by-2 diagonal block.
*>
*> If UPLO = 'L':
*> Only the first KB elements of IPIV are set.
*>
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
*> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
*> is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (LDW,NB)
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*> LDW is INTEGER
*> The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahef
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2013, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*> \endverbatim
*
* =====================================================================
SUBROUTINE DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KB, LDA, LDW, N, NB
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), W( LDW, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION EIGHT, SEVTEN
PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
* ..
* .. Local Scalars ..
INTEGER IMAX, J, JJ, JMAX, JP, K, KK, KKW, KP,
$ KSTEP, KW
DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1,
$ ROWMAX, T
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
EXTERNAL LSAME, IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMMTR, DGEMV, DSCAL, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Factorize the trailing columns of A using the upper triangle
* of A and working backwards, and compute the matrix W = U12*D
* for use in updating A11
*
* K is the main loop index, decreasing from N in steps of 1 or 2
*
* KW is the column of W which corresponds to column K of A
*
K = N
10 CONTINUE
KW = NB + K - N
*
* Exit from loop
*
IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
$ GO TO 30
*
* Copy column K of A to column KW of W and update it
*
CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
IF( K.LT.N )
$ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
$ LDA,
$ W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
*
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( W( K, KW ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
*
IF( K.GT.1 ) THEN
IMAX = IDAMAX( K-1, W( 1, KW ), 1 )
COLMAX = ABS( W( IMAX, KW ) )
ELSE
COLMAX = ZERO
END IF
*
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
* Column K is zero or underflow: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* Copy column IMAX to column KW-1 of W and update it
*
CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
$ W( IMAX+1, KW-1 ), 1 )
IF( K.LT.N )
$ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1,
$ K+1 ),
$ LDA, W( IMAX, KW+1 ), LDW, ONE,
$ W( 1, KW-1 ), 1 )
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
ROWMAX = ABS( W( JMAX, KW-1 ) )
IF( IMAX.GT.1 ) THEN
JMAX = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
ROWMAX = MAX( ROWMAX, ABS( W( JMAX, KW-1 ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
*
* copy column KW-1 of W to column KW of W
*
CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
ELSE
*
* interchange rows and columns K-1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
* ============================================================
*
* KK is the column of A where pivoting step stopped
*
KK = K - KSTEP + 1
*
* KKW is the column of W which corresponds to column KK of A
*
KKW = NB + KK - N
*
* Interchange rows and columns KP and KK.
* Updated column KP is already stored in column KKW of W.
*
IF( KP.NE.KK ) THEN
*
* Copy non-updated column KK to column KP of submatrix A
* at step K. No need to copy element into column K
* (or K and K-1 for 2-by-2 pivot) of A, since these columns
* will be later overwritten.
*
A( KP, KP ) = A( KK, KK )
CALL DCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
$ LDA )
IF( KP.GT.1 )
$ CALL DCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
*
* Interchange rows KK and KP in last K+1 to N columns of A
* (columns K (or K and K-1 for 2-by-2 pivot) of A will be
* later overwritten). Interchange rows KK and KP
* in last KKW to NB columns of W.
*
IF( K.LT.N )
$ CALL DSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
$ LDA )
CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
$ LDW )
END IF
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column kw of W now holds
*
* W(kw) = U(k)*D(k),
*
* where U(k) is the k-th column of U
*
* Store subdiag. elements of column U(k)
* and 1-by-1 block D(k) in column k of A.
* NOTE: Diagonal element U(k,k) is a UNIT element
* and not stored.
* A(k,k) := D(k,k) = W(k,kw)
* A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
*
CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
R1 = ONE / A( K, K )
CALL DSCAL( K-1, R1, A( 1, K ), 1 )
*
ELSE
*
* 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
*
* ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
*
* where U(k) and U(k-1) are the k-th and (k-1)-th columns
* of U
*
* Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
* block D(k-1:k,k-1:k) in columns k-1 and k of A.
* NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
* block and not stored.
* A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
* A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
* = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
*
IF( K.GT.2 ) THEN
*
* Compose the columns of the inverse of 2-by-2 pivot
* block D in the following way to reduce the number
* of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by
* this inverse
*
* D**(-1) = ( d11 d21 )**(-1) =
* ( d21 d22 )
*
* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
* ( (-d21 ) ( d11 ) )
*
* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
*
* * ( ( d22/d21 ) ( -1 ) ) =
* ( ( -1 ) ( d11/d21 ) )
*
* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
* ( ( -1 ) ( D22 ) )
*
* = 1/d21 * T * ( ( D11 ) ( -1 ) )
* ( ( -1 ) ( D22 ) )
*
* = D21 * ( ( D11 ) ( -1 ) )
* ( ( -1 ) ( D22 ) )
*
D21 = W( K-1, KW )
D11 = W( K, KW ) / D21
D22 = W( K-1, KW-1 ) / D21
T = ONE / ( D11*D22-ONE )
D21 = T / D21
*
* Update elements in columns A(k-1) and A(k) as
* dot products of rows of ( W(kw-1) W(kw) ) and columns
* of D**(-1)
*
DO 20 J = 1, K - 2
A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) )
20 CONTINUE
END IF
*
* Copy D(k) to A
*
A( K-1, K-1 ) = W( K-1, KW-1 )
A( K-1, K ) = W( K-1, KW )
A( K, K ) = W( K, KW )
*
END IF
*
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K-1 ) = -KP
END IF
*
* Decrease K and return to the start of the main loop
*
K = K - KSTEP
GO TO 10
*
30 CONTINUE
*
* Update the upper triangle of A11 (= A(1:k,1:k)) as
*
* A11 := A11 - U12*D*U12**T = A11 - U12*W**T
*
CALL DGEMMTR( 'Upper', 'No transpose', 'Transpose', K, N-K,
$ -ONE, A( 1, K+1 ), LDA, W( 1, KW+1 ), LDW,
$ ONE, A( 1, 1 ), LDA )
*
* Put U12 in standard form by partially undoing the interchanges
* in columns k+1:n looping backwards from k+1 to n
*
J = K + 1
60 CONTINUE
*
* Undo the interchanges (if any) of rows JJ and JP at each
* step J
*
* (Here, J is a diagonal index)
JJ = J
JP = IPIV( J )
IF( JP.LT.0 ) THEN
JP = -JP
* (Here, J is a diagonal index)
J = J + 1
END IF
* (NOTE: Here, J is used to determine row length. Length N-J+1
* of the rows to swap back doesn't include diagonal element)
J = J + 1
IF( JP.NE.JJ .AND. J.LE.N )
$ CALL DSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
IF( J.LT.N )
$ GO TO 60
*
* Set KB to the number of columns factorized
*
KB = N - K
*
ELSE
*
* Factorize the leading columns of A using the lower triangle
* of A and working forwards, and compute the matrix W = L21*D
* for use in updating A22
*
* K is the main loop index, increasing from 1 in steps of 1 or 2
*
K = 1
70 CONTINUE
*
* Exit from loop
*
IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
$ GO TO 90
*
* Copy column K of A to column K of W and update it
*
CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
$ LDA,
$ W( K, 1 ), LDW, ONE, W( K, K ), 1 )
*
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( W( K, K ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
*
IF( K.LT.N ) THEN
IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 )
COLMAX = ABS( W( IMAX, K ) )
ELSE
COLMAX = ZERO
END IF
*
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
* Column K is zero or underflow: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* Copy column IMAX to column K+1 of W and update it
*
CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ),
$ 1 )
CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX,
$ K+1 ),
$ 1 )
CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K,
$ 1 ),
$ LDA, W( IMAX, 1 ), LDW, ONE, W( K, K+1 ), 1 )
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
ROWMAX = ABS( W( JMAX, K+1 ) )
IF( IMAX.LT.N ) THEN
JMAX = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
ROWMAX = MAX( ROWMAX, ABS( W( JMAX, K+1 ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
*
* copy column K+1 of W to column K of W
*
CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
ELSE
*
* interchange rows and columns K+1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
* ============================================================
*
* KK is the column of A where pivoting step stopped
*
KK = K + KSTEP - 1
*
* Interchange rows and columns KP and KK.
* Updated column KP is already stored in column KK of W.
*
IF( KP.NE.KK ) THEN
*
* Copy non-updated column KK to column KP of submatrix A
* at step K. No need to copy element into column K
* (or K and K+1 for 2-by-2 pivot) of A, since these columns
* will be later overwritten.
*
A( KP, KP ) = A( KK, KK )
CALL DCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
$ LDA )
IF( KP.LT.N )
$ CALL DCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ),
$ 1 )
*
* Interchange rows KK and KP in first K-1 columns of A
* (columns K (or K and K+1 for 2-by-2 pivot) of A will be
* later overwritten). Interchange rows KK and KP
* in first KK columns of W.
*
IF( K.GT.1 )
$ CALL DSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
END IF
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k of W now holds
*
* W(k) = L(k)*D(k),
*
* where L(k) is the k-th column of L
*
* Store subdiag. elements of column L(k)
* and 1-by-1 block D(k) in column k of A.
* (NOTE: Diagonal element L(k,k) is a UNIT element
* and not stored)
* A(k,k) := D(k,k) = W(k,k)
* A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
*
CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
IF( K.LT.N ) THEN
R1 = ONE / A( K, K )
CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
END IF
*
ELSE
*
* 2-by-2 pivot block D(k): columns k and k+1 of W now hold
*
* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
*
* where L(k) and L(k+1) are the k-th and (k+1)-th columns
* of L
*
* Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
* block D(k:k+1,k:k+1) in columns k and k+1 of A.
* (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
* block and not stored)
* A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
* A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
* = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
*
IF( K.LT.N-1 ) THEN
*
* Compose the columns of the inverse of 2-by-2 pivot
* block D in the following way to reduce the number
* of FLOPS when we myltiply panel ( W(k) W(k+1) ) by
* this inverse
*
* D**(-1) = ( d11 d21 )**(-1) =
* ( d21 d22 )
*
* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
* ( (-d21 ) ( d11 ) )
*
* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
*
* * ( ( d22/d21 ) ( -1 ) ) =
* ( ( -1 ) ( d11/d21 ) )
*
* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
* ( ( -1 ) ( D22 ) )
*
* = 1/d21 * T * ( ( D11 ) ( -1 ) )
* ( ( -1 ) ( D22 ) )
*
* = D21 * ( ( D11 ) ( -1 ) )
* ( ( -1 ) ( D22 ) )
*
D21 = W( K+1, K )
D11 = W( K+1, K+1 ) / D21
D22 = W( K, K ) / D21
T = ONE / ( D11*D22-ONE )
D21 = T / D21
*
* Update elements in columns A(k) and A(k+1) as
* dot products of rows of ( W(k) W(k+1) ) and columns
* of D**(-1)
*
DO 80 J = K + 2, N
A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) )
A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
80 CONTINUE
END IF
*
* Copy D(k) to A
*
A( K, K ) = W( K, K )
A( K+1, K ) = W( K+1, K )
A( K+1, K+1 ) = W( K+1, K+1 )
*
END IF
*
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K+1 ) = -KP
END IF
*
* Increase K and return to the start of the main loop
*
K = K + KSTEP
GO TO 70
*
90 CONTINUE
*
* Update the lower triangle of A22 (= A(k:n,k:n)) as
*
* A22 := A22 - L21*D*L21**T = A22 - L21*W**T
*
CALL DGEMMTR( 'Lower', 'No transpose', 'Transpose', N-K+1,
$ K-1, -ONE, A( K, 1 ), LDA, W( K, 1 ), LDW,
$ ONE, A( K, K ), LDA )
*
* Put L21 in standard form by partially undoing the interchanges
* of rows in columns 1:k-1 looping backwards from k-1 to 1
*
J = K - 1
120 CONTINUE
*
* Undo the interchanges (if any) of rows JJ and JP at each
* step J
*
* (Here, J is a diagonal index)
JJ = J
JP = IPIV( J )
IF( JP.LT.0 ) THEN
JP = -JP
* (Here, J is a diagonal index)
J = J - 1
END IF
* (NOTE: Here, J is used to determine row length. Length J
* of the rows to swap back doesn't include diagonal element)
J = J - 1
IF( JP.NE.JJ .AND. J.GE.1 )
$ CALL DSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
IF( J.GT.1 )
$ GO TO 120
*
* Set KB to the number of columns factorized
*
KB = K - 1
*
END IF
RETURN
*
* End of DLASYF
*
END
*> \brief \b DLATBS solves a triangular banded system of equations.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATBS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatbs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatbs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatbs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
* SCALE, CNORM, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORMIN, TRANS, UPLO
* INTEGER INFO, KD, LDAB, N
* DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), CNORM( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATBS solves one of the triangular systems
*>
*> A *x = s*b or A**T*x = s*b
*>
*> with scaling to prevent overflow, where A is an upper or lower
*> triangular band matrix. Here A**T denotes the transpose of A, x and b
*> are n-element vectors, and s is a scaling factor, usually less than
*> or equal to 1, chosen so that the components of x will be less than
*> the overflow threshold. If the unscaled problem will not cause
*> overflow, the Level 2 BLAS routine DTBSV is called. If the matrix A
*> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
*> non-trivial solution to A*x = 0 is returned.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': Solve A * x = s*b (No transpose)
*> = 'T': Solve A**T* x = s*b (Transpose)
*> = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*> NORMIN is CHARACTER*1
*> Specifies whether CNORM has been set or not.
*> = 'Y': CNORM contains the column norms on entry
*> = 'N': CNORM is not set on entry. On exit, the norms will
*> be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of subdiagonals or superdiagonals in the
*> triangular matrix A. KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangular band matrix A, stored in the
*> first KD+1 rows of the array. The j-th column of A is stored
*> in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> On entry, the right hand side b of the triangular system.
*> On exit, X is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The scaling factor s for the triangular system
*> A * x = s*b or A**T* x = s*b.
*> If SCALE = 0, the matrix A is singular or badly scaled, and
*> the vector x is an exact or approximate solution to A*x = 0.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*> CNORM is DOUBLE PRECISION array, dimension (N)
*>
*> If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*> contains the norm of the off-diagonal part of the j-th column
*> of A. If TRANS = 'N', CNORM(j) must be greater than or equal
*> to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*> must be greater than or equal to the 1-norm.
*>
*> If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*> returns the 1-norm of the offdiagonal part of the j-th column
*> of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latbs
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> A rough bound on x is computed; if that is less than overflow, DTBSV
*> is called, otherwise, specific code is used which checks for possible
*> overflow or divide-by-zero at every operation.
*>
*> A columnwise scheme is used for solving A*x = b. The basic algorithm
*> if A is lower triangular is
*>
*> x[1:n] := b[1:n]
*> for j = 1, ..., n
*> x(j) := x(j) / A(j,j)
*> x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*> end
*>
*> Define bounds on the components of x after j iterations of the loop:
*> M(j) = bound on x[1:j]
*> G(j) = bound on x[j+1:n]
*> Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*>
*> Then for iteration j+1 we have
*> M(j+1) <= G(j) / | A(j+1,j+1) |
*> G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*> <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*>
*> where CNORM(j+1) is greater than or equal to the infinity-norm of
*> column j+1 of A, not counting the diagonal. Hence
*>
*> G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*> 1<=i<=j
*> and
*>
*> |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*> 1<=i< j
*>
*> Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTBSV if the
*> reciprocal of the largest M(j), j=1,..,n, is larger than
*> max(underflow, 1/overflow).
*>
*> The bound on x(j) is also used to determine when a step in the
*> columnwise method can be performed without fear of overflow. If
*> the computed bound is greater than a large constant, x is scaled to
*> prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*> 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*>
*> Similarly, a row-wise scheme is used to solve A**T*x = b. The basic
*> algorithm for A upper triangular is
*>
*> for j = 1, ..., n
*> x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
*> end
*>
*> We simultaneously compute two bounds
*> G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
*> M(j) = bound on x(i), 1<=i<=j
*>
*> The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*> add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*> Then the bound on x(j) is
*>
*> M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*>
*> <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*> 1<=i<=j
*>
*> and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater
*> than max(underflow, 1/overflow).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB,
$ X,
$ SCALE, CNORM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORMIN, TRANS, UPLO
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), CNORM( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
INTEGER I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND
DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
$ TMAX, TSCAL, USCAL, XBND, XJ, XMAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DTBSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
* Test the input parameters.
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
$ LSAME( NORMIN, 'N' ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( KD.LT.0 ) THEN
INFO = -6
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLATBS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
SCALE = ONE
IF( N.EQ.0 )
$ RETURN
*
* Determine machine dependent parameters to control overflow.
*
SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
BIGNUM = ONE / SMLNUM
*
IF( LSAME( NORMIN, 'N' ) ) THEN
*
* Compute the 1-norm of each column, not including the diagonal.
*
IF( UPPER ) THEN
*
* A is upper triangular.
*
DO 10 J = 1, N
JLEN = MIN( KD, J-1 )
CNORM( J ) = DASUM( JLEN, AB( KD+1-JLEN, J ), 1 )
10 CONTINUE
ELSE
*
* A is lower triangular.
*
DO 20 J = 1, N
JLEN = MIN( KD, N-J )
IF( JLEN.GT.0 ) THEN
CNORM( J ) = DASUM( JLEN, AB( 2, J ), 1 )
ELSE
CNORM( J ) = ZERO
END IF
20 CONTINUE
END IF
END IF
*
* Scale the column norms by TSCAL if the maximum element in CNORM is
* greater than BIGNUM.
*
IMAX = IDAMAX( N, CNORM, 1 )
TMAX = CNORM( IMAX )
IF( TMAX.LE.BIGNUM ) THEN
TSCAL = ONE
ELSE
TSCAL = ONE / ( SMLNUM*TMAX )
CALL DSCAL( N, TSCAL, CNORM, 1 )
END IF
*
* Compute a bound on the computed solution vector to see if the
* Level 2 BLAS routine DTBSV can be used.
*
J = IDAMAX( N, X, 1 )
XMAX = ABS( X( J ) )
XBND = XMAX
IF( NOTRAN ) THEN
*
* Compute the growth in A * x = b.
*
IF( UPPER ) THEN
JFIRST = N
JLAST = 1
JINC = -1
MAIND = KD + 1
ELSE
JFIRST = 1
JLAST = N
JINC = 1
MAIND = 1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 50
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, G(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
DO 30 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* M(j) = G(j-1) / abs(A(j,j))
*
TJJ = ABS( AB( MAIND, J ) )
XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
* G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
ELSE
*
* G(j) could overflow, set GROW to 0.
*
GROW = ZERO
END IF
30 CONTINUE
GROW = XBND
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 40 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* G(j) = G(j-1)*( 1 + CNORM(j) )
*
GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
40 CONTINUE
END IF
50 CONTINUE
*
ELSE
*
* Compute the growth in A**T * x = b.
*
IF( UPPER ) THEN
JFIRST = 1
JLAST = N
JINC = 1
MAIND = KD + 1
ELSE
JFIRST = N
JLAST = 1
JINC = -1
MAIND = 1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 80
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, M(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
DO 60 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
XJ = ONE + CNORM( J )
GROW = MIN( GROW, XBND / XJ )
*
* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
TJJ = ABS( AB( MAIND, J ) )
IF( XJ.GT.TJJ )
$ XBND = XBND*( TJJ / XJ )
60 CONTINUE
GROW = MIN( GROW, XBND )
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 70 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = ( 1 + CNORM(j) )*G(j-1)
*
XJ = ONE + CNORM( J )
GROW = GROW / XJ
70 CONTINUE
END IF
80 CONTINUE
END IF
*
IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
* Use the Level 2 BLAS solve if the reciprocal of the bound on
* elements of X is not too small.
*
CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, X, 1 )
ELSE
*
* Use a Level 1 BLAS solve, scaling intermediate results.
*
IF( XMAX.GT.BIGNUM ) THEN
*
* Scale X so that its components are less than or equal to
* BIGNUM in absolute value.
*
SCALE = BIGNUM / XMAX
CALL DSCAL( N, SCALE, X, 1 )
XMAX = BIGNUM
END IF
*
IF( NOTRAN ) THEN
*
* Solve A * x = b
*
DO 110 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = AB( MAIND, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 100
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by 1/b(j).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
* to avoid overflow when dividing by A(j,j).
*
REC = ( TJJ*BIGNUM ) / XJ
IF( CNORM( J ).GT.ONE ) THEN
*
* Scale by 1/CNORM(j) to avoid overflow when
* multiplying x(j) times column j.
*
REC = REC / CNORM( J )
END IF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A*x = 0.
*
DO 90 I = 1, N
X( I ) = ZERO
90 CONTINUE
X( J ) = ONE
XJ = ONE
SCALE = ZERO
XMAX = ZERO
END IF
100 CONTINUE
*
* Scale x if necessary to avoid overflow when adding a
* multiple of column j of A.
*
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
* Scale x by 1/(2*abs(x(j))).
*
REC = REC*HALF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
END IF
ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
* Scale x by 1/2.
*
CALL DSCAL( N, HALF, X, 1 )
SCALE = SCALE*HALF
END IF
*
IF( UPPER ) THEN
IF( J.GT.1 ) THEN
*
* Compute the update
* x(max(1,j-kd):j-1) := x(max(1,j-kd):j-1) -
* x(j)* A(max(1,j-kd):j-1,j)
*
JLEN = MIN( KD, J-1 )
CALL DAXPY( JLEN, -X( J )*TSCAL,
$ AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 )
I = IDAMAX( J-1, X, 1 )
XMAX = ABS( X( I ) )
END IF
ELSE IF( J.LT.N ) THEN
*
* Compute the update
* x(j+1:min(j+kd,n)) := x(j+1:min(j+kd,n)) -
* x(j) * A(j+1:min(j+kd,n),j)
*
JLEN = MIN( KD, N-J )
IF( JLEN.GT.0 )
$ CALL DAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1,
$ X( J+1 ), 1 )
I = J + IDAMAX( N-J, X( J+1 ), 1 )
XMAX = ABS( X( I ) )
END IF
110 CONTINUE
*
ELSE
*
* Solve A**T * x = b
*
DO 160 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) - sum A(k,j)*x(k).
* k<>j
*
XJ = ABS( X( J ) )
USCAL = TSCAL
REC = ONE / MAX( XMAX, ONE )
IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
* If x(j) could overflow, scale x by 1/(2*XMAX).
*
REC = REC*HALF
IF( NOUNIT ) THEN
TJJS = AB( MAIND, J )*TSCAL
ELSE
TJJS = TSCAL
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.ONE ) THEN
*
* Divide by A(j,j) when scaling x if A(j,j) > 1.
*
REC = MIN( ONE, REC*TJJ )
USCAL = USCAL / TJJS
END IF
IF( REC.LT.ONE ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
SUMJ = ZERO
IF( USCAL.EQ.ONE ) THEN
*
* If the scaling needed for A in the dot product is 1,
* call DDOT to perform the dot product.
*
IF( UPPER ) THEN
JLEN = MIN( KD, J-1 )
SUMJ = DDOT( JLEN, AB( KD+1-JLEN, J ), 1,
$ X( J-JLEN ), 1 )
ELSE
JLEN = MIN( KD, N-J )
IF( JLEN.GT.0 )
$ SUMJ = DDOT( JLEN, AB( 2, J ), 1, X( J+1 ),
$ 1 )
END IF
ELSE
*
* Otherwise, use in-line code for the dot product.
*
IF( UPPER ) THEN
JLEN = MIN( KD, J-1 )
DO 120 I = 1, JLEN
SUMJ = SUMJ + ( AB( KD+I-JLEN, J )*USCAL )*
$ X( J-JLEN-1+I )
120 CONTINUE
ELSE
JLEN = MIN( KD, N-J )
DO 130 I = 1, JLEN
SUMJ = SUMJ + ( AB( I+1, J )*USCAL )*X( J+I )
130 CONTINUE
END IF
END IF
*
IF( USCAL.EQ.TSCAL ) THEN
*
* Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
* was not used to scale the dotproduct.
*
X( J ) = X( J ) - SUMJ
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
*
* Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
TJJS = AB( MAIND, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 150
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale X by 1/abs(x(j)).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
REC = ( TJJ*BIGNUM ) / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A**T*x = 0.
*
DO 140 I = 1, N
X( I ) = ZERO
140 CONTINUE
X( J ) = ONE
SCALE = ZERO
XMAX = ZERO
END IF
150 CONTINUE
ELSE
*
* Compute x(j) := x(j) / A(j,j) - sumj if the dot
* product has already been divided by 1/A(j,j).
*
X( J ) = X( J ) / TJJS - SUMJ
END IF
XMAX = MAX( XMAX, ABS( X( J ) ) )
160 CONTINUE
END IF
SCALE = SCALE / TSCAL
END IF
*
* Scale the column norms by 1/TSCAL for return.
*
IF( TSCAL.NE.ONE ) THEN
CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
END IF
*
RETURN
*
* End of DLATBS
*
END
*> \brief \b DLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATDF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatdf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatdf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatdf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
* JPIV )
*
* .. Scalar Arguments ..
* INTEGER IJOB, LDZ, N
* DOUBLE PRECISION RDSCAL, RDSUM
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), JPIV( * )
* DOUBLE PRECISION RHS( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATDF uses the LU factorization of the n-by-n matrix Z computed by
*> DGETC2 and computes a contribution to the reciprocal Dif-estimate
*> by solving Z * x = b for x, and choosing the r.h.s. b such that
*> the norm of x is as large as possible. On entry RHS = b holds the
*> contribution from earlier solved sub-systems, and on return RHS = x.
*>
*> The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
*> where P and Q are permutation matrices. L is lower triangular with
*> unit diagonal elements and U is upper triangular.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] IJOB
*> \verbatim
*> IJOB is INTEGER
*> IJOB = 2: First compute an approximative null-vector e
*> of Z using DGECON, e is normalized and solve for
*> Zx = +-e - f with the sign giving the greater value
*> of 2-norm(x). About 5 times as expensive as Default.
*> IJOB .ne. 2: Local look ahead strategy where all entries of
*> the r.h.s. b is chosen as either +1 or -1 (Default).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Z.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, the LU part of the factorization of the n-by-n
*> matrix Z computed by DGETC2: Z = P * L * U * Q
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDA >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] RHS
*> \verbatim
*> RHS is DOUBLE PRECISION array, dimension (N)
*> On entry, RHS contains contributions from other subsystems.
*> On exit, RHS contains the solution of the subsystem with
*> entries according to the value of IJOB (see above).
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*> RDSUM is DOUBLE PRECISION
*> On entry, the sum of squares of computed contributions to
*> the Dif-estimate under computation by DTGSYL, where the
*> scaling factor RDSCAL (see below) has been factored out.
*> On exit, the corresponding sum of squares updated with the
*> contributions from the current sub-system.
*> If TRANS = 'T' RDSUM is not touched.
*> NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*> RDSCAL is DOUBLE PRECISION
*> On entry, scaling factor used to prevent overflow in RDSUM.
*> On exit, RDSCAL is updated w.r.t. the current contributions
*> in RDSUM.
*> If TRANS = 'T', RDSCAL is not touched.
*> NOTE: RDSCAL only makes sense when DTGSY2 is called by
*> DTGSYL.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N).
*> The pivot indices; for 1 <= i <= N, row i of the
*> matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] JPIV
*> \verbatim
*> JPIV is INTEGER array, dimension (N).
*> The pivot indices; for 1 <= j <= N, column j of the
*> matrix has been interchanged with column JPIV(j).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latdf
*
*> \par Further Details:
* =====================
*>
*> This routine is a further developed implementation of algorithm
*> BSOLVE in [1] using complete pivoting in the LU factorization.
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> \verbatim
*>
*>
*> [1] Bo Kagstrom and Lars Westin,
*> Generalized Schur Methods with Condition Estimators for
*> Solving the Generalized Sylvester Equation, IEEE Transactions
*> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
*>
*> [2] Peter Poromaa,
*> On Efficient and Robust Estimators for the Separation
*> between two Regular Matrix Pairs with Applications in
*> Condition Estimation. Report IMINF-95.05, Departement of
*> Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
$ JPIV )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IJOB, LDZ, N
DOUBLE PRECISION RDSCAL, RDSUM
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
DOUBLE PRECISION RHS( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXDIM
PARAMETER ( MAXDIM = 8 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J, K
DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP
* ..
* .. Local Arrays ..
INTEGER IWORK( MAXDIM )
DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ,
$ DLASWP,
$ DSCAL
* ..
* .. External Functions ..
DOUBLE PRECISION DASUM, DDOT
EXTERNAL DASUM, DDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( IJOB.NE.2 ) THEN
*
* Apply permutations IPIV to RHS
*
CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
*
* Solve for L-part choosing RHS either to +1 or -1.
*
PMONE = -ONE
*
DO 10 J = 1, N - 1
BP = RHS( J ) + ONE
BM = RHS( J ) - ONE
SPLUS = ONE
*
* Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
* SMIN computed more efficiently than in BSOLVE [1].
*
SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ),
$ 1 )
SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
SPLUS = SPLUS*RHS( J )
IF( SPLUS.GT.SMINU ) THEN
RHS( J ) = BP
ELSE IF( SMINU.GT.SPLUS ) THEN
RHS( J ) = BM
ELSE
*
* In this case the updating sums are equal and we can
* choose RHS(J) +1 or -1. The first time this happens
* we choose -1, thereafter +1. This is a simple way to
* get good estimates of matrices like Byers well-known
* example (see [1]). (Not done in BSOLVE.)
*
RHS( J ) = RHS( J ) + PMONE
PMONE = ONE
END IF
*
* Compute the remaining r.h.s.
*
TEMP = -RHS( J )
CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
*
10 CONTINUE
*
* Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
* in BSOLVE and will hopefully give us a better estimate because
* any ill-conditioning of the original matrix is transferred to U
* and not to L. U(N, N) is an approximation to sigma_min(LU).
*
CALL DCOPY( N-1, RHS, 1, XP, 1 )
XP( N ) = RHS( N ) + ONE
RHS( N ) = RHS( N ) - ONE
SPLUS = ZERO
SMINU = ZERO
DO 30 I = N, 1, -1
TEMP = ONE / Z( I, I )
XP( I ) = XP( I )*TEMP
RHS( I ) = RHS( I )*TEMP
DO 20 K = I + 1, N
XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
20 CONTINUE
SPLUS = SPLUS + ABS( XP( I ) )
SMINU = SMINU + ABS( RHS( I ) )
30 CONTINUE
IF( SPLUS.GT.SMINU )
$ CALL DCOPY( N, XP, 1, RHS, 1 )
*
* Apply the permutations JPIV to the computed solution (RHS)
*
CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
*
* Compute the sum of squares
*
CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
ELSE
*
* IJOB = 2, Compute approximate nullvector XM of Z
*
CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
*
* Compute RHS
*
CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
CALL DSCAL( N, TEMP, XM, 1 )
CALL DCOPY( N, XM, 1, XP, 1 )
CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
$ CALL DCOPY( N, XP, 1, RHS, 1 )
*
* Compute the sum of squares
*
CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
END IF
*
RETURN
*
* End of DLATDF
*
END
*> \brief \b DLATPS solves a triangular system of equations with the matrix held in packed storage.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATPS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatps.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatps.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatps.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
* CNORM, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORMIN, TRANS, UPLO
* INTEGER INFO, N
* DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), CNORM( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATPS solves one of the triangular systems
*>
*> A *x = s*b or A**T*x = s*b
*>
*> with scaling to prevent overflow, where A is an upper or lower
*> triangular matrix stored in packed form. Here A**T denotes the
*> transpose of A, x and b are n-element vectors, and s is a scaling
*> factor, usually less than or equal to 1, chosen so that the
*> components of x will be less than the overflow threshold. If the
*> unscaled problem will not cause overflow, the Level 2 BLAS routine
*> DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
*> then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': Solve A * x = s*b (No transpose)
*> = 'T': Solve A**T* x = s*b (Transpose)
*> = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*> NORMIN is CHARACTER*1
*> Specifies whether CNORM has been set or not.
*> = 'Y': CNORM contains the column norms on entry
*> = 'N': CNORM is not set on entry. On exit, the norms will
*> be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangular matrix A, packed columnwise in
*> a linear array. The j-th column of A is stored in the array
*> AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> On entry, the right hand side b of the triangular system.
*> On exit, X is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The scaling factor s for the triangular system
*> A * x = s*b or A**T* x = s*b.
*> If SCALE = 0, the matrix A is singular or badly scaled, and
*> the vector x is an exact or approximate solution to A*x = 0.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*> CNORM is DOUBLE PRECISION array, dimension (N)
*>
*> If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*> contains the norm of the off-diagonal part of the j-th column
*> of A. If TRANS = 'N', CNORM(j) must be greater than or equal
*> to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*> must be greater than or equal to the 1-norm.
*>
*> If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*> returns the 1-norm of the offdiagonal part of the j-th column
*> of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latps
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> A rough bound on x is computed; if that is less than overflow, DTPSV
*> is called, otherwise, specific code is used which checks for possible
*> overflow or divide-by-zero at every operation.
*>
*> A columnwise scheme is used for solving A*x = b. The basic algorithm
*> if A is lower triangular is
*>
*> x[1:n] := b[1:n]
*> for j = 1, ..., n
*> x(j) := x(j) / A(j,j)
*> x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*> end
*>
*> Define bounds on the components of x after j iterations of the loop:
*> M(j) = bound on x[1:j]
*> G(j) = bound on x[j+1:n]
*> Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*>
*> Then for iteration j+1 we have
*> M(j+1) <= G(j) / | A(j+1,j+1) |
*> G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*> <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*>
*> where CNORM(j+1) is greater than or equal to the infinity-norm of
*> column j+1 of A, not counting the diagonal. Hence
*>
*> G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*> 1<=i<=j
*> and
*>
*> |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*> 1<=i< j
*>
*> Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the
*> reciprocal of the largest M(j), j=1,..,n, is larger than
*> max(underflow, 1/overflow).
*>
*> The bound on x(j) is also used to determine when a step in the
*> columnwise method can be performed without fear of overflow. If
*> the computed bound is greater than a large constant, x is scaled to
*> prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*> 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*>
*> Similarly, a row-wise scheme is used to solve A**T*x = b. The basic
*> algorithm for A upper triangular is
*>
*> for j = 1, ..., n
*> x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
*> end
*>
*> We simultaneously compute two bounds
*> G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
*> M(j) = bound on x(i), 1<=i<=j
*>
*> The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*> add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*> Then the bound on x(j) is
*>
*> M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*>
*> <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*> 1<=i<=j
*>
*> and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater
*> than max(underflow, 1/overflow).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
$ CNORM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORMIN, TRANS, UPLO
INTEGER INFO, N
DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), CNORM( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
INTEGER I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN
DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
$ TMAX, TSCAL, USCAL, XBND, XJ, XMAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DTPSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
* Test the input parameters.
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
$ LSAME( NORMIN, 'N' ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLATPS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine machine dependent parameters to control overflow.
*
SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
BIGNUM = ONE / SMLNUM
SCALE = ONE
*
IF( LSAME( NORMIN, 'N' ) ) THEN
*
* Compute the 1-norm of each column, not including the diagonal.
*
IF( UPPER ) THEN
*
* A is upper triangular.
*
IP = 1
DO 10 J = 1, N
CNORM( J ) = DASUM( J-1, AP( IP ), 1 )
IP = IP + J
10 CONTINUE
ELSE
*
* A is lower triangular.
*
IP = 1
DO 20 J = 1, N - 1
CNORM( J ) = DASUM( N-J, AP( IP+1 ), 1 )
IP = IP + N - J + 1
20 CONTINUE
CNORM( N ) = ZERO
END IF
END IF
*
* Scale the column norms by TSCAL if the maximum element in CNORM is
* greater than BIGNUM.
*
IMAX = IDAMAX( N, CNORM, 1 )
TMAX = CNORM( IMAX )
IF( TMAX.LE.BIGNUM ) THEN
TSCAL = ONE
ELSE
TSCAL = ONE / ( SMLNUM*TMAX )
CALL DSCAL( N, TSCAL, CNORM, 1 )
END IF
*
* Compute a bound on the computed solution vector to see if the
* Level 2 BLAS routine DTPSV can be used.
*
J = IDAMAX( N, X, 1 )
XMAX = ABS( X( J ) )
XBND = XMAX
IF( NOTRAN ) THEN
*
* Compute the growth in A * x = b.
*
IF( UPPER ) THEN
JFIRST = N
JLAST = 1
JINC = -1
ELSE
JFIRST = 1
JLAST = N
JINC = 1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 50
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, G(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
IP = JFIRST*( JFIRST+1 ) / 2
JLEN = N
DO 30 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* M(j) = G(j-1) / abs(A(j,j))
*
TJJ = ABS( AP( IP ) )
XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
* G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
ELSE
*
* G(j) could overflow, set GROW to 0.
*
GROW = ZERO
END IF
IP = IP + JINC*JLEN
JLEN = JLEN - 1
30 CONTINUE
GROW = XBND
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 40 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* G(j) = G(j-1)*( 1 + CNORM(j) )
*
GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
40 CONTINUE
END IF
50 CONTINUE
*
ELSE
*
* Compute the growth in A**T * x = b.
*
IF( UPPER ) THEN
JFIRST = 1
JLAST = N
JINC = 1
ELSE
JFIRST = N
JLAST = 1
JINC = -1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 80
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, M(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
IP = JFIRST*( JFIRST+1 ) / 2
JLEN = 1
DO 60 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
XJ = ONE + CNORM( J )
GROW = MIN( GROW, XBND / XJ )
*
* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
TJJ = ABS( AP( IP ) )
IF( XJ.GT.TJJ )
$ XBND = XBND*( TJJ / XJ )
JLEN = JLEN + 1
IP = IP + JINC*JLEN
60 CONTINUE
GROW = MIN( GROW, XBND )
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 70 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = ( 1 + CNORM(j) )*G(j-1)
*
XJ = ONE + CNORM( J )
GROW = GROW / XJ
70 CONTINUE
END IF
80 CONTINUE
END IF
*
IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
* Use the Level 2 BLAS solve if the reciprocal of the bound on
* elements of X is not too small.
*
CALL DTPSV( UPLO, TRANS, DIAG, N, AP, X, 1 )
ELSE
*
* Use a Level 1 BLAS solve, scaling intermediate results.
*
IF( XMAX.GT.BIGNUM ) THEN
*
* Scale X so that its components are less than or equal to
* BIGNUM in absolute value.
*
SCALE = BIGNUM / XMAX
CALL DSCAL( N, SCALE, X, 1 )
XMAX = BIGNUM
END IF
*
IF( NOTRAN ) THEN
*
* Solve A * x = b
*
IP = JFIRST*( JFIRST+1 ) / 2
DO 110 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = AP( IP )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 100
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by 1/b(j).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
* to avoid overflow when dividing by A(j,j).
*
REC = ( TJJ*BIGNUM ) / XJ
IF( CNORM( J ).GT.ONE ) THEN
*
* Scale by 1/CNORM(j) to avoid overflow when
* multiplying x(j) times column j.
*
REC = REC / CNORM( J )
END IF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A*x = 0.
*
DO 90 I = 1, N
X( I ) = ZERO
90 CONTINUE
X( J ) = ONE
XJ = ONE
SCALE = ZERO
XMAX = ZERO
END IF
100 CONTINUE
*
* Scale x if necessary to avoid overflow when adding a
* multiple of column j of A.
*
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
* Scale x by 1/(2*abs(x(j))).
*
REC = REC*HALF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
END IF
ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
* Scale x by 1/2.
*
CALL DSCAL( N, HALF, X, 1 )
SCALE = SCALE*HALF
END IF
*
IF( UPPER ) THEN
IF( J.GT.1 ) THEN
*
* Compute the update
* x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
*
CALL DAXPY( J-1, -X( J )*TSCAL, AP( IP-J+1 ), 1,
$ X,
$ 1 )
I = IDAMAX( J-1, X, 1 )
XMAX = ABS( X( I ) )
END IF
IP = IP - J
ELSE
IF( J.LT.N ) THEN
*
* Compute the update
* x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
*
CALL DAXPY( N-J, -X( J )*TSCAL, AP( IP+1 ), 1,
$ X( J+1 ), 1 )
I = J + IDAMAX( N-J, X( J+1 ), 1 )
XMAX = ABS( X( I ) )
END IF
IP = IP + N - J + 1
END IF
110 CONTINUE
*
ELSE
*
* Solve A**T * x = b
*
IP = JFIRST*( JFIRST+1 ) / 2
JLEN = 1
DO 160 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) - sum A(k,j)*x(k).
* k<>j
*
XJ = ABS( X( J ) )
USCAL = TSCAL
REC = ONE / MAX( XMAX, ONE )
IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
* If x(j) could overflow, scale x by 1/(2*XMAX).
*
REC = REC*HALF
IF( NOUNIT ) THEN
TJJS = AP( IP )*TSCAL
ELSE
TJJS = TSCAL
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.ONE ) THEN
*
* Divide by A(j,j) when scaling x if A(j,j) > 1.
*
REC = MIN( ONE, REC*TJJ )
USCAL = USCAL / TJJS
END IF
IF( REC.LT.ONE ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
SUMJ = ZERO
IF( USCAL.EQ.ONE ) THEN
*
* If the scaling needed for A in the dot product is 1,
* call DDOT to perform the dot product.
*
IF( UPPER ) THEN
SUMJ = DDOT( J-1, AP( IP-J+1 ), 1, X, 1 )
ELSE IF( J.LT.N ) THEN
SUMJ = DDOT( N-J, AP( IP+1 ), 1, X( J+1 ), 1 )
END IF
ELSE
*
* Otherwise, use in-line code for the dot product.
*
IF( UPPER ) THEN
DO 120 I = 1, J - 1
SUMJ = SUMJ + ( AP( IP-J+I )*USCAL )*X( I )
120 CONTINUE
ELSE IF( J.LT.N ) THEN
DO 130 I = 1, N - J
SUMJ = SUMJ + ( AP( IP+I )*USCAL )*X( J+I )
130 CONTINUE
END IF
END IF
*
IF( USCAL.EQ.TSCAL ) THEN
*
* Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
* was not used to scale the dotproduct.
*
X( J ) = X( J ) - SUMJ
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
*
* Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
TJJS = AP( IP )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 150
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale X by 1/abs(x(j)).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
REC = ( TJJ*BIGNUM ) / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A**T*x = 0.
*
DO 140 I = 1, N
X( I ) = ZERO
140 CONTINUE
X( J ) = ONE
SCALE = ZERO
XMAX = ZERO
END IF
150 CONTINUE
ELSE
*
* Compute x(j) := x(j) / A(j,j) - sumj if the dot
* product has already been divided by 1/A(j,j).
*
X( J ) = X( J ) / TJJS - SUMJ
END IF
XMAX = MAX( XMAX, ABS( X( J ) ) )
JLEN = JLEN + 1
IP = IP + JINC*JLEN
160 CONTINUE
END IF
SCALE = SCALE / TSCAL
END IF
*
* Scale the column norms by 1/TSCAL for return.
*
IF( TSCAL.NE.ONE ) THEN
CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
END IF
*
RETURN
*
* End of DLATPS
*
END
*> \brief \b DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRD reduces NB rows and columns of a real symmetric matrix A to
*> symmetric tridiagonal form by an orthogonal similarity
*> transformation Q**T * A * Q, and returns the matrices V and W which are
*> needed to apply the transformation to the unreduced part of A.
*>
*> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
*> matrix, of which the upper triangle is supplied;
*> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
*> matrix, of which the lower triangle is supplied.
*>
*> This is an auxiliary routine called by DSYTRD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of rows and columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit:
*> if UPLO = 'U', the last NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements above the diagonal
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors;
*> if UPLO = 'L', the first NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements below the diagonal
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= (1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
*> elements of the last NB columns of the reduced matrix;
*> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
*> the first NB columns of the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors, stored in
*> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
*> See Further Details.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (LDW,NB)
*> The n-by-nb matrix W required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*> LDW is INTEGER
*> The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n) H(n-1) . . . H(n-nb+1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
*> and tau in TAU(i-1).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*> and tau in TAU(i).
*>
*> The elements of the vectors v together form the n-by-nb matrix V
*> which is needed, with W, to apply the transformation to the unreduced
*> part of the matrix, using a symmetric rank-2k update of the form:
*> A := A - V*W**T - W*V**T.
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5 and nb = 2:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( a a a v4 v5 ) ( d )
*> ( a a v4 v5 ) ( 1 d )
*> ( a 1 v5 ) ( v1 1 a )
*> ( d 1 ) ( v1 v2 a a )
*> ( d ) ( v1 v2 a a a )
*>
*> where d denotes a diagonal element of the reduced matrix, a denotes
*> an element of the original matrix that is unchanged, and vi denotes
*> an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, HALF
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IW
DOUBLE PRECISION ALPHA
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Reduce last NB columns of upper triangle
*
DO 10 I = N, N - NB + 1, -1
IW = I - N + NB
IF( I.LT.N ) THEN
*
* Update A(1:i,i)
*
CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
$ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1,
$ IW+1 ),
$ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
END IF
IF( I.GT.1 ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(1:i-2,i)
*
CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1,
$ TAU( I-1 ) )
E( I-1 ) = A( I-1, I )
A( I-1, I ) = ONE
*
* Compute W(1:i-1,i)
*
CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
$ ZERO, W( 1, IW ), 1 )
IF( I.LT.N ) THEN
CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1,
$ IW+1 ),
$ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
$ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1,
$ I+1 ),
$ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
$ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
END IF
CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
$ A( 1, I ), 1 )
CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
END IF
*
10 CONTINUE
ELSE
*
* Reduce first NB columns of lower triangle
*
DO 20 I = 1, NB
*
* Update A(i:n,i)
*
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
$ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
IF( I.LT.N ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:n,i)
*
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ),
$ 1,
$ TAU( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Compute W(i+1:n,i)
*
CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ),
$ LDW,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1,
$ 1 ),
$ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ),
$ LDA,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1,
$ 1 ),
$ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
$ A( I+1, I ), 1 )
CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ),
$ 1 )
END IF
*
20 CONTINUE
END IF
*
RETURN
*
* End of DLATRD
*
END
*> \brief \b DLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
* CNORM, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORMIN, TRANS, UPLO
* INTEGER INFO, LDA, N
* DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRS solves one of the triangular systems
*>
*> A *x = s*b or A**T *x = s*b
*>
*> with scaling to prevent overflow. Here A is an upper or lower
*> triangular matrix, A**T denotes the transpose of A, x and b are
*> n-element vectors, and s is a scaling factor, usually less than
*> or equal to 1, chosen so that the components of x will be less than
*> the overflow threshold. If the unscaled problem will not cause
*> overflow, the Level 2 BLAS routine DTRSV is called. If the matrix A
*> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
*> non-trivial solution to A*x = 0 is returned.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': Solve A * x = s*b (No transpose)
*> = 'T': Solve A**T* x = s*b (Transpose)
*> = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*> NORMIN is CHARACTER*1
*> Specifies whether CNORM has been set or not.
*> = 'Y': CNORM contains the column norms on entry
*> = 'N': CNORM is not set on entry. On exit, the norms will
*> be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The triangular matrix A. If UPLO = 'U', the leading n by n
*> upper triangular part of the array A contains the upper
*> triangular matrix, and the strictly lower triangular part of
*> A is not referenced. If UPLO = 'L', the leading n by n lower
*> triangular part of the array A contains the lower triangular
*> matrix, and the strictly upper triangular part of A is not
*> referenced. If DIAG = 'U', the diagonal elements of A are
*> also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max (1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> On entry, the right hand side b of the triangular system.
*> On exit, X is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The scaling factor s for the triangular system
*> A * x = s*b or A**T* x = s*b.
*> If SCALE = 0, the matrix A is singular or badly scaled, and
*> the vector x is an exact or approximate solution to A*x = 0.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*> CNORM is DOUBLE PRECISION array, dimension (N)
*>
*> If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*> contains the norm of the off-diagonal part of the j-th column
*> of A. If TRANS = 'N', CNORM(j) must be greater than or equal
*> to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*> must be greater than or equal to the 1-norm.
*>
*> If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*> returns the 1-norm of the offdiagonal part of the j-th column
*> of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrs
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> A rough bound on x is computed; if that is less than overflow, DTRSV
*> is called, otherwise, specific code is used which checks for possible
*> overflow or divide-by-zero at every operation.
*>
*> A columnwise scheme is used for solving A*x = b. The basic algorithm
*> if A is lower triangular is
*>
*> x[1:n] := b[1:n]
*> for j = 1, ..., n
*> x(j) := x(j) / A(j,j)
*> x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*> end
*>
*> Define bounds on the components of x after j iterations of the loop:
*> M(j) = bound on x[1:j]
*> G(j) = bound on x[j+1:n]
*> Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*>
*> Then for iteration j+1 we have
*> M(j+1) <= G(j) / | A(j+1,j+1) |
*> G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*> <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*>
*> where CNORM(j+1) is greater than or equal to the infinity-norm of
*> column j+1 of A, not counting the diagonal. Hence
*>
*> G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*> 1<=i<=j
*> and
*>
*> |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*> 1<=i< j
*>
*> Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
*> reciprocal of the largest M(j), j=1,..,n, is larger than
*> max(underflow, 1/overflow).
*>
*> The bound on x(j) is also used to determine when a step in the
*> columnwise method can be performed without fear of overflow. If
*> the computed bound is greater than a large constant, x is scaled to
*> prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*> 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*>
*> Similarly, a row-wise scheme is used to solve A**T*x = b. The basic
*> algorithm for A upper triangular is
*>
*> for j = 1, ..., n
*> x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
*> end
*>
*> We simultaneously compute two bounds
*> G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
*> M(j) = bound on x(i), 1<=i<=j
*>
*> The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*> add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*> Then the bound on x(j) is
*>
*> M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*>
*> <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*> 1<=i<=j
*>
*> and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
*> than max(underflow, 1/overflow).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X,
$ SCALE,
$ CNORM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORMIN, TRANS, UPLO
INTEGER INFO, LDA, N
DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
INTEGER I, IMAX, J, JFIRST, JINC, JLAST
DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
$ TMAX, TSCAL, USCAL, XBND, XJ, XMAX
* ..
* .. Local Arrays ..
DOUBLE PRECISION WORK(1)
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DDOT, DLAMCH, DLANGE
EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH,
$ DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DTRSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
* Test the input parameters.
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
$ LSAME( NORMIN, 'N' ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLATRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
SCALE = ONE
IF( N.EQ.0 )
$ RETURN
*
* Determine machine dependent parameters to control overflow.
*
SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
BIGNUM = ONE / SMLNUM
*
IF( LSAME( NORMIN, 'N' ) ) THEN
*
* Compute the 1-norm of each column, not including the diagonal.
*
IF( UPPER ) THEN
*
* A is upper triangular.
*
DO 10 J = 1, N
CNORM( J ) = DASUM( J-1, A( 1, J ), 1 )
10 CONTINUE
ELSE
*
* A is lower triangular.
*
DO 20 J = 1, N - 1
CNORM( J ) = DASUM( N-J, A( J+1, J ), 1 )
20 CONTINUE
CNORM( N ) = ZERO
END IF
END IF
*
* Scale the column norms by TSCAL if the maximum element in CNORM is
* greater than BIGNUM.
*
IMAX = IDAMAX( N, CNORM, 1 )
TMAX = CNORM( IMAX )
IF( TMAX.LE.BIGNUM ) THEN
TSCAL = ONE
ELSE
*
* Avoid NaN generation if entries in CNORM exceed the
* overflow threshold
*
IF( TMAX.LE.DLAMCH('Overflow') ) THEN
* Case 1: All entries in CNORM are valid floating-point numbers
TSCAL = ONE / ( SMLNUM*TMAX )
CALL DSCAL( N, TSCAL, CNORM, 1 )
ELSE
* Case 2: At least one column norm of A cannot be represented
* as floating-point number. Find the offdiagonal entry A( I, J )
* with the largest absolute value. If this entry is not +/- Infinity,
* use this value as TSCAL.
TMAX = ZERO
IF( UPPER ) THEN
*
* A is upper triangular.
*
DO J = 2, N
TMAX = MAX( DLANGE( 'M', J-1, 1, A( 1, J ), 1,
$ WORK ), TMAX )
END DO
ELSE
*
* A is lower triangular.
*
DO J = 1, N - 1
TMAX = MAX( DLANGE( 'M', N-J, 1, A( J+1, J ), 1,
$ WORK ), TMAX )
END DO
END IF
*
IF( TMAX.LE.DLAMCH('Overflow') ) THEN
TSCAL = ONE / ( SMLNUM*TMAX )
DO J = 1, N
IF( CNORM( J ).LE.DLAMCH('Overflow') ) THEN
CNORM( J ) = CNORM( J )*TSCAL
ELSE
* Recompute the 1-norm without introducing Infinity
* in the summation
CNORM( J ) = ZERO
IF( UPPER ) THEN
DO I = 1, J - 1
CNORM( J ) = CNORM( J ) +
$ TSCAL * ABS( A( I, J ) )
END DO
ELSE
DO I = J + 1, N
CNORM( J ) = CNORM( J ) +
$ TSCAL * ABS( A( I, J ) )
END DO
END IF
END IF
END DO
ELSE
* At least one entry of A is not a valid floating-point entry.
* Rely on TRSV to propagate Inf and NaN.
CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
RETURN
END IF
END IF
END IF
*
* Compute a bound on the computed solution vector to see if the
* Level 2 BLAS routine DTRSV can be used.
*
J = IDAMAX( N, X, 1 )
XMAX = ABS( X( J ) )
XBND = XMAX
IF( NOTRAN ) THEN
*
* Compute the growth in A * x = b.
*
IF( UPPER ) THEN
JFIRST = N
JLAST = 1
JINC = -1
ELSE
JFIRST = 1
JLAST = N
JINC = 1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 50
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, G(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
DO 30 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* M(j) = G(j-1) / abs(A(j,j))
*
TJJ = ABS( A( J, J ) )
XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
* G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
ELSE
*
* G(j) could overflow, set GROW to 0.
*
GROW = ZERO
END IF
30 CONTINUE
GROW = XBND
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 40 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* G(j) = G(j-1)*( 1 + CNORM(j) )
*
GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
40 CONTINUE
END IF
50 CONTINUE
*
ELSE
*
* Compute the growth in A**T * x = b.
*
IF( UPPER ) THEN
JFIRST = 1
JLAST = N
JINC = 1
ELSE
JFIRST = N
JLAST = 1
JINC = -1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 80
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, M(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
DO 60 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
XJ = ONE + CNORM( J )
GROW = MIN( GROW, XBND / XJ )
*
* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
TJJ = ABS( A( J, J ) )
IF( XJ.GT.TJJ )
$ XBND = XBND*( TJJ / XJ )
60 CONTINUE
GROW = MIN( GROW, XBND )
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 70 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = ( 1 + CNORM(j) )*G(j-1)
*
XJ = ONE + CNORM( J )
GROW = GROW / XJ
70 CONTINUE
END IF
80 CONTINUE
END IF
*
IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
* Use the Level 2 BLAS solve if the reciprocal of the bound on
* elements of X is not too small.
*
CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
ELSE
*
* Use a Level 1 BLAS solve, scaling intermediate results.
*
IF( XMAX.GT.BIGNUM ) THEN
*
* Scale X so that its components are less than or equal to
* BIGNUM in absolute value.
*
SCALE = BIGNUM / XMAX
CALL DSCAL( N, SCALE, X, 1 )
XMAX = BIGNUM
END IF
*
IF( NOTRAN ) THEN
*
* Solve A * x = b
*
DO 110 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 100
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by 1/b(j).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
* to avoid overflow when dividing by A(j,j).
*
REC = ( TJJ*BIGNUM ) / XJ
IF( CNORM( J ).GT.ONE ) THEN
*
* Scale by 1/CNORM(j) to avoid overflow when
* multiplying x(j) times column j.
*
REC = REC / CNORM( J )
END IF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A*x = 0.
*
DO 90 I = 1, N
X( I ) = ZERO
90 CONTINUE
X( J ) = ONE
XJ = ONE
SCALE = ZERO
XMAX = ZERO
END IF
100 CONTINUE
*
* Scale x if necessary to avoid overflow when adding a
* multiple of column j of A.
*
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
* Scale x by 1/(2*abs(x(j))).
*
REC = REC*HALF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
END IF
ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
* Scale x by 1/2.
*
CALL DSCAL( N, HALF, X, 1 )
SCALE = SCALE*HALF
END IF
*
IF( UPPER ) THEN
IF( J.GT.1 ) THEN
*
* Compute the update
* x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
*
CALL DAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X,
$ 1 )
I = IDAMAX( J-1, X, 1 )
XMAX = ABS( X( I ) )
END IF
ELSE
IF( J.LT.N ) THEN
*
* Compute the update
* x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
*
CALL DAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1,
$ X( J+1 ), 1 )
I = J + IDAMAX( N-J, X( J+1 ), 1 )
XMAX = ABS( X( I ) )
END IF
END IF
110 CONTINUE
*
ELSE
*
* Solve A**T * x = b
*
DO 160 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) - sum A(k,j)*x(k).
* k<>j
*
XJ = ABS( X( J ) )
USCAL = TSCAL
REC = ONE / MAX( XMAX, ONE )
IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
* If x(j) could overflow, scale x by 1/(2*XMAX).
*
REC = REC*HALF
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.ONE ) THEN
*
* Divide by A(j,j) when scaling x if A(j,j) > 1.
*
REC = MIN( ONE, REC*TJJ )
USCAL = USCAL / TJJS
END IF
IF( REC.LT.ONE ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
SUMJ = ZERO
IF( USCAL.EQ.ONE ) THEN
*
* If the scaling needed for A in the dot product is 1,
* call DDOT to perform the dot product.
*
IF( UPPER ) THEN
SUMJ = DDOT( J-1, A( 1, J ), 1, X, 1 )
ELSE IF( J.LT.N ) THEN
SUMJ = DDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
END IF
ELSE
*
* Otherwise, use in-line code for the dot product.
*
IF( UPPER ) THEN
DO 120 I = 1, J - 1
SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
120 CONTINUE
ELSE IF( J.LT.N ) THEN
DO 130 I = J + 1, N
SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
130 CONTINUE
END IF
END IF
*
IF( USCAL.EQ.TSCAL ) THEN
*
* Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
* was not used to scale the dotproduct.
*
X( J ) = X( J ) - SUMJ
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 150
END IF
*
* Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale X by 1/abs(x(j)).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
REC = ( TJJ*BIGNUM ) / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A**T*x = 0.
*
DO 140 I = 1, N
X( I ) = ZERO
140 CONTINUE
X( J ) = ONE
SCALE = ZERO
XMAX = ZERO
END IF
150 CONTINUE
ELSE
*
* Compute x(j) := x(j) / A(j,j) - sumj if the dot
* product has already been divided by 1/A(j,j).
*
X( J ) = X( J ) / TJJS - SUMJ
END IF
XMAX = MAX( XMAX, ABS( X( J ) ) )
160 CONTINUE
END IF
SCALE = SCALE / TSCAL
END IF
*
* Scale the column norms by 1/TSCAL for return.
*
IF( TSCAL.NE.ONE ) THEN
CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
END IF
*
RETURN
*
* End of DLATRS
*
END
*> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
*
* .. Scalar Arguments ..
* INTEGER L, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
*> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
*> of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
*> matrix and, R and A1 are M-by-M upper triangular matrices.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of columns of the matrix A containing the
*> meaningful part of the Householder vectors. N-M >= L >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the leading M-by-N upper trapezoidal part of the
*> array A must contain the matrix to be factorized.
*> On exit, the leading M-by-M upper triangular part of A
*> contains the upper triangular matrix R, and elements N-L+1 to
*> N of the first M rows of A, with the array TAU, represent the
*> orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrz
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The factorization is obtained by Householder's method. The kth
*> transformation matrix, Z( k ), which is used to introduce zeros into
*> the ( m - k + 1 )th row of A, is given in the form
*>
*> Z( k ) = ( I 0 ),
*> ( 0 T( k ) )
*>
*> where
*>
*> T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
*> ( 0 )
*> ( z( k ) )
*>
*> tau is a scalar and z( k ) is an l element vector. tau and z( k )
*> are chosen to annihilate the elements of the kth row of A2.
*>
*> The scalar tau is returned in the kth element of TAU and the vector
*> u( k ) in the kth row of A2, such that the elements of z( k ) are
*> in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
*> the upper triangular part of A1.
*>
*> Z is given by
*>
*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER L, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DLARZ
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
* Quick return if possible
*
IF( M.EQ.0 ) THEN
RETURN
ELSE IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
RETURN
END IF
*
DO 20 I = M, 1, -1
*
* Generate elementary reflector H(i) to annihilate
* [ A(i,i) A(i,n-l+1:n) ]
*
CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
*
* Apply H(i) to A(1:i-1,i:n) from the right
*
CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
$ TAU( I ), A( 1, I ), LDA, WORK )
*
20 CONTINUE
*
RETURN
*
* End of DLATRZ
*
END
*> \brief \b DLATZM
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATZM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatzm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatzm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatzm.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATZM( SIDE, M, N, V, INCV, TAU, C1, C2, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER INCV, LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C1( LDC, * ), C2( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DORMRZ.
*>
*> DLATZM applies a Householder matrix generated by DTZRQF to a matrix.
*>
*> Let P = I - tau*u*u**T, u = ( 1 ),
*> ( v )
*> where v is an (m-1) vector if SIDE = 'L', or a (n-1) vector if
*> SIDE = 'R'.
*>
*> If SIDE equals 'L', let
*> C = [ C1 ] 1
*> [ C2 ] m-1
*> n
*> Then C is overwritten by P*C.
*>
*> If SIDE equals 'R', let
*> C = [ C1, C2 ] m
*> 1 n-1
*> Then C is overwritten by C*P.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form P * C
*> = 'R': form C * P
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*> (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*> The vector v in the representation of P. V is not used
*> if TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*> INCV is INTEGER
*> The increment between elements of v. INCV <> 0
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of P.
*> \endverbatim
*>
*> \param[in,out] C1
*> \verbatim
*> C1 is DOUBLE PRECISION array, dimension
*> (LDC,N) if SIDE = 'L'
*> (M,1) if SIDE = 'R'
*> On entry, the n-vector C1 if SIDE = 'L', or the m-vector C1
*> if SIDE = 'R'.
*>
*> On exit, the first row of P*C if SIDE = 'L', or the first
*> column of C*P if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in,out] C2
*> \verbatim
*> C2 is DOUBLE PRECISION array, dimension
*> (LDC, N) if SIDE = 'L'
*> (LDC, N-1) if SIDE = 'R'
*> On entry, the (m - 1) x n matrix C2 if SIDE = 'L', or the
*> m x (n - 1) matrix C2 if SIDE = 'R'.
*>
*> On exit, rows 2:m of P*C if SIDE = 'L', or columns 2:m of C*P
*> if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the arrays C1 and C2. LDC >= (1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> (M) if SIDE = 'R'
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DLATZM( SIDE, M, N, V, INCV, TAU, C1, C2, LDC,
$ WORK )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C1( LDC, * ), C2( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGER
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( ( MIN( M, N ).EQ.0 ) .OR. ( TAU.EQ.ZERO ) )
$ RETURN
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* w := (C1 + v**T * C2)**T
*
CALL DCOPY( N, C1, LDC, WORK, 1 )
CALL DGEMV( 'Transpose', M-1, N, ONE, C2, LDC, V, INCV, ONE,
$ WORK, 1 )
*
* [ C1 ] := [ C1 ] - tau* [ 1 ] * w**T
* [ C2 ] [ C2 ] [ v ]
*
CALL DAXPY( N, -TAU, WORK, 1, C1, LDC )
CALL DGER( M-1, N, -TAU, V, INCV, WORK, 1, C2, LDC )
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* w := C1 + C2 * v
*
CALL DCOPY( M, C1, 1, WORK, 1 )
CALL DGEMV( 'No transpose', M, N-1, ONE, C2, LDC, V, INCV,
$ ONE, WORK, 1 )
*
* [ C1, C2 ] := [ C1, C2 ] - tau* w * [ 1 , v**T]
*
CALL DAXPY( M, -TAU, WORK, 1, C1, 1 )
CALL DGER( M, N-1, -TAU, WORK, 1, V, INCV, C2, LDC )
END IF
*
RETURN
*
* End of DLATZM
*
END
*> \brief \b DLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAUU2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlauu2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlauu2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlauu2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAUU2( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAUU2 computes the product U * U**T or L**T * L, where the triangular
*> factor U or L is stored in the upper or lower triangular part of
*> the array A.
*>
*> If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
*> overwriting the factor U in A.
*> If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
*> overwriting the factor L in A.
*>
*> This is the unblocked form of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the triangular factor stored in the array A
*> is upper or lower triangular:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the triangular factor U or L. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the triangular factor U or L.
*> On exit, if UPLO = 'U', the upper triangle of A is
*> overwritten with the upper triangle of the product U * U**T;
*> if UPLO = 'L', the lower triangle of A is overwritten with
*> the lower triangle of the product L**T * L.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lauu2
*
* =====================================================================
SUBROUTINE DLAUU2( UPLO, N, A, LDA, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I
DOUBLE PRECISION AII
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAUU2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Compute the product U * U**T.
*
DO 10 I = 1, N
AII = A( I, I )
IF( I.LT.N ) THEN
A( I, I ) = DDOT( N-I+1, A( I, I ), LDA, A( I, I ),
$ LDA )
CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1,
$ I+1 ),
$ LDA, A( I, I+1 ), LDA, AII, A( 1, I ), 1 )
ELSE
CALL DSCAL( I, AII, A( 1, I ), 1 )
END IF
10 CONTINUE
*
ELSE
*
* Compute the product L**T * L.
*
DO 20 I = 1, N
AII = A( I, I )
IF( I.LT.N ) THEN
A( I, I ) = DDOT( N-I+1, A( I, I ), 1, A( I, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ),
$ LDA,
$ A( I+1, I ), 1, AII, A( I, 1 ), LDA )
ELSE
CALL DSCAL( I, AII, A( I, 1 ), LDA )
END IF
20 CONTINUE
END IF
*
RETURN
*
* End of DLAUU2
*
END
*> \brief \b DLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAUUM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlauum.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlauum.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlauum.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAUUM( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAUUM computes the product U * U**T or L**T * L, where the triangular
*> factor U or L is stored in the upper or lower triangular part of
*> the array A.
*>
*> If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
*> overwriting the factor U in A.
*> If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
*> overwriting the factor L in A.
*>
*> This is the blocked form of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the triangular factor stored in the array A
*> is upper or lower triangular:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the triangular factor U or L. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the triangular factor U or L.
*> On exit, if UPLO = 'U', the upper triangle of A is
*> overwritten with the upper triangle of the product U * U**T;
*> if UPLO = 'L', the lower triangle of A is overwritten with
*> the lower triangle of the product L**T * L.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lauum
*
* =====================================================================
SUBROUTINE DLAUUM( UPLO, N, A, LDA, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IB, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLAUU2, DSYRK, DTRMM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAUUM', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DLAUUM', UPLO, N, -1, -1, -1 )
*
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code
*
CALL DLAUU2( UPLO, N, A, LDA, INFO )
ELSE
*
* Use blocked code
*
IF( UPPER ) THEN
*
* Compute the product U * U**T.
*
DO 10 I = 1, N, NB
IB = MIN( NB, N-I+1 )
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Non-unit',
$ I-1, IB, ONE, A( I, I ), LDA, A( 1, I ),
$ LDA )
CALL DLAUU2( 'Upper', IB, A( I, I ), LDA, INFO )
IF( I+IB.LE.N ) THEN
CALL DGEMM( 'No transpose', 'Transpose', I-1, IB,
$ N-I-IB+1, ONE, A( 1, I+IB ), LDA,
$ A( I, I+IB ), LDA, ONE, A( 1, I ), LDA )
CALL DSYRK( 'Upper', 'No transpose', IB, N-I-IB+1,
$ ONE, A( I, I+IB ), LDA, ONE, A( I, I ),
$ LDA )
END IF
10 CONTINUE
ELSE
*
* Compute the product L**T * L.
*
DO 20 I = 1, N, NB
IB = MIN( NB, N-I+1 )
CALL DTRMM( 'Left', 'Lower', 'Transpose', 'Non-unit',
$ IB,
$ I-1, ONE, A( I, I ), LDA, A( I, 1 ), LDA )
CALL DLAUU2( 'Lower', IB, A( I, I ), LDA, INFO )
IF( I+IB.LE.N ) THEN
CALL DGEMM( 'Transpose', 'No transpose', IB, I-1,
$ N-I-IB+1, ONE, A( I+IB, I ), LDA,
$ A( I+IB, 1 ), LDA, ONE, A( I, 1 ), LDA )
CALL DSYRK( 'Lower', 'Transpose', IB, N-I-IB+1,
$ ONE,
$ A( I+IB, I ), LDA, ONE, A( I, I ), LDA )
END IF
20 CONTINUE
END IF
END IF
*
RETURN
*
* End of DLAUUM
*
END
*> \brief \b DOPGTR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DOPGTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dopgtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dopgtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dopgtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DOPGTR( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDQ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), Q( LDQ, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DOPGTR generates a real orthogonal matrix Q which is defined as the
*> product of n-1 elementary reflectors H(i) of order n, as returned by
*> DSPTRD using packed storage:
*>
*> if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangular packed storage used in previous
*> call to DSPTRD;
*> = 'L': Lower triangular packed storage used in previous
*> call to DSPTRD.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix Q. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The vectors which define the elementary reflectors, as
*> returned by DSPTRD.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DSPTRD.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> The N-by-N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N-1)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup upgtr
*
* =====================================================================
SUBROUTINE DOPGTR( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDQ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), Q( LDQ, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IINFO, IJ, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DORG2L, DORG2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DOPGTR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Q was determined by a call to DSPTRD with UPLO = 'U'
*
* Unpack the vectors which define the elementary reflectors and
* set the last row and column of Q equal to those of the unit
* matrix
*
IJ = 2
DO 20 J = 1, N - 1
DO 10 I = 1, J - 1
Q( I, J ) = AP( IJ )
IJ = IJ + 1
10 CONTINUE
IJ = IJ + 2
Q( N, J ) = ZERO
20 CONTINUE
DO 30 I = 1, N - 1
Q( I, N ) = ZERO
30 CONTINUE
Q( N, N ) = ONE
*
* Generate Q(1:n-1,1:n-1)
*
CALL DORG2L( N-1, N-1, N-1, Q, LDQ, TAU, WORK, IINFO )
*
ELSE
*
* Q was determined by a call to DSPTRD with UPLO = 'L'.
*
* Unpack the vectors which define the elementary reflectors and
* set the first row and column of Q equal to those of the unit
* matrix
*
Q( 1, 1 ) = ONE
DO 40 I = 2, N
Q( I, 1 ) = ZERO
40 CONTINUE
IJ = 3
DO 60 J = 2, N
Q( 1, J ) = ZERO
DO 50 I = J + 1, N
Q( I, J ) = AP( IJ )
IJ = IJ + 1
50 CONTINUE
IJ = IJ + 2
60 CONTINUE
IF( N.GT.1 ) THEN
*
* Generate Q(2:n,2:n)
*
CALL DORG2R( N-1, N-1, N-1, Q( 2, 2 ), LDQ, TAU, WORK,
$ IINFO )
END IF
END IF
RETURN
*
* End of DOPGTR
*
END
*> \brief \b DOPMTR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DOPMTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dopmtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dopmtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dopmtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DOPMTR( SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS, UPLO
* INTEGER INFO, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DOPMTR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> nq-1 elementary reflectors, as returned by DSPTRD using packed
*> storage:
*>
*> if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangular packed storage used in previous
*> call to DSPTRD;
*> = 'L': Lower triangular packed storage used in previous
*> call to DSPTRD.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension
*> (M*(M+1)/2) if SIDE = 'L'
*> (N*(N+1)/2) if SIDE = 'R'
*> The vectors which define the elementary reflectors, as
*> returned by DSPTRD. AP is modified by the routine but
*> restored on exit.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L'
*> or (N-1) if SIDE = 'R'
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DSPTRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup upmtr
*
* =====================================================================
SUBROUTINE DOPMTR( SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC,
$ WORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL FORWRD, LEFT, NOTRAN, UPPER
INTEGER I, I1, I2, I3, IC, II, JC, MI, NI, NQ
DOUBLE PRECISION AII
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
UPPER = LSAME( UPLO, 'U' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DOPMTR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Q was determined by a call to DSPTRD with UPLO = 'U'
*
FORWRD = ( LEFT .AND. NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. .NOT.NOTRAN )
*
IF( FORWRD ) THEN
I1 = 1
I2 = NQ - 1
I3 = 1
II = 2
ELSE
I1 = NQ - 1
I2 = 1
I3 = -1
II = NQ*( NQ+1 ) / 2 - 1
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(1:i,1:n)
*
MI = I
ELSE
*
* H(i) is applied to C(1:m,1:i)
*
NI = I
END IF
*
* Apply H(i)
*
CALL DLARF1L( SIDE, MI, NI, AP( II-I+1 ), 1, TAU( I ), C,
$ LDC,
$ WORK )
*
IF( FORWRD ) THEN
II = II + I + 2
ELSE
II = II - I - 1
END IF
10 CONTINUE
ELSE
*
* Q was determined by a call to DSPTRD with UPLO = 'L'.
*
FORWRD = ( LEFT .AND. .NOT.NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. NOTRAN )
*
IF( FORWRD ) THEN
I1 = 1
I2 = NQ - 1
I3 = 1
II = 2
ELSE
I1 = NQ - 1
I2 = 1
I3 = -1
II = NQ*( NQ+1 ) / 2 - 1
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 20 I = I1, I2, I3
AII = AP( II )
AP( II ) = ONE
IF( LEFT ) THEN
*
* H(i) is applied to C(i+1:m,1:n)
*
MI = M - I
IC = I + 1
ELSE
*
* H(i) is applied to C(1:m,i+1:n)
*
NI = N - I
JC = I + 1
END IF
*
* Apply H(i)
*
CALL DLARF( SIDE, MI, NI, AP( II ), 1, TAU( I ),
$ C( IC, JC ), LDC, WORK )
AP( II ) = AII
*
IF( FORWRD ) THEN
II = II + NQ - I + 1
ELSE
II = II - NQ + I - 2
END IF
20 CONTINUE
END IF
RETURN
*
* End of DOPMTR
*
END
*> \brief \b DORBDB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORBDB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorbdb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorbdb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorbdb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
* X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
* TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIGNS, TRANS
* INTEGER INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
* $ Q
* ..
* .. Array Arguments ..
* DOUBLE PRECISION PHI( * ), THETA( * )
* DOUBLE PRECISION TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
* $ WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
* $ X21( LDX21, * ), X22( LDX22, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
*> partitioned orthogonal matrix X:
*>
*> [ B11 | B12 0 0 ]
*> [ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T
*> X = [-----------] = [---------] [----------------] [---------] .
*> [ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
*> [ 0 | 0 0 I ]
*>
*> X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
*> not the case, then X must be transposed and/or permuted. This can be
*> done in constant time using the TRANS and SIGNS options. See DORCSD
*> for details.)
*>
*> The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
*> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
*> represented implicitly by Householder vectors.
*>
*> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
*> implicitly by angles THETA, PHI.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER
*> = 'T': X, U1, U2, V1T, and V2T are stored in row-major
*> order;
*> otherwise: X, U1, U2, V1T, and V2T are stored in column-
*> major order.
*> \endverbatim
*>
*> \param[in] SIGNS
*> \verbatim
*> SIGNS is CHARACTER
*> = 'O': The lower-left block is made nonpositive (the
*> "other" convention);
*> otherwise: The upper-right block is made nonpositive (the
*> "default" convention).
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows and columns in X.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows in X11 and X12. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is INTEGER
*> The number of columns in X11 and X21. 0 <= Q <=
*> MIN(P,M-P,M-Q).
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*> X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
*> On entry, the top-left block of the orthogonal matrix to be
*> reduced. On exit, the form depends on TRANS:
*> If TRANS = 'N', then
*> the columns of tril(X11) specify reflectors for P1,
*> the rows of triu(X11,1) specify reflectors for Q1;
*> else TRANS = 'T', and
*> the rows of triu(X11) specify reflectors for P1,
*> the columns of tril(X11,-1) specify reflectors for Q1.
*> \endverbatim
*>
*> \param[in] LDX11
*> \verbatim
*> LDX11 is INTEGER
*> The leading dimension of X11. If TRANS = 'N', then LDX11 >=
*> P; else LDX11 >= Q.
*> \endverbatim
*>
*> \param[in,out] X12
*> \verbatim
*> X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
*> On entry, the top-right block of the orthogonal matrix to
*> be reduced. On exit, the form depends on TRANS:
*> If TRANS = 'N', then
*> the rows of triu(X12) specify the first P reflectors for
*> Q2;
*> else TRANS = 'T', and
*> the columns of tril(X12) specify the first P reflectors
*> for Q2.
*> \endverbatim
*>
*> \param[in] LDX12
*> \verbatim
*> LDX12 is INTEGER
*> The leading dimension of X12. If TRANS = 'N', then LDX12 >=
*> P; else LDX11 >= M-Q.
*> \endverbatim
*>
*> \param[in,out] X21
*> \verbatim
*> X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
*> On entry, the bottom-left block of the orthogonal matrix to
*> be reduced. On exit, the form depends on TRANS:
*> If TRANS = 'N', then
*> the columns of tril(X21) specify reflectors for P2;
*> else TRANS = 'T', and
*> the rows of triu(X21) specify reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX21
*> \verbatim
*> LDX21 is INTEGER
*> The leading dimension of X21. If TRANS = 'N', then LDX21 >=
*> M-P; else LDX21 >= Q.
*> \endverbatim
*>
*> \param[in,out] X22
*> \verbatim
*> X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
*> On entry, the bottom-right block of the orthogonal matrix to
*> be reduced. On exit, the form depends on TRANS:
*> If TRANS = 'N', then
*> the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
*> M-P-Q reflectors for Q2,
*> else TRANS = 'T', and
*> the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
*> M-P-Q reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX22
*> \verbatim
*> LDX22 is INTEGER
*> The leading dimension of X22. If TRANS = 'N', then LDX22 >=
*> M-P; else LDX22 >= M-Q.
*> \endverbatim
*>
*> \param[out] THETA
*> \verbatim
*> THETA is DOUBLE PRECISION array, dimension (Q)
*> The entries of the bidiagonal blocks B11, B12, B21, B22 can
*> be computed from the angles THETA and PHI. See Further
*> Details.
*> \endverbatim
*>
*> \param[out] PHI
*> \verbatim
*> PHI is DOUBLE PRECISION array, dimension (Q-1)
*> The entries of the bidiagonal blocks B11, B12, B21, B22 can
*> be computed from the angles THETA and PHI. See Further
*> Details.
*> \endverbatim
*>
*> \param[out] TAUP1
*> \verbatim
*> TAUP1 is DOUBLE PRECISION array, dimension (P)
*> The scalar factors of the elementary reflectors that define
*> P1.
*> \endverbatim
*>
*> \param[out] TAUP2
*> \verbatim
*> TAUP2 is DOUBLE PRECISION array, dimension (M-P)
*> The scalar factors of the elementary reflectors that define
*> P2.
*> \endverbatim
*>
*> \param[out] TAUQ1
*> \verbatim
*> TAUQ1 is DOUBLE PRECISION array, dimension (Q)
*> The scalar factors of the elementary reflectors that define
*> Q1.
*> \endverbatim
*>
*> \param[out] TAUQ2
*> \verbatim
*> TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
*> The scalar factors of the elementary reflectors that define
*> Q2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= M-Q.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unbdb
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The bidiagonal blocks B11, B12, B21, and B22 are represented
*> implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
*> PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
*> lower bidiagonal. Every entry in each bidiagonal band is a product
*> of a sine or cosine of a THETA with a sine or cosine of a PHI. See
*> [1] or DORCSD for details.
*>
*> P1, P2, Q1, and Q2 are represented as products of elementary
*> reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
*> using DORGQR and DORGLQ.
*> \endverbatim
*
*> \par References:
* ================
*>
*> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*> Algorithms, 50(1):33-65, 2009.
*>
* =====================================================================
SUBROUTINE DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12,
$ LDX12,
$ X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
$ TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIGNS, TRANS
INTEGER INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
$ Q
* ..
* .. Array Arguments ..
DOUBLE PRECISION PHI( * ), THETA( * )
DOUBLE PRECISION TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
$ WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
$ X21( LDX21, * ), X22( LDX22, * )
* ..
*
* ====================================================================
*
* .. Parameters ..
DOUBLE PRECISION REALONE
PARAMETER ( REALONE = 1.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL COLMAJOR, LQUERY
INTEGER I, LWORKMIN, LWORKOPT
DOUBLE PRECISION Z1, Z2, Z3, Z4
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLARF1F, DLARFGP, DSCAL,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DNRM2
LOGICAL LSAME
EXTERNAL DNRM2, LSAME
* ..
* .. Intrinsic Functions
INTRINSIC ATAN2, COS, MAX, SIN
* ..
* .. Executable Statements ..
*
* Test input arguments
*
INFO = 0
COLMAJOR = .NOT. LSAME( TRANS, 'T' )
IF( .NOT. LSAME( SIGNS, 'O' ) ) THEN
Z1 = REALONE
Z2 = REALONE
Z3 = REALONE
Z4 = REALONE
ELSE
Z1 = REALONE
Z2 = -REALONE
Z3 = REALONE
Z4 = -REALONE
END IF
LQUERY = LWORK .EQ. -1
*
IF( M .LT. 0 ) THEN
INFO = -3
ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
INFO = -4
ELSE IF( Q .LT. 0 .OR. Q .GT. P .OR. Q .GT. M-P .OR.
$ Q .GT. M-Q ) THEN
INFO = -5
ELSE IF( COLMAJOR .AND. LDX11 .LT. MAX( 1, P ) ) THEN
INFO = -7
ELSE IF( .NOT.COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN
INFO = -7
ELSE IF( COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN
INFO = -9
ELSE IF( .NOT.COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN
INFO = -9
ELSE IF( COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN
INFO = -11
ELSE IF( .NOT.COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN
INFO = -11
ELSE IF( COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN
INFO = -13
ELSE IF( .NOT.COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN
INFO = -13
END IF
*
* Compute workspace
*
IF( INFO .EQ. 0 ) THEN
LWORKOPT = M - Q
LWORKMIN = M - Q
WORK(1) = LWORKOPT
IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
INFO = -21
END IF
END IF
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'xORBDB', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Handle column-major and row-major separately
*
IF( COLMAJOR ) THEN
*
* Reduce columns 1, ..., Q of X11, X12, X21, and X22
*
DO I = 1, Q
*
IF( I .EQ. 1 ) THEN
CALL DSCAL( P-I+1, Z1, X11(I,I), 1 )
ELSE
CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), 1 )
CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I,
$ I-1),
$ 1, X11(I,I), 1 )
END IF
IF( I .EQ. 1 ) THEN
CALL DSCAL( M-P-I+1, Z2, X21(I,I), 1 )
ELSE
CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I), 1 )
CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I,
$ I-1),
$ 1, X21(I,I), 1 )
END IF
*
THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), 1 ),
$ DNRM2( P-I+1, X11(I,I), 1 ) )
*
IF( P .GT. I ) THEN
CALL DLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1,
$ TAUP1(I) )
ELSE IF( P .EQ. I ) THEN
CALL DLARFGP( P-I+1, X11(I,I), X11(I,I), 1, TAUP1(I) )
END IF
IF ( M-P .GT. I ) THEN
CALL DLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1,
$ TAUP2(I) )
ELSE IF ( M-P .EQ. I ) THEN
CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I), 1,
$ TAUP2(I) )
END IF
*
IF ( Q .GT. I ) THEN
CALL DLARF1F( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I),
$ X11(I,I+1), LDX11, WORK )
END IF
IF ( M-Q+1 .GT. I ) THEN
CALL DLARF1F( 'L', P-I+1, M-Q-I+1, X11(I,I), 1,
$ TAUP1(I),
$ X12(I,I), LDX12, WORK )
END IF
IF ( Q .GT. I ) THEN
CALL DLARF1F( 'L', M-P-I+1, Q-I, X21(I,I), 1,
$ TAUP2(I), X21(I,I+1), LDX21, WORK )
END IF
IF ( M-Q+1 .GT. I ) THEN
CALL DLARF1F( 'L', M-P-I+1, M-Q-I+1, X21(I,I), 1,
$ TAUP2(I), X22(I,I), LDX22, WORK )
END IF
*
IF( I .LT. Q ) THEN
CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I,I+1),
$ LDX11 )
CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I,I+1),
$ LDX21,
$ X11(I,I+1), LDX11 )
END IF
CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I),
$ LDX12 )
CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I),
$ LDX22,
$ X12(I,I), LDX12 )
*
IF( I .LT. Q )
$ PHI(I) = ATAN2( DNRM2( Q-I, X11(I,I+1), LDX11 ),
$ DNRM2( M-Q-I+1, X12(I,I), LDX12 ) )
*
IF( I .LT. Q ) THEN
IF ( Q-I .EQ. 1 ) THEN
CALL DLARFGP( Q-I, X11(I,I+1), X11(I,I+1), LDX11,
$ TAUQ1(I) )
ELSE
CALL DLARFGP( Q-I, X11(I,I+1), X11(I,I+2), LDX11,
$ TAUQ1(I) )
END IF
END IF
IF ( Q+I-1 .LT. M ) THEN
IF ( M-Q .EQ. I ) THEN
CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I), LDX12,
$ TAUQ2(I) )
ELSE
CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
$ TAUQ2(I) )
END IF
END IF
*
IF( I .LT. Q ) THEN
CALL DLARF1F( 'R', P-I, Q-I, X11(I,I+1), LDX11,
$ TAUQ1(I),
$ X11(I+1,I+1), LDX11, WORK )
CALL DLARF1F( 'R', M-P-I, Q-I, X11(I,I+1), LDX11,
$ TAUQ1(I),
$ X21(I+1,I+1), LDX21, WORK )
END IF
IF ( P .GT. I ) THEN
CALL DLARF1F( 'R', P-I, M-Q-I+1, X12(I,I), LDX12,
$ TAUQ2(I),
$ X12(I+1,I), LDX12, WORK )
END IF
IF ( M-P .GT. I ) THEN
CALL DLARF1F( 'R', M-P-I, M-Q-I+1, X12(I,I), LDX12,
$ TAUQ2(I), X22(I+1,I), LDX22, WORK )
END IF
*
END DO
*
* Reduce columns Q + 1, ..., P of X12, X22
*
DO I = Q + 1, P
*
CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), LDX12 )
IF ( I .GE. M-Q ) THEN
CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I), LDX12,
$ TAUQ2(I) )
ELSE
CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
$ TAUQ2(I) )
END IF
*
IF ( P .GT. I ) THEN
CALL DLARF1F( 'R', P-I, M-Q-I+1, X12(I,I), LDX12,
$ TAUQ2(I),
$ X12(I+1,I), LDX12, WORK )
END IF
IF( M-P-Q .GE. 1 )
$ CALL DLARF1F( 'R', M-P-Q, M-Q-I+1, X12(I,I), LDX12,
$ TAUQ2(I), X22(Q+1,I), LDX22, WORK )
*
END DO
*
* Reduce columns P + 1, ..., M - Q of X12, X22
*
DO I = 1, M - P - Q
*
CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(Q+I,P+I), LDX22 )
IF ( I .EQ. M-P-Q ) THEN
CALL DLARFGP( M-P-Q-I+1, X22(Q+I,P+I), X22(Q+I,P+I),
$ LDX22, TAUQ2(P+I) )
ELSE
CALL DLARFGP( M-P-Q-I+1, X22(Q+I,P+I), X22(Q+I,P+I+1),
$ LDX22, TAUQ2(P+I) )
END IF
IF ( I .LT. M-P-Q ) THEN
CALL DLARF1F( 'R', M-P-Q-I, M-P-Q-I+1, X22(Q+I,P+I),
$ LDX22,
$ TAUQ2(P+I), X22(Q+I+1,P+I), LDX22, WORK )
END IF
*
END DO
*
ELSE
*
* Reduce columns 1, ..., Q of X11, X12, X21, X22
*
DO I = 1, Q
*
IF( I .EQ. 1 ) THEN
CALL DSCAL( P-I+1, Z1, X11(I,I), LDX11 )
ELSE
CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), LDX11 )
CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I-1,
$ I),
$ LDX12, X11(I,I), LDX11 )
END IF
IF( I .EQ. 1 ) THEN
CALL DSCAL( M-P-I+1, Z2, X21(I,I), LDX21 )
ELSE
CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I),
$ LDX21 )
CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I-1,
$ I),
$ LDX22, X21(I,I), LDX21 )
END IF
*
THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), LDX21 ),
$ DNRM2( P-I+1, X11(I,I), LDX11 ) )
*
CALL DLARFGP( P-I+1, X11(I,I), X11(I,I+1), LDX11,
$ TAUP1(I) )
IF ( I .EQ. M-P ) THEN
CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I), LDX21,
$ TAUP2(I) )
ELSE
CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I+1), LDX21,
$ TAUP2(I) )
END IF
*
IF ( Q .GT. I ) THEN
CALL DLARF1F( 'R', Q-I, P-I+1, X11(I,I), LDX11,
$ TAUP1(I),
$ X11(I+1,I), LDX11, WORK )
END IF
IF ( M-Q+1 .GT. I ) THEN
CALL DLARF1F( 'R', M-Q-I+1, P-I+1, X11(I,I), LDX11,
$ TAUP1(I), X12(I,I), LDX12, WORK )
END IF
IF ( Q .GT. I ) THEN
CALL DLARF1F( 'R', Q-I, M-P-I+1, X21(I,I), LDX21,
$ TAUP2(I),
$ X21(I+1,I), LDX21, WORK )
END IF
IF ( M-Q+1 .GT. I ) THEN
CALL DLARF1F( 'R', M-Q-I+1, M-P-I+1, X21(I,I), LDX21,
$ TAUP2(I), X22(I,I), LDX22, WORK )
END IF
*
IF( I .LT. Q ) THEN
CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I+1,I), 1 )
CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I+1,I), 1,
$ X11(I+1,I), 1 )
END IF
CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I), 1 )
CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I), 1,
$ X12(I,I), 1 )
*
IF( I .LT. Q )
$ PHI(I) = ATAN2( DNRM2( Q-I, X11(I+1,I), 1 ),
$ DNRM2( M-Q-I+1, X12(I,I), 1 ) )
*
IF( I .LT. Q ) THEN
IF ( Q-I .EQ. 1) THEN
CALL DLARFGP( Q-I, X11(I+1,I), X11(I+1,I), 1,
$ TAUQ1(I) )
ELSE
CALL DLARFGP( Q-I, X11(I+1,I), X11(I+2,I), 1,
$ TAUQ1(I) )
END IF
END IF
IF ( M-Q .GT. I ) THEN
CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1,
$ TAUQ2(I) )
ELSE
CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I), 1,
$ TAUQ2(I) )
END IF
*
IF( I .LT. Q ) THEN
CALL DLARF1F( 'L', Q-I, P-I, X11(I+1,I), 1, TAUQ1(I),
$ X11(I+1,I+1), LDX11, WORK )
CALL DLARF1F( 'L', Q-I, M-P-I, X11(I+1,I), 1,
$ TAUQ1(I), X21(I+1,I+1), LDX21, WORK )
END IF
CALL DLARF1F( 'L', M-Q-I+1, P-I, X12(I,I), 1, TAUQ2(I),
$ X12(I,I+1), LDX12, WORK )
IF ( M-P-I .GT. 0 ) THEN
CALL DLARF1F( 'L', M-Q-I+1, M-P-I, X12(I,I), 1,
$ TAUQ2(I), X22(I,I+1), LDX22, WORK )
END IF
*
END DO
*
* Reduce columns Q + 1, ..., P of X12, X22
*
DO I = Q + 1, P
*
CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), 1 )
CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1,
$ TAUQ2(I) )
*
IF ( P .GT. I ) THEN
CALL DLARF1F( 'L', M-Q-I+1, P-I, X12(I,I), 1,
$ TAUQ2(I), X12(I,I+1), LDX12, WORK )
END IF
IF( M-P-Q .GE. 1 )
$ CALL DLARF1F( 'L', M-Q-I+1, M-P-Q, X12(I,I), 1,
$ TAUQ2(I), X22(I,Q+1), LDX22, WORK )
*
END DO
*
* Reduce columns P + 1, ..., M - Q of X12, X22
*
DO I = 1, M - P - Q
*
CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(P+I,Q+I), 1 )
IF ( M-P-Q .EQ. I ) THEN
CALL DLARFGP( M-P-Q-I+1, X22(P+I,Q+I), X22(P+I,Q+I),
$ 1,
$ TAUQ2(P+I) )
ELSE
CALL DLARFGP( M-P-Q-I+1, X22(P+I,Q+I), X22(P+I+1,Q+I),
$ 1,
$ TAUQ2(P+I) )
CALL DLARF1F( 'L', M-P-Q-I+1, M-P-Q-I, X22(P+I,Q+I),
$ 1, TAUQ2(P+I), X22(P+I,Q+I+1), LDX22,
$ WORK )
END IF
*
END DO
*
END IF
*
RETURN
*
* End of DORBDB
*
END
*> \brief \b DORCSD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORCSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorcsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorcsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorcsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS,
* SIGNS, M, P, Q, X11, LDX11, X12,
* LDX12, X21, LDX21, X22, LDX22, THETA,
* U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
* LDV2T, WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
* INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
* $ LDX21, LDX22, LWORK, M, P, Q
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION THETA( * )
* DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
* $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
* $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
* $ * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORCSD computes the CS decomposition of an M-by-M partitioned
*> orthogonal matrix X:
*>
*> [ I 0 0 | 0 0 0 ]
*> [ 0 C 0 | 0 -S 0 ]
*> [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
*> X = [-----------] = [---------] [---------------------] [---------] .
*> [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
*> [ 0 S 0 | 0 C 0 ]
*> [ 0 0 I | 0 0 0 ]
*>
*> X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
*> (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
*> R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
*> which R = MIN(P,M-P,Q,M-Q).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU1
*> \verbatim
*> JOBU1 is CHARACTER
*> = 'Y': U1 is computed;
*> otherwise: U1 is not computed.
*> \endverbatim
*>
*> \param[in] JOBU2
*> \verbatim
*> JOBU2 is CHARACTER
*> = 'Y': U2 is computed;
*> otherwise: U2 is not computed.
*> \endverbatim
*>
*> \param[in] JOBV1T
*> \verbatim
*> JOBV1T is CHARACTER
*> = 'Y': V1T is computed;
*> otherwise: V1T is not computed.
*> \endverbatim
*>
*> \param[in] JOBV2T
*> \verbatim
*> JOBV2T is CHARACTER
*> = 'Y': V2T is computed;
*> otherwise: V2T is not computed.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER
*> = 'T': X, U1, U2, V1T, and V2T are stored in row-major
*> order;
*> otherwise: X, U1, U2, V1T, and V2T are stored in column-
*> major order.
*> \endverbatim
*>
*> \param[in] SIGNS
*> \verbatim
*> SIGNS is CHARACTER
*> = 'O': The lower-left block is made nonpositive (the
*> "other" convention);
*> otherwise: The upper-right block is made nonpositive (the
*> "default" convention).
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows and columns in X.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows in X11 and X12. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is INTEGER
*> The number of columns in X11 and X21. 0 <= Q <= M.
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*> X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
*> On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX11
*> \verbatim
*> LDX11 is INTEGER
*> The leading dimension of X11. LDX11 >= MAX(1,P).
*> \endverbatim
*>
*> \param[in,out] X12
*> \verbatim
*> X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
*> On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX12
*> \verbatim
*> LDX12 is INTEGER
*> The leading dimension of X12. LDX12 >= MAX(1,P).
*> \endverbatim
*>
*> \param[in,out] X21
*> \verbatim
*> X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
*> On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX21
*> \verbatim
*> LDX21 is INTEGER
*> The leading dimension of X11. LDX21 >= MAX(1,M-P).
*> \endverbatim
*>
*> \param[in,out] X22
*> \verbatim
*> X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
*> On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX22
*> \verbatim
*> LDX22 is INTEGER
*> The leading dimension of X11. LDX22 >= MAX(1,M-P).
*> \endverbatim
*>
*> \param[out] THETA
*> \verbatim
*> THETA is DOUBLE PRECISION array, dimension (R), in which R =
*> MIN(P,M-P,Q,M-Q).
*> C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
*> S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
*> \endverbatim
*>
*> \param[out] U1
*> \verbatim
*> U1 is DOUBLE PRECISION array, dimension (LDU1,P)
*> If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
*> \endverbatim
*>
*> \param[in] LDU1
*> \verbatim
*> LDU1 is INTEGER
*> The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
*> MAX(1,P).
*> \endverbatim
*>
*> \param[out] U2
*> \verbatim
*> U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
*> If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
*> matrix U2.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*> LDU2 is INTEGER
*> The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
*> MAX(1,M-P).
*> \endverbatim
*>
*> \param[out] V1T
*> \verbatim
*> V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
*> If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
*> matrix V1**T.
*> \endverbatim
*>
*> \param[in] LDV1T
*> \verbatim
*> LDV1T is INTEGER
*> The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
*> MAX(1,Q).
*> \endverbatim
*>
*> \param[out] V2T
*> \verbatim
*> V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
*> If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
*> matrix V2**T.
*> \endverbatim
*>
*> \param[in] LDV2T
*> \verbatim
*> LDV2T is INTEGER
*> The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
*> MAX(1,M-Q).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
*> ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
*> define the matrix in intermediate bidiagonal-block form
*> remaining after nonconvergence. INFO specifies the number
*> of nonzero PHI's.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the work array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: DBBCSD did not converge. See the description of WORK
*> above for details.
*> \endverbatim
*
*> \par References:
* ================
*>
*> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*> Algorithms, 50(1):33-65, 2009.
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup uncsd
*
* =====================================================================
RECURSIVE SUBROUTINE DORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T,
$ TRANS,
$ SIGNS, M, P, Q, X11, LDX11, X12,
$ LDX12, X21, LDX21, X22, LDX22, THETA,
$ U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
$ LDV2T, WORK, LWORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
$ LDX21, LDX22, LWORK, M, P, Q
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION THETA( * )
DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
$ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
$ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
$ * )
* ..
*
* ===================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0,
$ ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
CHARACTER TRANST, SIGNST
INTEGER CHILDINFO, I, IB11D, IB11E, IB12D, IB12E,
$ IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB,
$ IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1,
$ ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN,
$ LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN,
$ LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN,
$ LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN,
$ LORGQRWORKOPT, LWORKMIN, LWORKOPT
LOGICAL COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2,
$ WANTV1T, WANTV2T
* ..
* .. External Subroutines ..
EXTERNAL DBBCSD, DLACPY, DLAPMR, DLAPMT,
$ DORBDB, DORGLQ, DORGQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test input arguments
*
INFO = 0
WANTU1 = LSAME( JOBU1, 'Y' )
WANTU2 = LSAME( JOBU2, 'Y' )
WANTV1T = LSAME( JOBV1T, 'Y' )
WANTV2T = LSAME( JOBV2T, 'Y' )
COLMAJOR = .NOT. LSAME( TRANS, 'T' )
DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' )
LQUERY = LWORK .EQ. -1
IF( M .LT. 0 ) THEN
INFO = -7
ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
INFO = -8
ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
INFO = -9
ELSE IF ( COLMAJOR .AND. LDX11 .LT. MAX( 1, P ) ) THEN
INFO = -11
ELSE IF (.NOT. COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN
INFO = -11
ELSE IF (COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN
INFO = -13
ELSE IF (.NOT. COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN
INFO = -13
ELSE IF (COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN
INFO = -15
ELSE IF (.NOT. COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN
INFO = -15
ELSE IF (COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN
INFO = -17
ELSE IF (.NOT. COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN
INFO = -17
ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
INFO = -20
ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
INFO = -22
ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
INFO = -24
ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
INFO = -26
END IF
*
* Work with transpose if convenient
*
IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN
IF( COLMAJOR ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
IF( DEFAULTSIGNS ) THEN
SIGNST = 'O'
ELSE
SIGNST = 'D'
END IF
CALL DORCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST,
$ M,
$ Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22,
$ LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1,
$ U2, LDU2, WORK, LWORK, IWORK, INFO )
RETURN
END IF
*
* Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
* convenient
*
IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN
IF( DEFAULTSIGNS ) THEN
SIGNST = 'O'
ELSE
SIGNST = 'D'
END IF
CALL DORCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M,
$ M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11,
$ LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T,
$ LDV1T, WORK, LWORK, IWORK, INFO )
RETURN
END IF
*
* Compute workspace
*
IF( INFO .EQ. 0 ) THEN
*
IPHI = 2
ITAUP1 = IPHI + MAX( 1, Q - 1 )
ITAUP2 = ITAUP1 + MAX( 1, P )
ITAUQ1 = ITAUP2 + MAX( 1, M - P )
ITAUQ2 = ITAUQ1 + MAX( 1, Q )
IORGQR = ITAUQ2 + MAX( 1, M - Q )
CALL DORGQR( M-Q, M-Q, M-Q, U1, MAX(1,M-Q), U1, WORK, -1,
$ CHILDINFO )
LORGQRWORKOPT = INT( WORK(1) )
LORGQRWORKMIN = MAX( 1, M - Q )
IORGLQ = ITAUQ2 + MAX( 1, M - Q )
CALL DORGLQ( M-Q, M-Q, M-Q, U1, MAX(1,M-Q), U1, WORK, -1,
$ CHILDINFO )
LORGLQWORKOPT = INT( WORK(1) )
LORGLQWORKMIN = MAX( 1, M - Q )
IORBDB = ITAUQ2 + MAX( 1, M - Q )
CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
$ X21, LDX21, X22, LDX22, THETA, V1T, U1, U2, V1T,
$ V2T, WORK, -1, CHILDINFO )
LORBDBWORKOPT = INT( WORK(1) )
LORBDBWORKMIN = LORBDBWORKOPT
IB11D = ITAUQ2 + MAX( 1, M - Q )
IB11E = IB11D + MAX( 1, Q )
IB12D = IB11E + MAX( 1, Q - 1 )
IB12E = IB12D + MAX( 1, Q )
IB21D = IB12E + MAX( 1, Q - 1 )
IB21E = IB21D + MAX( 1, Q )
IB22D = IB21E + MAX( 1, Q - 1 )
IB22E = IB22D + MAX( 1, Q )
IBBCSD = IB22E + MAX( 1, Q - 1 )
CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
$ THETA, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
$ LDV2T, U1, U1, U1, U1, U1, U1, U1, U1, WORK, -1,
$ CHILDINFO )
LBBCSDWORKOPT = INT( WORK(1) )
LBBCSDWORKMIN = LBBCSDWORKOPT
LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT,
$ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKOPT ) - 1
LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN,
$ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKMIN ) - 1
WORK(1) = MAX(LWORKOPT,LWORKMIN)
*
IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
INFO = -22
ELSE
LORGQRWORK = LWORK - IORGQR + 1
LORGLQWORK = LWORK - IORGLQ + 1
LORBDBWORK = LWORK - IORBDB + 1
LBBCSDWORK = LWORK - IBBCSD + 1
END IF
END IF
*
* Abort if any illegal arguments
*
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'DORCSD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Transform to bidiagonal block form
*
CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
$ X21,
$ LDX21, X22, LDX22, THETA, WORK(IPHI), WORK(ITAUP1),
$ WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2),
$ WORK(IORBDB), LORBDBWORK, CHILDINFO )
*
* Accumulate Householder reflectors
*
IF( COLMAJOR ) THEN
IF( WANTU1 .AND. P .GT. 0 ) THEN
CALL DLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 )
CALL DORGQR( P, P, Q, U1, LDU1, WORK(ITAUP1),
$ WORK(IORGQR),
$ LORGQRWORK, INFO)
END IF
IF( WANTU2 .AND. M-P .GT. 0 ) THEN
CALL DLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 )
CALL DORGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
$ WORK(IORGQR), LORGQRWORK, INFO )
END IF
IF( WANTV1T .AND. Q .GT. 0 ) THEN
CALL DLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2),
$ LDV1T )
V1T(1, 1) = ONE
DO J = 2, Q
V1T(1,J) = ZERO
V1T(J,1) = ZERO
END DO
CALL DORGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T,
$ WORK(ITAUQ1),
$ WORK(IORGLQ), LORGLQWORK, INFO )
END IF
IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
CALL DLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T )
IF (M-P .GT. Q) Then
CALL DLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22,
$ V2T(P+1,P+1), LDV2T )
END IF
IF (M .GT. Q) THEN
CALL DORGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
$ WORK(IORGLQ), LORGLQWORK, INFO )
END IF
END IF
ELSE
IF( WANTU1 .AND. P .GT. 0 ) THEN
CALL DLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 )
CALL DORGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1),
$ WORK(IORGLQ),
$ LORGLQWORK, INFO)
END IF
IF( WANTU2 .AND. M-P .GT. 0 ) THEN
CALL DLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 )
CALL DORGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
$ WORK(IORGLQ), LORGLQWORK, INFO )
END IF
IF( WANTV1T .AND. Q .GT. 0 ) THEN
CALL DLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2),
$ LDV1T )
V1T(1, 1) = ONE
DO J = 2, Q
V1T(1,J) = ZERO
V1T(J,1) = ZERO
END DO
CALL DORGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T,
$ WORK(ITAUQ1),
$ WORK(IORGQR), LORGQRWORK, INFO )
END IF
IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
CALL DLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T )
CALL DLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22,
$ V2T(P+1,P+1), LDV2T )
CALL DORGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
$ WORK(IORGQR), LORGQRWORK, INFO )
END IF
END IF
*
* Compute the CSD of the matrix in bidiagonal-block form
*
CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
$ THETA,
$ WORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
$ LDV2T, WORK(IB11D), WORK(IB11E), WORK(IB12D),
$ WORK(IB12E), WORK(IB21D), WORK(IB21E), WORK(IB22D),
$ WORK(IB22E), WORK(IBBCSD), LBBCSDWORK, INFO )
*
* Permute rows and columns to place identity submatrices in top-
* left corner of (1,1)-block and/or bottom-right corner of (1,2)-
* block and/or bottom-right corner of (2,1)-block and/or top-left
* corner of (2,2)-block
*
IF( Q .GT. 0 .AND. WANTU2 ) THEN
DO I = 1, Q
IWORK(I) = M - P - Q + I
END DO
DO I = Q + 1, M - P
IWORK(I) = I - Q
END DO
IF( COLMAJOR ) THEN
CALL DLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK )
ELSE
CALL DLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK )
END IF
END IF
IF( M .GT. 0 .AND. WANTV2T ) THEN
DO I = 1, P
IWORK(I) = M - P - Q + I
END DO
DO I = P + 1, M - Q
IWORK(I) = I - P
END DO
IF( .NOT. COLMAJOR ) THEN
CALL DLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
ELSE
CALL DLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
END IF
END IF
*
RETURN
*
* End DORCSD
*
END
*> \brief \b DORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORG2L + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorg2l.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorg2l.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorg2l.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORG2L( M, N, K, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORG2L generates an m by n real matrix Q with orthonormal columns,
*> which is defined as the last n columns of a product of k elementary
*> reflectors of order m
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the (n-k+i)-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQLF in the last k columns of its array
*> argument A.
*> On exit, the m by n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ung2l
*
* =====================================================================
SUBROUTINE DORG2L( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, II, J, L
* ..
* .. External Subroutines ..
EXTERNAL DLARF1L, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORG2L', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Initialise columns 1:n-k to columns of the unit matrix
*
DO 20 J = 1, N - K
DO 10 L = 1, M
A( L, J ) = ZERO
10 CONTINUE
A( M-N+J, J ) = ONE
20 CONTINUE
*
DO 40 I = 1, K
II = N - K + I
*
* Apply H(i) to A(1:m-k+i,1:n-k+i) from the left
*
!A(M-N+II, II) = ONE
CALL DLARF1L( 'Left', M-N+II, II-1, A( 1, II ), 1, TAU( I ),
$ A,
$ LDA, WORK )
CALL DSCAL( M-N+II-1, -TAU( I ), A( 1, II ), 1 )
A( M-N+II, II ) = ONE - TAU( I )
*
* Set A(m-k+i+1:m,n-k+i) to zero
*
DO 30 L = M - N + II + 1, M
A( L, II ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of DORG2L
*
END
*> \brief \b DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORG2R + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorg2r.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorg2r.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorg2r.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORG2R generates an m by n real matrix Q with orthonormal columns,
*> which is defined as the first n columns of a product of k elementary
*> reflectors of order m
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQRF in the first k columns of its array
*> argument A.
*> On exit, the m-by-n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ung2r
*
* =====================================================================
SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, L
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORG2R', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Initialise columns k+1:n to columns of the unit matrix
*
DO 20 J = K + 1, N
DO 10 L = 1, M
A( L, J ) = ZERO
10 CONTINUE
A( J, J ) = ONE
20 CONTINUE
*
DO 40 I = K, 1, -1
*
* Apply H(i) to A(i:m,i:n) from the left
*
IF( I.LT.N ) THEN
CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK )
END IF
IF( I.LT.M )
$ CALL DSCAL( M-I, -TAU( I ), A( I+1, I ), 1 )
A( I, I ) = ONE - TAU( I )
*
* Set A(1:i-1,i) to zero
*
DO 30 L = 1, I - 1
A( L, I ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of DORG2R
*
END
*> \brief \b DORGBR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGBR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgbr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgbr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgbr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER VECT
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGBR generates one of the real orthogonal matrices Q or P**T
*> determined by DGEBRD when reducing a real matrix A to bidiagonal
*> form: A = Q * B * P**T. Q and P**T are defined as products of
*> elementary reflectors H(i) or G(i) respectively.
*>
*> If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
*> is of order M:
*> if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n
*> columns of Q, where m >= n >= k;
*> if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an
*> M-by-M matrix.
*>
*> If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
*> is of order N:
*> if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m
*> rows of P**T, where n >= m >= k;
*> if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as
*> an N-by-N matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> Specifies whether the matrix Q or the matrix P**T is
*> required, as defined in the transformation applied by DGEBRD:
*> = 'Q': generate Q;
*> = 'P': generate P**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q or P**T to be returned.
*> M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q or P**T to be returned.
*> N >= 0.
*> If VECT = 'Q', M >= N >= min(M,K);
*> if VECT = 'P', N >= M >= min(N,K).
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> If VECT = 'Q', the number of columns in the original M-by-K
*> matrix reduced by DGEBRD.
*> If VECT = 'P', the number of rows in the original K-by-N
*> matrix reduced by DGEBRD.
*> K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the vectors which define the elementary reflectors,
*> as returned by DGEBRD.
*> On exit, the M-by-N matrix Q or P**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension
*> (min(M,K)) if VECT = 'Q'
*> (min(N,K)) if VECT = 'P'
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i) or G(i), which determines Q or P**T, as
*> returned by DGEBRD in its array argument TAUQ or TAUP.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,min(M,N)).
*> For optimum performance LWORK >= min(M,N)*NB, where NB
*> is the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungbr
*
* =====================================================================
SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER VECT
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WANTQ
INTEGER I, IINFO, J, LWKOPT, MN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DORGLQ, DORGQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
WANTQ = LSAME( VECT, 'Q' )
MN = MIN( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 .OR. ( WANTQ .AND. ( N.GT.M .OR. N.LT.MIN( M,
$ K ) ) ) .OR. ( .NOT.WANTQ .AND. ( M.GT.N .OR. M.LT.
$ MIN( N, K ) ) ) ) THEN
INFO = -3
ELSE IF( K.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LWORK.LT.MAX( 1, MN ) .AND. .NOT.LQUERY ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = 1
IF( WANTQ ) THEN
IF( M.GE.K ) THEN
CALL DORGQR( M, N, K, A, LDA, TAU, WORK, -1, IINFO )
ELSE
IF( M.GT.1 ) THEN
CALL DORGQR( M-1, M-1, M-1, A, LDA, TAU, WORK, -1,
$ IINFO )
END IF
END IF
ELSE
IF( K.LT.N ) THEN
CALL DORGLQ( M, N, K, A, LDA, TAU, WORK, -1, IINFO )
ELSE
IF( N.GT.1 ) THEN
CALL DORGLQ( N-1, N-1, N-1, A, LDA, TAU, WORK, -1,
$ IINFO )
END IF
END IF
END IF
LWKOPT = INT( WORK( 1 ) )
LWKOPT = MAX (LWKOPT, MN)
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGBR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
WORK( 1 ) = LWKOPT
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( WANTQ ) THEN
*
* Form Q, determined by a call to DGEBRD to reduce an m-by-k
* matrix
*
IF( M.GE.K ) THEN
*
* If m >= k, assume m >= n >= k
*
CALL DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO )
*
ELSE
*
* If m < k, assume m = n
*
* Shift the vectors which define the elementary reflectors one
* column to the right, and set the first row and column of Q
* to those of the unit matrix
*
DO 20 J = M, 2, -1
A( 1, J ) = ZERO
DO 10 I = J + 1, M
A( I, J ) = A( I, J-1 )
10 CONTINUE
20 CONTINUE
A( 1, 1 ) = ONE
DO 30 I = 2, M
A( I, 1 ) = ZERO
30 CONTINUE
IF( M.GT.1 ) THEN
*
* Form Q(2:m,2:m)
*
CALL DORGQR( M-1, M-1, M-1, A( 2, 2 ), LDA, TAU, WORK,
$ LWORK, IINFO )
END IF
END IF
ELSE
*
* Form P**T, determined by a call to DGEBRD to reduce a k-by-n
* matrix
*
IF( K.LT.N ) THEN
*
* If k < n, assume k <= m <= n
*
CALL DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO )
*
ELSE
*
* If k >= n, assume m = n
*
* Shift the vectors which define the elementary reflectors one
* row downward, and set the first row and column of P**T to
* those of the unit matrix
*
A( 1, 1 ) = ONE
DO 40 I = 2, N
A( I, 1 ) = ZERO
40 CONTINUE
DO 60 J = 2, N
DO 50 I = J - 1, 2, -1
A( I, J ) = A( I-1, J )
50 CONTINUE
A( 1, J ) = ZERO
60 CONTINUE
IF( N.GT.1 ) THEN
*
* Form P**T(2:n,2:n)
*
CALL DORGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK,
$ LWORK, IINFO )
END IF
END IF
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORGBR
*
END
*> \brief \b DORGHR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGHR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorghr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorghr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorghr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGHR generates a real orthogonal matrix Q which is defined as the
*> product of IHI-ILO elementary reflectors of order N, as returned by
*> DGEHRD:
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix Q. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> ILO and IHI must have the same values as in the previous call
*> of DGEHRD. Q is equal to the unit matrix except in the
*> submatrix Q(ilo+1:ihi,ilo+1:ihi).
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the vectors which define the elementary reflectors,
*> as returned by DGEHRD.
*> On exit, the N-by-N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEHRD.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= IHI-ILO.
*> For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unghr
*
* =====================================================================
SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IINFO, J, LWKOPT, NB, NH
* ..
* .. External Subroutines ..
EXTERNAL DORGQR, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NH = IHI - ILO
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, NH ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
*
IF( INFO.EQ.0 ) THEN
NB = ILAENV( 1, 'DORGQR', ' ', NH, NH, NH, -1 )
LWKOPT = MAX( 1, NH )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGHR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Shift the vectors which define the elementary reflectors one
* column to the right, and set the first ilo and the last n-ihi
* rows and columns to those of the unit matrix
*
DO 40 J = IHI, ILO + 1, -1
DO 10 I = 1, J - 1
A( I, J ) = ZERO
10 CONTINUE
DO 20 I = J + 1, IHI
A( I, J ) = A( I, J-1 )
20 CONTINUE
DO 30 I = IHI + 1, N
A( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
DO 60 J = 1, ILO
DO 50 I = 1, N
A( I, J ) = ZERO
50 CONTINUE
A( J, J ) = ONE
60 CONTINUE
DO 80 J = IHI + 1, N
DO 70 I = 1, N
A( I, J ) = ZERO
70 CONTINUE
A( J, J ) = ONE
80 CONTINUE
*
IF( NH.GT.0 ) THEN
*
* Generate Q(ilo+1:ihi,ilo+1:ihi)
*
CALL DORGQR( NH, NH, NH, A( ILO+1, ILO+1 ), LDA, TAU( ILO ),
$ WORK, LWORK, IINFO )
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORGHR
*
END
*> \brief \b DORGL2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgl2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgl2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgl2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGL2 generates an m by n real matrix Q with orthonormal rows,
*> which is defined as the first m rows of a product of k elementary
*> reflectors of order n
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th row must contain the vector which defines
*> the elementary reflector H(i), for i = 1,2,...,k, as returned
*> by DGELQF in the first k rows of its array argument A.
*> On exit, the m-by-n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungl2
*
* =====================================================================
SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, L
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGL2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.LE.0 )
$ RETURN
*
IF( K.LT.M ) THEN
*
* Initialise rows k+1:m to rows of the unit matrix
*
DO 20 J = 1, N
DO 10 L = K + 1, M
A( L, J ) = ZERO
10 CONTINUE
IF( J.GT.K .AND. J.LE.M )
$ A( J, J ) = ONE
20 CONTINUE
END IF
*
DO 40 I = K, 1, -1
*
* Apply H(i) to A(i:m,i:n) from the right
*
IF( I.LT.N ) THEN
IF( I.LT.M ) THEN
CALL DLARF1F( 'Right', M-I, N-I+1, A( I, I ), LDA,
$ TAU( I ), A( I+1, I ), LDA, WORK )
END IF
CALL DSCAL( N-I, -TAU( I ), A( I, I+1 ), LDA )
END IF
A( I, I ) = ONE - TAU( I )
*
* Set A(i,1:i-1) to zero
*
DO 30 L = 1, I - 1
A( I, L ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of DORGL2
*
END
*> \brief \b DORGLQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGLQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorglq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorglq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorglq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
*> which is defined as the first M rows of a product of K elementary
*> reflectors of order N
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th row must contain the vector which defines
*> the elementary reflector H(i), for i = 1,2,...,k, as returned
*> by DGELQF in the first k rows of its array argument A.
*> On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M).
*> For optimum performance LWORK >= M*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unglq
*
* =====================================================================
SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK,
$ LWKOPT, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORGL2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DORGLQ', ' ', M, N, K, -1 )
LWKOPT = MAX( 1, M )*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGLQ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.LE.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGLQ', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGLQ', ' ', M, N, K,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the last block.
* The first kk rows are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
* Set A(kk+1:m,1:kk) to zero.
*
DO 20 J = 1, KK
DO 10 I = KK + 1, M
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the last or only block.
*
IF( KK.LT.M )
$ CALL DORGL2( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
$ TAU( KK+1 ), WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF( I+IB.LE.M ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I,
$ I ),
$ LDA, TAU( I ), WORK, LDWORK )
*
* Apply H**T to A(i+ib:m,i:n) from the right
*
CALL DLARFB( 'Right', 'Transpose', 'Forward',
$ 'Rowwise',
$ M-I-IB+1, N-I+1, IB, A( I, I ), LDA, WORK,
$ LDWORK, A( I+IB, I ), LDA, WORK( IB+1 ),
$ LDWORK )
END IF
*
* Apply H**T to columns i:n of current block
*
CALL DORGL2( IB, N-I+1, IB, A( I, I ), LDA, TAU( I ),
$ WORK,
$ IINFO )
*
* Set columns 1:i-1 of current block to zero
*
DO 40 J = 1, I - 1
DO 30 L = I, I + IB - 1
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGLQ
*
END
*> \brief \b DORGQL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGQL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgql.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgql.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgql.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGQL( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGQL generates an M-by-N real matrix Q with orthonormal columns,
*> which is defined as the last N columns of a product of K elementary
*> reflectors of order M
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the (n-k+i)-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQLF in the last k columns of its array
*> argument A.
*> On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungql
*
* =====================================================================
SUBROUTINE DORGQL( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, KK, L, LDWORK, LWKOPT,
$ NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORG2L, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = ILAENV( 1, 'DORGQL', ' ', M, N, K, -1 )
LWKOPT = N*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGQL', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGQL', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGQL', ' ', M, N, K,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the first block.
* The last kk columns are handled by the block method.
*
KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB )
*
* Set A(m-kk+1:m,1:n-kk) to zero.
*
DO 20 J = 1, N - KK
DO 10 I = M - KK + 1, M
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the first or only block.
*
CALL DORG2L( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = K - KK + 1, K, NB
IB = MIN( NB, K-I+1 )
IF( N-K+I.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
$ A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
*
CALL DLARFB( 'Left', 'No transpose', 'Backward',
$ 'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
$ A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
$ WORK( IB+1 ), LDWORK )
END IF
*
* Apply H to rows 1:m-k+i+ib-1 of current block
*
CALL DORG2L( M-K+I+IB-1, IB, IB, A( 1, N-K+I ), LDA,
$ TAU( I ), WORK, IINFO )
*
* Set rows m-k+i+ib:m of current block to zero
*
DO 40 J = N - K + I, N - K + I + IB - 1
DO 30 L = M - K + I + IB, M
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGQL
*
END
*> \brief \b DORGQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGQR generates an M-by-N real matrix Q with orthonormal columns,
*> which is defined as the first N columns of a product of K elementary
*> reflectors of order M
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQRF in the first k columns of its array
*> argument A.
*> On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungqr
*
* =====================================================================
SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK,
$ LWKOPT, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORG2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DORGQR', ' ', M, N, K, -1 )
LWKOPT = MAX( 1, N )*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGQR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGQR', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGQR', ' ', M, N, K,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the last block.
* The first kk columns are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
* Set A(1:kk,kk+1:n) to zero.
*
DO 20 J = KK + 1, N
DO 10 I = 1, KK
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the last or only block.
*
IF( KK.LT.N )
$ CALL DORG2R( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
$ TAU( KK+1 ), WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF( I+IB.LE.N ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
$ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(i:m,i+ib:n) from the left
*
CALL DLARFB( 'Left', 'No transpose', 'Forward',
$ 'Columnwise', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
$ LDA, WORK( IB+1 ), LDWORK )
END IF
*
* Apply H to rows i:m of current block
*
CALL DORG2R( M-I+1, IB, IB, A( I, I ), LDA, TAU( I ),
$ WORK,
$ IINFO )
*
* Set rows 1:i-1 of current block to zero
*
DO 40 J = I, I + IB - 1
DO 30 L = 1, I - 1
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGQR
*
END
*> \brief \b DORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGR2( M, N, K, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGR2 generates an m by n real matrix Q with orthonormal rows,
*> which is defined as the last m rows of a product of k elementary
*> reflectors of order n
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the (m-k+i)-th row must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGERQF in the last k rows of its array argument
*> A.
*> On exit, the m by n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungr2
*
* =====================================================================
SUBROUTINE DORGR2( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, II, J, L
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGR2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.LE.0 )
$ RETURN
*
IF( K.LT.M ) THEN
*
* Initialise rows 1:m-k to rows of the unit matrix
*
DO 20 J = 1, N
DO 10 L = 1, M - K
A( L, J ) = ZERO
10 CONTINUE
IF( J.GT.N-M .AND. J.LE.N-K )
$ A( M-N+J, J ) = ONE
20 CONTINUE
END IF
*
DO 40 I = 1, K
II = M - K + I
*
* Apply H(i) to A(1:m-k+i,1:n-k+i) from the right
*
!A( II, N-M+II ) = ONE
CALL DLARF1L( 'Right', II-1, N-M+II, A( II, 1 ), LDA,
$ TAU( I ),
$ A, LDA, WORK )
CALL DSCAL( N-M+II-1, -TAU( I ), A( II, 1 ), LDA )
A( II, N-M+II ) = ONE - TAU( I )
*
* Set A(m-k+i,n-k+i+1:n) to zero
*
DO 30 L = N - M + II + 1, N
A( II, L ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of DORGR2
*
END
*> \brief \b DORGRQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGRQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgrq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgrq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgrq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGRQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
*> which is defined as the last M rows of a product of K elementary
*> reflectors of order N
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the (m-k+i)-th row must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGERQF in the last k rows of its array argument
*> A.
*> On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M).
*> For optimum performance LWORK >= M*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungrq
*
* =====================================================================
SUBROUTINE DORGRQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, II, IINFO, IWS, J, KK, L, LDWORK,
$ LWKOPT, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORGR2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
IF( M.LE.0 ) THEN
LWKOPT = 1
ELSE
NB = ILAENV( 1, 'DORGRQ', ' ', M, N, K, -1 )
LWKOPT = M*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGRQ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.LE.0 ) THEN
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGRQ', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGRQ', ' ', M, N, K,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the first block.
* The last kk rows are handled by the block method.
*
KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB )
*
* Set A(1:m-kk,n-kk+1:n) to zero.
*
DO 20 J = N - KK + 1, N
DO 10 I = 1, M - KK
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the first or only block.
*
CALL DORGR2( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = K - KK + 1, K, NB
IB = MIN( NB, K-I+1 )
II = M - K + I
IF( II.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
$ A( II, 1 ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H**T to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
CALL DLARFB( 'Right', 'Transpose', 'Backward',
$ 'Rowwise',
$ II-1, N-K+I+IB-1, IB, A( II, 1 ), LDA, WORK,
$ LDWORK, A, LDA, WORK( IB+1 ), LDWORK )
END IF
*
* Apply H**T to columns 1:n-k+i+ib-1 of current block
*
CALL DORGR2( IB, N-K+I+IB-1, IB, A( II, 1 ), LDA,
$ TAU( I ),
$ WORK, IINFO )
*
* Set columns n-k+i+ib:n of current block to zero
*
DO 40 L = N - K + I + IB, N
DO 30 J = II, II + IB - 1
A( J, L ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGRQ
*
END
*> \brief \b DORGTR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGTR( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGTR generates a real orthogonal matrix Q which is defined as the
*> product of n-1 elementary reflectors of order N, as returned by
*> DSYTRD:
*>
*> if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A contains elementary reflectors
*> from DSYTRD;
*> = 'L': Lower triangle of A contains elementary reflectors
*> from DSYTRD.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix Q. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the vectors which define the elementary reflectors,
*> as returned by DSYTRD.
*> On exit, the N-by-N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DSYTRD.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N-1).
*> For optimum performance LWORK >= (N-1)*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungtr
*
* =====================================================================
SUBROUTINE DORGTR( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER I, IINFO, J, LWKOPT, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORGQL, DORGQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.MAX( 1, N-1 ) .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
*
IF( INFO.EQ.0 ) THEN
IF( UPPER ) THEN
NB = ILAENV( 1, 'DORGQL', ' ', N-1, N-1, N-1, -1 )
ELSE
NB = ILAENV( 1, 'DORGQR', ' ', N-1, N-1, N-1, -1 )
END IF
LWKOPT = MAX( 1, N-1 )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGTR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( UPPER ) THEN
*
* Q was determined by a call to DSYTRD with UPLO = 'U'
*
* Shift the vectors which define the elementary reflectors one
* column to the left, and set the last row and column of Q to
* those of the unit matrix
*
DO 20 J = 1, N - 1
DO 10 I = 1, J - 1
A( I, J ) = A( I, J+1 )
10 CONTINUE
A( N, J ) = ZERO
20 CONTINUE
DO 30 I = 1, N - 1
A( I, N ) = ZERO
30 CONTINUE
A( N, N ) = ONE
*
* Generate Q(1:n-1,1:n-1)
*
CALL DORGQL( N-1, N-1, N-1, A, LDA, TAU, WORK, LWORK,
$ IINFO )
*
ELSE
*
* Q was determined by a call to DSYTRD with UPLO = 'L'.
*
* Shift the vectors which define the elementary reflectors one
* column to the right, and set the first row and column of Q to
* those of the unit matrix
*
DO 50 J = N, 2, -1
A( 1, J ) = ZERO
DO 40 I = J + 1, N
A( I, J ) = A( I, J-1 )
40 CONTINUE
50 CONTINUE
A( 1, 1 ) = ONE
DO 60 I = 2, N
A( I, 1 ) = ZERO
60 CONTINUE
IF( N.GT.1 ) THEN
*
* Generate Q(2:n,2:n)
*
CALL DORGQR( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK,
$ LWORK, IINFO )
END IF
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORGTR
*
END
*> \brief \b DORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORM2L + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorm2l.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorm2l.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorm2l.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORM2L overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T * C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQLF in the last k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unm2l
*
* =====================================================================
SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, MI, NI, NQ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF1L, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORM2L', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(1:m-k+i,1:n)
*
MI = M - K + I
ELSE
*
* H(i) is applied to C(1:m,1:n-k+i)
*
NI = N - K + I
END IF
*
* Apply H(i)
*
CALL DLARF1L( SIDE, MI, NI, A( 1, I ), 1, TAU( I ), C, LDC,
$ WORK )
10 CONTINUE
RETURN
*
* End of DORM2L
*
END
*> \brief \b DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORM2R + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorm2r.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorm2r.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorm2r.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORM2R overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T* C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQRF in the first k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unm2r
*
* =====================================================================
SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, IC, JC, MI, NI, NQ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DLARF1F
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORM2R', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H(i) is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H(i)
*
CALL DLARF1F( SIDE, MI, NI, A( I, I ), 1, TAU( I ), C( IC,
$ JC ),
$ LDC, WORK )
10 CONTINUE
RETURN
*
* End of DORM2R
*
END
*> \brief \b DORMBR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMBR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormbr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormbr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormbr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C,
* LDC, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS, VECT
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C
*> with
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
*> with
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': P * C C * P
*> TRANS = 'T': P**T * C C * P**T
*>
*> Here Q and P**T are the orthogonal matrices determined by DGEBRD when
*> reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
*> P**T are defined as products of elementary reflectors H(i) and G(i)
*> respectively.
*>
*> Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
*> order of the orthogonal matrix Q or P**T that is applied.
*>
*> If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
*> if nq >= k, Q = H(1) H(2) . . . H(k);
*> if nq < k, Q = H(1) H(2) . . . H(nq-1).
*>
*> If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
*> if k < nq, P = G(1) G(2) . . . G(k);
*> if k >= nq, P = G(1) G(2) . . . G(nq-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> = 'Q': apply Q or Q**T;
*> = 'P': apply P or P**T.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q, Q**T, P or P**T from the Left;
*> = 'R': apply Q, Q**T, P or P**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q or P;
*> = 'T': Transpose, apply Q**T or P**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> If VECT = 'Q', the number of columns in the original
*> matrix reduced by DGEBRD.
*> If VECT = 'P', the number of rows in the original
*> matrix reduced by DGEBRD.
*> K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,min(nq,K)) if VECT = 'Q'
*> (LDA,nq) if VECT = 'P'
*> The vectors which define the elementary reflectors H(i) and
*> G(i), whose products determine the matrices Q and P, as
*> returned by DGEBRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If VECT = 'Q', LDA >= max(1,nq);
*> if VECT = 'P', LDA >= max(1,min(nq,K)).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(nq,K))
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i) or G(i) which determines Q or P, as returned
*> by DGEBRD in the array argument TAUQ or TAUP.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
*> or P*C or P**T*C or C*P or C*P**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmbr
*
* =====================================================================
SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C,
$ LDC, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS, VECT
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL APPLYQ, LEFT, LQUERY, NOTRAN
CHARACTER TRANST
INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORMLQ, DORMQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
APPLYQ = LSAME( VECT, 'Q' )
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q or P and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.APPLYQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN
INFO = -1
ELSE IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( K.LT.0 ) THEN
INFO = -6
ELSE IF( ( APPLYQ .AND. LDA.LT.MAX( 1, NQ ) ) .OR.
$ ( .NOT.APPLYQ .AND. LDA.LT.MAX( 1, MIN( NQ, K ) ) ) )
$ THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.EQ.0 ) THEN
IF( APPLYQ ) THEN
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
ELSE
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
END IF
LWKOPT = NW*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMBR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
WORK( 1 ) = 1
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
IF( APPLYQ ) THEN
*
* Apply Q
*
IF( NQ.GE.K ) THEN
*
* Q was determined by a call to DGEBRD with nq >= k
*
CALL DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, IINFO )
ELSE IF( NQ.GT.1 ) THEN
*
* Q was determined by a call to DGEBRD with nq < k
*
IF( LEFT ) THEN
MI = M - 1
NI = N
I1 = 2
I2 = 1
ELSE
MI = M
NI = N - 1
I1 = 1
I2 = 2
END IF
CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA,
$ TAU,
$ C( I1, I2 ), LDC, WORK, LWORK, IINFO )
END IF
ELSE
*
* Apply P
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
IF( NQ.GT.K ) THEN
*
* P was determined by a call to DGEBRD with nq > k
*
CALL DORMLQ( SIDE, TRANST, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, IINFO )
ELSE IF( NQ.GT.1 ) THEN
*
* P was determined by a call to DGEBRD with nq <= k
*
IF( LEFT ) THEN
MI = M - 1
NI = N
I1 = 2
I2 = 1
ELSE
MI = M
NI = N - 1
I1 = 1
I2 = 2
END IF
CALL DORMLQ( SIDE, TRANST, MI, NI, NQ-1, A( 1, 2 ), LDA,
$ TAU, C( I1, I2 ), LDC, WORK, LWORK, IINFO )
END IF
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMBR
*
END
*> \brief \b DORMHR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMHR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormhr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormhr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormhr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMHR( SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C,
* LDC, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER IHI, ILO, INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMHR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> IHI-ILO elementary reflectors, as returned by DGEHRD:
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> ILO and IHI must have the same values as in the previous call
*> of DGEHRD. Q is equal to the unit matrix except in the
*> submatrix Q(ilo+1:ihi,ilo+1:ihi).
*> If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
*> ILO = 1 and IHI = 0, if M = 0;
*> if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
*> ILO = 1 and IHI = 0, if N = 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L'
*> (LDA,N) if SIDE = 'R'
*> The vectors which define the elementary reflectors, as
*> returned by DGEHRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEHRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmhr
*
* =====================================================================
SUBROUTINE DORMHR( SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C,
$ LDC, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER IHI, ILO, INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LEFT, LQUERY
INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NH, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORMQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NH = IHI - ILO
LEFT = LSAME( SIDE, 'L' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND.
$ .NOT.LSAME( TRANS, 'T' ) )
$ THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, NQ ) ) THEN
INFO = -5
ELSE IF( IHI.LT.MIN( ILO, NQ ) .OR. IHI.GT.NQ ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.EQ.0 ) THEN
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, NH, N, NH, -1 )
ELSE
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, NH, NH, -1 )
END IF
LWKOPT = NW*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMHR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. NH.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( LEFT ) THEN
MI = NH
NI = N
I1 = ILO + 1
I2 = 1
ELSE
MI = M
NI = NH
I1 = 1
I2 = ILO + 1
END IF
*
CALL DORMQR( SIDE, TRANS, MI, NI, NH, A( ILO+1, ILO ), LDA,
$ TAU( ILO ), C( I1, I2 ), LDC, WORK, LWORK, IINFO )
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMHR
*
END
*> \brief \b DORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORML2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorml2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorml2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorml2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORML2 overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T* C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGELQF in the first k rows of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unml2
*
* =====================================================================
SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, IC, JC, MI, NI, NQ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF1F, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORML2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H(i) is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H(i)
*
CALL DLARF1F( SIDE, MI, NI, A( I, I ), LDA, TAU( I ),
$ C( IC, JC ), LDC, WORK )
10 CONTINUE
RETURN
*
* End of DORML2
*
END
*> \brief \b DORMLQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMLQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormlq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormlq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormlq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMLQ overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGELQF in the first k rows of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmlq
*
* =====================================================================
SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
CHARACTER TRANST
INTEGER I, I1, I2, I3, IB, IC, IINFO, IWT, JC, LDWORK,
$ LWKOPT, MI, NB, NBMIN, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORML2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
NB = MIN( NBMAX, ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N,
$ K,
$ -1 ) )
LWKOPT = NW*NB + TSIZE
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMLQ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IF( LWORK.LT.LWKOPT ) THEN
NB = (LWORK-TSIZE) / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMLQ', SIDE // TRANS, M, N,
$ K,
$ -1 ) )
END IF
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IWT = 1 + NW*NB
IF( ( LEFT .AND. NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Rowwise', NQ-I+1, IB, A( I, I ),
$ LDA, TAU( I ), WORK( IWT ), LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H or H**T is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANST, 'Forward', 'Rowwise', MI, NI,
$ IB,
$ A( I, I ), LDA, WORK( IWT ), LDT,
$ C( IC, JC ), LDC, WORK, LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMLQ
*
END
*> \brief \b DORMQL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMQL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormql.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormql.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormql.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMQL overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQLF in the last k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmql
*
* =====================================================================
SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
INTEGER I, I1, I2, I3, IB, IINFO, IWT, LDWORK, LWKOPT,
$ MI, NB, NBMIN, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORM2L, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = MIN( NBMAX, ILAENV( 1, 'DORMQL', SIDE // TRANS, M,
$ N,
$ K, -1 ) )
LWKOPT = NW*NB + TSIZE
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMQL', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IF( LWORK.LT.LWKOPT ) THEN
NB = (LWORK-TSIZE) / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMQL', SIDE // TRANS, M, N,
$ K,
$ -1 ) )
END IF
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IWT = 1 + NW*NB
IF( ( LEFT .AND. NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Columnwise', NQ-K+I+IB-1, IB,
$ A( 1, I ), LDA, TAU( I ), WORK( IWT ), LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(1:m-k+i+ib-1,1:n)
*
MI = M - K + I + IB - 1
ELSE
*
* H or H**T is applied to C(1:m,1:n-k+i+ib-1)
*
NI = N - K + I + IB - 1
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANS, 'Backward', 'Columnwise', MI,
$ NI,
$ IB, A( 1, I ), LDA, WORK( IWT ), LDT, C, LDC,
$ WORK, LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMQL
*
END
*> \brief \b DORMQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMQR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQRF in the first k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmqr
*
* =====================================================================
SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
INTEGER I, I1, I2, I3, IB, IC, IINFO, IWT, JC, LDWORK,
$ LWKOPT, MI, NB, NBMIN, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORM2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
NB = MIN( NBMAX, ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N,
$ K,
$ -1 ) )
LWKOPT = NW*NB + TSIZE
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMQR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IF( LWORK.LT.LWKOPT ) THEN
NB = (LWORK-TSIZE) / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMQR', SIDE // TRANS, M, N,
$ K,
$ -1 ) )
END IF
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IWT = 1 + NW*NB
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', NQ-I+1, IB, A( I,
$ I ),
$ LDA, TAU( I ), WORK( IWT ), LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H or H**T is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANS, 'Forward', 'Columnwise', MI,
$ NI,
$ IB, A( I, I ), LDA, WORK( IWT ), LDT,
$ C( IC, JC ), LDC, WORK, LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMQR
*
END
*> \brief \b DORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMR2 overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T* C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q' (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGERQF in the last k rows of its array argument A.
*> A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmr2
*
* =====================================================================
SUBROUTINE DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, MI, NI, NQ
DOUBLE PRECISION AII
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMR2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(1:m-k+i,1:n)
*
MI = M - K + I
ELSE
*
* H(i) is applied to C(1:m,1:n-k+i)
*
NI = N - K + I
END IF
*
* Apply H(i)
*
AII = A( I, NQ-K+I )
A( I, NQ-K+I ) = ONE
CALL DLARF( SIDE, MI, NI, A( I, 1 ), LDA, TAU( I ), C, LDC,
$ WORK )
A( I, NQ-K+I ) = AII
10 CONTINUE
RETURN
*
* End of DORMR2
*
END
*> \brief \b DORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMR3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormr3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormr3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormr3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, L, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMR3 overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T* C if SIDE = 'L' and TRANS = 'C', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'C',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DTZRZF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of columns of the matrix A containing
*> the meaningful part of the Householder reflectors.
*> If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DTZRZF in the last k rows of its array argument A.
*> A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DTZRZF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m-by-n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmr3
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C,
$ LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, L, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, IC, JA, JC, MI, NI, NQ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARZ, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( L.LT.0 .OR. ( LEFT .AND. ( L.GT.M ) ) .OR.
$ ( .NOT.LEFT .AND. ( L.GT.N ) ) ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMR3', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. .NOT.NOTRAN .OR. .NOT.LEFT .AND. NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
JA = M - L + 1
JC = 1
ELSE
MI = M
JA = N - L + 1
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) or H(i)**T is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H(i) or H(i)**T is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H(i) or H(i)**T
*
CALL DLARZ( SIDE, MI, NI, L, A( I, JA ), LDA, TAU( I ),
$ C( IC, JC ), LDC, WORK )
*
10 CONTINUE
*
RETURN
*
* End of DORMR3
*
END
*> \brief \b DORMRQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMRQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormrq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormrq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormrq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMRQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMRQ overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGERQF in the last k rows of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmrq
*
* =====================================================================
SUBROUTINE DORMRQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
CHARACTER TRANST
INTEGER I, I1, I2, I3, IB, IINFO, IWT, LDWORK, LWKOPT,
$ MI, NB, NBMIN, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORMR2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = MIN( NBMAX, ILAENV( 1, 'DORMRQ', SIDE // TRANS, M,
$ N,
$ K, -1 ) )
LWKOPT = NW*NB + TSIZE
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMRQ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IF( LWORK.LT.LWKOPT ) THEN
NB = (LWORK-TSIZE) / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMRQ', SIDE // TRANS, M, N,
$ K,
$ -1 ) )
END IF
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IWT = 1 + NW*NB
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Rowwise', NQ-K+I+IB-1, IB,
$ A( I, 1 ), LDA, TAU( I ), WORK( IWT ), LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(1:m-k+i+ib-1,1:n)
*
MI = M - K + I + IB - 1
ELSE
*
* H or H**T is applied to C(1:m,1:n-k+i+ib-1)
*
NI = N - K + I + IB - 1
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANST, 'Backward', 'Rowwise', MI, NI,
$ IB, A( I, 1 ), LDA, WORK( IWT ), LDT, C, LDC,
$ WORK, LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMRQ
*
END
*> \brief \b DORMRZ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMRZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormrz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormrz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormrz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMRZ( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, L, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMRZ overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DTZRZF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of columns of the matrix A containing
*> the meaningful part of the Householder reflectors.
*> If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DTZRZF in the last k rows of its array argument A.
*> A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DTZRZF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmrz
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DORMRZ( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C,
$ LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, L, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
CHARACTER TRANST
INTEGER I, I1, I2, I3, IB, IC, IINFO, IWT, JA, JC,
$ LDWORK, LWKOPT, MI, NB, NBMIN, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARZB, DLARZT, DORMR3, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( L.LT.0 .OR. ( LEFT .AND. ( L.GT.M ) ) .OR.
$ ( .NOT.LEFT .AND. ( L.GT.N ) ) ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = MIN( NBMAX, ILAENV( 1, 'DORMRQ', SIDE // TRANS, M,
$ N,
$ K, -1 ) )
LWKOPT = NW*NB + TSIZE
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMRZ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IF( LWORK.LT.LWKOPT ) THEN
NB = (LWORK-TSIZE) / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMRQ', SIDE // TRANS, M, N,
$ K,
$ -1 ) )
END IF
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC,
$ WORK, IINFO )
ELSE
*
* Use blocked code
*
IWT = 1 + NW*NB
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
JA = M - L + 1
ELSE
MI = M
IC = 1
JA = N - L + 1
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARZT( 'Backward', 'Rowwise', L, IB, A( I, JA ),
$ LDA,
$ TAU( I ), WORK( IWT ), LDT )
*
IF( LEFT ) THEN
*
* H or H**T is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H or H**T is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H or H**T
*
CALL DLARZB( SIDE, TRANST, 'Backward', 'Rowwise', MI, NI,
$ IB, L, A( I, JA ), LDA, WORK( IWT ), LDT,
$ C( IC, JC ), LDC, WORK, LDWORK )
10 CONTINUE
*
END IF
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DORMRZ
*
END
*> \brief \b DORMTR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS, UPLO
* INTEGER INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMTR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> nq-1 elementary reflectors, as returned by DSYTRD:
*>
*> if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A contains elementary reflectors
*> from DSYTRD;
*> = 'L': Lower triangle of A contains elementary reflectors
*> from DSYTRD.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L'
*> (LDA,N) if SIDE = 'R'
*> The vectors which define the elementary reflectors, as
*> returned by DSYTRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DSYTRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmtr
*
* =====================================================================
SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C,
$ LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, UPPER
INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORMQL, DORMQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND.
$ .NOT.LSAME( TRANS, 'T' ) )
$ THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
IF( UPPER ) THEN
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
ELSE
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
END IF
LWKOPT = NW*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMTR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. NQ.EQ.1 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( LEFT ) THEN
MI = M - 1
NI = N
ELSE
MI = M
NI = N - 1
END IF
*
IF( UPPER ) THEN
*
* Q was determined by a call to DSYTRD with UPLO = 'U'
*
CALL DORMQL( SIDE, TRANS, MI, NI, NQ-1, A( 1, 2 ), LDA, TAU,
$ C,
$ LDC, WORK, LWORK, IINFO )
ELSE
*
* Q was determined by a call to DSYTRD with UPLO = 'L'
*
IF( LEFT ) THEN
I1 = 2
I2 = 1
ELSE
I1 = 1
I2 = 2
END IF
CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA, TAU,
$ C( I1, I2 ), LDC, WORK, LWORK, IINFO )
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMTR
*
END
*> \brief \b DPBCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite band matrix using the
*> Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangular factor stored in AB;
*> = 'L': Lower triangular factor stored in AB.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T of the band matrix A, stored in the
*> first KD+1 rows of the array. The j-th column of U or L is
*> stored in the j-th column of the array AB as follows:
*> if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm (or infinity-norm) of the symmetric band matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*> estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbcon
*
* =====================================================================
SUBROUTINE DPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
CHARACTER NORMIN
INTEGER IX, KASE
DOUBLE PRECISION AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLATBS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
SMLNUM = DLAMCH( 'Safe minimum' )
*
* Estimate the 1-norm of the inverse.
*
KASE = 0
NORMIN = 'N'
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( UPPER ) THEN
*
* Multiply by inv(U**T).
*
CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
$ KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
$ INFO )
NORMIN = 'Y'
*
* Multiply by inv(U).
*
CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
$ INFO )
ELSE
*
* Multiply by inv(L).
*
CALL DLATBS( 'Lower', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
$ INFO )
NORMIN = 'Y'
*
* Multiply by inv(L**T).
*
CALL DLATBS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
$ KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
$ INFO )
END IF
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
SCALE = SCALEL*SCALEU
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
20 CONTINUE
*
RETURN
*
* End of DPBCON
*
END
*> \brief \b DPBEQU
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBEQU computes row and column scalings intended to equilibrate a
*> symmetric positive definite band matrix A and reduce its condition
*> number (with respect to the two-norm). S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangular of A is stored;
*> = 'L': Lower triangular of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangle of the symmetric band matrix A,
*> stored in the first KD+1 rows of the array. The j-th column
*> of A is stored in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array A. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> If INFO = 0, S contains the ratio of the smallest S(i) to
*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
*> large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbequ
*
* =====================================================================
SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, J
DOUBLE PRECISION SMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBEQU', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SCOND = ONE
AMAX = ZERO
RETURN
END IF
*
IF( UPPER ) THEN
J = KD + 1
ELSE
J = 1
END IF
*
* Initialize SMIN and AMAX.
*
S( 1 ) = AB( J, 1 )
SMIN = S( 1 )
AMAX = S( 1 )
*
* Find the minimum and maximum diagonal elements.
*
DO 10 I = 2, N
S( I ) = AB( J, I )
SMIN = MIN( SMIN, S( I ) )
AMAX = MAX( AMAX, S( I ) )
10 CONTINUE
*
IF( SMIN.LE.ZERO ) THEN
*
* Find the first non-positive diagonal element and return.
*
DO 20 I = 1, N
IF( S( I ).LE.ZERO ) THEN
INFO = I
RETURN
END IF
20 CONTINUE
ELSE
*
* Set the scale factors to the reciprocals
* of the diagonal elements.
*
DO 30 I = 1, N
S( I ) = ONE / SQRT( S( I ) )
30 CONTINUE
*
* Compute SCOND = min(S(I)) / max(S(I))
*
SCOND = SQRT( SMIN ) / SQRT( AMAX )
END IF
RETURN
*
* End of DPBEQU
*
END
*> \brief \b DPBRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
* LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite
*> and banded, and provides error bounds and backward error estimates
*> for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangle of the symmetric band matrix A,
*> stored in the first KD+1 rows of the array. The j-th column
*> of A is stored in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] AFB
*> \verbatim
*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T of the band matrix A as computed by
*> DPBTRF, in the same storage format as A (see AB).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*> LDAFB is INTEGER
*> The leading dimension of the array AFB. LDAFB >= KD+1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DPBTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbrfs
*
* =====================================================================
SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
$ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER COUNT, I, J, K, KASE, L, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DPBTRS, DSBMV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDAFB.LT.KD+1 ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = MIN( N+1, 2*KD+2 )
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
$ WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
L = KD + 1 - K
DO 40 I = MAX( 1, K-KD ), K - 1
WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
40 CONTINUE
WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
L = 1 - K
DO 60 I = K + 1, MIN( N, K+KD )
WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
60 CONTINUE
WORK( K ) = WORK( K ) + S
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
$ INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(A))*
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(A) is the inverse of A
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(A)*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(A) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(A**T).
*
CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ),
$ N,
$ INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( N+I )*WORK( I )
110 CONTINUE
ELSE IF( KASE.EQ.2 ) THEN
*
* Multiply by inv(A)*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( N+I )*WORK( I )
120 CONTINUE
CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ),
$ N,
$ INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DPBRFS
*
END
*> \brief \b DPBSTF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBSTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbstf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbstf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbstf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBSTF computes a split Cholesky factorization of a real
*> symmetric positive definite band matrix A.
*>
*> This routine is designed to be used in conjunction with DSBGST.
*>
*> The factorization has the form A = S**T*S where S is a band matrix
*> of the same bandwidth as A and the following structure:
*>
*> S = ( U )
*> ( M L )
*>
*> where U is upper triangular of order m = (n+kd)/2, and L is lower
*> triangular of order n-m.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first kd+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, if INFO = 0, the factor S from the split Cholesky
*> factorization A = S**T*S. See Further Details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the factorization could not be completed,
*> because the updated element a(i,i) was negative; the
*> matrix A is not positive definite.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbstf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 7, KD = 2:
*>
*> S = ( s11 s12 s13 )
*> ( s22 s23 s24 )
*> ( s33 s34 )
*> ( s44 )
*> ( s53 s54 s55 )
*> ( s64 s65 s66 )
*> ( s75 s76 s77 )
*>
*> If UPLO = 'U', the array AB holds:
*>
*> on entry: on exit:
*>
*> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
*> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
*>
*> If UPLO = 'L', the array AB holds:
*>
*> on entry: on exit:
*>
*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
*> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
*> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, KLD, KM, M
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSYR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBSTF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
KLD = MAX( 1, LDAB-1 )
*
* Set the splitting point m.
*
M = ( N+KD ) / 2
*
IF( UPPER ) THEN
*
* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
*
DO 10 J = N, M + 1, -1
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( KD+1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( KD+1, J ) = AJJ
KM = MIN( J-1, KD )
*
* Compute elements j-km:j-1 of the j-th column and update the
* the leading submatrix within the band.
*
CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
$ AB( KD+1, J-KM ), KLD )
10 CONTINUE
*
* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
*
DO 20 J = 1, M
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( KD+1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( KD+1, J ) = AJJ
KM = MIN( KD, M-J )
*
* Compute elements j+1:j+km of the j-th row and update the
* trailing submatrix within the band.
*
IF( KM.GT.0 ) THEN
CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
$ AB( KD+1, J+1 ), KLD )
END IF
20 CONTINUE
ELSE
*
* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
*
DO 30 J = N, M + 1, -1
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( 1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( 1, J ) = AJJ
KM = MIN( J-1, KD )
*
* Compute elements j-km:j-1 of the j-th row and update the
* trailing submatrix within the band.
*
CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
$ AB( 1, J-KM ), KLD )
30 CONTINUE
*
* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
*
DO 40 J = 1, M
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( 1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( 1, J ) = AJJ
KM = MIN( KD, M-J )
*
* Compute elements j+1:j+km of the j-th column and update the
* trailing submatrix within the band.
*
IF( KM.GT.0 ) THEN
CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1,
$ AB( 1, J+1 ), KLD )
END IF
40 CONTINUE
END IF
RETURN
*
50 CONTINUE
INFO = J
RETURN
*
* End of DPBSTF
*
END
*> \brief <b> DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBSV computes the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric positive definite band matrix and X
*> and B are N-by-NRHS matrices.
*>
*> The Cholesky decomposition is used to factor A as
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular band matrix, and L is a lower
*> triangular band matrix, with the same number of superdiagonals or
*> subdiagonals as A. The factored form of A is then used to solve the
*> system of equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
*> See below for further details.
*>
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**T*U or A = L*L**T of the band
*> matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> of A is not positive, so the factorization could not
*> be completed, and the solution has not been computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbsv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 6, KD = 2, and UPLO = 'U':
*>
*> On entry: On exit:
*>
*> * * a13 a24 a35 a46 * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*>
*> Similarly, if UPLO = 'L' the format of A is as follows:
*>
*> On entry: On exit:
*>
*> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
*> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
*> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPBTRF, DPBTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBSV ', -INFO )
RETURN
END IF
*
* Compute the Cholesky factorization A = U**T*U or A = L*L**T.
*
CALL DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
END IF
RETURN
*
* End of DPBSV
*
END
*> \brief <b> DPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
* EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, UPLO
* INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), S( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
*> compute the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric positive definite band matrix and X
*> and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
*> Whether or not the system will be equilibrated depends on the
*> scaling of the matrix A, but if equilibration is used, A is
*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*> factor the matrix A (after equilibration if FACT = 'E') as
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular band matrix, and L is a lower
*> triangular band matrix.
*>
*> 3. If the leading principal minor of order i is not positive,
*> then the routine returns with INFO = i. Otherwise, the factored
*> form of A is used to estimate the condition number of the matrix
*> A. If the reciprocal of the condition number is less than machine
*> precision, INFO = N+1 is returned as a warning, but the routine
*> still goes on to solve for X and compute error bounds as
*> described below.
*>
*> 4. The system of equations is solved for X using the factored form
*> of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*> diag(S) so that it solves the original system before
*> equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of the matrix A is
*> supplied on entry, and if not, whether the matrix A should be
*> equilibrated before it is factored.
*> = 'F': On entry, AFB contains the factored form of A.
*> If EQUED = 'Y', the matrix A has been equilibrated
*> with scaling factors given by S. AB and AFB will not
*> be modified.
*> = 'N': The matrix A will be copied to AFB and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> copied to AFB and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right-hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array, except
*> if FACT = 'F' and EQUED = 'Y', then A must contain the
*> equilibrated matrix diag(S)*A*diag(S). The j-th column of A
*> is stored in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
*> See below for further details.
*>
*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*> diag(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array A. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] AFB
*> \verbatim
*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*> If FACT = 'F', then AFB is an input argument and on entry
*> contains the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T of the band matrix
*> A, in the same storage format as A (see AB). If EQUED = 'Y',
*> then AFB is the factored form of the equilibrated matrix A.
*>
*> If FACT = 'N', then AFB is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T.
*>
*> If FACT = 'E', then AFB is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T of the equilibrated
*> matrix A (see the description of A for the form of the
*> equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*> LDAFB is INTEGER
*> The leading dimension of the array AFB. LDAFB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration (always true if FACT = 'N').
*> = 'Y': Equilibration was done, i.e., A has been replaced by
*> diag(S) * A * diag(S).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A; not accessed if EQUED = 'N'. S is
*> an input argument if FACT = 'F'; otherwise, S is an output
*> argument. If FACT = 'F' and EQUED = 'Y', each element of S
*> must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*> B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*> the original system of equations. Note that if EQUED = 'Y',
*> A and B are modified on exit, and the solution to the
*> equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A after equilibration (if done). If RCOND is less than the
*> machine precision (in particular, if RCOND = 0), the matrix
*> is singular to working precision. This condition is
*> indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: the leading principal minor of order i of A
*> is not positive, so the factorization could not
*> be completed, and the solution has not been
*> computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbsvx
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 6, KD = 2, and UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13
*> a22 a23 a24
*> a33 a34 a35
*> a44 a45 a46
*> a55 a56
*> (aij=conjg(aji)) a66
*>
*> Band storage of the upper triangle of A:
*>
*> * * a13 a24 a35 a46
*> * a12 a23 a34 a45 a56
*> a11 a22 a33 a44 a55 a66
*>
*> Similarly, if UPLO = 'L' the format of A is as follows:
*>
*> a11 a22 a33 a44 a55 a66
*> a21 a32 a43 a54 a65 *
*> a31 a42 a53 a64 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB,
$ LDAFB,
$ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
$ WORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), S( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL EQUIL, NOFACT, RCEQU, UPPER
INTEGER I, INFEQU, J, J1, J2
DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSB
EXTERNAL LSAME, DLAMCH, DLANSB
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU,
$ DPBRFS,
$ DPBTRF, DPBTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
UPPER = LSAME( UPLO, 'U' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
RCEQU = .FALSE.
ELSE
RCEQU = LSAME( EQUED, 'Y' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND.
$ .NOT.EQUIL .AND.
$ .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KD.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -7
ELSE IF( LDAFB.LT.KD+1 ) THEN
INFO = -9
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -10
ELSE
IF( RCEQU ) THEN
SMIN = BIGNUM
SMAX = ZERO
DO 10 J = 1, N
SMIN = MIN( SMIN, S( J ) )
SMAX = MAX( SMAX, S( J ) )
10 CONTINUE
IF( SMIN.LE.ZERO ) THEN
INFO = -11
ELSE IF( N.GT.0 ) THEN
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
ELSE
SCOND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -15
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
$ EQUED )
RCEQU = LSAME( EQUED, 'Y' )
END IF
END IF
*
* Scale the right-hand side.
*
IF( RCEQU ) THEN
DO 30 J = 1, NRHS
DO 20 I = 1, N
B( I, J ) = S( I )*B( I, J )
20 CONTINUE
30 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the Cholesky factorization A = U**T *U or A = L*L**T.
*
IF( UPPER ) THEN
DO 40 J = 1, N
J1 = MAX( J-KD, 1 )
CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
$ AFB( KD+1-J+J1, J ), 1 )
40 CONTINUE
ELSE
DO 50 J = 1, N
J2 = MIN( J+KD, N )
CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
50 CONTINUE
END IF
*
CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK,
$ IWORK,
$ INFO )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB,
$ X,
$ LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( RCEQU ) THEN
DO 70 J = 1, NRHS
DO 60 I = 1, N
X( I, J ) = S( I )*X( I, J )
60 CONTINUE
70 CONTINUE
DO 80 J = 1, NRHS
FERR( J ) = FERR( J ) / SCOND
80 CONTINUE
END IF
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DPBSVX
*
END
*> \brief \b DPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbtf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbtf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbtf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBTF2 computes the Cholesky factorization of a real symmetric
*> positive definite band matrix A.
*>
*> The factorization has the form
*> A = U**T * U , if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix, U**T is the transpose of U, and
*> L is lower triangular.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of super-diagonals of the matrix A if UPLO = 'U',
*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**T*U or A = L*L**T of the band
*> matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, the leading principal minor of order k
*> is not positive, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbtf2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 6, KD = 2, and UPLO = 'U':
*>
*> On entry: On exit:
*>
*> * * a13 a24 a35 a46 * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*>
*> Similarly, if UPLO = 'L' the format of A is as follows:
*>
*> On entry: On exit:
*>
*> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
*> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
*> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, KLD, KN
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSYR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
KLD = MAX( 1, LDAB-1 )
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T*U.
*
DO 10 J = 1, N
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = AB( KD+1, J )
IF( AJJ.LE.ZERO )
$ GO TO 30
AJJ = SQRT( AJJ )
AB( KD+1, J ) = AJJ
*
* Compute elements J+1:J+KN of row J and update the
* trailing submatrix within the band.
*
KN = MIN( KD, N-J )
IF( KN.GT.0 ) THEN
CALL DSCAL( KN, ONE / AJJ, AB( KD, J+1 ), KLD )
CALL DSYR( 'Upper', KN, -ONE, AB( KD, J+1 ), KLD,
$ AB( KD+1, J+1 ), KLD )
END IF
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = AB( 1, J )
IF( AJJ.LE.ZERO )
$ GO TO 30
AJJ = SQRT( AJJ )
AB( 1, J ) = AJJ
*
* Compute elements J+1:J+KN of column J and update the
* trailing submatrix within the band.
*
KN = MIN( KD, N-J )
IF( KN.GT.0 ) THEN
CALL DSCAL( KN, ONE / AJJ, AB( 2, J ), 1 )
CALL DSYR( 'Lower', KN, -ONE, AB( 2, J ), 1,
$ AB( 1, J+1 ), KLD )
END IF
20 CONTINUE
END IF
RETURN
*
30 CONTINUE
INFO = J
RETURN
*
* End of DPBTF2
*
END
*> \brief \b DPBTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbtrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbtrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbtrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBTRF computes the Cholesky factorization of a real symmetric
*> positive definite band matrix A.
*>
*> The factorization has the form
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**T*U or A = L*L**T of the band
*> matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> is not positive, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbtrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 6, KD = 2, and UPLO = 'U':
*>
*> On entry: On exit:
*>
*> * * a13 a24 a35 a46 * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*>
*> Similarly, if UPLO = 'L' the format of A is as follows:
*>
*> On entry: On exit:
*>
*> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
*> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
*> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
*
* =====================================================================
SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
INTEGER NBMAX, LDWORK
PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 )
* ..
* .. Local Scalars ..
INTEGER I, I2, I3, IB, II, J, JJ, NB
* ..
* .. Local Arrays ..
DOUBLE PRECISION WORK( LDWORK, NBMAX )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DPBTF2, DPOTF2, DSYRK, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
$ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment
*
NB = ILAENV( 1, 'DPBTRF', UPLO, N, KD, -1, -1 )
*
* The block size must not exceed the semi-bandwidth KD, and must not
* exceed the limit set by the size of the local array WORK.
*
NB = MIN( NB, NBMAX )
*
IF( NB.LE.1 .OR. NB.GT.KD ) THEN
*
* Use unblocked code
*
CALL DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
ELSE
*
* Use blocked code
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Compute the Cholesky factorization of a symmetric band
* matrix, given the upper triangle of the matrix in band
* storage.
*
* Zero the upper triangle of the work array.
*
DO 20 J = 1, NB
DO 10 I = 1, J - 1
WORK( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* Process the band matrix one diagonal block at a time.
*
DO 70 I = 1, N, NB
IB = MIN( NB, N-I+1 )
*
* Factorize the diagonal block
*
CALL DPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
IF( II.NE.0 ) THEN
INFO = I + II - 1
GO TO 150
END IF
IF( I+IB.LE.N ) THEN
*
* Update the relevant part of the trailing submatrix.
* If A11 denotes the diagonal block which has just been
* factorized, then we need to update the remaining
* blocks in the diagram:
*
* A11 A12 A13
* A22 A23
* A33
*
* The numbers of rows and columns in the partitioning
* are IB, I2, I3 respectively. The blocks A12, A22 and
* A23 are empty if IB = KD. The upper triangle of A13
* lies outside the band.
*
I2 = MIN( KD-IB, N-I-IB+1 )
I3 = MIN( IB, N-I-KD+1 )
*
IF( I2.GT.0 ) THEN
*
* Update A12
*
CALL DTRSM( 'Left', 'Upper', 'Transpose',
$ 'Non-unit', IB, I2, ONE, AB( KD+1, I ),
$ LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 )
*
* Update A22
*
CALL DSYRK( 'Upper', 'Transpose', I2, IB, -ONE,
$ AB( KD+1-IB, I+IB ), LDAB-1, ONE,
$ AB( KD+1, I+IB ), LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Copy the lower triangle of A13 into the work array.
*
DO 40 JJ = 1, I3
DO 30 II = JJ, IB
WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
30 CONTINUE
40 CONTINUE
*
* Update A13 (in the work array).
*
CALL DTRSM( 'Left', 'Upper', 'Transpose',
$ 'Non-unit', IB, I3, ONE, AB( KD+1, I ),
$ LDAB-1, WORK, LDWORK )
*
* Update A23
*
IF( I2.GT.0 )
$ CALL DGEMM( 'Transpose', 'No Transpose', I2,
$ I3,
$ IB, -ONE, AB( KD+1-IB, I+IB ),
$ LDAB-1, WORK, LDWORK, ONE,
$ AB( 1+IB, I+KD ), LDAB-1 )
*
* Update A33
*
CALL DSYRK( 'Upper', 'Transpose', I3, IB, -ONE,
$ WORK, LDWORK, ONE, AB( KD+1, I+KD ),
$ LDAB-1 )
*
* Copy the lower triangle of A13 back into place.
*
DO 60 JJ = 1, I3
DO 50 II = JJ, IB
AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
50 CONTINUE
60 CONTINUE
END IF
END IF
70 CONTINUE
ELSE
*
* Compute the Cholesky factorization of a symmetric band
* matrix, given the lower triangle of the matrix in band
* storage.
*
* Zero the lower triangle of the work array.
*
DO 90 J = 1, NB
DO 80 I = J + 1, NB
WORK( I, J ) = ZERO
80 CONTINUE
90 CONTINUE
*
* Process the band matrix one diagonal block at a time.
*
DO 140 I = 1, N, NB
IB = MIN( NB, N-I+1 )
*
* Factorize the diagonal block
*
CALL DPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
IF( II.NE.0 ) THEN
INFO = I + II - 1
GO TO 150
END IF
IF( I+IB.LE.N ) THEN
*
* Update the relevant part of the trailing submatrix.
* If A11 denotes the diagonal block which has just been
* factorized, then we need to update the remaining
* blocks in the diagram:
*
* A11
* A21 A22
* A31 A32 A33
*
* The numbers of rows and columns in the partitioning
* are IB, I2, I3 respectively. The blocks A21, A22 and
* A32 are empty if IB = KD. The lower triangle of A31
* lies outside the band.
*
I2 = MIN( KD-IB, N-I-IB+1 )
I3 = MIN( IB, N-I-KD+1 )
*
IF( I2.GT.0 ) THEN
*
* Update A21
*
CALL DTRSM( 'Right', 'Lower', 'Transpose',
$ 'Non-unit', I2, IB, ONE, AB( 1, I ),
$ LDAB-1, AB( 1+IB, I ), LDAB-1 )
*
* Update A22
*
CALL DSYRK( 'Lower', 'No Transpose', I2, IB,
$ -ONE,
$ AB( 1+IB, I ), LDAB-1, ONE,
$ AB( 1, I+IB ), LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Copy the upper triangle of A31 into the work array.
*
DO 110 JJ = 1, IB
DO 100 II = 1, MIN( JJ, I3 )
WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
100 CONTINUE
110 CONTINUE
*
* Update A31 (in the work array).
*
CALL DTRSM( 'Right', 'Lower', 'Transpose',
$ 'Non-unit', I3, IB, ONE, AB( 1, I ),
$ LDAB-1, WORK, LDWORK )
*
* Update A32
*
IF( I2.GT.0 )
$ CALL DGEMM( 'No transpose', 'Transpose', I3,
$ I2,
$ IB, -ONE, WORK, LDWORK,
$ AB( 1+IB, I ), LDAB-1, ONE,
$ AB( 1+KD-IB, I+IB ), LDAB-1 )
*
* Update A33
*
CALL DSYRK( 'Lower', 'No Transpose', I3, IB,
$ -ONE,
$ WORK, LDWORK, ONE, AB( 1, I+KD ),
$ LDAB-1 )
*
* Copy the upper triangle of A31 back into place.
*
DO 130 JJ = 1, IB
DO 120 II = 1, MIN( JJ, I3 )
AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
120 CONTINUE
130 CONTINUE
END IF
END IF
140 CONTINUE
END IF
END IF
RETURN
*
150 CONTINUE
RETURN
*
* End of DPBTRF
*
END
*> \brief \b DPBTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite band matrix A using the Cholesky factorization
*> A = U**T*U or A = L*L**T computed by DPBTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangular factor stored in AB;
*> = 'L': Lower triangular factor stored in AB.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T of the band matrix A, stored in the
*> first KD+1 rows of the array. The j-th column of U or L is
*> stored in the j-th column of the array AB as follows:
*> if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbtrs
*
* =====================================================================
SUBROUTINE DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DTBSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Solve A*X = B where A = U**T *U.
*
DO 10 J = 1, NRHS
*
* Solve U**T *X = B, overwriting B with X.
*
CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KD, AB,
$ LDAB, B( 1, J ), 1 )
*
* Solve U*X = B, overwriting B with X.
*
CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N, KD,
$ AB,
$ LDAB, B( 1, J ), 1 )
10 CONTINUE
ELSE
*
* Solve A*X = B where A = L*L**T.
*
DO 20 J = 1, NRHS
*
* Solve L*X = B, overwriting B with X.
*
CALL DTBSV( 'Lower', 'No transpose', 'Non-unit', N, KD,
$ AB,
$ LDAB, B( 1, J ), 1 )
*
* Solve L**T *X = B, overwriting B with X.
*
CALL DTBSV( 'Lower', 'Transpose', 'Non-unit', N, KD, AB,
$ LDAB, B( 1, J ), 1 )
20 CONTINUE
END IF
*
RETURN
*
* End of DPBTRS
*
END
*> \brief \b DPFTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPFTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpftrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpftrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPFTRF( TRANSR, UPLO, N, A, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER N, INFO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: * )
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPFTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the block version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': The Normal TRANSR of RFP A is stored;
*> = 'T': The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of RFP A is stored;
*> = 'L': Lower triangle of RFP A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
*> On entry, the symmetric matrix A in RFP format. RFP format is
*> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
*> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
*> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
*> the transpose of RFP A as defined when
*> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
*> follows: If UPLO = 'U' the RFP A contains the NT elements of
*> upper packed A. If UPLO = 'L' the RFP A contains the elements
*> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
*> 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
*> is odd. See the Note below for more details.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization RFP A = U**T*U or RFP A = L*L**T.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> is not positive, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pftrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPFTRF( TRANSR, UPLO, N, A, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO
INTEGER N, INFO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( 0: * )
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, NISODD, NORMALTRANSR
INTEGER N1, N2, K
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DSYRK, DPOTRF, DTRSM
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPFTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* If N is odd, set NISODD = .TRUE.
* If N is even, set K = N/2 and NISODD = .FALSE.
*
IF( MOD( N, 2 ).EQ.0 ) THEN
K = N / 2
NISODD = .FALSE.
ELSE
NISODD = .TRUE.
END IF
*
* Set N1 and N2 depending on LOWER
*
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
*
* start execution: there are eight cases
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
* T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
* T1 -> a(0), T2 -> a(n), S -> a(n1)
*
CALL DPOTRF( 'L', N1, A( 0 ), N, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'R', 'L', 'T', 'N', N2, N1, ONE, A( 0 ),
$ N,
$ A( N1 ), N )
CALL DSYRK( 'U', 'N', N2, N1, -ONE, A( N1 ), N, ONE,
$ A( N ), N )
CALL DPOTRF( 'U', N2, A( N ), N, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
*
ELSE
*
* SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
* T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
* T1 -> a(n2), T2 -> a(n1), S -> a(0)
*
CALL DPOTRF( 'L', N1, A( N2 ), N, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'L', 'L', 'N', 'N', N1, N2, ONE, A( N2 ),
$ N,
$ A( 0 ), N )
CALL DSYRK( 'U', 'T', N2, N1, -ONE, A( 0 ), N, ONE,
$ A( N1 ), N )
CALL DPOTRF( 'U', N2, A( N1 ), N, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
*
END IF
*
ELSE
*
* N is odd and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE and N is odd
* T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
* T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
*
CALL DPOTRF( 'U', N1, A( 0 ), N1, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'L', 'U', 'T', 'N', N1, N2, ONE, A( 0 ),
$ N1,
$ A( N1*N1 ), N1 )
CALL DSYRK( 'L', 'T', N2, N1, -ONE, A( N1*N1 ), N1,
$ ONE,
$ A( 1 ), N1 )
CALL DPOTRF( 'L', N2, A( 1 ), N1, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
*
ELSE
*
* SRPA for UPPER, TRANSPOSE and N is odd
* T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
* T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
*
CALL DPOTRF( 'U', N1, A( N2*N2 ), N2, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'R', 'U', 'N', 'N', N2, N1, ONE,
$ A( N2*N2 ),
$ N2, A( 0 ), N2 )
CALL DSYRK( 'L', 'N', N2, N1, -ONE, A( 0 ), N2, ONE,
$ A( N1*N2 ), N2 )
CALL DPOTRF( 'L', N2, A( N1*N2 ), N2, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
*
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
* T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
CALL DPOTRF( 'L', K, A( 1 ), N+1, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'R', 'L', 'T', 'N', K, K, ONE, A( 1 ),
$ N+1,
$ A( K+1 ), N+1 )
CALL DSYRK( 'U', 'N', K, K, -ONE, A( K+1 ), N+1, ONE,
$ A( 0 ), N+1 )
CALL DPOTRF( 'U', K, A( 0 ), N+1, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
*
ELSE
*
* SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
* T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
CALL DPOTRF( 'L', K, A( K+1 ), N+1, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'L', 'L', 'N', 'N', K, K, ONE, A( K+1 ),
$ N+1, A( 0 ), N+1 )
CALL DSYRK( 'U', 'T', K, K, -ONE, A( 0 ), N+1, ONE,
$ A( K ), N+1 )
CALL DPOTRF( 'U', K, A( K ), N+1, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
*
END IF
*
ELSE
*
* N is even and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE and N is even (see paper)
* T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
* T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
CALL DPOTRF( 'U', K, A( 0+K ), K, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'L', 'U', 'T', 'N', K, K, ONE, A( K ), N1,
$ A( K*( K+1 ) ), K )
CALL DSYRK( 'L', 'T', K, K, -ONE, A( K*( K+1 ) ), K,
$ ONE,
$ A( 0 ), K )
CALL DPOTRF( 'L', K, A( 0 ), K, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
*
ELSE
*
* SRPA for UPPER, TRANSPOSE and N is even (see paper)
* T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
* T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
CALL DPOTRF( 'U', K, A( K*( K+1 ) ), K, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRSM( 'R', 'U', 'N', 'N', K, K, ONE,
$ A( K*( K+1 ) ), K, A( 0 ), K )
CALL DSYRK( 'L', 'N', K, K, -ONE, A( 0 ), K, ONE,
$ A( K*K ), K )
CALL DPOTRF( 'L', K, A( K*K ), K, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of DPFTRF
*
END
*> \brief \b DPFTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPFTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpftri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpftri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, N
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPFTRI computes the inverse of a (real) symmetric positive definite
*> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
*> computed by DPFTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': The Normal TRANSR of RFP A is stored;
*> = 'T': The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
*> On entry, the symmetric matrix A in RFP format. RFP format is
*> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
*> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
*> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
*> the transpose of RFP A as defined when
*> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
*> follows: If UPLO = 'U' the RFP A contains the nt elements of
*> upper packed A. If UPLO = 'L' the RFP A contains the elements
*> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
*> 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
*> is odd. See the Note below for more details.
*>
*> On exit, the symmetric inverse of the original matrix, in the
*> same storage format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the (i,i) element of the factor U or L is
*> zero, and the inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pftri
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO
INTEGER INFO, N
* .. Array Arguments ..
DOUBLE PRECISION A( 0: * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, NISODD, NORMALTRANSR
INTEGER N1, N2, K
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DTFTRI, DLAUUM, DTRMM,
$ DSYRK
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPFTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Invert the triangular Cholesky factor U or L.
*
CALL DTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
IF( INFO.GT.0 )
$ RETURN
*
* If N is odd, set NISODD = .TRUE.
* If N is even, set K = N/2 and NISODD = .FALSE.
*
IF( MOD( N, 2 ).EQ.0 ) THEN
K = N / 2
NISODD = .FALSE.
ELSE
NISODD = .TRUE.
END IF
*
* Set N1 and N2 depending on LOWER
*
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
*
* Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
* inv(L)^C*inv(L). There are eight cases.
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
* T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
* T1 -> a(0), T2 -> a(n), S -> a(N1)
*
CALL DLAUUM( 'L', N1, A( 0 ), N, INFO )
CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE,
$ A( 0 ), N )
CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ),
$ N,
$ A( N1 ), N )
CALL DLAUUM( 'U', N2, A( N ), N, INFO )
*
ELSE
*
* SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
* T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
* T1 -> a(N2), T2 -> a(N1), S -> a(0)
*
CALL DLAUUM( 'L', N1, A( N2 ), N, INFO )
CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
$ A( N2 ), N )
CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ),
$ N,
$ A( 0 ), N )
CALL DLAUUM( 'U', N2, A( N1 ), N, INFO )
*
END IF
*
ELSE
*
* N is odd and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE, and N is odd
* T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
*
CALL DLAUUM( 'U', N1, A( 0 ), N1, INFO )
CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1,
$ ONE,
$ A( 0 ), N1 )
CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ),
$ N1,
$ A( N1*N1 ), N1 )
CALL DLAUUM( 'L', N2, A( 1 ), N1, INFO )
*
ELSE
*
* SRPA for UPPER, TRANSPOSE, and N is odd
* T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
*
CALL DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE,
$ A( N2*N2 ), N2 )
CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE,
$ A( N1*N2 ),
$ N2, A( 0 ), N2 )
CALL DLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
*
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
* T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
CALL DLAUUM( 'L', K, A( 1 ), N+1, INFO )
CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE,
$ A( 1 ), N+1 )
CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ),
$ N+1,
$ A( K+1 ), N+1 )
CALL DLAUUM( 'U', K, A( 0 ), N+1, INFO )
*
ELSE
*
* SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
* T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
CALL DLAUUM( 'L', K, A( K+1 ), N+1, INFO )
CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
$ A( K+1 ), N+1 )
CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ),
$ N+1,
$ A( 0 ), N+1 )
CALL DLAUUM( 'U', K, A( K ), N+1, INFO )
*
END IF
*
ELSE
*
* N is even and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE, and N is even (see paper)
* T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
* T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
CALL DLAUUM( 'U', K, A( K ), K, INFO )
CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K,
$ ONE,
$ A( K ), K )
CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K,
$ A( K*( K+1 ) ), K )
CALL DLAUUM( 'L', K, A( 0 ), K, INFO )
*
ELSE
*
* SRPA for UPPER, TRANSPOSE, and N is even (see paper)
* T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
* T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
CALL DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE,
$ A( K*( K+1 ) ), K )
CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ),
$ K,
$ A( 0 ), K )
CALL DLAUUM( 'L', K, A( K*K ), K, INFO )
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of DPFTRI
*
END
*> \brief \b DPFTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPFTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpftrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpftrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPFTRS( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPFTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite matrix A using the Cholesky factorization
*> A = U**T*U or A = L*L**T computed by DPFTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': The Normal TRANSR of RFP A is stored;
*> = 'T': The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of RFP A is stored;
*> = 'L': Lower triangle of RFP A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ).
*> The triangular factor U or L from the Cholesky factorization
*> of RFP A = U**T*U or RFP A = L*L**T, as computed by DPFTRF.
*> See note below for more details about RFP A.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pftrs
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPFTRS( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( 0: * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, NORMALTRANSR
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DTFSM
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPFTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
* start execution: there are two triangular solves
*
IF( LOWER ) THEN
CALL DTFSM( TRANSR, 'L', UPLO, 'N', 'N', N, NRHS, ONE, A, B,
$ LDB )
CALL DTFSM( TRANSR, 'L', UPLO, 'T', 'N', N, NRHS, ONE, A, B,
$ LDB )
ELSE
CALL DTFSM( TRANSR, 'L', UPLO, 'T', 'N', N, NRHS, ONE, A, B,
$ LDB )
CALL DTFSM( TRANSR, 'L', UPLO, 'N', 'N', N, NRHS, ONE, A, B,
$ LDB )
END IF
*
RETURN
*
* End of DPFTRS
*
END
*> \brief \b DPOCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpocon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpocon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpocon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite matrix using the
*> Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, as computed by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm (or infinity-norm) of the symmetric matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*> estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pocon
*
* =====================================================================
SUBROUTINE DPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
CHARACTER NORMIN
INTEGER IX, KASE
DOUBLE PRECISION AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLATRS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
SMLNUM = DLAMCH( 'Safe minimum' )
*
* Estimate the 1-norm of inv(A).
*
KASE = 0
NORMIN = 'N'
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( UPPER ) THEN
*
* Multiply by inv(U**T).
*
CALL DLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
$ A,
$ LDA, WORK, SCALEL, WORK( 2*N+1 ), INFO )
NORMIN = 'Y'
*
* Multiply by inv(U).
*
CALL DLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ A, LDA, WORK, SCALEU, WORK( 2*N+1 ), INFO )
ELSE
*
* Multiply by inv(L).
*
CALL DLATRS( 'Lower', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ A, LDA, WORK, SCALEL, WORK( 2*N+1 ), INFO )
NORMIN = 'Y'
*
* Multiply by inv(L**T).
*
CALL DLATRS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
$ A,
$ LDA, WORK, SCALEU, WORK( 2*N+1 ), INFO )
END IF
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
SCALE = SCALEL*SCALEU
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
20 CONTINUE
RETURN
*
* End of DPOCON
*
END
*> \brief \b DPOEQU
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOEQU computes row and column scalings intended to equilibrate a
*> symmetric positive definite matrix A and reduce its condition number
*> (with respect to the two-norm). S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The N-by-N symmetric positive definite matrix whose scaling
*> factors are to be computed. Only the diagonal elements of A
*> are referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> If INFO = 0, S contains the ratio of the smallest S(i) to
*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
*> large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup poequ
*
* =====================================================================
SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION SMIN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOEQU', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SCOND = ONE
AMAX = ZERO
RETURN
END IF
*
* Find the minimum and maximum diagonal elements.
*
S( 1 ) = A( 1, 1 )
SMIN = S( 1 )
AMAX = S( 1 )
DO 10 I = 2, N
S( I ) = A( I, I )
SMIN = MIN( SMIN, S( I ) )
AMAX = MAX( AMAX, S( I ) )
10 CONTINUE
*
IF( SMIN.LE.ZERO ) THEN
*
* Find the first non-positive diagonal element and return.
*
DO 20 I = 1, N
IF( S( I ).LE.ZERO ) THEN
INFO = I
RETURN
END IF
20 CONTINUE
ELSE
*
* Set the scale factors to the reciprocals
* of the diagonal elements.
*
DO 30 I = 1, N
S( I ) = ONE / SQRT( S( I ) )
30 CONTINUE
*
* Compute SCOND = min(S(I)) / max(S(I))
*
SCOND = SQRT( SMIN ) / SQRT( AMAX )
END IF
RETURN
*
* End of DPOEQU
*
END
*> \brief \b DPOEQUB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOEQUB computes row and column scalings intended to equilibrate a
*> symmetric positive definite matrix A and reduce its condition number
*> (with respect to the two-norm). S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*>
*> This routine differs from DPOEQU by restricting the scaling factors
*> to a power of the radix. Barring over- and underflow, scaling by
*> these factors introduces no additional rounding errors. However, the
*> scaled diagonal entries are no longer approximately 1 but lie
*> between sqrt(radix) and 1/sqrt(radix).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The N-by-N symmetric positive definite matrix whose scaling
*> factors are to be computed. Only the diagonal elements of A
*> are referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> If INFO = 0, S contains the ratio of the smallest S(i) to
*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
*> large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup poequb
*
* =====================================================================
SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION SMIN, BASE, TMP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT, LOG, INT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
* Positive definite only performs 1 pass of equilibration.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOEQUB', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 ) THEN
SCOND = ONE
AMAX = ZERO
RETURN
END IF
BASE = DLAMCH( 'B' )
TMP = -0.5D+0 / LOG ( BASE )
*
* Find the minimum and maximum diagonal elements.
*
S( 1 ) = A( 1, 1 )
SMIN = S( 1 )
AMAX = S( 1 )
DO 10 I = 2, N
S( I ) = A( I, I )
SMIN = MIN( SMIN, S( I ) )
AMAX = MAX( AMAX, S( I ) )
10 CONTINUE
*
IF( SMIN.LE.ZERO ) THEN
*
* Find the first non-positive diagonal element and return.
*
DO 20 I = 1, N
IF( S( I ).LE.ZERO ) THEN
INFO = I
RETURN
END IF
20 CONTINUE
ELSE
*
* Set the scale factors to the reciprocals
* of the diagonal elements.
*
DO 30 I = 1, N
S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
30 CONTINUE
*
* Compute SCOND = min(S(I)) / max(S(I)).
*
SCOND = SQRT( SMIN ) / SQRT( AMAX )
END IF
*
RETURN
*
* End of DPOEQUB
*
END
*> \brief \b DPORFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPORFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dporfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dporfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dporfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
* LDX, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPORFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite,
*> and provides error bounds and backward error estimates for the
*> solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
*> upper triangular part of A contains the upper triangular part
*> of the matrix A, and the strictly lower triangular part of A
*> is not referenced. If UPLO = 'L', the leading N-by-N lower
*> triangular part of A contains the lower triangular part of
*> the matrix A, and the strictly upper triangular part of A is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, as computed by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DPOTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup porfs
*
* =====================================================================
SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
$ LDX, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER COUNT, I, J, K, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPORFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
$ WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
DO 40 I = 1, K - 1
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
40 CONTINUE
WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
DO 60 I = K + 1, N
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
60 CONTINUE
WORK( K ) = WORK( K ) + S
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(A))*
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(A) is the inverse of A
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(A)*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(A) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(A**T).
*
CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N,
$ INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
110 CONTINUE
ELSE IF( KASE.EQ.2 ) THEN
*
* Multiply by inv(A)*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
120 CONTINUE
CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N,
$ INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DPORFS
*
END
*> \brief <b> DPOSV computes the solution to system of linear equations A * X = B for PO matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOSV computes the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric positive definite matrix and X and B
*> are N-by-NRHS matrices.
*>
*> The Cholesky decomposition is used to factor A as
*> A = U**T* U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is a lower triangular
*> matrix. The factored form of A is then used to solve the system of
*> equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> of A is not positive, so the factorization could not
*> be completed, and the solution has not been computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup posv
*
* =====================================================================
SUBROUTINE DPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPOTRF, DPOTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOSV ', -INFO )
RETURN
END IF
*
* Compute the Cholesky factorization A = U**T*U or A = L*L**T.
*
CALL DPOTRF( UPLO, N, A, LDA, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
END IF
RETURN
*
* End of DPOSV
*
END
*> \brief <b> DPOSVX computes the solution to system of linear equations A * X = B for PO matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
* S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, UPLO
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), S( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
*> compute the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric positive definite matrix and X and B
*> are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
*> Whether or not the system will be equilibrated depends on the
*> scaling of the matrix A, but if equilibration is used, A is
*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*> factor the matrix A (after equilibration if FACT = 'E') as
*> A = U**T* U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is a lower triangular
*> matrix.
*>
*> 3. If the leading principal minor of order i is not positive,
*> then the routine returns with INFO = i. Otherwise, the factored
*> form of A is used to estimate the condition number of the matrix
*> A. If the reciprocal of the condition number is less than machine
*> precision, INFO = N+1 is returned as a warning, but the routine
*> still goes on to solve for X and compute error bounds as
*> described below.
*>
*> 4. The system of equations is solved for X using the factored form
*> of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*> diag(S) so that it solves the original system before
*> equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of the matrix A is
*> supplied on entry, and if not, whether the matrix A should be
*> equilibrated before it is factored.
*> = 'F': On entry, AF contains the factored form of A.
*> If EQUED = 'Y', the matrix A has been equilibrated
*> with scaling factors given by S. A and AF will not
*> be modified.
*> = 'N': The matrix A will be copied to AF and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> copied to AF and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A, except if FACT = 'F' and
*> EQUED = 'Y', then A must contain the equilibrated matrix
*> diag(S)*A*diag(S). If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced. A is not modified if
*> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*>
*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*> diag(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> If FACT = 'F', then AF is an input argument and on entry
*> contains the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T, in the same storage
*> format as A. If EQUED .ne. 'N', then AF is the factored form
*> of the equilibrated matrix diag(S)*A*diag(S).
*>
*> If FACT = 'N', then AF is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T of the original
*> matrix A.
*>
*> If FACT = 'E', then AF is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T of the equilibrated
*> matrix A (see the description of A for the form of the
*> equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration (always true if FACT = 'N').
*> = 'Y': Equilibration was done, i.e., A has been replaced by
*> diag(S) * A * diag(S).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A; not accessed if EQUED = 'N'. S is
*> an input argument if FACT = 'F'; otherwise, S is an output
*> argument. If FACT = 'F' and EQUED = 'Y', each element of S
*> must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*> B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*> the original system of equations. Note that if EQUED = 'Y',
*> A and B are modified on exit, and the solution to the
*> equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A after equilibration (if done). If RCOND is less than the
*> machine precision (in particular, if RCOND = 0), the matrix
*> is singular to working precision. This condition is
*> indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: the leading principal minor of order i of A
*> is not positive, so the factorization could not
*> be completed, and the solution has not been
*> computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup posvx
*
* =====================================================================
SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF,
$ EQUED,
$ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
$ IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), S( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL EQUIL, NOFACT, RCEQU
INTEGER I, INFEQU, J
DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAQSY, DPOCON, DPOEQU, DPORFS,
$ DPOTRF,
$ DPOTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
RCEQU = .FALSE.
ELSE
RCEQU = LSAME( EQUED, 'Y' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND.
$ .NOT.EQUIL .AND.
$ .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -9
ELSE
IF( RCEQU ) THEN
SMIN = BIGNUM
SMAX = ZERO
DO 10 J = 1, N
SMIN = MIN( SMIN, S( J ) )
SMAX = MAX( SMAX, S( J ) )
10 CONTINUE
IF( SMIN.LE.ZERO ) THEN
INFO = -10
ELSE IF( N.GT.0 ) THEN
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
ELSE
SCOND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
RCEQU = LSAME( EQUED, 'Y' )
END IF
END IF
*
* Scale the right hand side.
*
IF( RCEQU ) THEN
DO 30 J = 1, NRHS
DO 20 I = 1, N
B( I, J ) = S( I )*B( I, J )
20 CONTINUE
30 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the Cholesky factorization A = U**T *U or A = L*L**T.
*
CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
CALL DPOTRF( UPLO, N, AF, LDAF, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANSY( '1', UPLO, N, A, LDA, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
$ FERR, BERR, WORK, IWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( RCEQU ) THEN
DO 50 J = 1, NRHS
DO 40 I = 1, N
X( I, J ) = S( I )*X( I, J )
40 CONTINUE
50 CONTINUE
DO 60 J = 1, NRHS
FERR( J ) = FERR( J ) / SCOND
60 CONTINUE
END IF
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DPOSVX
*
END
*> \brief \b DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOTF2 computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*> A = U**T * U , if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization A = U**T *U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, the leading principal minor of order k
*> is not positive, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potf2
*
* =====================================================================
SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T *U.
*
DO 10 J = 1, N
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = A( J, J ) - DDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 )
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of row J.
*
IF( J.LT.N ) THEN
CALL DGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ),
$ LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
END IF
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = A( J, J ) - DDOT( J-1, A( J, 1 ), LDA, A( J, 1 ),
$ LDA )
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of column J.
*
IF( J.LT.N ) THEN
CALL DGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1,
$ 1 ),
$ LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
END IF
20 CONTINUE
END IF
GO TO 40
*
30 CONTINUE
INFO = J
*
40 CONTINUE
RETURN
*
* End of DPOTF2
*
END
*> \brief \b DPOTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the block version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> is not positive, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potrf
*
* =====================================================================
SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, JB, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DPOTRF2, DSYRK, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code.
*
CALL DPOTRF2( UPLO, N, A, LDA, INFO )
ELSE
*
* Use blocked code.
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T*U.
*
DO 10 J = 1, N, NB
*
* Update and factorize the current diagonal block and test
* for non-positive-definiteness.
*
JB = MIN( NB, N-J+1 )
CALL DSYRK( 'Upper', 'Transpose', JB, J-1, -ONE,
$ A( 1, J ), LDA, ONE, A( J, J ), LDA )
CALL DPOTRF2( 'Upper', JB, A( J, J ), LDA, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( J+JB.LE.N ) THEN
*
* Compute the current block row.
*
CALL DGEMM( 'Transpose', 'No transpose', JB,
$ N-J-JB+1,
$ J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ),
$ LDA, ONE, A( J, J+JB ), LDA )
CALL DTRSM( 'Left', 'Upper', 'Transpose',
$ 'Non-unit',
$ JB, N-J-JB+1, ONE, A( J, J ), LDA,
$ A( J, J+JB ), LDA )
END IF
10 CONTINUE
*
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
DO 20 J = 1, N, NB
*
* Update and factorize the current diagonal block and test
* for non-positive-definiteness.
*
JB = MIN( NB, N-J+1 )
CALL DSYRK( 'Lower', 'No transpose', JB, J-1, -ONE,
$ A( J, 1 ), LDA, ONE, A( J, J ), LDA )
CALL DPOTRF2( 'Lower', JB, A( J, J ), LDA, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( J+JB.LE.N ) THEN
*
* Compute the current block column.
*
CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1,
$ JB,
$ J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ),
$ LDA, ONE, A( J+JB, J ), LDA )
CALL DTRSM( 'Right', 'Lower', 'Transpose',
$ 'Non-unit',
$ N-J-JB+1, JB, ONE, A( J, J ), LDA,
$ A( J+JB, J ), LDA )
END IF
20 CONTINUE
END IF
END IF
GO TO 40
*
30 CONTINUE
INFO = INFO + J - 1
*
40 CONTINUE
RETURN
*
* End of DPOTRF
*
END
*> \brief \b DPOTRF2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DPOTRF2( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOTRF2 computes the Cholesky factorization of a real symmetric
*> positive definite matrix A using the recursive algorithm.
*>
*> The factorization has the form
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the recursive version of the algorithm. It divides
*> the matrix into four submatrices:
*>
*> [ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2
*> A = [ -----|----- ] with n1 = n/2
*> [ A21 | A22 ] n2 = n-n1
*>
*> The subroutine calls itself to factor A11. Update and scale A21
*> or A12, update A22 then calls itself to factor A22.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> is not positive, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potrf2
*
* =====================================================================
RECURSIVE SUBROUTINE DPOTRF2( UPLO, N, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER N1, N2, IINFO
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DSYRK, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOTRF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* N=1 case
*
IF( N.EQ.1 ) THEN
*
* Test for non-positive-definiteness
*
IF( A( 1, 1 ).LE.ZERO.OR.DISNAN( A( 1, 1 ) ) ) THEN
INFO = 1
RETURN
END IF
*
* Factor
*
A( 1, 1 ) = SQRT( A( 1, 1 ) )
*
* Use recursive code
*
ELSE
N1 = N/2
N2 = N-N1
*
* Factor A11
*
CALL DPOTRF2( UPLO, N1, A( 1, 1 ), LDA, IINFO )
IF ( IINFO.NE.0 ) THEN
INFO = IINFO
RETURN
END IF
*
* Compute the Cholesky factorization A = U**T*U
*
IF( UPPER ) THEN
*
* Update and scale A12
*
CALL DTRSM( 'L', 'U', 'T', 'N', N1, N2, ONE,
$ A( 1, 1 ), LDA, A( 1, N1+1 ), LDA )
*
* Update and factor A22
*
CALL DSYRK( UPLO, 'T', N2, N1, -ONE, A( 1, N1+1 ), LDA,
$ ONE, A( N1+1, N1+1 ), LDA )
CALL DPOTRF2( UPLO, N2, A( N1+1, N1+1 ), LDA, IINFO )
IF ( IINFO.NE.0 ) THEN
INFO = IINFO + N1
RETURN
END IF
*
* Compute the Cholesky factorization A = L*L**T
*
ELSE
*
* Update and scale A21
*
CALL DTRSM( 'R', 'L', 'T', 'N', N2, N1, ONE,
$ A( 1, 1 ), LDA, A( N1+1, 1 ), LDA )
*
* Update and factor A22
*
CALL DSYRK( UPLO, 'N', N2, N1, -ONE, A( N1+1, 1 ), LDA,
$ ONE, A( N1+1, N1+1 ), LDA )
CALL DPOTRF2( UPLO, N2, A( N1+1, N1+1 ), LDA, IINFO )
IF ( IINFO.NE.0 ) THEN
INFO = IINFO + N1
RETURN
END IF
END IF
END IF
RETURN
*
* End of DPOTRF2
*
END
*> \brief \b DPOTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOTRI( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOTRI computes the inverse of a real symmetric positive definite
*> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
*> computed by DPOTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T, as computed by
*> DPOTRF.
*> On exit, the upper or lower triangle of the (symmetric)
*> inverse of A, overwriting the input factor U or L.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the (i,i) element of the factor U or L is
*> zero, and the inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potri
*
* =====================================================================
SUBROUTINE DPOTRI( UPLO, N, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLAUUM, DTRTRI, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Invert the triangular Cholesky factor U or L.
*
CALL DTRTRI( UPLO, 'Non-unit', N, A, LDA, INFO )
IF( INFO.GT.0 )
$ RETURN
*
* Form inv(U) * inv(U)**T or inv(L)**T * inv(L).
*
CALL DLAUUM( UPLO, N, A, LDA, INFO )
*
RETURN
*
* End of DPOTRI
*
END
*> \brief \b DPOTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite matrix A using the Cholesky factorization
*> A = U**T*U or A = L*L**T computed by DPOTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, as computed by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potrs
*
* =====================================================================
SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Solve A*X = B where A = U**T *U.
*
* Solve U**T *X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N,
$ NRHS,
$ ONE, A, LDA, B, LDB )
*
* Solve U*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
$ NRHS, ONE, A, LDA, B, LDB )
ELSE
*
* Solve A*X = B where A = L*L**T.
*
* Solve L*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', N,
$ NRHS, ONE, A, LDA, B, LDB )
*
* Solve L**T *X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', N,
$ NRHS,
$ ONE, A, LDA, B, LDB )
END IF
*
RETURN
*
* End of DPOTRS
*
END
*> \brief \b DPPCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPCON( UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite packed matrix using
*> the Cholesky factorization A = U**T*U or A = L*L**T computed by
*> DPPTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, packed columnwise in a linear
*> array. The j-th column of U or L is stored in the array AP
*> as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm (or infinity-norm) of the symmetric matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*> estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppcon
*
* =====================================================================
SUBROUTINE DPPCON( UPLO, N, AP, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
CHARACTER NORMIN
INTEGER IX, KASE
DOUBLE PRECISION AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLATPS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
SMLNUM = DLAMCH( 'Safe minimum' )
*
* Estimate the 1-norm of the inverse.
*
KASE = 0
NORMIN = 'N'
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( UPPER ) THEN
*
* Multiply by inv(U**T).
*
CALL DLATPS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
$ AP, WORK, SCALEL, WORK( 2*N+1 ), INFO )
NORMIN = 'Y'
*
* Multiply by inv(U).
*
CALL DLATPS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ AP, WORK, SCALEU, WORK( 2*N+1 ), INFO )
ELSE
*
* Multiply by inv(L).
*
CALL DLATPS( 'Lower', 'No transpose', 'Non-unit', NORMIN,
$ N,
$ AP, WORK, SCALEL, WORK( 2*N+1 ), INFO )
NORMIN = 'Y'
*
* Multiply by inv(L**T).
*
CALL DLATPS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
$ AP, WORK, SCALEU, WORK( 2*N+1 ), INFO )
END IF
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
SCALE = SCALEL*SCALEU
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
20 CONTINUE
RETURN
*
* End of DPPCON
*
END
*> \brief \b DPPEQU
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPEQU computes row and column scalings intended to equilibrate a
*> symmetric positive definite matrix A in packed storage and reduce
*> its condition number (with respect to the two-norm). S contains the
*> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
*> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
*> This choice of S puts the condition number of B within a factor N of
*> the smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangle of the symmetric matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> If INFO = 0, S contains the ratio of the smallest S(i) to
*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
*> large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppequ
*
* =====================================================================
SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, JJ
DOUBLE PRECISION SMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPEQU', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SCOND = ONE
AMAX = ZERO
RETURN
END IF
*
* Initialize SMIN and AMAX.
*
S( 1 ) = AP( 1 )
SMIN = S( 1 )
AMAX = S( 1 )
*
IF( UPPER ) THEN
*
* UPLO = 'U': Upper triangle of A is stored.
* Find the minimum and maximum diagonal elements.
*
JJ = 1
DO 10 I = 2, N
JJ = JJ + I
S( I ) = AP( JJ )
SMIN = MIN( SMIN, S( I ) )
AMAX = MAX( AMAX, S( I ) )
10 CONTINUE
*
ELSE
*
* UPLO = 'L': Lower triangle of A is stored.
* Find the minimum and maximum diagonal elements.
*
JJ = 1
DO 20 I = 2, N
JJ = JJ + N - I + 2
S( I ) = AP( JJ )
SMIN = MIN( SMIN, S( I ) )
AMAX = MAX( AMAX, S( I ) )
20 CONTINUE
END IF
*
IF( SMIN.LE.ZERO ) THEN
*
* Find the first non-positive diagonal element and return.
*
DO 30 I = 1, N
IF( S( I ).LE.ZERO ) THEN
INFO = I
RETURN
END IF
30 CONTINUE
ELSE
*
* Set the scale factors to the reciprocals
* of the diagonal elements.
*
DO 40 I = 1, N
S( I ) = ONE / SQRT( S( I ) )
40 CONTINUE
*
* Compute SCOND = min(S(I)) / max(S(I))
*
SCOND = SQRT( SMIN ) / SQRT( AMAX )
END IF
RETURN
*
* End of DPPEQU
*
END
*> \brief \b DPPRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpprfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpprfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpprfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
* BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite
*> and packed, and provides error bounds and backward error estimates
*> for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangle of the symmetric matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] AFP
*> \verbatim
*> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
*> packed columnwise in a linear array in the same format as A
*> (see AP).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DPPTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pprfs
*
* =====================================================================
SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX,
$ FERR,
$ BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE,
$ WORK( N+1 ),
$ 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(A)*abs(X) + abs(B).
*
KK = 1
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
IK = KK
DO 40 I = 1, K - 1
WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
IK = IK + 1
40 CONTINUE
WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
KK = KK + K
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
IK = KK + 1
DO 60 I = K + 1, N
WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
IK = IK + 1
60 CONTINUE
WORK( K ) = WORK( K ) + S
KK = KK + ( N-K+1 )
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(A))*
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(A) is the inverse of A
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(A)*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(A) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(A**T).
*
CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
110 CONTINUE
ELSE IF( KASE.EQ.2 ) THEN
*
* Multiply by inv(A)*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
120 CONTINUE
CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DPPRFS
*
END
*> \brief <b> DPPSV computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPSV( UPLO, N, NRHS, AP, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPSV computes the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric positive definite matrix stored in
*> packed format and X and B are N-by-NRHS matrices.
*>
*> The Cholesky decomposition is used to factor A as
*> A = U**T* U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is a lower triangular
*> matrix. The factored form of A is then used to solve the system of
*> equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> See below for further details.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T, in the same storage
*> format as A.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> of A is not positive, so the factorization could not
*> be completed, and the solution has not been computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppsv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The packed storage scheme is illustrated by the following example
*> when N = 4, UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13 a14
*> a22 a23 a24
*> a33 a34 (aij = conjg(aji))
*> a44
*>
*> Packed storage of the upper triangle of A:
*>
*> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPPSV( UPLO, N, NRHS, AP, B, LDB, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPPTRF, DPPTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPSV ', -INFO )
RETURN
END IF
*
* Compute the Cholesky factorization A = U**T*U or A = L*L**T.
*
CALL DPPTRF( UPLO, N, AP, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO )
*
END IF
RETURN
*
* End of DPPSV
*
END
*> \brief <b> DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
* X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, UPLO
* INTEGER INFO, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
* $ FERR( * ), S( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
*> compute the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric positive definite matrix stored in
*> packed format and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
*> Whether or not the system will be equilibrated depends on the
*> scaling of the matrix A, but if equilibration is used, A is
*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*> factor the matrix A (after equilibration if FACT = 'E') as
*> A = U**T* U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is a lower triangular
*> matrix.
*>
*> 3. If the leading principal minor of order i is not positive,
*> then the routine returns with INFO = i. Otherwise, the factored
*> form of A is used to estimate the condition number of the matrix
*> A. If the reciprocal of the condition number is less than machine
*> precision, INFO = N+1 is returned as a warning, but the routine
*> still goes on to solve for X and compute error bounds as
*> described below.
*>
*> 4. The system of equations is solved for X using the factored form
*> of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*> diag(S) so that it solves the original system before
*> equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of the matrix A is
*> supplied on entry, and if not, whether the matrix A should be
*> equilibrated before it is factored.
*> = 'F': On entry, AFP contains the factored form of A.
*> If EQUED = 'Y', the matrix A has been equilibrated
*> with scaling factors given by S. AP and AFP will not
*> be modified.
*> = 'N': The matrix A will be copied to AFP and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> copied to AFP and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array, except if FACT = 'F'
*> and EQUED = 'Y', then A must contain the equilibrated matrix
*> diag(S)*A*diag(S). The j-th column of A is stored in the
*> array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> See below for further details. A is not modified if
*> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*>
*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*> diag(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in,out] AFP
*> \verbatim
*> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> If FACT = 'F', then AFP is an input argument and on entry
*> contains the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T, in the same storage
*> format as A. If EQUED .ne. 'N', then AFP is the factored
*> form of the equilibrated matrix A.
*>
*> If FACT = 'N', then AFP is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T * U or A = L * L**T of the original
*> matrix A.
*>
*> If FACT = 'E', then AFP is an output argument and on exit
*> returns the triangular factor U or L from the Cholesky
*> factorization A = U**T * U or A = L * L**T of the equilibrated
*> matrix A (see the description of AP for the form of the
*> equilibrated matrix).
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration (always true if FACT = 'N').
*> = 'Y': Equilibration was done, i.e., A has been replaced by
*> diag(S) * A * diag(S).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A; not accessed if EQUED = 'N'. S is
*> an input argument if FACT = 'F'; otherwise, S is an output
*> argument. If FACT = 'F' and EQUED = 'Y', each element of S
*> must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*> B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*> the original system of equations. Note that if EQUED = 'Y',
*> A and B are modified on exit, and the solution to the
*> equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A after equilibration (if done). If RCOND is less than the
*> machine precision (in particular, if RCOND = 0), the matrix
*> is singular to working precision. This condition is
*> indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: the leading principal minor of order i of A
*> is not positive, so the factorization could not
*> be completed, and the solution has not been
*> computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppsvx
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The packed storage scheme is illustrated by the following example
*> when N = 4, UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13 a14
*> a22 a23 a24
*> a33 a34 (aij = conjg(aji))
*> a44
*>
*> Packed storage of the upper triangle of A:
*>
*> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B,
$ LDB,
$ X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
$ FERR( * ), S( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL EQUIL, NOFACT, RCEQU
INTEGER I, INFEQU, J
DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSP
EXTERNAL LSAME, DLAMCH, DLANSP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAQSP, DPPCON, DPPEQU,
$ DPPRFS,
$ DPPTRF, DPPTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
RCEQU = .FALSE.
ELSE
RCEQU = LSAME( EQUED, 'Y' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND.
$ .NOT.EQUIL .AND.
$ .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -7
ELSE
IF( RCEQU ) THEN
SMIN = BIGNUM
SMAX = ZERO
DO 10 J = 1, N
SMIN = MIN( SMIN, S( J ) )
SMAX = MAX( SMAX, S( J ) )
10 CONTINUE
IF( SMIN.LE.ZERO ) THEN
INFO = -8
ELSE IF( N.GT.0 ) THEN
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
ELSE
SCOND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -12
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
RCEQU = LSAME( EQUED, 'Y' )
END IF
END IF
*
* Scale the right-hand side.
*
IF( RCEQU ) THEN
DO 30 J = 1, NRHS
DO 20 I = 1, N
B( I, J ) = S( I )*B( I, J )
20 CONTINUE
30 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the Cholesky factorization A = U**T * U or A = L * L**T.
*
CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
CALL DPPTRF( UPLO, N, AFP, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANSP( 'I', UPLO, N, AP, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, IWORK, INFO )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
$ BERR,
$ WORK, IWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( RCEQU ) THEN
DO 50 J = 1, NRHS
DO 40 I = 1, N
X( I, J ) = S( I )*X( I, J )
40 CONTINUE
50 CONTINUE
DO 60 J = 1, NRHS
FERR( J ) = FERR( J ) / SCOND
60 CONTINUE
END IF
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DPPSVX
*
END
*> \brief \b DPPTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpptrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpptrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpptrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPTRF( UPLO, N, AP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A stored in packed format.
*>
*> The factorization has the form
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> See below for further details.
*>
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**T*U or A = L*L**T, in the same
*> storage format as A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> is not positive, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pptrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The packed storage scheme is illustrated by the following example
*> when N = 4, UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13 a14
*> a22 a23 a24
*> a33 a34 (aij = aji)
*> a44
*>
*> Packed storage of the upper triangle of A:
*>
*> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPPTRF( UPLO, N, AP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, JC, JJ
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSPR, DTPSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T*U.
*
JJ = 0
DO 10 J = 1, N
JC = JJ + 1
JJ = JJ + J
*
* Compute elements 1:J-1 of column J.
*
IF( J.GT.1 )
$ CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', J-1, AP,
$ AP( JC ), 1 )
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = AP( JJ ) - DDOT( J-1, AP( JC ), 1, AP( JC ), 1 )
IF( AJJ.LE.ZERO ) THEN
AP( JJ ) = AJJ
GO TO 30
END IF
AP( JJ ) = SQRT( AJJ )
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
JJ = 1
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = AP( JJ )
IF( AJJ.LE.ZERO ) THEN
AP( JJ ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
AP( JJ ) = AJJ
*
* Compute elements J+1:N of column J and update the trailing
* submatrix.
*
IF( J.LT.N ) THEN
CALL DSCAL( N-J, ONE / AJJ, AP( JJ+1 ), 1 )
CALL DSPR( 'Lower', N-J, -ONE, AP( JJ+1 ), 1,
$ AP( JJ+N-J+1 ) )
JJ = JJ + N - J + 1
END IF
20 CONTINUE
END IF
GO TO 40
*
30 CONTINUE
INFO = J
*
40 CONTINUE
RETURN
*
* End of DPPTRF
*
END
*> \brief \b DPPTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpptri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpptri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpptri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPTRI( UPLO, N, AP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPTRI computes the inverse of a real symmetric positive definite
*> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
*> computed by DPPTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangular factor is stored in AP;
*> = 'L': Lower triangular factor is stored in AP.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the triangular factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T, packed columnwise as
*> a linear array. The j-th column of U or L is stored in the
*> array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
*>
*> On exit, the upper or lower triangle of the (symmetric)
*> inverse of A, overwriting the input factor U or L.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the (i,i) element of the factor U or L is
*> zero, and the inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pptri
*
* =====================================================================
SUBROUTINE DPPTRI( UPLO, N, AP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, JC, JJ, JJN
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSPR, DTPMV, DTPTRI, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Invert the triangular Cholesky factor U or L.
*
CALL DTPTRI( UPLO, 'Non-unit', N, AP, INFO )
IF( INFO.GT.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Compute the product inv(U) * inv(U)**T.
*
JJ = 0
DO 10 J = 1, N
JC = JJ + 1
JJ = JJ + J
IF( J.GT.1 )
$ CALL DSPR( 'Upper', J-1, ONE, AP( JC ), 1, AP )
AJJ = AP( JJ )
CALL DSCAL( J, AJJ, AP( JC ), 1 )
10 CONTINUE
*
ELSE
*
* Compute the product inv(L)**T * inv(L).
*
JJ = 1
DO 20 J = 1, N
JJN = JJ + N - J + 1
AP( JJ ) = DDOT( N-J+1, AP( JJ ), 1, AP( JJ ), 1 )
IF( J.LT.N )
$ CALL DTPMV( 'Lower', 'Transpose', 'Non-unit', N-J,
$ AP( JJN ), AP( JJ+1 ), 1 )
JJ = JJN
20 CONTINUE
END IF
*
RETURN
*
* End of DPPTRI
*
END
*> \brief \b DPPTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpptrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpptrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpptrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite matrix A in packed storage using the Cholesky
*> factorization A = U**T*U or A = L*L**T computed by DPPTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, packed columnwise in a linear
*> array. The j-th column of U or L is stored in the array AP
*> as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pptrs
*
* =====================================================================
SUBROUTINE DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DTPSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Solve A*X = B where A = U**T * U.
*
DO 10 I = 1, NRHS
*
* Solve U**T *X = B, overwriting B with X.
*
CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', N, AP,
$ B( 1, I ), 1 )
*
* Solve U*X = B, overwriting B with X.
*
CALL DTPSV( 'Upper', 'No transpose', 'Non-unit', N, AP,
$ B( 1, I ), 1 )
10 CONTINUE
ELSE
*
* Solve A*X = B where A = L * L**T.
*
DO 20 I = 1, NRHS
*
* Solve L*Y = B, overwriting B with X.
*
CALL DTPSV( 'Lower', 'No transpose', 'Non-unit', N, AP,
$ B( 1, I ), 1 )
*
* Solve L**T *X = Y, overwriting B with X.
*
CALL DTPSV( 'Lower', 'Transpose', 'Non-unit', N, AP,
$ B( 1, I ), 1 )
20 CONTINUE
END IF
*
RETURN
*
* End of DPPTRS
*
END
*> \brief \b DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPSTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpstf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpstf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpstf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION TOL
* INTEGER INFO, LDA, N, RANK
* CHARACTER UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
* INTEGER PIV( N )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPSTF2 computes the Cholesky factorization with complete
*> pivoting of a real symmetric positive semidefinite matrix A.
*>
*> The factorization has the form
*> P**T * A * P = U**T * U , if UPLO = 'U',
*> P**T * A * P = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular, and
*> P is stored as vector PIV.
*>
*> This algorithm does not attempt to check that A is positive
*> semidefinite. This version of the algorithm calls level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization as above.
*> \endverbatim
*>
*> \param[out] PIV
*> \verbatim
*> PIV is INTEGER array, dimension (N)
*> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The rank of A given by the number of steps the algorithm
*> completed.
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
*> will be used. The algorithm terminates at the (K-1)st step
*> if the pivot <= TOL.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> Work space.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> < 0: If INFO = -K, the K-th argument had an illegal value,
*> = 0: algorithm completed successfully, and
*> > 0: the matrix A is either rank deficient with computed rank
*> as returned in RANK, or is not positive semidefinite. See
*> Section 7 of LAPACK Working Note #161 for further
*> information.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pstf2
*
* =====================================================================
SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, LDA, N, RANK
CHARACTER UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
INTEGER PIV( N )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AJJ, DSTOP, DTEMP
INTEGER I, ITEMP, J, PVT
LOGICAL UPPER
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
LOGICAL LSAME, DISNAN
EXTERNAL DLAMCH, LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT, MAXLOC
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPSTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Initialize PIV
*
DO 100 I = 1, N
PIV( I ) = I
100 CONTINUE
*
* Compute stopping value
*
PVT = 1
AJJ = A( PVT, PVT )
DO I = 2, N
IF( A( I, I ).GT.AJJ ) THEN
PVT = I
AJJ = A( PVT, PVT )
END IF
END DO
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
RANK = 0
INFO = 1
GO TO 170
END IF
*
* Compute stopping value if not supplied
*
IF( TOL.LT.ZERO ) THEN
DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
ELSE
DSTOP = TOL
END IF
*
* Set first half of WORK to zero, holds dot products
*
DO 110 I = 1, N
WORK( I ) = 0
110 CONTINUE
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization P**T * A * P = U**T * U
*
DO 130 J = 1, N
*
* Find pivot, test for exit, else swap rows and columns
* Update dot products, compute possible pivots which are
* stored in the second half of WORK
*
DO 120 I = J, N
*
IF( J.GT.1 ) THEN
WORK( I ) = WORK( I ) + A( J-1, I )**2
END IF
WORK( N+I ) = A( I, I ) - WORK( I )
*
120 CONTINUE
*
IF( J.GT.1 ) THEN
ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
PVT = ITEMP + J - 1
AJJ = WORK( N+PVT )
IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 160
END IF
END IF
*
IF( J.NE.PVT ) THEN
*
* Pivot OK, so can now swap pivot rows and columns
*
A( PVT, PVT ) = A( J, J )
CALL DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
IF( PVT.LT.N )
$ CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
$ A( PVT, PVT+1 ), LDA )
CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA, A( J+1, PVT ),
$ 1 )
*
* Swap dot products and PIV
*
DTEMP = WORK( J )
WORK( J ) = WORK( PVT )
WORK( PVT ) = DTEMP
ITEMP = PIV( PVT )
PIV( PVT ) = PIV( J )
PIV( J ) = ITEMP
END IF
*
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of row J
*
IF( J.LT.N ) THEN
CALL DGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
$ A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
END IF
*
130 CONTINUE
*
ELSE
*
* Compute the Cholesky factorization P**T * A * P = L * L**T
*
DO 150 J = 1, N
*
* Find pivot, test for exit, else swap rows and columns
* Update dot products, compute possible pivots which are
* stored in the second half of WORK
*
DO 140 I = J, N
*
IF( J.GT.1 ) THEN
WORK( I ) = WORK( I ) + A( I, J-1 )**2
END IF
WORK( N+I ) = A( I, I ) - WORK( I )
*
140 CONTINUE
*
IF( J.GT.1 ) THEN
ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
PVT = ITEMP + J - 1
AJJ = WORK( N+PVT )
IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 160
END IF
END IF
*
IF( J.NE.PVT ) THEN
*
* Pivot OK, so can now swap pivot rows and columns
*
A( PVT, PVT ) = A( J, J )
CALL DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
IF( PVT.LT.N )
$ CALL DSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1,
$ PVT ),
$ 1 )
CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ),
$ LDA )
*
* Swap dot products and PIV
*
DTEMP = WORK( J )
WORK( J ) = WORK( PVT )
WORK( PVT ) = DTEMP
ITEMP = PIV( PVT )
PIV( PVT ) = PIV( J )
PIV( J ) = ITEMP
END IF
*
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of column J
*
IF( J.LT.N ) THEN
CALL DGEMV( 'No Trans', N-J, J-1, -ONE, A( J+1, 1 ),
$ LDA,
$ A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
END IF
*
150 CONTINUE
*
END IF
*
* Ran to completion, A has full rank
*
RANK = N
*
GO TO 170
160 CONTINUE
*
* Rank is number of steps completed. Set INFO = 1 to signal
* that the factorization cannot be used to solve a system.
*
RANK = J - 1
INFO = 1
*
170 CONTINUE
RETURN
*
* End of DPSTF2
*
END
*> \brief \b DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
*
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPSTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpstrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpstrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpstrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION TOL
* INTEGER INFO, LDA, N, RANK
* CHARACTER UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
* INTEGER PIV( N )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPSTRF computes the Cholesky factorization with complete
*> pivoting of a real symmetric positive semidefinite matrix A.
*>
*> The factorization has the form
*> P**T * A * P = U**T * U , if UPLO = 'U',
*> P**T * A * P = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular, and
*> P is stored as vector PIV.
*>
*> This algorithm does not attempt to check that A is positive
*> semidefinite. This version of the algorithm calls level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization as above.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] PIV
*> \verbatim
*> PIV is INTEGER array, dimension (N)
*> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The rank of A given by the number of steps the algorithm
*> completed.
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
*> will be used. The algorithm terminates at the (K-1)st step
*> if the pivot <= TOL.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> Work space.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> < 0: If INFO = -K, the K-th argument had an illegal value,
*> = 0: algorithm completed successfully, and
*> > 0: the matrix A is either rank deficient with computed rank
*> as returned in RANK, or is not positive semidefinite. See
*> Section 7 of LAPACK Working Note #161 for further
*> information.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pstrf
*
* =====================================================================
SUBROUTINE DPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, LDA, N, RANK
CHARACTER UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
INTEGER PIV( N )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AJJ, DSTOP, DTEMP
INTEGER I, ITEMP, J, JB, K, NB, PVT
LOGICAL UPPER
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
INTEGER ILAENV
LOGICAL LSAME, DISNAN
EXTERNAL DLAMCH, ILAENV, LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DPSTF2, DSCAL, DSWAP, DSYRK,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT, MAXLOC
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPSTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get block size
*
NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code
*
CALL DPSTF2( UPLO, N, A( 1, 1 ), LDA, PIV, RANK, TOL, WORK,
$ INFO )
GO TO 200
*
ELSE
*
* Initialize PIV
*
DO 100 I = 1, N
PIV( I ) = I
100 CONTINUE
*
* Compute stopping value
*
PVT = 1
AJJ = A( PVT, PVT )
DO I = 2, N
IF( A( I, I ).GT.AJJ ) THEN
PVT = I
AJJ = A( PVT, PVT )
END IF
END DO
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
RANK = 0
INFO = 1
GO TO 200
END IF
*
* Compute stopping value if not supplied
*
IF( TOL.LT.ZERO ) THEN
DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
ELSE
DSTOP = TOL
END IF
*
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization P**T * A * P = U**T * U
*
DO 140 K = 1, N, NB
*
* Account for last block not being NB wide
*
JB = MIN( NB, N-K+1 )
*
* Set relevant part of first half of WORK to zero,
* holds dot products
*
DO 110 I = K, N
WORK( I ) = 0
110 CONTINUE
*
DO 130 J = K, K + JB - 1
*
* Find pivot, test for exit, else swap rows and columns
* Update dot products, compute possible pivots which are
* stored in the second half of WORK
*
DO 120 I = J, N
*
IF( J.GT.K ) THEN
WORK( I ) = WORK( I ) + A( J-1, I )**2
END IF
WORK( N+I ) = A( I, I ) - WORK( I )
*
120 CONTINUE
*
IF( J.GT.1 ) THEN
ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
PVT = ITEMP + J - 1
AJJ = WORK( N+PVT )
IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 190
END IF
END IF
*
IF( J.NE.PVT ) THEN
*
* Pivot OK, so can now swap pivot rows and columns
*
A( PVT, PVT ) = A( J, J )
CALL DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
IF( PVT.LT.N )
$ CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
$ A( PVT, PVT+1 ), LDA )
CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA,
$ A( J+1, PVT ), 1 )
*
* Swap dot products and PIV
*
DTEMP = WORK( J )
WORK( J ) = WORK( PVT )
WORK( PVT ) = DTEMP
ITEMP = PIV( PVT )
PIV( PVT ) = PIV( J )
PIV( J ) = ITEMP
END IF
*
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of row J.
*
IF( J.LT.N ) THEN
CALL DGEMV( 'Trans', J-K, N-J, -ONE, A( K,
$ J+1 ),
$ LDA, A( K, J ), 1, ONE, A( J, J+1 ),
$ LDA )
CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
END IF
*
130 CONTINUE
*
* Update trailing matrix, J already incremented
*
IF( K+JB.LE.N ) THEN
CALL DSYRK( 'Upper', 'Trans', N-J+1, JB, -ONE,
$ A( K, J ), LDA, ONE, A( J, J ), LDA )
END IF
*
140 CONTINUE
*
ELSE
*
* Compute the Cholesky factorization P**T * A * P = L * L**T
*
DO 180 K = 1, N, NB
*
* Account for last block not being NB wide
*
JB = MIN( NB, N-K+1 )
*
* Set relevant part of first half of WORK to zero,
* holds dot products
*
DO 150 I = K, N
WORK( I ) = 0
150 CONTINUE
*
DO 170 J = K, K + JB - 1
*
* Find pivot, test for exit, else swap rows and columns
* Update dot products, compute possible pivots which are
* stored in the second half of WORK
*
DO 160 I = J, N
*
IF( J.GT.K ) THEN
WORK( I ) = WORK( I ) + A( I, J-1 )**2
END IF
WORK( N+I ) = A( I, I ) - WORK( I )
*
160 CONTINUE
*
IF( J.GT.1 ) THEN
ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
PVT = ITEMP + J - 1
AJJ = WORK( N+PVT )
IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 190
END IF
END IF
*
IF( J.NE.PVT ) THEN
*
* Pivot OK, so can now swap pivot rows and columns
*
A( PVT, PVT ) = A( J, J )
CALL DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ),
$ LDA )
IF( PVT.LT.N )
$ CALL DSWAP( N-PVT, A( PVT+1, J ), 1,
$ A( PVT+1, PVT ), 1 )
CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT,
$ J+1 ),
$ LDA )
*
* Swap dot products and PIV
*
DTEMP = WORK( J )
WORK( J ) = WORK( PVT )
WORK( PVT ) = DTEMP
ITEMP = PIV( PVT )
PIV( PVT ) = PIV( J )
PIV( J ) = ITEMP
END IF
*
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of column J.
*
IF( J.LT.N ) THEN
CALL DGEMV( 'No Trans', N-J, J-K, -ONE,
$ A( J+1, K ), LDA, A( J, K ), LDA, ONE,
$ A( J+1, J ), 1 )
CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
END IF
*
170 CONTINUE
*
* Update trailing matrix, J already incremented
*
IF( K+JB.LE.N ) THEN
CALL DSYRK( 'Lower', 'No Trans', N-J+1, JB, -ONE,
$ A( J, K ), LDA, ONE, A( J, J ), LDA )
END IF
*
180 CONTINUE
*
END IF
END IF
*
* Ran to completion, A has full rank
*
RANK = N
*
GO TO 200
190 CONTINUE
*
* Rank is the number of steps completed. Set INFO = 1 to signal
* that the factorization cannot be used to solve a system.
*
RANK = J - 1
INFO = 1
*
200 CONTINUE
RETURN
*
* End of DPSTRF
*
END
*> \brief \b DPTCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTCON computes the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite tridiagonal matrix
*> using the factorization A = L*D*L**T or A = U**T*D*U computed by
*> DPTTRF.
*>
*> Norm(inv(A)) is computed by a direct method, and the reciprocal of
*> the condition number is computed as
*> RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the diagonal matrix D from the
*> factorization of A, as computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) off-diagonal elements of the unit bidiagonal factor
*> U or L from the factorization of A, as computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
*> 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptcon
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The method used is described in Nicholas J. Higham, "Efficient
*> Algorithms for Computing the Condition Number of a Tridiagonal
*> Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IX
DOUBLE PRECISION AINVNM
* ..
* .. External Functions ..
INTEGER IDAMAX
EXTERNAL IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
* Check that D(1:N) is positive.
*
DO 10 I = 1, N
IF( D( I ).LE.ZERO )
$ RETURN
10 CONTINUE
*
* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
* m(i,j) = abs(A(i,j)), i = j,
* m(i,j) = -abs(A(i,j)), i .ne. j,
*
* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
*
* Solve M(L) * x = e.
*
WORK( 1 ) = ONE
DO 20 I = 2, N
WORK( I ) = ONE + WORK( I-1 )*ABS( E( I-1 ) )
20 CONTINUE
*
* Solve D * M(L)**T * x = b.
*
WORK( N ) = WORK( N ) / D( N )
DO 30 I = N - 1, 1, -1
WORK( I ) = WORK( I ) / D( I ) + WORK( I+1 )*ABS( E( I ) )
30 CONTINUE
*
* Compute AINVNM = max(x(i)), 1<=i<=n.
*
IX = IDAMAX( N, WORK, 1 )
AINVNM = ABS( WORK( IX ) )
*
* Compute the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
RETURN
*
* End of DPTCON
*
END
*> \brief \b DPTEQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpteqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpteqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpteqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric positive definite tridiagonal matrix by first factoring the
*> matrix using DPTTRF, and then calling DBDSQR to compute the singular
*> values of the bidiagonal factor.
*>
*> This routine computes the eigenvalues of the positive definite
*> tridiagonal matrix to high relative accuracy. This means that if the
*> eigenvalues range over many orders of magnitude in size, then the
*> small eigenvalues and corresponding eigenvectors will be computed
*> more accurately than, for example, with the standard QR method.
*>
*> The eigenvectors of a full or band symmetric positive definite matrix
*> can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to
*> reduce this matrix to tridiagonal form. (The reduction to tridiagonal
*> form, however, may preclude the possibility of obtaining high
*> relative accuracy in the small eigenvalues of the original matrix, if
*> these eigenvalues range over many orders of magnitude.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'V': Compute eigenvectors of original symmetric
*> matrix also. Array Z contains the orthogonal
*> matrix used to reduce the original matrix to
*> tridiagonal form.
*> = 'I': Compute eigenvectors of tridiagonal matrix also.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal
*> matrix.
*> On normal exit, D contains the eigenvalues, in descending
*> order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', the orthogonal matrix used in the
*> reduction to tridiagonal form.
*> On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
*> original symmetric matrix;
*> if COMPZ = 'I', the orthonormal eigenvectors of the
*> tridiagonal matrix.
*> If INFO > 0 on exit, Z contains the eigenvectors associated
*> with only the stored eigenvalues.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> COMPZ = 'V' or 'I', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, and i is:
*> <= N the Cholesky factorization of the matrix could
*> not be performed because the leading principal
*> minor of order i was not positive.
*> > N the SVD algorithm failed to converge;
*> if INFO = N+i, i off-diagonal elements of the
*> bidiagonal factor did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pteqr
*
* =====================================================================
SUBROUTINE DPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DBDSQR, DLASET, DPTTRF, XERBLA
* ..
* .. Local Arrays ..
DOUBLE PRECISION C( 1, 1 ), VT( 1, 1 )
* ..
* .. Local Scalars ..
INTEGER I, ICOMPZ, NRU
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
$ N ) ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTEQR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ICOMPZ.GT.0 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
IF( ICOMPZ.EQ.2 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
* Call DPTTRF to factor the matrix.
*
CALL DPTTRF( N, D, E, INFO )
IF( INFO.NE.0 )
$ RETURN
DO 10 I = 1, N
D( I ) = SQRT( D( I ) )
10 CONTINUE
DO 20 I = 1, N - 1
E( I ) = E( I )*D( I )
20 CONTINUE
*
* Call DBDSQR to compute the singular values/vectors of the
* bidiagonal factor.
*
IF( ICOMPZ.GT.0 ) THEN
NRU = N
ELSE
NRU = 0
END IF
CALL DBDSQR( 'Lower', N, 0, NRU, 0, D, E, VT, 1, Z, LDZ, C, 1,
$ WORK, INFO )
*
* Square the singular values.
*
IF( INFO.EQ.0 ) THEN
DO 30 I = 1, N
D( I ) = D( I )*D( I )
30 CONTINUE
ELSE
INFO = N + INFO
END IF
*
RETURN
*
* End of DPTEQR
*
END
*> \brief \b DPTRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
* BERR, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
* $ E( * ), EF( * ), FERR( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite
*> and tridiagonal, and provides error bounds and backward error
*> estimates for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] DF
*> \verbatim
*> DF is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the diagonal matrix D from the
*> factorization computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] EF
*> \verbatim
*> EF is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the unit bidiagonal factor
*> L from the factorization computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DPTTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j).
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptrfs
*
* =====================================================================
SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
$ BERR, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
$ E( * ), EF( * ), FERR( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
INTEGER COUNT, I, IX, J, NZ
DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
$ SAFMIN
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DPTTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL IDAMAX, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( NRHS.LT.0 ) THEN
INFO = -2
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = 4
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 90 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X. Also compute
* abs(A)*abs(x) + abs(b) for use in the backward error bound.
*
IF( N.EQ.1 ) THEN
BI = B( 1, J )
DX = D( 1 )*X( 1, J )
WORK( N+1 ) = BI - DX
WORK( 1 ) = ABS( BI ) + ABS( DX )
ELSE
BI = B( 1, J )
DX = D( 1 )*X( 1, J )
EX = E( 1 )*X( 2, J )
WORK( N+1 ) = BI - DX - EX
WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
DO 30 I = 2, N - 1
BI = B( I, J )
CX = E( I-1 )*X( I-1, J )
DX = D( I )*X( I, J )
EX = E( I )*X( I+1, J )
WORK( N+I ) = BI - CX - DX - EX
WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
30 CONTINUE
BI = B( N, J )
CX = E( N-1 )*X( N-1, J )
DX = D( N )*X( N, J )
WORK( N+N ) = BI - CX - DX
WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
END IF
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
S = ZERO
DO 40 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
40 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(A))*
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(A) is the inverse of A
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(A)*abs(X) + abs(B) is less than SAFE2.
*
DO 50 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
50 CONTINUE
IX = IDAMAX( N, WORK, 1 )
FERR( J ) = WORK( IX )
*
* Estimate the norm of inv(A).
*
* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
* m(i,j) = abs(A(i,j)), i = j,
* m(i,j) = -abs(A(i,j)), i .ne. j,
*
* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
*
* Solve M(L) * x = e.
*
WORK( 1 ) = ONE
DO 60 I = 2, N
WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
60 CONTINUE
*
* Solve D * M(L)**T * x = b.
*
WORK( N ) = WORK( N ) / DF( N )
DO 70 I = N - 1, 1, -1
WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
70 CONTINUE
*
* Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
*
IX = IDAMAX( N, WORK, 1 )
FERR( J ) = FERR( J )*ABS( WORK( IX ) )
*
* Normalize error.
*
LSTRES = ZERO
DO 80 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
80 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
90 CONTINUE
*
RETURN
*
* End of DPTRFS
*
END
*> \brief <b> DPTSV computes the solution to system of linear equations A * X = B for PT matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTSV( N, NRHS, D, E, B, LDB, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTSV computes the solution to a real system of linear equations
*> A*X = B, where A is an N-by-N symmetric positive definite tridiagonal
*> matrix, and X and B are N-by-NRHS matrices.
*>
*> A is factored as A = L*D*L**T, and the factored form of A is then
*> used to solve the system of equations.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A. On exit, the n diagonal elements of the diagonal matrix
*> D from the factorization A = L*D*L**T.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A. On exit, the (n-1) subdiagonal elements of the
*> unit bidiagonal factor L from the L*D*L**T factorization of
*> A. (E can also be regarded as the superdiagonal of the unit
*> bidiagonal factor U from the U**T*D*U factorization of A.)
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading principal minor of order i
*> is not positive, and the solution has not been
*> computed. The factorization has not been completed
*> unless i = N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptsv
*
* =====================================================================
SUBROUTINE DPTSV( N, NRHS, D, E, B, LDB, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), D( * ), E( * )
* ..
*
* =====================================================================
*
* .. External Subroutines ..
EXTERNAL DPTTRF, DPTTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( NRHS.LT.0 ) THEN
INFO = -2
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTSV ', -INFO )
RETURN
END IF
*
* Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
CALL DPTTRF( N, D, E, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DPTTRS( N, NRHS, D, E, B, LDB, INFO )
END IF
RETURN
*
* End of DPTSV
*
END
*> \brief <b> DPTSVX computes the solution to system of linear equations A * X = B for PT matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
* RCOND, FERR, BERR, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER FACT
* INTEGER INFO, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
* $ E( * ), EF( * ), FERR( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTSVX uses the factorization A = L*D*L**T to compute the solution
*> to a real system of linear equations A*X = B, where A is an N-by-N
*> symmetric positive definite tridiagonal matrix and X and B are
*> N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
*> is a unit lower bidiagonal matrix and D is diagonal. The
*> factorization can also be regarded as having the form
*> A = U**T*D*U.
*>
*> 2. If the leading principal minor of order i is not positive,
*> then the routine returns with INFO = i. Otherwise, the factored
*> form of A is used to estimate the condition number of the matrix
*> A. If the reciprocal of the condition number is less than machine
*> precision, INFO = N+1 is returned as a warning, but the routine
*> still goes on to solve for X and compute error bounds as
*> described below.
*>
*> 3. The system of equations is solved for X using the factored form
*> of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of A has been
*> supplied on entry.
*> = 'F': On entry, DF and EF contain the factored form of A.
*> D, E, DF, and EF will not be modified.
*> = 'N': The matrix A will be copied to DF and EF and
*> factored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in,out] DF
*> \verbatim
*> DF is DOUBLE PRECISION array, dimension (N)
*> If FACT = 'F', then DF is an input argument and on entry
*> contains the n diagonal elements of the diagonal matrix D
*> from the L*D*L**T factorization of A.
*> If FACT = 'N', then DF is an output argument and on exit
*> contains the n diagonal elements of the diagonal matrix D
*> from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in,out] EF
*> \verbatim
*> EF is DOUBLE PRECISION array, dimension (N-1)
*> If FACT = 'F', then EF is an input argument and on entry
*> contains the (n-1) subdiagonal elements of the unit
*> bidiagonal factor L from the L*D*L**T factorization of A.
*> If FACT = 'N', then EF is an output argument and on exit
*> contains the (n-1) subdiagonal elements of the unit
*> bidiagonal factor L from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal condition number of the matrix A. If RCOND
*> is less than the machine precision (in particular, if
*> RCOND = 0), the matrix is singular to working precision.
*> This condition is indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j).
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in any
*> element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: the leading principal minor of order i of A
*> is not positive, so the factorization could not
*> be completed, and the solution has not been
*> computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptsvx
*
* =====================================================================
SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
$ RCOND, FERR, BERR, WORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER FACT
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
$ E( * ), EF( * ), FERR( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOFACT
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DPTCON, DPTRFS, DPTTRF,
$ DPTTRS,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTSVX', -INFO )
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
CALL DCOPY( N, D, 1, DF, 1 )
IF( N.GT.1 )
$ CALL DCOPY( N-1, E, 1, EF, 1 )
CALL DPTTRF( N, DF, EF, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANST( '1', N, D, E )
*
* Compute the reciprocal of the condition number of A.
*
CALL DPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO )
*
* Compute the solution vectors X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DPTTRS( N, NRHS, DF, EF, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR,
$ WORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DPTSVX
*
END
*> \brief \b DPTTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpttrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpttrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpttrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTTRF( N, D, E, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTTRF computes the L*D*L**T factorization of a real symmetric
*> positive definite tridiagonal matrix A. The factorization may also
*> be regarded as having the form A = U**T*D*U.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A. On exit, the n diagonal elements of the diagonal matrix
*> D from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A. On exit, the (n-1) subdiagonal elements of the
*> unit bidiagonal factor L from the L*D*L**T factorization of A.
*> E can also be regarded as the superdiagonal of the unit
*> bidiagonal factor U from the U**T*D*U factorization of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, the leading principal minor of order k
*> is not positive; if k < N, the factorization could not
*> be completed, while if k = N, the factorization was
*> completed, but D(N) <= 0.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pttrf
*
* =====================================================================
SUBROUTINE DPTTRF( N, D, E, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, I4
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DPTTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
I4 = MOD( N-1, 4 )
DO 10 I = 1, I4
IF( D( I ).LE.ZERO ) THEN
INFO = I
GO TO 30
END IF
EI = E( I )
E( I ) = EI / D( I )
D( I+1 ) = D( I+1 ) - E( I )*EI
10 CONTINUE
*
DO 20 I = I4 + 1, N - 4, 4
*
* Drop out of the loop if d(i) <= 0: the matrix is not positive
* definite.
*
IF( D( I ).LE.ZERO ) THEN
INFO = I
GO TO 30
END IF
*
* Solve for e(i) and d(i+1).
*
EI = E( I )
E( I ) = EI / D( I )
D( I+1 ) = D( I+1 ) - E( I )*EI
*
IF( D( I+1 ).LE.ZERO ) THEN
INFO = I + 1
GO TO 30
END IF
*
* Solve for e(i+1) and d(i+2).
*
EI = E( I+1 )
E( I+1 ) = EI / D( I+1 )
D( I+2 ) = D( I+2 ) - E( I+1 )*EI
*
IF( D( I+2 ).LE.ZERO ) THEN
INFO = I + 2
GO TO 30
END IF
*
* Solve for e(i+2) and d(i+3).
*
EI = E( I+2 )
E( I+2 ) = EI / D( I+2 )
D( I+3 ) = D( I+3 ) - E( I+2 )*EI
*
IF( D( I+3 ).LE.ZERO ) THEN
INFO = I + 3
GO TO 30
END IF
*
* Solve for e(i+3) and d(i+4).
*
EI = E( I+3 )
E( I+3 ) = EI / D( I+3 )
D( I+4 ) = D( I+4 ) - E( I+3 )*EI
20 CONTINUE
*
* Check d(n) for positive definiteness.
*
IF( D( N ).LE.ZERO )
$ INFO = N
*
30 CONTINUE
RETURN
*
* End of DPTTRF
*
END
*> \brief \b DPTTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpttrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpttrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpttrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTTRS( N, NRHS, D, E, B, LDB, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTTRS solves a tridiagonal system of the form
*> A * X = B
*> using the L*D*L**T factorization of A computed by DPTTRF. D is a
*> diagonal matrix specified in the vector D, L is a unit bidiagonal
*> matrix whose subdiagonal is specified in the vector E, and X and B
*> are N by NRHS matrices.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the tridiagonal matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the diagonal matrix D from the
*> L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the unit bidiagonal factor
*> L from the L*D*L**T factorization of A. E can also be regarded
*> as the superdiagonal of the unit bidiagonal factor U from the
*> factorization A = U**T*D*U.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side vectors B for the system of
*> linear equations.
*> On exit, the solution vectors, X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pttrs
*
* =====================================================================
SUBROUTINE DPTTRS( N, NRHS, D, E, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER J, JB, NB
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DPTTS2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( NRHS.LT.0 ) THEN
INFO = -2
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
* Determine the number of right-hand sides to solve at a time.
*
IF( NRHS.EQ.1 ) THEN
NB = 1
ELSE
NB = MAX( 1, ILAENV( 1, 'DPTTRS', ' ', N, NRHS, -1, -1 ) )
END IF
*
IF( NB.GE.NRHS ) THEN
CALL DPTTS2( N, NRHS, D, E, B, LDB )
ELSE
DO 10 J = 1, NRHS, NB
JB = MIN( NRHS-J+1, NB )
CALL DPTTS2( N, JB, D, E, B( 1, J ), LDB )
10 CONTINUE
END IF
*
RETURN
*
* End of DPTTRS
*
END
*> \brief \b DPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTTS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptts2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptts2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptts2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTTS2( N, NRHS, D, E, B, LDB )
*
* .. Scalar Arguments ..
* INTEGER LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTTS2 solves a tridiagonal system of the form
*> A * X = B
*> using the L*D*L**T factorization of A computed by DPTTRF. D is a
*> diagonal matrix specified in the vector D, L is a unit bidiagonal
*> matrix whose subdiagonal is specified in the vector E, and X and B
*> are N by NRHS matrices.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the tridiagonal matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the diagonal matrix D from the
*> L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the unit bidiagonal factor
*> L from the L*D*L**T factorization of A. E can also be regarded
*> as the superdiagonal of the unit bidiagonal factor U from the
*> factorization A = U**T*D*U.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side vectors B for the system of
*> linear equations.
*> On exit, the solution vectors, X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptts2
*
* =====================================================================
SUBROUTINE DPTTS2( N, NRHS, D, E, B, LDB )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Subroutines ..
EXTERNAL DSCAL
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.1 ) THEN
IF( N.EQ.1 )
$ CALL DSCAL( NRHS, 1.D0 / D( 1 ), B, LDB )
RETURN
END IF
*
* Solve A * X = B using the factorization A = L*D*L**T,
* overwriting each right hand side vector with its solution.
*
DO 30 J = 1, NRHS
*
* Solve L * x = b.
*
DO 10 I = 2, N
B( I, J ) = B( I, J ) - B( I-1, J )*E( I-1 )
10 CONTINUE
*
* Solve D * L**T * x = b.
*
B( N, J ) = B( N, J ) / D( N )
DO 20 I = N - 1, 1, -1
B( I, J ) = B( I, J ) / D( I ) - B( I+1, J )*E( I )
20 CONTINUE
30 CONTINUE
*
RETURN
*
* End of DPTTS2
*
END
*> \brief \b DRSCL multiplies a vector by the reciprocal of a real scalar.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DRSCL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/drscl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/drscl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/drscl.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DRSCL( N, SA, SX, INCX )
*
* .. Scalar Arguments ..
* INTEGER INCX, N
* DOUBLE PRECISION SA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION SX( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DRSCL multiplies an n-element real vector x by the real scalar 1/a.
*> This is done without overflow or underflow as long as
*> the final result x/a does not overflow or underflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of components of the vector x.
*> \endverbatim
*>
*> \param[in] SA
*> \verbatim
*> SA is DOUBLE PRECISION
*> The scalar a which is used to divide each component of x.
*> SA must be >= 0, or the subroutine will divide by zero.
*> \endverbatim
*>
*> \param[in,out] SX
*> \verbatim
*> SX is DOUBLE PRECISION array, dimension
*> (1+(N-1)*abs(INCX))
*> The n-element vector x.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between successive values of the vector SX.
*> > 0: SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i), 1< i<= n
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup rscl
*
* =====================================================================
SUBROUTINE DRSCL( N, SA, SX, INCX )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX, N
DOUBLE PRECISION SA
* ..
* .. Array Arguments ..
DOUBLE PRECISION SX( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL DONE
DOUBLE PRECISION BIGNUM, CDEN, CDEN1, CNUM, CNUM1, MUL, SMLNUM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
* Initialize the denominator to SA and the numerator to 1.
*
CDEN = SA
CNUM = ONE
*
10 CONTINUE
CDEN1 = CDEN*SMLNUM
CNUM1 = CNUM / BIGNUM
IF( ABS( CDEN1 ).GT.ABS( CNUM ) .AND. CNUM.NE.ZERO ) THEN
*
* Pre-multiply X by SMLNUM if CDEN is large compared to CNUM.
*
MUL = SMLNUM
DONE = .FALSE.
CDEN = CDEN1
ELSE IF( ABS( CNUM1 ).GT.ABS( CDEN ) ) THEN
*
* Pre-multiply X by BIGNUM if CDEN is small compared to CNUM.
*
MUL = BIGNUM
DONE = .FALSE.
CNUM = CNUM1
ELSE
*
* Multiply X by CNUM / CDEN and return.
*
MUL = CNUM / CDEN
DONE = .TRUE.
END IF
*
* Scale the vector X by MUL
*
CALL DSCAL( N, MUL, SX, INCX )
*
IF( .NOT.DONE )
$ GO TO 10
*
RETURN
*
* End of DRSCL
*
END
*> \brief <b> DSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbev.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBEV( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, KD, LDAB, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBEV computes all the eigenvalues and, optionally, eigenvectors of
*> a real symmetric band matrix A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, AB is overwritten by values generated during the
*> reduction to tridiagonal form. If UPLO = 'U', the first
*> superdiagonal and the diagonal of the tridiagonal matrix T
*> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*> the diagonal and first subdiagonal of T are returned in the
*> first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*> eigenvectors of the matrix A, with the i-th column of Z
*> holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(1,3*N-2))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the algorithm failed to converge; i
*> off-diagonal elements of an intermediate tridiagonal
*> form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbev
*
* =====================================================================
SUBROUTINE DSBEV( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, KD, LDAB, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, WANTZ
INTEGER IINFO, IMAX, INDE, INDWRK, ISCALE
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSB
EXTERNAL LSAME, DLAMCH, DLANSB
* ..
* .. External Subroutines ..
EXTERNAL DLASCL, DSBTRD, DSCAL, DSTEQR, DSTERF,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KD.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -9
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBEV ', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( LOWER ) THEN
W( 1 ) = AB( 1, 1 )
ELSE
W( 1 ) = AB( KD+1, 1 )
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
IF( LOWER ) THEN
CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
$ INFO )
ELSE
CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
$ INFO )
END IF
END IF
*
* Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
*
INDE = 1
INDWRK = INDE + N
CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z,
$ LDZ,
$ WORK( INDWRK ), IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, call SSTEQR.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ,
$ WORK( INDWRK ),
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
RETURN
*
* End of DSBEV
*
END
*> \brief <b> DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
* LWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBEVD computes all the eigenvalues and, optionally, eigenvectors of
*> a real symmetric band matrix A. If eigenvectors are desired, it uses
*> a divide and conquer algorithm.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, AB is overwritten by values generated during the
*> reduction to tridiagonal form. If UPLO = 'U', the first
*> superdiagonal and the diagonal of the tridiagonal matrix T
*> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*> the diagonal and first subdiagonal of T are returned in the
*> first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*> eigenvectors of the matrix A, with the i-th column of Z
*> holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> IF N <= 1, LWORK must be at least 1.
*> If JOBZ = 'N' and N > 2, LWORK must be at least 2*N.
*> If JOBZ = 'V' and N > 2, LWORK must be at least
*> ( 1 + 5*N + 2*N**2 ).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
*> If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the algorithm failed to converge; i
*> off-diagonal elements of an intermediate tridiagonal
*> form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbevd
*
* =====================================================================
SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
$ WORK,
$ LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, LQUERY, WANTZ
INTEGER IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
$ LLWRK2, LWMIN
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSB
EXTERNAL LSAME, DLAMCH, DLANSB
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL,
$ DSTEDC,
$ DSTERF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE
IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 5*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N
END IF
END IF
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KD.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBEVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
W( 1 ) = AB( 1, 1 )
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
IF( LOWER ) THEN
CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
$ INFO )
ELSE
CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
$ INFO )
END IF
END IF
*
* Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
*
INDE = 1
INDWRK = INDE + N
INDWK2 = INDWRK + N*N
LLWRK2 = LWORK - INDWK2 + 1
CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z,
$ LDZ,
$ WORK( INDWRK ), IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
$ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ),
$ N,
$ ZERO, WORK( INDWK2 ), N )
CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 )
$ CALL DSCAL( N, ONE / SIGMA, W, 1 )
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
RETURN
*
* End of DSBEVD
*
END
*> \brief <b> DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
* VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
* IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric band matrix A. Eigenvalues and eigenvectors can
*> be selected by specifying either a range of values or a range of
*> indices for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found;
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found;
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, AB is overwritten by values generated during the
*> reduction to tridiagonal form. If UPLO = 'U', the first
*> superdiagonal and the diagonal of the tridiagonal matrix T
*> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*> the diagonal and first subdiagonal of T are returned in the
*> first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> If JOBZ = 'V', the N-by-N orthogonal matrix used in the
*> reduction to tridiagonal form.
*> If JOBZ = 'N', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. If JOBZ = 'V', then
*> LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing AB to tridiagonal form.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*DLAMCH('S').
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices
*> with Guaranteed High Relative Accuracy," by Demmel and
*> Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the selected eigenvalues in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If an eigenvector fails to converge, then that column of Z
*> contains the latest approximation to the eigenvector, and the
*> index of the eigenvector is returned in IFAIL.
*> If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (N)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvectors that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, then i eigenvectors failed to converge.
*> Their indices are stored in array IFAIL.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbevx
*
* =====================================================================
SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ,
$ VL,
$ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
$ IFAIL, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
CHARACTER ORDER
INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
$ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
$ NSPLIT
DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
$ SIGMA, SMLNUM, TMP1, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSB
EXTERNAL LSAME, DLAMCH, DLANSB
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD,
$ DSCAL,
$ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
LOWER = LSAME( UPLO, 'L' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KD.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -7
ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -11
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -13
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
$ INFO = -18
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBEVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
M = 1
IF( LOWER ) THEN
TMP1 = AB( 1, 1 )
ELSE
TMP1 = AB( KD+1, 1 )
END IF
IF( VALEIG ) THEN
IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
$ M = 0
END IF
IF( M.EQ.1 ) THEN
W( 1 ) = TMP1
IF( WANTZ )
$ Z( 1, 1 ) = ONE
END IF
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
ABSTLL = ABSTOL
IF( VALEIG ) THEN
VLL = VL
VUU = VU
ELSE
VLL = ZERO
VUU = ZERO
END IF
ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
IF( LOWER ) THEN
CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
$ INFO )
ELSE
CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
$ INFO )
END IF
IF( ABSTOL.GT.0 )
$ ABSTLL = ABSTOL*SIGMA
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
*
* Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
*
INDD = 1
INDE = INDD + N
INDWRK = INDE + N
CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ),
$ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
*
* If all eigenvalues are desired and ABSTOL is less than or equal
* to zero, then call DSTERF or SSTEQR. If this fails for some
* eigenvalue, then try DSTEBZ.
*
TEST = .FALSE.
IF (INDEIG) THEN
IF (IL.EQ.1 .AND. IU.EQ.N) THEN
TEST = .TRUE.
END IF
END IF
IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
INDEE = INDWRK + 2*N
IF( .NOT.WANTZ ) THEN
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
$ WORK( INDWRK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 10 I = 1, N
IFAIL( I ) = 0
10 CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 30
END IF
INFO = 0
END IF
*
* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDIBL = 1
INDISP = INDIBL + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
$ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
$ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
* Apply orthogonal matrix used in reduction to tridiagonal
* form to eigenvectors returned by DSTEIN.
*
DO 20 J = 1, M
CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
$ Z( 1, J ), 1 )
20 CONTINUE
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
30 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 50 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 40 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
40 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( INDIBL+I-1 )
W( I ) = W( J )
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
W( J ) = TMP1
IWORK( INDIBL+J-1 ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
50 CONTINUE
END IF
*
RETURN
*
* End of DSBEVX
*
END
*> \brief \b DSBGST
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgst.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X,
* LDX, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO, VECT
* INTEGER INFO, KA, KB, LDAB, LDBB, LDX, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBGST reduces a real symmetric-definite banded generalized
*> eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y,
*> such that C has the same bandwidth as A.
*>
*> B must have been previously factorized as S**T*S by DPBSTF, using a
*> split Cholesky factorization. A is overwritten by C = X**T*A*X, where
*> X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
*> bandwidth of A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> = 'N': do not form the transformation matrix X;
*> = 'V': form X.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*> KA is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*> KB is INTEGER
*> The number of superdiagonals of the matrix B if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first ka+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
*>
*> On exit, the transformed matrix X**T*A*X, stored in the same
*> format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in] BB
*> \verbatim
*> BB is DOUBLE PRECISION array, dimension (LDBB,N)
*> The banded factor S from the split Cholesky factorization of
*> B, as returned by DPBSTF, stored in the first KB+1 rows of
*> the array.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*> LDBB is INTEGER
*> The leading dimension of the array BB. LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,N)
*> If VECT = 'V', the n-by-n matrix X.
*> If VECT = 'N', the array X is not referenced.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X.
*> LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgst
*
* =====================================================================
SUBROUTINE DSBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB,
$ X,
$ LDX, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO, VECT
INTEGER INFO, KA, KB, LDAB, LDBB, LDX, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPDATE, UPPER, WANTX
INTEGER I, I0, I1, I2, INCA, J, J1, J1T, J2, J2T, K,
$ KA1, KB1, KBT, L, M, NR, NRT, NX
DOUBLE PRECISION BII, RA, RA1, T
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGER, DLAR2V, DLARGV, DLARTG, DLARTV,
$ DLASET,
$ DROT, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTX = LSAME( VECT, 'V' )
UPPER = LSAME( UPLO, 'U' )
KA1 = KA + 1
KB1 = KB + 1
INFO = 0
IF( .NOT.WANTX .AND. .NOT.LSAME( VECT, 'N' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KA.LT.0 ) THEN
INFO = -4
ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
INFO = -5
ELSE IF( LDAB.LT.KA+1 ) THEN
INFO = -7
ELSE IF( LDBB.LT.KB+1 ) THEN
INFO = -9
ELSE IF( LDX.LT.1 .OR. WANTX .AND. LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBGST', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
INCA = LDAB*KA1
*
* Initialize X to the unit matrix, if needed
*
IF( WANTX )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, X, LDX )
*
* Set M to the splitting point m. It must be the same value as is
* used in DPBSTF. The chosen value allows the arrays WORK and RWORK
* to be of dimension (N).
*
M = ( N+KB ) / 2
*
* The routine works in two phases, corresponding to the two halves
* of the split Cholesky factorization of B as S**T*S where
*
* S = ( U )
* ( M L )
*
* with U upper triangular of order m, and L lower triangular of
* order n-m. S has the same bandwidth as B.
*
* S is treated as a product of elementary matrices:
*
* S = S(m)*S(m-1)*...*S(2)*S(1)*S(m+1)*S(m+2)*...*S(n-1)*S(n)
*
* where S(i) is determined by the i-th row of S.
*
* In phase 1, the index i takes the values n, n-1, ... , m+1;
* in phase 2, it takes the values 1, 2, ... , m.
*
* For each value of i, the current matrix A is updated by forming
* inv(S(i))**T*A*inv(S(i)). This creates a triangular bulge outside
* the band of A. The bulge is then pushed down toward the bottom of
* A in phase 1, and up toward the top of A in phase 2, by applying
* plane rotations.
*
* There are kb*(kb+1)/2 elements in the bulge, but at most 2*kb-1
* of them are linearly independent, so annihilating a bulge requires
* only 2*kb-1 plane rotations. The rotations are divided into a 1st
* set of kb-1 rotations, and a 2nd set of kb rotations.
*
* Wherever possible, rotations are generated and applied in vector
* operations of length NR between the indices J1 and J2 (sometimes
* replaced by modified values NRT, J1T or J2T).
*
* The cosines and sines of the rotations are stored in the array
* WORK. The cosines of the 1st set of rotations are stored in
* elements n+2:n+m-kb-1 and the sines of the 1st set in elements
* 2:m-kb-1; the cosines of the 2nd set are stored in elements
* n+m-kb+1:2*n and the sines of the second set in elements m-kb+1:n.
*
* The bulges are not formed explicitly; nonzero elements outside the
* band are created only when they are required for generating new
* rotations; they are stored in the array WORK, in positions where
* they are later overwritten by the sines of the rotations which
* annihilate them.
*
* **************************** Phase 1 *****************************
*
* The logical structure of this phase is:
*
* UPDATE = .TRUE.
* DO I = N, M + 1, -1
* use S(i) to update A and create a new bulge
* apply rotations to push all bulges KA positions downward
* END DO
* UPDATE = .FALSE.
* DO I = M + KA + 1, N - 1
* apply rotations to push all bulges KA positions downward
* END DO
*
* To avoid duplicating code, the two loops are merged.
*
UPDATE = .TRUE.
I = N + 1
10 CONTINUE
IF( UPDATE ) THEN
I = I - 1
KBT = MIN( KB, I-1 )
I0 = I - 1
I1 = MIN( N, I+KA )
I2 = I - KBT + KA1
IF( I.LT.M+1 ) THEN
UPDATE = .FALSE.
I = I + 1
I0 = M
IF( KA.EQ.0 )
$ GO TO 480
GO TO 10
END IF
ELSE
I = I + KA
IF( I.GT.N-1 )
$ GO TO 480
END IF
*
IF( UPPER ) THEN
*
* Transform A, working with the upper triangle
*
IF( UPDATE ) THEN
*
* Form inv(S(i))**T * A * inv(S(i))
*
BII = BB( KB1, I )
DO 20 J = I, I1
AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII
20 CONTINUE
DO 30 J = MAX( 1, I-KA ), I
AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII
30 CONTINUE
DO 60 K = I - KBT, I - 1
DO 40 J = I - KBT, K
AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
$ BB( J-I+KB1, I )*AB( K-I+KA1, I ) -
$ BB( K-I+KB1, I )*AB( J-I+KA1, I ) +
$ AB( KA1, I )*BB( J-I+KB1, I )*
$ BB( K-I+KB1, I )
40 CONTINUE
DO 50 J = MAX( 1, I-KA ), I - KBT - 1
AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
$ BB( K-I+KB1, I )*AB( J-I+KA1, I )
50 CONTINUE
60 CONTINUE
DO 80 J = I, I1
DO 70 K = MAX( J-KA, I-KBT ), I - 1
AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
$ BB( K-I+KB1, I )*AB( I-J+KA1, J )
70 CONTINUE
80 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by inv(S(i))
*
CALL DSCAL( N-M, ONE / BII, X( M+1, I ), 1 )
IF( KBT.GT.0 )
$ CALL DGER( N-M, KBT, -ONE, X( M+1, I ), 1,
$ BB( KB1-KBT, I ), 1, X( M+1, I-KBT ), LDX )
END IF
*
* store a(i,i1) in RA1 for use in next loop over K
*
RA1 = AB( I-I1+KA1, I1 )
END IF
*
* Generate and apply vectors of rotations to chase all the
* existing bulges KA positions down toward the bottom of the
* band
*
DO 130 K = 1, KB - 1
IF( UPDATE ) THEN
*
* Determine the rotations which would annihilate the bulge
* which has in theory just been created
*
IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN
*
* generate rotation to annihilate a(i,i-k+ka+1)
*
CALL DLARTG( AB( K+1, I-K+KA ), RA1,
$ WORK( N+I-K+KA-M ), WORK( I-K+KA-M ),
$ RA )
*
* create nonzero element a(i-k,i-k+ka+1) outside the
* band and store it in WORK(i-k)
*
T = -BB( KB1-K, I )*RA1
WORK( I-K ) = WORK( N+I-K+KA-M )*T -
$ WORK( I-K+KA-M )*AB( 1, I-K+KA )
AB( 1, I-K+KA ) = WORK( I-K+KA-M )*T +
$ WORK( N+I-K+KA-M )*AB( 1, I-K+KA )
RA1 = RA
END IF
END IF
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( UPDATE ) THEN
J2T = MAX( J2, I+2*KA-K+1 )
ELSE
J2T = J2
END IF
NRT = ( N-J2T+KA ) / KA1
DO 90 J = J2T, J1, KA1
*
* create nonzero element a(j-ka,j+1) outside the band
* and store it in WORK(j-m)
*
WORK( J-M ) = WORK( J-M )*AB( 1, J+1 )
AB( 1, J+1 ) = WORK( N+J-M )*AB( 1, J+1 )
90 CONTINUE
*
* generate rotations in 1st set to annihilate elements which
* have been created outside the band
*
IF( NRT.GT.0 )
$ CALL DLARGV( NRT, AB( 1, J2T ), INCA, WORK( J2T-M ),
$ KA1,
$ WORK( N+J2T-M ), KA1 )
IF( NR.GT.0 ) THEN
*
* apply rotations in 1st set from the right
*
DO 100 L = 1, KA - 1
CALL DLARTV( NR, AB( KA1-L, J2 ), INCA,
$ AB( KA-L, J2+1 ), INCA, WORK( N+J2-M ),
$ WORK( J2-M ), KA1 )
100 CONTINUE
*
* apply rotations in 1st set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ),
$ AB( KA, J2+1 ), INCA, WORK( N+J2-M ),
$ WORK( J2-M ), KA1 )
*
END IF
*
* start applying rotations in 1st set from the left
*
DO 110 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA,
$ AB( L+1, J2+KA1-L ), INCA,
$ WORK( N+J2-M ), WORK( J2-M ), KA1 )
110 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 1st set
*
DO 120 J = J2, J1, KA1
CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ WORK( N+J-M ), WORK( J-M ) )
120 CONTINUE
END IF
130 CONTINUE
*
IF( UPDATE ) THEN
IF( I2.LE.N .AND. KBT.GT.0 ) THEN
*
* create nonzero element a(i-kbt,i-kbt+ka+1) outside the
* band and store it in WORK(i-kbt)
*
WORK( I-KBT ) = -BB( KB1-KBT, I )*RA1
END IF
END IF
*
DO 170 K = KB, 1, -1
IF( UPDATE ) THEN
J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1
ELSE
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
END IF
*
* finish applying rotations in 2nd set from the left
*
DO 140 L = KB - K, 1, -1
NRT = ( N-J2+KA+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J2-L+1 ), INCA,
$ AB( L+1, J2-L+1 ), INCA, WORK( N+J2-KA ),
$ WORK( J2-KA ), KA1 )
140 CONTINUE
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
DO 150 J = J1, J2, -KA1
WORK( J ) = WORK( J-KA )
WORK( N+J ) = WORK( N+J-KA )
150 CONTINUE
DO 160 J = J2, J1, KA1
*
* create nonzero element a(j-ka,j+1) outside the band
* and store it in WORK(j)
*
WORK( J ) = WORK( J )*AB( 1, J+1 )
AB( 1, J+1 ) = WORK( N+J )*AB( 1, J+1 )
160 CONTINUE
IF( UPDATE ) THEN
IF( I-K.LT.N-KA .AND. K.LE.KBT )
$ WORK( I-K+KA ) = WORK( I-K )
END IF
170 CONTINUE
*
DO 210 K = KB, 1, -1
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( NR.GT.0 ) THEN
*
* generate rotations in 2nd set to annihilate elements
* which have been created outside the band
*
CALL DLARGV( NR, AB( 1, J2 ), INCA, WORK( J2 ), KA1,
$ WORK( N+J2 ), KA1 )
*
* apply rotations in 2nd set from the right
*
DO 180 L = 1, KA - 1
CALL DLARTV( NR, AB( KA1-L, J2 ), INCA,
$ AB( KA-L, J2+1 ), INCA, WORK( N+J2 ),
$ WORK( J2 ), KA1 )
180 CONTINUE
*
* apply rotations in 2nd set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ),
$ AB( KA, J2+1 ), INCA, WORK( N+J2 ),
$ WORK( J2 ), KA1 )
*
END IF
*
* start applying rotations in 2nd set from the left
*
DO 190 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA,
$ AB( L+1, J2+KA1-L ), INCA, WORK( N+J2 ),
$ WORK( J2 ), KA1 )
190 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 2nd set
*
DO 200 J = J2, J1, KA1
CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ WORK( N+J ), WORK( J ) )
200 CONTINUE
END IF
210 CONTINUE
*
DO 230 K = 1, KB - 1
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
*
* finish applying rotations in 1st set from the left
*
DO 220 L = KB - K, 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA,
$ AB( L+1, J2+KA1-L ), INCA,
$ WORK( N+J2-M ), WORK( J2-M ), KA1 )
220 CONTINUE
230 CONTINUE
*
IF( KB.GT.1 ) THEN
DO 240 J = N - 1, I - KB + 2*KA + 1, -1
WORK( N+J-M ) = WORK( N+J-KA-M )
WORK( J-M ) = WORK( J-KA-M )
240 CONTINUE
END IF
*
ELSE
*
* Transform A, working with the lower triangle
*
IF( UPDATE ) THEN
*
* Form inv(S(i))**T * A * inv(S(i))
*
BII = BB( 1, I )
DO 250 J = I, I1
AB( J-I+1, I ) = AB( J-I+1, I ) / BII
250 CONTINUE
DO 260 J = MAX( 1, I-KA ), I
AB( I-J+1, J ) = AB( I-J+1, J ) / BII
260 CONTINUE
DO 290 K = I - KBT, I - 1
DO 270 J = I - KBT, K
AB( K-J+1, J ) = AB( K-J+1, J ) -
$ BB( I-J+1, J )*AB( I-K+1, K ) -
$ BB( I-K+1, K )*AB( I-J+1, J ) +
$ AB( 1, I )*BB( I-J+1, J )*
$ BB( I-K+1, K )
270 CONTINUE
DO 280 J = MAX( 1, I-KA ), I - KBT - 1
AB( K-J+1, J ) = AB( K-J+1, J ) -
$ BB( I-K+1, K )*AB( I-J+1, J )
280 CONTINUE
290 CONTINUE
DO 310 J = I, I1
DO 300 K = MAX( J-KA, I-KBT ), I - 1
AB( J-K+1, K ) = AB( J-K+1, K ) -
$ BB( I-K+1, K )*AB( J-I+1, I )
300 CONTINUE
310 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by inv(S(i))
*
CALL DSCAL( N-M, ONE / BII, X( M+1, I ), 1 )
IF( KBT.GT.0 )
$ CALL DGER( N-M, KBT, -ONE, X( M+1, I ), 1,
$ BB( KBT+1, I-KBT ), LDBB-1,
$ X( M+1, I-KBT ), LDX )
END IF
*
* store a(i1,i) in RA1 for use in next loop over K
*
RA1 = AB( I1-I+1, I )
END IF
*
* Generate and apply vectors of rotations to chase all the
* existing bulges KA positions down toward the bottom of the
* band
*
DO 360 K = 1, KB - 1
IF( UPDATE ) THEN
*
* Determine the rotations which would annihilate the bulge
* which has in theory just been created
*
IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN
*
* generate rotation to annihilate a(i-k+ka+1,i)
*
CALL DLARTG( AB( KA1-K, I ), RA1,
$ WORK( N+I-K+KA-M ),
$ WORK( I-K+KA-M ), RA )
*
* create nonzero element a(i-k+ka+1,i-k) outside the
* band and store it in WORK(i-k)
*
T = -BB( K+1, I-K )*RA1
WORK( I-K ) = WORK( N+I-K+KA-M )*T -
$ WORK( I-K+KA-M )*AB( KA1, I-K )
AB( KA1, I-K ) = WORK( I-K+KA-M )*T +
$ WORK( N+I-K+KA-M )*AB( KA1, I-K )
RA1 = RA
END IF
END IF
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( UPDATE ) THEN
J2T = MAX( J2, I+2*KA-K+1 )
ELSE
J2T = J2
END IF
NRT = ( N-J2T+KA ) / KA1
DO 320 J = J2T, J1, KA1
*
* create nonzero element a(j+1,j-ka) outside the band
* and store it in WORK(j-m)
*
WORK( J-M ) = WORK( J-M )*AB( KA1, J-KA+1 )
AB( KA1, J-KA+1 ) = WORK( N+J-M )*AB( KA1, J-KA+1 )
320 CONTINUE
*
* generate rotations in 1st set to annihilate elements which
* have been created outside the band
*
IF( NRT.GT.0 )
$ CALL DLARGV( NRT, AB( KA1, J2T-KA ), INCA,
$ WORK( J2T-M ),
$ KA1, WORK( N+J2T-M ), KA1 )
IF( NR.GT.0 ) THEN
*
* apply rotations in 1st set from the left
*
DO 330 L = 1, KA - 1
CALL DLARTV( NR, AB( L+1, J2-L ), INCA,
$ AB( L+2, J2-L ), INCA, WORK( N+J2-M ),
$ WORK( J2-M ), KA1 )
330 CONTINUE
*
* apply rotations in 1st set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2,
$ J2 ),
$ INCA, WORK( N+J2-M ), WORK( J2-M ), KA1 )
*
END IF
*
* start applying rotations in 1st set from the right
*
DO 340 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
$ AB( KA1-L, J2+1 ), INCA, WORK( N+J2-M ),
$ WORK( J2-M ), KA1 )
340 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 1st set
*
DO 350 J = J2, J1, KA1
CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ WORK( N+J-M ), WORK( J-M ) )
350 CONTINUE
END IF
360 CONTINUE
*
IF( UPDATE ) THEN
IF( I2.LE.N .AND. KBT.GT.0 ) THEN
*
* create nonzero element a(i-kbt+ka+1,i-kbt) outside the
* band and store it in WORK(i-kbt)
*
WORK( I-KBT ) = -BB( KBT+1, I-KBT )*RA1
END IF
END IF
*
DO 400 K = KB, 1, -1
IF( UPDATE ) THEN
J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1
ELSE
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
END IF
*
* finish applying rotations in 2nd set from the right
*
DO 370 L = KB - K, 1, -1
NRT = ( N-J2+KA+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J2-KA ), INCA,
$ AB( KA1-L, J2-KA+1 ), INCA,
$ WORK( N+J2-KA ), WORK( J2-KA ), KA1 )
370 CONTINUE
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
DO 380 J = J1, J2, -KA1
WORK( J ) = WORK( J-KA )
WORK( N+J ) = WORK( N+J-KA )
380 CONTINUE
DO 390 J = J2, J1, KA1
*
* create nonzero element a(j+1,j-ka) outside the band
* and store it in WORK(j)
*
WORK( J ) = WORK( J )*AB( KA1, J-KA+1 )
AB( KA1, J-KA+1 ) = WORK( N+J )*AB( KA1, J-KA+1 )
390 CONTINUE
IF( UPDATE ) THEN
IF( I-K.LT.N-KA .AND. K.LE.KBT )
$ WORK( I-K+KA ) = WORK( I-K )
END IF
400 CONTINUE
*
DO 440 K = KB, 1, -1
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( NR.GT.0 ) THEN
*
* generate rotations in 2nd set to annihilate elements
* which have been created outside the band
*
CALL DLARGV( NR, AB( KA1, J2-KA ), INCA, WORK( J2 ),
$ KA1,
$ WORK( N+J2 ), KA1 )
*
* apply rotations in 2nd set from the left
*
DO 410 L = 1, KA - 1
CALL DLARTV( NR, AB( L+1, J2-L ), INCA,
$ AB( L+2, J2-L ), INCA, WORK( N+J2 ),
$ WORK( J2 ), KA1 )
410 CONTINUE
*
* apply rotations in 2nd set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2,
$ J2 ),
$ INCA, WORK( N+J2 ), WORK( J2 ), KA1 )
*
END IF
*
* start applying rotations in 2nd set from the right
*
DO 420 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
$ AB( KA1-L, J2+1 ), INCA, WORK( N+J2 ),
$ WORK( J2 ), KA1 )
420 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 2nd set
*
DO 430 J = J2, J1, KA1
CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ WORK( N+J ), WORK( J ) )
430 CONTINUE
END IF
440 CONTINUE
*
DO 460 K = 1, KB - 1
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
*
* finish applying rotations in 1st set from the right
*
DO 450 L = KB - K, 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
$ AB( KA1-L, J2+1 ), INCA, WORK( N+J2-M ),
$ WORK( J2-M ), KA1 )
450 CONTINUE
460 CONTINUE
*
IF( KB.GT.1 ) THEN
DO 470 J = N - 1, I - KB + 2*KA + 1, -1
WORK( N+J-M ) = WORK( N+J-KA-M )
WORK( J-M ) = WORK( J-KA-M )
470 CONTINUE
END IF
*
END IF
*
GO TO 10
*
480 CONTINUE
*
* **************************** Phase 2 *****************************
*
* The logical structure of this phase is:
*
* UPDATE = .TRUE.
* DO I = 1, M
* use S(i) to update A and create a new bulge
* apply rotations to push all bulges KA positions upward
* END DO
* UPDATE = .FALSE.
* DO I = M - KA - 1, 2, -1
* apply rotations to push all bulges KA positions upward
* END DO
*
* To avoid duplicating code, the two loops are merged.
*
UPDATE = .TRUE.
I = 0
490 CONTINUE
IF( UPDATE ) THEN
I = I + 1
KBT = MIN( KB, M-I )
I0 = I + 1
I1 = MAX( 1, I-KA )
I2 = I + KBT - KA1
IF( I.GT.M ) THEN
UPDATE = .FALSE.
I = I - 1
I0 = M + 1
IF( KA.EQ.0 )
$ RETURN
GO TO 490
END IF
ELSE
I = I - KA
IF( I.LT.2 )
$ RETURN
END IF
*
IF( I.LT.M-KBT ) THEN
NX = M
ELSE
NX = N
END IF
*
IF( UPPER ) THEN
*
* Transform A, working with the upper triangle
*
IF( UPDATE ) THEN
*
* Form inv(S(i))**T * A * inv(S(i))
*
BII = BB( KB1, I )
DO 500 J = I1, I
AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII
500 CONTINUE
DO 510 J = I, MIN( N, I+KA )
AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII
510 CONTINUE
DO 540 K = I + 1, I + KBT
DO 520 J = K, I + KBT
AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
$ BB( I-J+KB1, J )*AB( I-K+KA1, K ) -
$ BB( I-K+KB1, K )*AB( I-J+KA1, J ) +
$ AB( KA1, I )*BB( I-J+KB1, J )*
$ BB( I-K+KB1, K )
520 CONTINUE
DO 530 J = I + KBT + 1, MIN( N, I+KA )
AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
$ BB( I-K+KB1, K )*AB( I-J+KA1, J )
530 CONTINUE
540 CONTINUE
DO 560 J = I1, I
DO 550 K = I + 1, MIN( J+KA, I+KBT )
AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
$ BB( I-K+KB1, K )*AB( J-I+KA1, I )
550 CONTINUE
560 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by inv(S(i))
*
CALL DSCAL( NX, ONE / BII, X( 1, I ), 1 )
IF( KBT.GT.0 )
$ CALL DGER( NX, KBT, -ONE, X( 1, I ), 1, BB( KB,
$ I+1 ),
$ LDBB-1, X( 1, I+1 ), LDX )
END IF
*
* store a(i1,i) in RA1 for use in next loop over K
*
RA1 = AB( I1-I+KA1, I )
END IF
*
* Generate and apply vectors of rotations to chase all the
* existing bulges KA positions up toward the top of the band
*
DO 610 K = 1, KB - 1
IF( UPDATE ) THEN
*
* Determine the rotations which would annihilate the bulge
* which has in theory just been created
*
IF( I+K-KA1.GT.0 .AND. I+K.LT.M ) THEN
*
* generate rotation to annihilate a(i+k-ka-1,i)
*
CALL DLARTG( AB( K+1, I ), RA1, WORK( N+I+K-KA ),
$ WORK( I+K-KA ), RA )
*
* create nonzero element a(i+k-ka-1,i+k) outside the
* band and store it in WORK(m-kb+i+k)
*
T = -BB( KB1-K, I+K )*RA1
WORK( M-KB+I+K ) = WORK( N+I+K-KA )*T -
$ WORK( I+K-KA )*AB( 1, I+K )
AB( 1, I+K ) = WORK( I+K-KA )*T +
$ WORK( N+I+K-KA )*AB( 1, I+K )
RA1 = RA
END IF
END IF
J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
NR = ( J2+KA-1 ) / KA1
J1 = J2 - ( NR-1 )*KA1
IF( UPDATE ) THEN
J2T = MIN( J2, I-2*KA+K-1 )
ELSE
J2T = J2
END IF
NRT = ( J2T+KA-1 ) / KA1
DO 570 J = J1, J2T, KA1
*
* create nonzero element a(j-1,j+ka) outside the band
* and store it in WORK(j)
*
WORK( J ) = WORK( J )*AB( 1, J+KA-1 )
AB( 1, J+KA-1 ) = WORK( N+J )*AB( 1, J+KA-1 )
570 CONTINUE
*
* generate rotations in 1st set to annihilate elements which
* have been created outside the band
*
IF( NRT.GT.0 )
$ CALL DLARGV( NRT, AB( 1, J1+KA ), INCA, WORK( J1 ),
$ KA1,
$ WORK( N+J1 ), KA1 )
IF( NR.GT.0 ) THEN
*
* apply rotations in 1st set from the left
*
DO 580 L = 1, KA - 1
CALL DLARTV( NR, AB( KA1-L, J1+L ), INCA,
$ AB( KA-L, J1+L ), INCA, WORK( N+J1 ),
$ WORK( J1 ), KA1 )
580 CONTINUE
*
* apply rotations in 1st set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( KA1, J1 ), AB( KA1, J1-1 ),
$ AB( KA, J1 ), INCA, WORK( N+J1 ),
$ WORK( J1 ), KA1 )
*
END IF
*
* start applying rotations in 1st set from the right
*
DO 590 L = KA - 1, KB - K + 1, -1
NRT = ( J2+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J1T ), INCA,
$ AB( L+1, J1T-1 ), INCA, WORK( N+J1T ),
$ WORK( J1T ), KA1 )
590 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 1st set
*
DO 600 J = J1, J2, KA1
CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
$ WORK( N+J ), WORK( J ) )
600 CONTINUE
END IF
610 CONTINUE
*
IF( UPDATE ) THEN
IF( I2.GT.0 .AND. KBT.GT.0 ) THEN
*
* create nonzero element a(i+kbt-ka-1,i+kbt) outside the
* band and store it in WORK(m-kb+i+kbt)
*
WORK( M-KB+I+KBT ) = -BB( KB1-KBT, I+KBT )*RA1
END IF
END IF
*
DO 650 K = KB, 1, -1
IF( UPDATE ) THEN
J2 = I + K + 1 - MAX( 2, K+I0-M )*KA1
ELSE
J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
END IF
*
* finish applying rotations in 2nd set from the right
*
DO 620 L = KB - K, 1, -1
NRT = ( J2+KA+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J1T+KA ), INCA,
$ AB( L+1, J1T+KA-1 ), INCA,
$ WORK( N+M-KB+J1T+KA ),
$ WORK( M-KB+J1T+KA ), KA1 )
620 CONTINUE
NR = ( J2+KA-1 ) / KA1
J1 = J2 - ( NR-1 )*KA1
DO 630 J = J1, J2, KA1
WORK( M-KB+J ) = WORK( M-KB+J+KA )
WORK( N+M-KB+J ) = WORK( N+M-KB+J+KA )
630 CONTINUE
DO 640 J = J1, J2, KA1
*
* create nonzero element a(j-1,j+ka) outside the band
* and store it in WORK(m-kb+j)
*
WORK( M-KB+J ) = WORK( M-KB+J )*AB( 1, J+KA-1 )
AB( 1, J+KA-1 ) = WORK( N+M-KB+J )*AB( 1, J+KA-1 )
640 CONTINUE
IF( UPDATE ) THEN
IF( I+K.GT.KA1 .AND. K.LE.KBT )
$ WORK( M-KB+I+K-KA ) = WORK( M-KB+I+K )
END IF
650 CONTINUE
*
DO 690 K = KB, 1, -1
J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
NR = ( J2+KA-1 ) / KA1
J1 = J2 - ( NR-1 )*KA1
IF( NR.GT.0 ) THEN
*
* generate rotations in 2nd set to annihilate elements
* which have been created outside the band
*
CALL DLARGV( NR, AB( 1, J1+KA ), INCA,
$ WORK( M-KB+J1 ),
$ KA1, WORK( N+M-KB+J1 ), KA1 )
*
* apply rotations in 2nd set from the left
*
DO 660 L = 1, KA - 1
CALL DLARTV( NR, AB( KA1-L, J1+L ), INCA,
$ AB( KA-L, J1+L ), INCA,
$ WORK( N+M-KB+J1 ), WORK( M-KB+J1 ), KA1 )
660 CONTINUE
*
* apply rotations in 2nd set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( KA1, J1 ), AB( KA1, J1-1 ),
$ AB( KA, J1 ), INCA, WORK( N+M-KB+J1 ),
$ WORK( M-KB+J1 ), KA1 )
*
END IF
*
* start applying rotations in 2nd set from the right
*
DO 670 L = KA - 1, KB - K + 1, -1
NRT = ( J2+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J1T ), INCA,
$ AB( L+1, J1T-1 ), INCA,
$ WORK( N+M-KB+J1T ), WORK( M-KB+J1T ),
$ KA1 )
670 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 2nd set
*
DO 680 J = J1, J2, KA1
CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
$ WORK( N+M-KB+J ), WORK( M-KB+J ) )
680 CONTINUE
END IF
690 CONTINUE
*
DO 710 K = 1, KB - 1
J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
*
* finish applying rotations in 1st set from the right
*
DO 700 L = KB - K, 1, -1
NRT = ( J2+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L, J1T ), INCA,
$ AB( L+1, J1T-1 ), INCA, WORK( N+J1T ),
$ WORK( J1T ), KA1 )
700 CONTINUE
710 CONTINUE
*
IF( KB.GT.1 ) THEN
DO 720 J = 2, MIN( I+KB, M ) - 2*KA - 1
WORK( N+J ) = WORK( N+J+KA )
WORK( J ) = WORK( J+KA )
720 CONTINUE
END IF
*
ELSE
*
* Transform A, working with the lower triangle
*
IF( UPDATE ) THEN
*
* Form inv(S(i))**T * A * inv(S(i))
*
BII = BB( 1, I )
DO 730 J = I1, I
AB( I-J+1, J ) = AB( I-J+1, J ) / BII
730 CONTINUE
DO 740 J = I, MIN( N, I+KA )
AB( J-I+1, I ) = AB( J-I+1, I ) / BII
740 CONTINUE
DO 770 K = I + 1, I + KBT
DO 750 J = K, I + KBT
AB( J-K+1, K ) = AB( J-K+1, K ) -
$ BB( J-I+1, I )*AB( K-I+1, I ) -
$ BB( K-I+1, I )*AB( J-I+1, I ) +
$ AB( 1, I )*BB( J-I+1, I )*
$ BB( K-I+1, I )
750 CONTINUE
DO 760 J = I + KBT + 1, MIN( N, I+KA )
AB( J-K+1, K ) = AB( J-K+1, K ) -
$ BB( K-I+1, I )*AB( J-I+1, I )
760 CONTINUE
770 CONTINUE
DO 790 J = I1, I
DO 780 K = I + 1, MIN( J+KA, I+KBT )
AB( K-J+1, J ) = AB( K-J+1, J ) -
$ BB( K-I+1, I )*AB( I-J+1, J )
780 CONTINUE
790 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by inv(S(i))
*
CALL DSCAL( NX, ONE / BII, X( 1, I ), 1 )
IF( KBT.GT.0 )
$ CALL DGER( NX, KBT, -ONE, X( 1, I ), 1, BB( 2, I ),
$ 1,
$ X( 1, I+1 ), LDX )
END IF
*
* store a(i,i1) in RA1 for use in next loop over K
*
RA1 = AB( I-I1+1, I1 )
END IF
*
* Generate and apply vectors of rotations to chase all the
* existing bulges KA positions up toward the top of the band
*
DO 840 K = 1, KB - 1
IF( UPDATE ) THEN
*
* Determine the rotations which would annihilate the bulge
* which has in theory just been created
*
IF( I+K-KA1.GT.0 .AND. I+K.LT.M ) THEN
*
* generate rotation to annihilate a(i,i+k-ka-1)
*
CALL DLARTG( AB( KA1-K, I+K-KA ), RA1,
$ WORK( N+I+K-KA ), WORK( I+K-KA ), RA )
*
* create nonzero element a(i+k,i+k-ka-1) outside the
* band and store it in WORK(m-kb+i+k)
*
T = -BB( K+1, I )*RA1
WORK( M-KB+I+K ) = WORK( N+I+K-KA )*T -
$ WORK( I+K-KA )*AB( KA1, I+K-KA )
AB( KA1, I+K-KA ) = WORK( I+K-KA )*T +
$ WORK( N+I+K-KA )*AB( KA1, I+K-KA )
RA1 = RA
END IF
END IF
J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
NR = ( J2+KA-1 ) / KA1
J1 = J2 - ( NR-1 )*KA1
IF( UPDATE ) THEN
J2T = MIN( J2, I-2*KA+K-1 )
ELSE
J2T = J2
END IF
NRT = ( J2T+KA-1 ) / KA1
DO 800 J = J1, J2T, KA1
*
* create nonzero element a(j+ka,j-1) outside the band
* and store it in WORK(j)
*
WORK( J ) = WORK( J )*AB( KA1, J-1 )
AB( KA1, J-1 ) = WORK( N+J )*AB( KA1, J-1 )
800 CONTINUE
*
* generate rotations in 1st set to annihilate elements which
* have been created outside the band
*
IF( NRT.GT.0 )
$ CALL DLARGV( NRT, AB( KA1, J1 ), INCA, WORK( J1 ),
$ KA1,
$ WORK( N+J1 ), KA1 )
IF( NR.GT.0 ) THEN
*
* apply rotations in 1st set from the right
*
DO 810 L = 1, KA - 1
CALL DLARTV( NR, AB( L+1, J1 ), INCA, AB( L+2,
$ J1-1 ),
$ INCA, WORK( N+J1 ), WORK( J1 ), KA1 )
810 CONTINUE
*
* apply rotations in 1st set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( 1, J1 ), AB( 1, J1-1 ),
$ AB( 2, J1-1 ), INCA, WORK( N+J1 ),
$ WORK( J1 ), KA1 )
*
END IF
*
* start applying rotations in 1st set from the left
*
DO 820 L = KA - 1, KB - K + 1, -1
NRT = ( J2+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA,
$ AB( KA1-L, J1T-KA1+L ), INCA,
$ WORK( N+J1T ), WORK( J1T ), KA1 )
820 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 1st set
*
DO 830 J = J1, J2, KA1
CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
$ WORK( N+J ), WORK( J ) )
830 CONTINUE
END IF
840 CONTINUE
*
IF( UPDATE ) THEN
IF( I2.GT.0 .AND. KBT.GT.0 ) THEN
*
* create nonzero element a(i+kbt,i+kbt-ka-1) outside the
* band and store it in WORK(m-kb+i+kbt)
*
WORK( M-KB+I+KBT ) = -BB( KBT+1, I )*RA1
END IF
END IF
*
DO 880 K = KB, 1, -1
IF( UPDATE ) THEN
J2 = I + K + 1 - MAX( 2, K+I0-M )*KA1
ELSE
J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
END IF
*
* finish applying rotations in 2nd set from the left
*
DO 850 L = KB - K, 1, -1
NRT = ( J2+KA+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J1T+L-1 ), INCA,
$ AB( KA1-L, J1T+L-1 ), INCA,
$ WORK( N+M-KB+J1T+KA ),
$ WORK( M-KB+J1T+KA ), KA1 )
850 CONTINUE
NR = ( J2+KA-1 ) / KA1
J1 = J2 - ( NR-1 )*KA1
DO 860 J = J1, J2, KA1
WORK( M-KB+J ) = WORK( M-KB+J+KA )
WORK( N+M-KB+J ) = WORK( N+M-KB+J+KA )
860 CONTINUE
DO 870 J = J1, J2, KA1
*
* create nonzero element a(j+ka,j-1) outside the band
* and store it in WORK(m-kb+j)
*
WORK( M-KB+J ) = WORK( M-KB+J )*AB( KA1, J-1 )
AB( KA1, J-1 ) = WORK( N+M-KB+J )*AB( KA1, J-1 )
870 CONTINUE
IF( UPDATE ) THEN
IF( I+K.GT.KA1 .AND. K.LE.KBT )
$ WORK( M-KB+I+K-KA ) = WORK( M-KB+I+K )
END IF
880 CONTINUE
*
DO 920 K = KB, 1, -1
J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
NR = ( J2+KA-1 ) / KA1
J1 = J2 - ( NR-1 )*KA1
IF( NR.GT.0 ) THEN
*
* generate rotations in 2nd set to annihilate elements
* which have been created outside the band
*
CALL DLARGV( NR, AB( KA1, J1 ), INCA, WORK( M-KB+J1 ),
$ KA1, WORK( N+M-KB+J1 ), KA1 )
*
* apply rotations in 2nd set from the right
*
DO 890 L = 1, KA - 1
CALL DLARTV( NR, AB( L+1, J1 ), INCA, AB( L+2,
$ J1-1 ),
$ INCA, WORK( N+M-KB+J1 ), WORK( M-KB+J1 ),
$ KA1 )
890 CONTINUE
*
* apply rotations in 2nd set from both sides to diagonal
* blocks
*
CALL DLAR2V( NR, AB( 1, J1 ), AB( 1, J1-1 ),
$ AB( 2, J1-1 ), INCA, WORK( N+M-KB+J1 ),
$ WORK( M-KB+J1 ), KA1 )
*
END IF
*
* start applying rotations in 2nd set from the left
*
DO 900 L = KA - 1, KB - K + 1, -1
NRT = ( J2+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA,
$ AB( KA1-L, J1T-KA1+L ), INCA,
$ WORK( N+M-KB+J1T ), WORK( M-KB+J1T ),
$ KA1 )
900 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 2nd set
*
DO 910 J = J1, J2, KA1
CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
$ WORK( N+M-KB+J ), WORK( M-KB+J ) )
910 CONTINUE
END IF
920 CONTINUE
*
DO 940 K = 1, KB - 1
J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
*
* finish applying rotations in 1st set from the left
*
DO 930 L = KB - K, 1, -1
NRT = ( J2+L-1 ) / KA1
J1T = J2 - ( NRT-1 )*KA1
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA,
$ AB( KA1-L, J1T-KA1+L ), INCA,
$ WORK( N+J1T ), WORK( J1T ), KA1 )
930 CONTINUE
940 CONTINUE
*
IF( KB.GT.1 ) THEN
DO 950 J = 2, MIN( I+KB, M ) - 2*KA - 1
WORK( N+J ) = WORK( N+J+KA )
WORK( J ) = WORK( J+KA )
950 CONTINUE
END IF
*
END IF
*
GO TO 490
*
* End of DSBGST
*
END
*> \brief \b DSBGV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBGV( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z,
* LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ),
* $ WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBGV computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of
*> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
*> and banded, and B is also positive definite.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*> KA is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*> KB is INTEGER
*> The number of superdiagonals of the matrix B if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first ka+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
*>
*> On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*> BB is DOUBLE PRECISION array, dimension (LDBB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix B, stored in the first kb+1 rows of the array. The
*> j-th column of B is stored in the j-th column of the array BB
*> as follows:
*> if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
*>
*> On exit, the factor S from the split Cholesky factorization
*> B = S**T*S, as returned by DPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*> LDBB is INTEGER
*> The leading dimension of the array BB. LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*> eigenvectors, with the i-th column of Z holding the
*> eigenvector associated with W(i). The eigenvectors are
*> normalized so that Z**T*B*Z = I.
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is:
*> <= N: the algorithm failed to converge:
*> i off-diagonal elements of an intermediate
*> tridiagonal form did not converge to zero;
*> > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF
*> returned INFO = i: B is not positive definite.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgv
*
* =====================================================================
SUBROUTINE DSBGV( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
$ Z,
$ LDZ, WORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL UPPER, WANTZ
CHARACTER VECT
INTEGER IINFO, INDE, INDWRK
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPBSTF, DSBGST, DSBTRD, DSTEQR, DSTERF,
$ XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KA.LT.0 ) THEN
INFO = -4
ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
INFO = -5
ELSE IF( LDAB.LT.KA+1 ) THEN
INFO = -7
ELSE IF( LDBB.LT.KB+1 ) THEN
INFO = -9
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBGV', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form a split Cholesky factorization of B.
*
CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem.
*
INDE = 1
INDWRK = INDE + N
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,
$ WORK( INDWRK ), IINFO )
*
* Reduce to tridiagonal form.
*
IF( WANTZ ) THEN
VECT = 'U'
ELSE
VECT = 'N'
END IF
CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z,
$ LDZ,
$ WORK( INDWRK ), IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, call SSTEQR.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ,
$ WORK( INDWRK ),
$ INFO )
END IF
RETURN
*
* End of DSBGV
*
END
*> \brief \b DSBGVD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
* Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ),
* $ WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of the
*> form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and
*> banded, and B is also positive definite. If eigenvectors are
*> desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*> KA is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*> KB is INTEGER
*> The number of superdiagonals of the matrix B if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first ka+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
*>
*> On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*> BB is DOUBLE PRECISION array, dimension (LDBB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix B, stored in the first kb+1 rows of the array. The
*> j-th column of B is stored in the j-th column of the array BB
*> as follows:
*> if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
*>
*> On exit, the factor S from the split Cholesky factorization
*> B = S**T*S, as returned by DPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*> LDBB is INTEGER
*> The leading dimension of the array BB. LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*> eigenvectors, with the i-th column of Z holding the
*> eigenvector associated with W(i). The eigenvectors are
*> normalized so Z**T*B*Z = I.
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK >= 1.
*> If JOBZ = 'N' and N > 1, LWORK >= 2*N.
*> If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If JOBZ = 'N' or N <= 1, LIWORK >= 1.
*> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is:
*> <= N: the algorithm failed to converge:
*> i off-diagonal elements of an intermediate
*> tridiagonal form did not converge to zero;
*> > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF
*> returned INFO = i: B is not positive definite.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgvd
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB,
$ W,
$ Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER, WANTZ
CHARACTER VECT
INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLWRK2,
$ LWMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DPBSTF, DSBGST, DSBTRD,
$ DSTEDC,
$ DSTERF, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 5*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N
END IF
*
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KA.LT.0 ) THEN
INFO = -4
ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
INFO = -5
ELSE IF( LDAB.LT.KA+1 ) THEN
INFO = -7
ELSE IF( LDBB.LT.KB+1 ) THEN
INFO = -9
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -14
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBGVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form a split Cholesky factorization of B.
*
CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem.
*
INDE = 1
INDWRK = INDE + N
INDWK2 = INDWRK + N*N
LLWRK2 = LWORK - INDWK2 + 1
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,
$ WORK, IINFO )
*
* Reduce to tridiagonal form.
*
IF( WANTZ ) THEN
VECT = 'U'
ELSE
VECT = 'N'
END IF
CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z,
$ LDZ,
$ WORK( INDWRK ), IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
$ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ),
$ N,
$ ZERO, WORK( INDWK2 ), N )
CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
END IF
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSBGVD
*
END
*> \brief \b DSBGVX
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
* LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
* LDZ, WORK, IWORK, IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
* $ N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
* $ W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of
*> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
*> and banded, and B is also positive definite. Eigenvalues and
*> eigenvectors can be selected by specifying either all eigenvalues,
*> a range of values or a range of indices for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*> KA is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*> KB is INTEGER
*> The number of superdiagonals of the matrix B if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first ka+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
*>
*> On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*> BB is DOUBLE PRECISION array, dimension (LDBB, N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix B, stored in the first kb+1 rows of the array. The
*> j-th column of B is stored in the j-th column of the array BB
*> as follows:
*> if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
*>
*> On exit, the factor S from the split Cholesky factorization
*> B = S**T*S, as returned by DPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*> LDBB is INTEGER
*> The leading dimension of the array BB. LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> If JOBZ = 'V', the n-by-n matrix used in the reduction of
*> A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
*> and consequently C to tridiagonal form.
*> If JOBZ = 'N', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. If JOBZ = 'N',
*> LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*>
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*>
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*>
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*>
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing A to tridiagonal form.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*DLAMCH('S').
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*> eigenvectors, with the i-th column of Z holding the
*> eigenvector associated with W(i). The eigenvectors are
*> normalized so Z**T*B*Z = I.
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (M)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvalues that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> <= N: if INFO = i, then i eigenvectors failed to converge.
*> Their indices are stored in IFAIL.
*> > N: DPBSTF returned an error code; i.e.,
*> if INFO = N + i, for 1 <= i <= N, then the leading
*> principal minor of order i of B is not positive.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgvx
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
$ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
$ LDZ, WORK, IWORK, IFAIL, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
$ N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
$ W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
CHARACTER ORDER, VECT
INTEGER I, IINFO, INDD, INDE, INDEE, INDISP,
$ INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT
DOUBLE PRECISION TMP1
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DLACPY, DPBSTF, DSBGST,
$ DSBTRD,
$ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KA.LT.0 ) THEN
INFO = -5
ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
INFO = -6
ELSE IF( LDAB.LT.KA+1 ) THEN
INFO = -8
ELSE IF( LDBB.LT.KB+1 ) THEN
INFO = -10
ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
INFO = -12
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -14
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -16
END IF
END IF
END IF
IF( INFO.EQ.0) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -21
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBGVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
* Form a split Cholesky factorization of B.
*
CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem.
*
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
$ WORK, IINFO )
*
* Reduce symmetric band matrix to tridiagonal form.
*
INDD = 1
INDE = INDD + N
INDWRK = INDE + N
IF( WANTZ ) THEN
VECT = 'U'
ELSE
VECT = 'N'
END IF
CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, WORK( INDD ),
$ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
*
* If all eigenvalues are desired and ABSTOL is less than or equal
* to zero, then call DSTERF or SSTEQR. If this fails for some
* eigenvalue, then try DSTEBZ.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
INDEE = INDWRK + 2*N
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
$ WORK( INDWRK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 10 I = 1, N
IFAIL( I ) = 0
10 CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 30
END IF
INFO = 0
END IF
*
* Otherwise, call DSTEBZ and, if eigenvectors are desired,
* call DSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDISP = 1 + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
$ IWORK( 1 ), IWORK( INDISP ), WORK( INDWRK ),
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
$ IWORK( 1 ), IWORK( INDISP ), Z, LDZ,
$ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
* Apply transformation matrix used in reduction to tridiagonal
* form to eigenvectors returned by DSTEIN.
*
DO 20 J = 1, M
CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
$ Z( 1, J ), 1 )
20 CONTINUE
END IF
*
30 CONTINUE
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 50 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 40 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
40 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( 1 + I-1 )
W( I ) = W( J )
IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
W( J ) = TMP1
IWORK( 1 + J-1 ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
50 CONTINUE
END IF
*
RETURN
*
* End of DSBGVX
*
END
*> \brief \b DSBTRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbtrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbtrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbtrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO, VECT
* INTEGER INFO, KD, LDAB, LDQ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBTRD reduces a real symmetric band matrix A to symmetric
*> tridiagonal form T by an orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> = 'N': do not form Q;
*> = 'V': form Q;
*> = 'U': update a matrix X, by forming X*Q.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> On exit, the diagonal elements of AB are overwritten by the
*> diagonal elements of the tridiagonal matrix T; if KD > 0, the
*> elements on the first superdiagonal (if UPLO = 'U') or the
*> first subdiagonal (if UPLO = 'L') are overwritten by the
*> off-diagonal elements of T; the rest of AB is overwritten by
*> values generated during the reduction.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if VECT = 'U', then Q must contain an N-by-N
*> matrix X; if VECT = 'N' or 'V', then Q need not be set.
*>
*> On exit:
*> if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
*> if VECT = 'U', Q contains the product X*Q;
*> if VECT = 'N', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbtrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified by Linda Kaufman, Bell Labs.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
$ WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO, VECT
INTEGER INFO, KD, LDAB, LDQ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL INITQ, UPPER, WANTQ
INTEGER I, I2, IBL, INCA, INCX, IQAEND, IQB, IQEND, J,
$ J1, J1END, J1INC, J2, JEND, JIN, JINC, K, KD1,
$ KDM1, KDN, L, LAST, LEND, NQ, NR, NRT
DOUBLE PRECISION TEMP
* ..
* .. External Subroutines ..
EXTERNAL DLAR2V, DLARGV, DLARTG, DLARTV, DLASET,
$ DROT,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INITQ = LSAME( VECT, 'V' )
WANTQ = INITQ .OR. LSAME( VECT, 'U' )
UPPER = LSAME( UPLO, 'U' )
KD1 = KD + 1
KDM1 = KD - 1
INCX = LDAB - 1
IQEND = 1
*
INFO = 0
IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'N' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KD.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD1 ) THEN
INFO = -6
ELSE IF( LDQ.LT.MAX( 1, N ) .AND. WANTQ ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBTRD', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Initialize Q to the unit matrix, if needed
*
IF( INITQ )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
*
* Wherever possible, plane rotations are generated and applied in
* vector operations of length NR over the index set J1:J2:KD1.
*
* The cosines and sines of the plane rotations are stored in the
* arrays D and WORK.
*
INCA = KD1*LDAB
KDN = MIN( N-1, KD )
IF( UPPER ) THEN
*
IF( KD.GT.1 ) THEN
*
* Reduce to tridiagonal form, working with upper triangle
*
NR = 0
J1 = KDN + 2
J2 = 1
*
DO 90 I = 1, N - 2
*
* Reduce i-th row of matrix to tridiagonal form
*
DO 80 K = KDN + 1, 2, -1
J1 = J1 + KDN
J2 = J2 + KDN
*
IF( NR.GT.0 ) THEN
*
* generate plane rotations to annihilate nonzero
* elements which have been created outside the band
*
CALL DLARGV( NR, AB( 1, J1-1 ), INCA,
$ WORK( J1 ),
$ KD1, D( J1 ), KD1 )
*
* apply rotations from the right
*
*
* Dependent on the the number of diagonals either
* DLARTV or DROT is used
*
IF( NR.GE.2*KD-1 ) THEN
DO 10 L = 1, KD - 1
CALL DLARTV( NR, AB( L+1, J1-1 ), INCA,
$ AB( L, J1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
10 CONTINUE
*
ELSE
JEND = J1 + ( NR-1 )*KD1
DO 20 JINC = J1, JEND, KD1
CALL DROT( KDM1, AB( 2, JINC-1 ), 1,
$ AB( 1, JINC ), 1, D( JINC ),
$ WORK( JINC ) )
20 CONTINUE
END IF
END IF
*
*
IF( K.GT.2 ) THEN
IF( K.LE.N-I+1 ) THEN
*
* generate plane rotation to annihilate a(i,i+k-1)
* within the band
*
CALL DLARTG( AB( KD-K+3, I+K-2 ),
$ AB( KD-K+2, I+K-1 ), D( I+K-1 ),
$ WORK( I+K-1 ), TEMP )
AB( KD-K+3, I+K-2 ) = TEMP
*
* apply rotation from the right
*
CALL DROT( K-3, AB( KD-K+4, I+K-2 ), 1,
$ AB( KD-K+3, I+K-1 ), 1, D( I+K-1 ),
$ WORK( I+K-1 ) )
END IF
NR = NR + 1
J1 = J1 - KDN - 1
END IF
*
* apply plane rotations from both sides to diagonal
* blocks
*
IF( NR.GT.0 )
$ CALL DLAR2V( NR, AB( KD1, J1-1 ), AB( KD1, J1 ),
$ AB( KD, J1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
*
* apply plane rotations from the left
*
IF( NR.GT.0 ) THEN
IF( 2*KD-1.LT.NR ) THEN
*
* Dependent on the the number of diagonals either
* DLARTV or DROT is used
*
DO 30 L = 1, KD - 1
IF( J2+L.GT.N ) THEN
NRT = NR - 1
ELSE
NRT = NR
END IF
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( KD-L, J1+L ),
$ INCA,
$ AB( KD-L+1, J1+L ), INCA,
$ D( J1 ), WORK( J1 ), KD1 )
30 CONTINUE
ELSE
J1END = J1 + KD1*( NR-2 )
IF( J1END.GE.J1 ) THEN
DO 40 JIN = J1, J1END, KD1
CALL DROT( KD-1, AB( KD-1, JIN+1 ),
$ INCX,
$ AB( KD, JIN+1 ), INCX,
$ D( JIN ), WORK( JIN ) )
40 CONTINUE
END IF
LEND = MIN( KDM1, N-J2 )
LAST = J1END + KD1
IF( LEND.GT.0 )
$ CALL DROT( LEND, AB( KD-1, LAST+1 ), INCX,
$ AB( KD, LAST+1 ), INCX, D( LAST ),
$ WORK( LAST ) )
END IF
END IF
*
IF( WANTQ ) THEN
*
* accumulate product of plane rotations in Q
*
IF( INITQ ) THEN
*
* take advantage of the fact that Q was
* initially the Identity matrix
*
IQEND = MAX( IQEND, J2 )
I2 = MAX( 0, K-3 )
IQAEND = 1 + I*KD
IF( K.EQ.2 )
$ IQAEND = IQAEND + KD
IQAEND = MIN( IQAEND, IQEND )
DO 50 J = J1, J2, KD1
IBL = I - I2 / KDM1
I2 = I2 + 1
IQB = MAX( 1, J-IBL )
NQ = 1 + IQAEND - IQB
IQAEND = MIN( IQAEND+KD, IQEND )
CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB,
$ J ),
$ 1, D( J ), WORK( J ) )
50 CONTINUE
ELSE
*
DO 60 J = J1, J2, KD1
CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ),
$ 1,
$ D( J ), WORK( J ) )
60 CONTINUE
END IF
*
END IF
*
IF( J2+KDN.GT.N ) THEN
*
* adjust J2 to keep within the bounds of the matrix
*
NR = NR - 1
J2 = J2 - KDN - 1
END IF
*
DO 70 J = J1, J2, KD1
*
* create nonzero element a(j-1,j+kd) outside the band
* and store it in WORK
*
WORK( J+KD ) = WORK( J )*AB( 1, J+KD )
AB( 1, J+KD ) = D( J )*AB( 1, J+KD )
70 CONTINUE
80 CONTINUE
90 CONTINUE
END IF
*
IF( KD.GT.0 ) THEN
*
* copy off-diagonal elements to E
*
DO 100 I = 1, N - 1
E( I ) = AB( KD, I+1 )
100 CONTINUE
ELSE
*
* set E to zero if original matrix was diagonal
*
DO 110 I = 1, N - 1
E( I ) = ZERO
110 CONTINUE
END IF
*
* copy diagonal elements to D
*
DO 120 I = 1, N
D( I ) = AB( KD1, I )
120 CONTINUE
*
ELSE
*
IF( KD.GT.1 ) THEN
*
* Reduce to tridiagonal form, working with lower triangle
*
NR = 0
J1 = KDN + 2
J2 = 1
*
DO 210 I = 1, N - 2
*
* Reduce i-th column of matrix to tridiagonal form
*
DO 200 K = KDN + 1, 2, -1
J1 = J1 + KDN
J2 = J2 + KDN
*
IF( NR.GT.0 ) THEN
*
* generate plane rotations to annihilate nonzero
* elements which have been created outside the band
*
CALL DLARGV( NR, AB( KD1, J1-KD1 ), INCA,
$ WORK( J1 ), KD1, D( J1 ), KD1 )
*
* apply plane rotations from one side
*
*
* Dependent on the the number of diagonals either
* DLARTV or DROT is used
*
IF( NR.GT.2*KD-1 ) THEN
DO 130 L = 1, KD - 1
CALL DLARTV( NR, AB( KD1-L, J1-KD1+L ),
$ INCA,
$ AB( KD1-L+1, J1-KD1+L ), INCA,
$ D( J1 ), WORK( J1 ), KD1 )
130 CONTINUE
ELSE
JEND = J1 + KD1*( NR-1 )
DO 140 JINC = J1, JEND, KD1
CALL DROT( KDM1, AB( KD, JINC-KD ), INCX,
$ AB( KD1, JINC-KD ), INCX,
$ D( JINC ), WORK( JINC ) )
140 CONTINUE
END IF
*
END IF
*
IF( K.GT.2 ) THEN
IF( K.LE.N-I+1 ) THEN
*
* generate plane rotation to annihilate a(i+k-1,i)
* within the band
*
CALL DLARTG( AB( K-1, I ), AB( K, I ),
$ D( I+K-1 ), WORK( I+K-1 ), TEMP )
AB( K-1, I ) = TEMP
*
* apply rotation from the left
*
CALL DROT( K-3, AB( K-2, I+1 ), LDAB-1,
$ AB( K-1, I+1 ), LDAB-1, D( I+K-1 ),
$ WORK( I+K-1 ) )
END IF
NR = NR + 1
J1 = J1 - KDN - 1
END IF
*
* apply plane rotations from both sides to diagonal
* blocks
*
IF( NR.GT.0 )
$ CALL DLAR2V( NR, AB( 1, J1-1 ), AB( 1, J1 ),
$ AB( 2, J1-1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
*
* apply plane rotations from the right
*
*
* Dependent on the the number of diagonals either
* DLARTV or DROT is used
*
IF( NR.GT.0 ) THEN
IF( NR.GT.2*KD-1 ) THEN
DO 150 L = 1, KD - 1
IF( J2+L.GT.N ) THEN
NRT = NR - 1
ELSE
NRT = NR
END IF
IF( NRT.GT.0 )
$ CALL DLARTV( NRT, AB( L+2, J1-1 ),
$ INCA,
$ AB( L+1, J1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
150 CONTINUE
ELSE
J1END = J1 + KD1*( NR-2 )
IF( J1END.GE.J1 ) THEN
DO 160 J1INC = J1, J1END, KD1
CALL DROT( KDM1, AB( 3, J1INC-1 ), 1,
$ AB( 2, J1INC ), 1, D( J1INC ),
$ WORK( J1INC ) )
160 CONTINUE
END IF
LEND = MIN( KDM1, N-J2 )
LAST = J1END + KD1
IF( LEND.GT.0 )
$ CALL DROT( LEND, AB( 3, LAST-1 ), 1,
$ AB( 2, LAST ), 1, D( LAST ),
$ WORK( LAST ) )
END IF
END IF
*
*
*
IF( WANTQ ) THEN
*
* accumulate product of plane rotations in Q
*
IF( INITQ ) THEN
*
* take advantage of the fact that Q was
* initially the Identity matrix
*
IQEND = MAX( IQEND, J2 )
I2 = MAX( 0, K-3 )
IQAEND = 1 + I*KD
IF( K.EQ.2 )
$ IQAEND = IQAEND + KD
IQAEND = MIN( IQAEND, IQEND )
DO 170 J = J1, J2, KD1
IBL = I - I2 / KDM1
I2 = I2 + 1
IQB = MAX( 1, J-IBL )
NQ = 1 + IQAEND - IQB
IQAEND = MIN( IQAEND+KD, IQEND )
CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB,
$ J ),
$ 1, D( J ), WORK( J ) )
170 CONTINUE
ELSE
*
DO 180 J = J1, J2, KD1
CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ),
$ 1,
$ D( J ), WORK( J ) )
180 CONTINUE
END IF
END IF
*
IF( J2+KDN.GT.N ) THEN
*
* adjust J2 to keep within the bounds of the matrix
*
NR = NR - 1
J2 = J2 - KDN - 1
END IF
*
DO 190 J = J1, J2, KD1
*
* create nonzero element a(j+kd,j-1) outside the
* band and store it in WORK
*
WORK( J+KD ) = WORK( J )*AB( KD1, J )
AB( KD1, J ) = D( J )*AB( KD1, J )
190 CONTINUE
200 CONTINUE
210 CONTINUE
END IF
*
IF( KD.GT.0 ) THEN
*
* copy off-diagonal elements to E
*
DO 220 I = 1, N - 1
E( I ) = AB( 2, I )
220 CONTINUE
ELSE
*
* set E to zero if original matrix was diagonal
*
DO 230 I = 1, N - 1
E( I ) = ZERO
230 CONTINUE
END IF
*
* copy diagonal elements to D
*
DO 240 I = 1, N
D( I ) = AB( 1, I )
240 CONTINUE
END IF
*
RETURN
*
* End of DSBTRD
*
END
*> \brief \b DSFRK performs a symmetric rank-k operation for matrix in RFP format.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSFRK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsfrk.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsfrk.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsfrk.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSFRK( TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA, BETA,
* C )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION ALPHA, BETA
* INTEGER K, LDA, N
* CHARACTER TRANS, TRANSR, UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Level 3 BLAS like routine for C in RFP Format.
*>
*> DSFRK performs one of the symmetric rank--k operations
*>
*> C := alpha*A*A**T + beta*C,
*>
*> or
*>
*> C := alpha*A**T*A + beta*C,
*>
*> where alpha and beta are real scalars, C is an n--by--n symmetric
*> matrix and A is an n--by--k matrix in the first case and a k--by--n
*> matrix in the second case.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': The Normal Form of RFP A is stored;
*> = 'T': The Transpose Form of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the upper or lower
*> triangular part of the array C is to be referenced as
*> follows:
*>
*> UPLO = 'U' or 'u' Only the upper triangular part of C
*> is to be referenced.
*>
*> UPLO = 'L' or 'l' Only the lower triangular part of C
*> is to be referenced.
*>
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> On entry, TRANS specifies the operation to be performed as
*> follows:
*>
*> TRANS = 'N' or 'n' C := alpha*A*A**T + beta*C.
*>
*> TRANS = 'T' or 't' C := alpha*A**T*A + beta*C.
*>
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of the matrix C. N must be
*> at least zero.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> On entry with TRANS = 'N' or 'n', K specifies the number
*> of columns of the matrix A, and on entry with TRANS = 'T'
*> or 't', K specifies the number of rows of the matrix A. K
*> must be at least zero.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> On entry, ALPHA specifies the scalar alpha.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,ka)
*> where KA
*> is K when TRANS = 'N' or 'n', and is N otherwise. Before
*> entry with TRANS = 'N' or 'n', the leading N--by--K part of
*> the array A must contain the matrix A, otherwise the leading
*> K--by--N part of the array A must contain the matrix A.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> On entry, LDA specifies the first dimension of A as declared
*> in the calling (sub) program. When TRANS = 'N' or 'n'
*> then LDA must be at least max( 1, n ), otherwise LDA must
*> be at least max( 1, k ).
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> On entry, BETA specifies the scalar beta.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (NT)
*> NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
*> Format. RFP Format is described by TRANSR, UPLO and N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hfrk
*
* =====================================================================
SUBROUTINE DSFRK( TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA,
$ BETA,
$ C )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA, BETA
INTEGER K, LDA, N
CHARACTER TRANS, TRANSR, UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( * )
* ..
*
* =====================================================================
*
* ..
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, NORMALTRANSR, NISODD, NOTRANS
INTEGER INFO, NROWA, J, NK, N1, N2
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DGEMM, DSYRK
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
NOTRANS = LSAME( TRANS, 'N' )
*
IF( NOTRANS ) THEN
NROWA = N
ELSE
NROWA = K
END IF
*
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOTRANS .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NROWA ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSFRK ', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
* The quick return case: ((ALPHA.EQ.0).AND.(BETA.NE.ZERO)) is not
* done (it is in DSYRK for example) and left in the general case.
*
IF( ( N.EQ.0 ) .OR. ( ( ( ALPHA.EQ.ZERO ) .OR. ( K.EQ.0 ) ) .AND.
$ ( BETA.EQ.ONE ) ) )RETURN
*
IF( ( ALPHA.EQ.ZERO ) .AND. ( BETA.EQ.ZERO ) ) THEN
DO J = 1, ( ( N*( N+1 ) ) / 2 )
C( J ) = ZERO
END DO
RETURN
END IF
*
* C is N-by-N.
* If N is odd, set NISODD = .TRUE., and N1 and N2.
* If N is even, NISODD = .FALSE., and NK.
*
IF( MOD( N, 2 ).EQ.0 ) THEN
NISODD = .FALSE.
NK = N / 2
ELSE
NISODD = .TRUE.
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
END IF
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'N', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* N is odd, TRANSR = 'N', UPLO = 'L', and TRANS = 'N'
*
CALL DSYRK( 'L', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( 1 ), N )
CALL DSYRK( 'U', 'N', N2, K, ALPHA, A( N1+1, 1 ),
$ LDA,
$ BETA, C( N+1 ), N )
CALL DGEMM( 'N', 'T', N2, N1, K, ALPHA, A( N1+1,
$ 1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( N1+1 ), N )
*
ELSE
*
* N is odd, TRANSR = 'N', UPLO = 'L', and TRANS = 'T'
*
CALL DSYRK( 'L', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( 1 ), N )
CALL DSYRK( 'U', 'T', N2, K, ALPHA, A( 1, N1+1 ),
$ LDA,
$ BETA, C( N+1 ), N )
CALL DGEMM( 'T', 'N', N2, N1, K, ALPHA, A( 1,
$ N1+1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( N1+1 ), N )
*
END IF
*
ELSE
*
* N is odd, TRANSR = 'N', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* N is odd, TRANSR = 'N', UPLO = 'U', and TRANS = 'N'
*
CALL DSYRK( 'L', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( N2+1 ), N )
CALL DSYRK( 'U', 'N', N2, K, ALPHA, A( N2, 1 ),
$ LDA,
$ BETA, C( N1+1 ), N )
CALL DGEMM( 'N', 'T', N1, N2, K, ALPHA, A( 1, 1 ),
$ LDA, A( N2, 1 ), LDA, BETA, C( 1 ), N )
*
ELSE
*
* N is odd, TRANSR = 'N', UPLO = 'U', and TRANS = 'T'
*
CALL DSYRK( 'L', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( N2+1 ), N )
CALL DSYRK( 'U', 'T', N2, K, ALPHA, A( 1, N2 ),
$ LDA,
$ BETA, C( N1+1 ), N )
CALL DGEMM( 'T', 'N', N1, N2, K, ALPHA, A( 1, 1 ),
$ LDA, A( 1, N2 ), LDA, BETA, C( 1 ), N )
*
END IF
*
END IF
*
ELSE
*
* N is odd, and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'T', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* N is odd, TRANSR = 'T', UPLO = 'L', and TRANS = 'N'
*
CALL DSYRK( 'U', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( 1 ), N1 )
CALL DSYRK( 'L', 'N', N2, K, ALPHA, A( N1+1, 1 ),
$ LDA,
$ BETA, C( 2 ), N1 )
CALL DGEMM( 'N', 'T', N1, N2, K, ALPHA, A( 1, 1 ),
$ LDA, A( N1+1, 1 ), LDA, BETA,
$ C( N1*N1+1 ), N1 )
*
ELSE
*
* N is odd, TRANSR = 'T', UPLO = 'L', and TRANS = 'T'
*
CALL DSYRK( 'U', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( 1 ), N1 )
CALL DSYRK( 'L', 'T', N2, K, ALPHA, A( 1, N1+1 ),
$ LDA,
$ BETA, C( 2 ), N1 )
CALL DGEMM( 'T', 'N', N1, N2, K, ALPHA, A( 1, 1 ),
$ LDA, A( 1, N1+1 ), LDA, BETA,
$ C( N1*N1+1 ), N1 )
*
END IF
*
ELSE
*
* N is odd, TRANSR = 'T', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* N is odd, TRANSR = 'T', UPLO = 'U', and TRANS = 'N'
*
CALL DSYRK( 'U', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( N2*N2+1 ), N2 )
CALL DSYRK( 'L', 'N', N2, K, ALPHA, A( N1+1, 1 ),
$ LDA,
$ BETA, C( N1*N2+1 ), N2 )
CALL DGEMM( 'N', 'T', N2, N1, K, ALPHA, A( N1+1,
$ 1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( 1 ), N2 )
*
ELSE
*
* N is odd, TRANSR = 'T', UPLO = 'U', and TRANS = 'T'
*
CALL DSYRK( 'U', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( N2*N2+1 ), N2 )
CALL DSYRK( 'L', 'T', N2, K, ALPHA, A( 1, N1+1 ),
$ LDA,
$ BETA, C( N1*N2+1 ), N2 )
CALL DGEMM( 'T', 'N', N2, N1, K, ALPHA, A( 1,
$ N1+1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( 1 ), N2 )
*
END IF
*
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'N', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* N is even, TRANSR = 'N', UPLO = 'L', and TRANS = 'N'
*
CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( 2 ), N+1 )
CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( NK+1, 1 ),
$ LDA,
$ BETA, C( 1 ), N+1 )
CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( NK+1,
$ 1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( NK+2 ),
$ N+1 )
*
ELSE
*
* N is even, TRANSR = 'N', UPLO = 'L', and TRANS = 'T'
*
CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( 2 ), N+1 )
CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, NK+1 ),
$ LDA,
$ BETA, C( 1 ), N+1 )
CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1,
$ NK+1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( NK+2 ),
$ N+1 )
*
END IF
*
ELSE
*
* N is even, TRANSR = 'N', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* N is even, TRANSR = 'N', UPLO = 'U', and TRANS = 'N'
*
CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( NK+2 ), N+1 )
CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( NK+1, 1 ),
$ LDA,
$ BETA, C( NK+1 ), N+1 )
CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( 1, 1 ),
$ LDA, A( NK+1, 1 ), LDA, BETA, C( 1 ),
$ N+1 )
*
ELSE
*
* N is even, TRANSR = 'N', UPLO = 'U', and TRANS = 'T'
*
CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( NK+2 ), N+1 )
CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, NK+1 ),
$ LDA,
$ BETA, C( NK+1 ), N+1 )
CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1, 1 ),
$ LDA, A( 1, NK+1 ), LDA, BETA, C( 1 ),
$ N+1 )
*
END IF
*
END IF
*
ELSE
*
* N is even, and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'T', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* N is even, TRANSR = 'T', UPLO = 'L', and TRANS = 'N'
*
CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( NK+1 ), NK )
CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( NK+1, 1 ),
$ LDA,
$ BETA, C( 1 ), NK )
CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( 1, 1 ),
$ LDA, A( NK+1, 1 ), LDA, BETA,
$ C( ( ( NK+1 )*NK )+1 ), NK )
*
ELSE
*
* N is even, TRANSR = 'T', UPLO = 'L', and TRANS = 'T'
*
CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( NK+1 ), NK )
CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, NK+1 ),
$ LDA,
$ BETA, C( 1 ), NK )
CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1, 1 ),
$ LDA, A( 1, NK+1 ), LDA, BETA,
$ C( ( ( NK+1 )*NK )+1 ), NK )
*
END IF
*
ELSE
*
* N is even, TRANSR = 'T', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* N is even, TRANSR = 'T', UPLO = 'U', and TRANS = 'N'
*
CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( NK*( NK+1 )+1 ), NK )
CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( NK+1, 1 ),
$ LDA,
$ BETA, C( NK*NK+1 ), NK )
CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( NK+1,
$ 1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( 1 ), NK )
*
ELSE
*
* N is even, TRANSR = 'T', UPLO = 'U', and TRANS = 'T'
*
CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
$ BETA, C( NK*( NK+1 )+1 ), NK )
CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, NK+1 ),
$ LDA,
$ BETA, C( NK*NK+1 ), NK )
CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1,
$ NK+1 ),
$ LDA, A( 1, 1 ), LDA, BETA, C( 1 ), NK )
*
END IF
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of DSFRK
*
END
*> \brief \b DSPCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPCON( UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric packed matrix A using the factorization
*> A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by DSPTRF, stored as a
*> packed triangular matrix.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*> estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpcon
*
* =====================================================================
SUBROUTINE DSPCON( UPLO, N, AP, IPIV, ANORM, RCOND, WORK,
$ IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IP, KASE
DOUBLE PRECISION AINVNM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DSPTRS, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.LE.ZERO ) THEN
RETURN
END IF
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
IP = N*( N+1 ) / 2
DO 10 I = N, 1, -1
IF( IPIV( I ).GT.0 .AND. AP( IP ).EQ.ZERO )
$ RETURN
IP = IP - I
10 CONTINUE
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
IP = 1
DO 20 I = 1, N
IF( IPIV( I ).GT.0 .AND. AP( IP ).EQ.ZERO )
$ RETURN
IP = IP + N - I + 1
20 CONTINUE
END IF
*
* Estimate the 1-norm of the inverse.
*
KASE = 0
30 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
*
* Multiply by inv(L*D*L**T) or inv(U*D*U**T).
*
CALL DSPTRS( UPLO, N, 1, AP, IPIV, WORK, N, INFO )
GO TO 30
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
RETURN
*
* End of DSPCON
*
END
*> \brief <b> DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspev.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPEV computes all the eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A in packed storage.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, AP is overwritten by values generated during the
*> reduction to tridiagonal form. If UPLO = 'U', the diagonal
*> and first superdiagonal of the tridiagonal matrix T overwrite
*> the corresponding elements of A, and if UPLO = 'L', the
*> diagonal and first subdiagonal of T overwrite the
*> corresponding elements of A.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*> eigenvectors of the matrix A, with the i-th column of Z
*> holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, the algorithm failed to converge; i
*> off-diagonal elements of an intermediate tridiagonal
*> form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpev
*
* =====================================================================
SUBROUTINE DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL WANTZ
INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSP
EXTERNAL LSAME, DLAMCH, DLANSP
* ..
* .. External Subroutines ..
EXTERNAL DOPGTR, DSCAL, DSPTRD, DSTEQR, DSTERF,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR.
$ LSAME( UPLO, 'L' ) ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -7
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPEV ', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
W( 1 ) = AP( 1 )
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
END IF
*
* Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
CALL DSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ),
$ IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, first call
* DOPGTR to generate the orthogonal matrix, then call DSTEQR.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
INDWRK = INDTAU + N
CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
$ WORK( INDWRK ), IINFO )
CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ,
$ WORK( INDTAU ),
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
RETURN
*
* End of DSPEV
*
END
*> \brief <b> DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
* IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPEVD computes all the eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A in packed storage. If eigenvectors are
*> desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, AP is overwritten by values generated during the
*> reduction to tridiagonal form. If UPLO = 'U', the diagonal
*> and first superdiagonal of the tridiagonal matrix T overwrite
*> the corresponding elements of A, and if UPLO = 'L', the
*> diagonal and first subdiagonal of T overwrite the
*> corresponding elements of A.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*> eigenvectors of the matrix A, with the i-th column of Z
*> holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the required LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK must be at least 1.
*> If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
*> If JOBZ = 'V' and N > 1, LWORK must be at least
*> 1 + 6*N + N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the required sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
*> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the required sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, the algorithm failed to converge; i
*> off-diagonal elements of an intermediate tridiagonal
*> form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpevd
*
* =====================================================================
SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
$ IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WANTZ
INTEGER IINFO, INDE, INDTAU, INDWRK, ISCALE, LIWMIN,
$ LLWORK, LWMIN
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSP
EXTERNAL LSAME, DLAMCH, DLANSP
* ..
* .. External Subroutines ..
EXTERNAL DOPMTR, DSCAL, DSPTRD, DSTEDC, DSTERF,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR.
$ LSAME( UPLO, 'L' ) ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -7
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE
IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 6*N + N**2
ELSE
LIWMIN = 1
LWMIN = 2*N
END IF
END IF
IWORK( 1 ) = LIWMIN
WORK( 1 ) = LWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -9
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPEVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
W( 1 ) = AP( 1 )
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
END IF
*
* Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
CALL DSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ),
$ IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, first call
* DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
* tridiagonal matrix, then call DOPMTR to multiply it by the
* Householder transformations represented in AP.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
INDWRK = INDTAU + N
LLWORK = LWORK - INDWRK + 1
CALL DSTEDC( 'I', N, W, WORK( INDE ), Z, LDZ,
$ WORK( INDWRK ),
$ LLWORK, IWORK, LIWORK, INFO )
CALL DOPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z,
$ LDZ,
$ WORK( INDWRK ), IINFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 )
$ CALL DSCAL( N, ONE / SIGMA, W, 1 )
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
RETURN
*
* End of DSPEVD
*
END
*> \brief <b> DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
* ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, IU, LDZ, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A in packed storage. Eigenvalues/vectors
*> can be selected by specifying either a range of values or a range of
*> indices for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found;
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found;
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, AP is overwritten by values generated during the
*> reduction to tridiagonal form. If UPLO = 'U', the diagonal
*> and first superdiagonal of the tridiagonal matrix T overwrite
*> the corresponding elements of A, and if UPLO = 'L', the
*> diagonal and first subdiagonal of T overwrite the
*> corresponding elements of A.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing AP to tridiagonal form.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*DLAMCH('S').
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices
*> with Guaranteed High Relative Accuracy," by Demmel and
*> Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the selected eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If an eigenvector fails to converge, then that column of Z
*> contains the latest approximation to the eigenvector, and the
*> index of the eigenvector is returned in IFAIL.
*> If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (N)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvectors that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, then i eigenvectors failed to converge.
*> Their indices are stored in array IFAIL.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpevx
*
* =====================================================================
SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
$ ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
CHARACTER ORDER
INTEGER I, IINFO, IMAX, INDD, INDE, INDEE,
$ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
$ J, JJ, NSPLIT
DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
$ SIGMA, SMLNUM, TMP1, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSP
EXTERNAL LSAME, DLAMCH, DLANSP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DOPGTR, DOPMTR, DSCAL, DSPTRD,
$ DSTEBZ,
$ DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR.
$ LSAME( UPLO, 'U' ) ) )
$ THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -7
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -9
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
$ INFO = -14
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPEVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = AP( 1 )
ELSE
IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
M = 1
W( 1 ) = AP( 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
ABSTLL = ABSTOL
IF( VALEIG ) THEN
VLL = VL
VUU = VU
ELSE
VLL = ZERO
VUU = ZERO
END IF
ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
IF( ABSTOL.GT.0 )
$ ABSTLL = ABSTOL*SIGMA
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
*
* Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
INDTAU = 1
INDE = INDTAU + N
INDD = INDE + N
INDWRK = INDD + N
CALL DSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
$ WORK( INDTAU ), IINFO )
*
* If all eigenvalues are desired and ABSTOL is less than or equal
* to zero, then call DSTERF or DOPGTR and SSTEQR. If this fails
* for some eigenvalue, then try DSTEBZ.
*
TEST = .FALSE.
IF (INDEIG) THEN
IF (IL.EQ.1 .AND. IU.EQ.N) THEN
TEST = .TRUE.
END IF
END IF
IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
INDEE = INDWRK + 2*N
IF( .NOT.WANTZ ) THEN
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
$ WORK( INDWRK ), IINFO )
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
$ WORK( INDWRK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 10 I = 1, N
IFAIL( I ) = 0
10 CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 20
END IF
INFO = 0
END IF
*
* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDISP = 1 + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
$ IWORK( 1 ), IWORK( INDISP ), WORK( INDWRK ),
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
$ IWORK( 1 ), IWORK( INDISP ), Z, LDZ,
$ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
* Apply orthogonal matrix used in reduction to tridiagonal
* form to eigenvectors returned by DSTEIN.
*
CALL DOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z,
$ LDZ,
$ WORK( INDWRK ), IINFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
20 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 40 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 30 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
30 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( 1 + I-1 )
W( I ) = W( J )
IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
W( J ) = TMP1
IWORK( 1 + J-1 ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
40 CONTINUE
END IF
*
RETURN
*
* End of DSPEVX
*
END
*> \brief \b DSPGST
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgst.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, ITYPE, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), BP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPGST reduces a real symmetric-definite generalized eigenproblem
*> to standard form, using packed storage.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*>
*> B must have been previously factorized as U**T*U or L*L**T by DPPTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*> = 2 or 3: compute U*A*U**T or L**T*A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored and B is factored as
*> U**T*U;
*> = 'L': Lower triangle of A is stored and B is factored as
*> L*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, if INFO = 0, the transformed matrix, stored in the
*> same format as A.
*> \endverbatim
*>
*> \param[in] BP
*> \verbatim
*> BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The triangular factor from the Cholesky factorization of B,
*> stored in the same format as A, as returned by DPPTRF.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgst
*
* =====================================================================
SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), BP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, HALF
PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, J1, J1J1, JJ, K, K1, K1K1, KK
DOUBLE PRECISION AJJ, AKK, BJJ, BKK, CT
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DSPMV, DSPR2, DTPMV,
$ DTPSV,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPGST', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
* J1 and JJ are the indices of A(1,j) and A(j,j)
*
JJ = 0
DO 10 J = 1, N
J1 = JJ + 1
JJ = JJ + J
*
* Compute the j-th column of the upper triangle of A
*
BJJ = BP( JJ )
CALL DTPSV( UPLO, 'Transpose', 'Nonunit', J, BP,
$ AP( J1 ), 1 )
CALL DSPMV( UPLO, J-1, -ONE, AP, BP( J1 ), 1, ONE,
$ AP( J1 ), 1 )
CALL DSCAL( J-1, ONE / BJJ, AP( J1 ), 1 )
AP( JJ ) = ( AP( JJ )-DDOT( J-1, AP( J1 ), 1,
$ BP( J1 ),
$ 1 ) ) / BJJ
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
* KK and K1K1 are the indices of A(k,k) and A(k+1,k+1)
*
KK = 1
DO 20 K = 1, N
K1K1 = KK + N - K + 1
*
* Update the lower triangle of A(k:n,k:n)
*
AKK = AP( KK )
BKK = BP( KK )
AKK = AKK / BKK**2
AP( KK ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, AP( KK+1 ), 1 )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
CALL DSPR2( UPLO, N-K, -ONE, AP( KK+1 ), 1,
$ BP( KK+1 ), 1, AP( K1K1 ) )
CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
CALL DTPSV( UPLO, 'No transpose', 'Non-unit', N-K,
$ BP( K1K1 ), AP( KK+1 ), 1 )
END IF
KK = K1K1
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
* K1 and KK are the indices of A(1,k) and A(k,k)
*
KK = 0
DO 30 K = 1, N
K1 = KK + 1
KK = KK + K
*
* Update the upper triangle of A(1:k,1:k)
*
AKK = AP( KK )
BKK = BP( KK )
CALL DTPMV( UPLO, 'No transpose', 'Non-unit', K-1, BP,
$ AP( K1 ), 1 )
CT = HALF*AKK
CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
CALL DSPR2( UPLO, K-1, ONE, AP( K1 ), 1, BP( K1 ), 1,
$ AP )
CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
CALL DSCAL( K-1, BKK, AP( K1 ), 1 )
AP( KK ) = AKK*BKK**2
30 CONTINUE
ELSE
*
* Compute L**T *A*L
*
* JJ and J1J1 are the indices of A(j,j) and A(j+1,j+1)
*
JJ = 1
DO 40 J = 1, N
J1J1 = JJ + N - J + 1
*
* Compute the j-th column of the lower triangle of A
*
AJJ = AP( JJ )
BJJ = BP( JJ )
AP( JJ ) = AJJ*BJJ + DDOT( N-J, AP( JJ+1 ), 1,
$ BP( JJ+1 ), 1 )
CALL DSCAL( N-J, BJJ, AP( JJ+1 ), 1 )
CALL DSPMV( UPLO, N-J, ONE, AP( J1J1 ), BP( JJ+1 ), 1,
$ ONE, AP( JJ+1 ), 1 )
CALL DTPMV( UPLO, 'Transpose', 'Non-unit', N-J+1,
$ BP( JJ ), AP( JJ ), 1 )
JJ = J1J1
40 CONTINUE
END IF
END IF
RETURN
*
* End of DSPGST
*
END
*> \brief \b DSPGV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, ITYPE, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPGV computes all the eigenvalues and, optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.
*> Here A and B are assumed to be symmetric, stored in packed format,
*> and B is also positive definite.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> Specifies the problem type to be solved:
*> = 1: A*x = (lambda)*B*x
*> = 2: A*B*x = (lambda)*x
*> = 3: B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*> BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> B, packed columnwise in a linear array. The j-th column of B
*> is stored in the array BP as follows:
*> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*> On exit, the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T, in the same storage
*> format as B.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*> eigenvectors. The eigenvectors are normalized as follows:
*> if ITYPE = 1 or 2, Z**T*B*Z = I;
*> if ITYPE = 3, Z**T*inv(B)*Z = I.
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: DPPTRF or DSPEV returned an error code:
*> <= N: if INFO = i, DSPEV failed to converge;
*> i off-diagonal elements of an intermediate
*> tridiagonal form did not converge to zero.
*> > N: if INFO = n + i, for 1 <= i <= n, then the leading
*> principal minor of order i of B is not positive.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgv
*
* =====================================================================
SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ,
$ WORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL UPPER, WANTZ
CHARACTER TRANS
INTEGER J, NEIG
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPPTRF, DSPEV, DSPGST, DTPMV, DTPSV,
$ XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
*
INFO = 0
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPGV ', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form a Cholesky factorization of B.
*
CALL DPPTRF( UPLO, N, BP, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
CALL DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO )
*
IF( WANTZ ) THEN
*
* Backtransform eigenvectors to the original problem.
*
NEIG = N
IF( INFO.GT.0 )
$ NEIG = INFO - 1
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
DO 10 J = 1, NEIG
CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
10 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* For B*A*x=(lambda)*x;
* backtransform eigenvectors: x = L*y or U**T*y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
DO 20 J = 1, NEIG
CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
20 CONTINUE
END IF
END IF
RETURN
*
* End of DSPGV
*
END
*> \brief \b DSPGVD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK,
* LWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
*> B are assumed to be symmetric, stored in packed format, and B is also
*> positive definite.
*> If eigenvectors are desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> Specifies the problem type to be solved:
*> = 1: A*x = (lambda)*B*x
*> = 2: A*B*x = (lambda)*x
*> = 3: B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*> BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> B, packed columnwise in a linear array. The j-th column of B
*> is stored in the array BP as follows:
*> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*> On exit, the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T, in the same storage
*> format as B.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*> eigenvectors. The eigenvectors are normalized as follows:
*> if ITYPE = 1 or 2, Z**T*B*Z = I;
*> if ITYPE = 3, Z**T*inv(B)*Z = I.
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the required LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK >= 1.
*> If JOBZ = 'N' and N > 1, LWORK >= 2*N.
*> If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the required sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If JOBZ = 'N' or N <= 1, LIWORK >= 1.
*> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the required sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: DPPTRF or DSPEVD returned an error code:
*> <= N: if INFO = i, DSPEVD failed to converge;
*> i off-diagonal elements of an intermediate
*> tridiagonal form did not converge to zero;
*> > N: if INFO = N + i, for 1 <= i <= N, then the leading
*> principal minor of order i of B is not positive.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgvd
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ,
$ WORK,
$ LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY, UPPER, WANTZ
CHARACTER TRANS
INTEGER J, LIWMIN, LWMIN, NEIG
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPPTRF, DSPEVD, DSPGST, DTPMV, DTPSV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE
IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 6*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N
END IF
END IF
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPGVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form a Cholesky factorization of BP.
*
CALL DPPTRF( UPLO, N, BP, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
CALL DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
LWMIN = INT( MAX( DBLE( LWMIN ), DBLE( WORK( 1 ) ) ) )
LIWMIN = INT( MAX( DBLE( LIWMIN ), DBLE( IWORK( 1 ) ) ) )
*
IF( WANTZ ) THEN
*
* Backtransform eigenvectors to the original problem.
*
NEIG = N
IF( INFO.GT.0 )
$ NEIG = INFO - 1
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
* backtransform eigenvectors: x = inv(L)**T *y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
DO 10 J = 1, NEIG
CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
10 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* For B*A*x=(lambda)*x;
* backtransform eigenvectors: x = L*y or U**T *y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
DO 20 J = 1, NEIG
CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
20 CONTINUE
END IF
END IF
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSPGVD
*
END
*> \brief \b DSPGVX
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
* IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
* IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
*> and B are assumed to be symmetric, stored in packed storage, and B
*> is also positive definite. Eigenvalues and eigenvectors can be
*> selected by specifying either a range of values or a range of indices
*> for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> Specifies the problem type to be solved:
*> = 1: A*x = (lambda)*B*x
*> = 2: A*B*x = (lambda)*x
*> = 3: B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A and B are stored;
*> = 'L': Lower triangle of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix pencil (A,B). N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*> BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> B, packed columnwise in a linear array. The j-th column of B
*> is stored in the array BP as follows:
*> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*> On exit, the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T, in the same storage
*> format as B.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*>
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*>
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*>
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*>
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing A to tridiagonal form.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*DLAMCH('S').
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> On normal exit, the first M elements contain the selected
*> eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*> If JOBZ = 'N', then Z is not referenced.
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> The eigenvectors are normalized as follows:
*> if ITYPE = 1 or 2, Z**T*B*Z = I;
*> if ITYPE = 3, Z**T*inv(B)*Z = I.
*>
*> If an eigenvector fails to converge, then that column of Z
*> contains the latest approximation to the eigenvector, and the
*> index of the eigenvector is returned in IFAIL.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (N)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvectors that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: DPPTRF or DSPEVX returned an error code:
*> <= N: if INFO = i, DSPEVX failed to converge;
*> i eigenvectors failed to converge. Their indices
*> are stored in array IFAIL.
*> > N: if INFO = N + i, for 1 <= i <= N, then the leading
*> principal minor of order i of B is not positive.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgvx
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
$ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
$ IFAIL, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
CHARACTER TRANS
INTEGER J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
UPPER = LSAME( UPLO, 'U' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
INFO = 0
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -3
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL ) THEN
INFO = -9
END IF
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 ) THEN
INFO = -10
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -11
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -16
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPGVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
* Form a Cholesky factorization of B.
*
CALL DPPTRF( UPLO, N, BP, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL,
$ M,
$ W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
IF( WANTZ ) THEN
*
* Backtransform eigenvectors to the original problem.
*
IF( INFO.GT.0 )
$ M = INFO - 1
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
DO 10 J = 1, M
CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
10 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* For B*A*x=(lambda)*x;
* backtransform eigenvectors: x = L*y or U**T*y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
DO 20 J = 1, M
CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
20 CONTINUE
END IF
END IF
*
RETURN
*
* End of DSPGVX
*
END
*> \brief \b DSPRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsprfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsprfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsprfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
* FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric indefinite
*> and packed, and provides error bounds and backward error estimates
*> for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangle of the symmetric matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] AFP
*> \verbatim
*> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The factored form of the matrix A. AFP contains the block
*> diagonal matrix D and the multipliers used to obtain the
*> factor U or L from the factorization A = U*D*U**T or
*> A = L*D*L**T as computed by DSPTRF, stored as a packed
*> triangular matrix.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DSPTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hprfs
*
* =====================================================================
SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
$ LDX,
$ FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE,
$ WORK( N+1 ),
$ 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(A)*abs(X) + abs(B).
*
KK = 1
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
IK = KK
DO 40 I = 1, K - 1
WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
IK = IK + 1
40 CONTINUE
WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
KK = KK + K
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
IK = KK + 1
DO 60 I = K + 1, N
WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
IK = IK + 1
60 CONTINUE
WORK( K ) = WORK( K ) + S
KK = KK + ( N-K+1 )
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
$ INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(A))*
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(A) is the inverse of A
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(A)*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(A) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(A**T).
*
CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
$ INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
110 CONTINUE
ELSE IF( KASE.EQ.2 ) THEN
*
* Multiply by inv(A)*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
120 CONTINUE
CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
$ INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DSPRFS
*
END
*> \brief <b> DSPSV computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPSV computes the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric matrix stored in packed format and X
*> and B are N-by-NRHS matrices.
*>
*> The diagonal pivoting method is used to factor A as
*> A = U * D * U**T, if UPLO = 'U', or
*> A = L * D * L**T, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, D is symmetric and block diagonal with 1-by-1
*> and 2-by-2 diagonal blocks. The factored form of A is then used to
*> solve the system of equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> See below for further details.
*>
*> On exit, the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L from the factorization
*> A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*> a packed triangular matrix in the same storage format as A.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D, as
*> determined by DSPTRF. If IPIV(k) > 0, then rows and columns
*> k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
*> diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
*> then rows and columns k-1 and -IPIV(k) were interchanged and
*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
*> IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
*> -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
*> diagonal block.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, so the solution could not be
*> computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpsv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The packed storage scheme is illustrated by the following example
*> when N = 4, UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13 a14
*> a22 a23 a24
*> a33 a34 (aij = aji)
*> a44
*>
*> Packed storage of the upper triangle of A:
*>
*> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSPTRF, DSPTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPSV ', -INFO )
RETURN
END IF
*
* Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
CALL DSPTRF( UPLO, N, AP, IPIV, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
END IF
RETURN
*
* End of DSPSV
*
END
*> \brief <b> DSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
* LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER FACT, UPLO
* INTEGER INFO, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
*> A = L*D*L**T to compute the solution to a real system of linear
*> equations A * X = B, where A is an N-by-N symmetric matrix stored
*> in packed format and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the diagonal pivoting method is used to factor A as
*> A = U * D * U**T, if UPLO = 'U', or
*> A = L * D * L**T, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
*> returns with INFO = i. Otherwise, the factored form of A is used
*> to estimate the condition number of the matrix A. If the
*> reciprocal of the condition number is less than machine precision,
*> INFO = N+1 is returned as a warning, but the routine still goes on
*> to solve for X and compute error bounds as described below.
*>
*> 3. The system of equations is solved for X using the factored form
*> of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of A has been
*> supplied on entry.
*> = 'F': On entry, AFP and IPIV contain the factored form of
*> A. AP, AFP and IPIV will not be modified.
*> = 'N': The matrix A will be copied to AFP and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangle of the symmetric matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> See below for further details.
*> \endverbatim
*>
*> \param[in,out] AFP
*> \verbatim
*> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> If FACT = 'F', then AFP is an input argument and on entry
*> contains the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L from the factorization
*> A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*> a packed triangular matrix in the same storage format as A.
*>
*> If FACT = 'N', then AFP is an output argument and on exit
*> contains the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L from the factorization
*> A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*> a packed triangular matrix in the same storage format as A.
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains details of the interchanges and the block structure
*> of D, as determined by DSPTRF.
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*>
*> If FACT = 'N', then IPIV is an output argument and on exit
*> contains details of the interchanges and the block structure
*> of D, as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A. If RCOND is less than the machine precision (in
*> particular, if RCOND = 0), the matrix is singular to working
*> precision. This condition is indicated by a return code of
*> INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: D(i,i) is exactly zero. The factorization
*> has been completed but the factor D is exactly
*> singular, so the solution and error bounds could
*> not be computed. RCOND = 0 is returned.
*> = N+1: D is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpsvx
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The packed storage scheme is illustrated by the following example
*> when N = 4, UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13 a14
*> a22 a23 a24
*> a33 a34 (aij = aji)
*> a44
*>
*> Packed storage of the upper triangle of A:
*>
*> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB,
$ X,
$ LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOFACT
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSP
EXTERNAL LSAME, DLAMCH, DLANSP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DSPCON, DSPRFS, DSPTRF,
$ DSPTRS,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPSVX', -INFO )
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
CALL DSPTRF( UPLO, N, AFP, IPIV, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANSP( 'I', UPLO, N, AP, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DSPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* Compute the solution vectors X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DSPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
$ FERR,
$ BERR, WORK, IWORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DSPSVX
*
END
*> \brief \b DSPTRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPTRD reduces a real symmetric matrix A stored in packed form to
*> symmetric tridiagonal form T by an orthogonal similarity
*> transformation: Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> On exit, if UPLO = 'U', the diagonal and first superdiagonal
*> of A are overwritten by the corresponding elements of the
*> tridiagonal matrix T, and the elements above the first
*> superdiagonal, with the array TAU, represent the orthogonal
*> matrix Q as a product of elementary reflectors; if UPLO
*> = 'L', the diagonal and first subdiagonal of A are over-
*> written by the corresponding elements of the tridiagonal
*> matrix T, and the elements below the first subdiagonal, with
*> the array TAU, represent the orthogonal matrix Q as a product
*> of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n-1) . . . H(2) H(1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
*> overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(n-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
*> overwriting A(i+2:n,i), and tau is stored in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, HALF
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
$ HALF = 1.0D0 / 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, I1, I1I1, II
DOUBLE PRECISION ALPHA, TAUI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPTRD', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* I1 is the index in AP of A(1,I+1).
*
I1 = N*( N-1 ) / 2 + 1
DO 10 I = N - 1, 1, -1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(1:i-1,i+1)
*
CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
E( I ) = AP( I1+I-1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(1:i,1:i)
*
AP( I1+I-1 ) = ONE
*
* Compute y := tau * A * v storing y in TAU(1:i)
*
CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
$ 1 )
*
* Compute w := y - 1/2 * tau * (y**T *v) * v
*
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
*
AP( I1+I-1 ) = E( I )
END IF
D( I+1 ) = AP( I1+I )
TAU( I ) = TAUI
I1 = I1 - I
10 CONTINUE
D( 1 ) = AP( 1 )
ELSE
*
* Reduce the lower triangle of A. II is the index in AP of
* A(i,i) and I1I1 is the index of A(i+1,i+1).
*
II = 1
DO 20 I = 1, N - 1
I1I1 = II + N - I + 1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(i+2:n,i)
*
CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
E( I ) = AP( II+1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(i+1:n,i+1:n)
*
AP( II+1 ) = ONE
*
* Compute y := tau * A * v storing y in TAU(i:n-1)
*
CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ),
$ 1,
$ ZERO, TAU( I ), 1 )
*
* Compute w := y - 1/2 * tau * (y**T *v) * v
*
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
$ 1 )
CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ),
$ 1,
$ AP( I1I1 ) )
*
AP( II+1 ) = E( I )
END IF
D( I ) = AP( II )
TAU( I ) = TAUI
II = I1I1
20 CONTINUE
D( N ) = AP( II )
END IF
*
RETURN
*
* End of DSPTRD
*
END
*> \brief \b DSPTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPTRF computes the factorization of a real symmetric matrix A stored
*> in packed format using the Bunch-Kaufman diagonal pivoting method:
*>
*> A = U*D*U**T or A = L*D*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L, stored as a packed triangular
*> matrix overwriting A (see below for further details).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D.
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, and division by zero will occur if it
*> is used to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', then A = U*D*U**T, where
*> U = P(n)*U(n)* ... *P(k)U(k)* ...,
*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I v 0 ) k-s
*> U(k) = ( 0 I 0 ) s
*> ( 0 0 I ) n-k
*> k-s s n-k
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*> and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*> If UPLO = 'L', then A = L*D*L**T, where
*> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I 0 0 ) k-1
*> L(k) = ( 0 I 0 ) s
*> ( 0 v I ) n-k-s+1
*> k-1 s n-k-s+1
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> J. Lewis, Boeing Computer Services Company
*>
* =====================================================================
SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION EIGHT, SEVTEN
PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
$ KSTEP, KX, NPP
DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
$ ROWMAX, T, WK, WKM1, WKP1
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
EXTERNAL LSAME, IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSPR, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPTRF', -INFO )
RETURN
END IF
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
IF( UPPER ) THEN
*
* Factorize A as U*D*U**T using the upper triangle of A
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2
*
K = N
KC = ( N-1 )*N / 2 + 1
10 CONTINUE
KNC = KC
*
* If K < 1, exit from loop
*
IF( K.LT.1 )
$ GO TO 110
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( AP( KC+K-1 ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value
*
IF( K.GT.1 ) THEN
IMAX = IDAMAX( K-1, AP( KC ), 1 )
COLMAX = ABS( AP( KC+IMAX-1 ) )
ELSE
COLMAX = ZERO
END IF
*
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
* Column K is zero: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
ROWMAX = ZERO
JMAX = IMAX
KX = IMAX*( IMAX+1 ) / 2 + IMAX
DO 20 J = IMAX + 1, K
IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
ROWMAX = ABS( AP( KX ) )
JMAX = J
END IF
KX = KX + J
20 CONTINUE
KPC = ( IMAX-1 )*IMAX / 2 + 1
IF( IMAX.GT.1 ) THEN
JMAX = IDAMAX( IMAX-1, AP( KPC ), 1 )
ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-1 ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K-1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
KK = K - KSTEP + 1
IF( KSTEP.EQ.2 )
$ KNC = KNC - K + 1
IF( KP.NE.KK ) THEN
*
* Interchange rows and columns KK and KP in the leading
* submatrix A(1:k,1:k)
*
CALL DSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
KX = KPC + KP - 1
DO 30 J = KP + 1, KK - 1
KX = KX + J - 1
T = AP( KNC+J-1 )
AP( KNC+J-1 ) = AP( KX )
AP( KX ) = T
30 CONTINUE
T = AP( KNC+KK-1 )
AP( KNC+KK-1 ) = AP( KPC+KP-1 )
AP( KPC+KP-1 ) = T
IF( KSTEP.EQ.2 ) THEN
T = AP( KC+K-2 )
AP( KC+K-2 ) = AP( KC+KP-1 )
AP( KC+KP-1 ) = T
END IF
END IF
*
* Update the leading submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = U(k)*D(k)
*
* where U(k) is the k-th column of U
*
* Perform a rank-1 update of A(1:k-1,1:k-1) as
*
* A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
*
R1 = ONE / AP( KC+K-1 )
CALL DSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
*
* Store U(k) in column k
*
CALL DSCAL( K-1, R1, AP( KC ), 1 )
ELSE
*
* 2-by-2 pivot block D(k): columns k and k-1 now hold
*
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
* where U(k) and U(k-1) are the k-th and (k-1)-th columns
* of U
*
* Perform a rank-2 update of A(1:k-2,1:k-2) as
*
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
*
IF( K.GT.2 ) THEN
*
D12 = AP( K-1+( K-1 )*K / 2 )
D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
D11 = AP( K+( K-1 )*K / 2 ) / D12
T = ONE / ( D11*D22-ONE )
D12 = T / D12
*
DO 50 J = K - 2, 1, -1
WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
$ AP( J+( K-1 )*K / 2 ) )
WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
$ AP( J+( K-2 )*( K-1 ) / 2 ) )
DO 40 I = J, 1, -1
AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
$ AP( I+( K-1 )*K / 2 )*WK -
$ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
40 CONTINUE
AP( J+( K-1 )*K / 2 ) = WK
AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
50 CONTINUE
*
END IF
*
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K-1 ) = -KP
END IF
*
* Decrease K and return to the start of the main loop
*
K = K - KSTEP
KC = KNC - K
GO TO 10
*
ELSE
*
* Factorize A as L*D*L**T using the lower triangle of A
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2
*
K = 1
KC = 1
NPP = N*( N+1 ) / 2
60 CONTINUE
KNC = KC
*
* If K > N, exit from loop
*
IF( K.GT.N )
$ GO TO 110
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( AP( KC ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value
*
IF( K.LT.N ) THEN
IMAX = K + IDAMAX( N-K, AP( KC+1 ), 1 )
COLMAX = ABS( AP( KC+IMAX-K ) )
ELSE
COLMAX = ZERO
END IF
*
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
* Column K is zero: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
ROWMAX = ZERO
KX = KC + IMAX - K
DO 70 J = K, IMAX - 1
IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
ROWMAX = ABS( AP( KX ) )
JMAX = J
END IF
KX = KX + N - J
70 CONTINUE
KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
IF( IMAX.LT.N ) THEN
JMAX = IMAX + IDAMAX( N-IMAX, AP( KPC+1 ), 1 )
ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-IMAX ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K+1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
KK = K + KSTEP - 1
IF( KSTEP.EQ.2 )
$ KNC = KNC + N - K + 1
IF( KP.NE.KK ) THEN
*
* Interchange rows and columns KK and KP in the trailing
* submatrix A(k:n,k:n)
*
IF( KP.LT.N )
$ CALL DSWAP( N-KP, AP( KNC+KP-KK+1 ), 1,
$ AP( KPC+1 ),
$ 1 )
KX = KNC + KP - KK
DO 80 J = KK + 1, KP - 1
KX = KX + N - J + 1
T = AP( KNC+J-KK )
AP( KNC+J-KK ) = AP( KX )
AP( KX ) = T
80 CONTINUE
T = AP( KNC )
AP( KNC ) = AP( KPC )
AP( KPC ) = T
IF( KSTEP.EQ.2 ) THEN
T = AP( KC+1 )
AP( KC+1 ) = AP( KC+KP-K )
AP( KC+KP-K ) = T
END IF
END IF
*
* Update the trailing submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = L(k)*D(k)
*
* where L(k) is the k-th column of L
*
IF( K.LT.N ) THEN
*
* Perform a rank-1 update of A(k+1:n,k+1:n) as
*
* A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
*
R1 = ONE / AP( KC )
CALL DSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
$ AP( KC+N-K+1 ) )
*
* Store L(k) in column K
*
CALL DSCAL( N-K, R1, AP( KC+1 ), 1 )
END IF
ELSE
*
* 2-by-2 pivot block D(k): columns K and K+1 now hold
*
* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
*
* where L(k) and L(k+1) are the k-th and (k+1)-th columns
* of L
*
IF( K.LT.N-1 ) THEN
*
* Perform a rank-2 update of A(k+2:n,k+2:n) as
*
* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
*
* where L(k) and L(k+1) are the k-th and (k+1)-th
* columns of L
*
D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
T = ONE / ( D11*D22-ONE )
D21 = T / D21
*
DO 100 J = K + 2, N
WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
$ AP( J+K*( 2*N-K-1 ) / 2 ) )
WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
$ AP( J+( K-1 )*( 2*N-K ) / 2 ) )
*
DO 90 I = J, N
AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
$ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
$ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
90 CONTINUE
*
AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
*
100 CONTINUE
END IF
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K+1 ) = -KP
END IF
*
* Increase K and return to the start of the main loop
*
K = K + KSTEP
KC = KNC + N - K + 2
GO TO 60
*
END IF
*
110 CONTINUE
RETURN
*
* End of DSPTRF
*
END
*> \brief \b DSPTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPTRI( UPLO, N, AP, IPIV, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPTRI computes the inverse of a real symmetric indefinite matrix
*> A in packed storage using the factorization A = U*D*U**T or
*> A = L*D*L**T computed by DSPTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the block diagonal matrix D and the multipliers
*> used to obtain the factor U or L as computed by DSPTRF,
*> stored as a packed triangular matrix.
*>
*> On exit, if INFO = 0, the (symmetric) inverse of the original
*> matrix, stored as a packed triangular matrix. The j-th column
*> of inv(A) is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
*> if UPLO = 'L',
*> AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSPTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*> inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptri
*
* =====================================================================
SUBROUTINE DSPTRI( UPLO, N, AP, IPIV, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, K, KC, KCNEXT, KP, KPC, KSTEP, KX, NPP
DOUBLE PRECISION AK, AKKP1, AKP1, D, T, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSPMV, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
KP = N*( N+1 ) / 2
DO 10 INFO = N, 1, -1
IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO )
$ RETURN
KP = KP - INFO
10 CONTINUE
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
KP = 1
DO 20 INFO = 1, N
IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO )
$ RETURN
KP = KP + N - INFO + 1
20 CONTINUE
END IF
INFO = 0
*
IF( UPPER ) THEN
*
* Compute inv(A) from the factorization A = U*D*U**T.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
KC = 1
30 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 50
*
KCNEXT = KC + K
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
AP( KC+K-1 ) = ONE / AP( KC+K-1 )
*
* Compute column K of the inverse.
*
IF( K.GT.1 ) THEN
CALL DCOPY( K-1, AP( KC ), 1, WORK, 1 )
CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO,
$ AP( KC ),
$ 1 )
AP( KC+K-1 ) = AP( KC+K-1 ) -
$ DDOT( K-1, WORK, 1, AP( KC ), 1 )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( AP( KCNEXT+K-1 ) )
AK = AP( KC+K-1 ) / T
AKP1 = AP( KCNEXT+K ) / T
AKKP1 = AP( KCNEXT+K-1 ) / T
D = T*( AK*AKP1-ONE )
AP( KC+K-1 ) = AKP1 / D
AP( KCNEXT+K ) = AK / D
AP( KCNEXT+K-1 ) = -AKKP1 / D
*
* Compute columns K and K+1 of the inverse.
*
IF( K.GT.1 ) THEN
CALL DCOPY( K-1, AP( KC ), 1, WORK, 1 )
CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO,
$ AP( KC ),
$ 1 )
AP( KC+K-1 ) = AP( KC+K-1 ) -
$ DDOT( K-1, WORK, 1, AP( KC ), 1 )
AP( KCNEXT+K-1 ) = AP( KCNEXT+K-1 ) -
$ DDOT( K-1, AP( KC ), 1,
$ AP( KCNEXT ),
$ 1 )
CALL DCOPY( K-1, AP( KCNEXT ), 1, WORK, 1 )
CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO,
$ AP( KCNEXT ), 1 )
AP( KCNEXT+K ) = AP( KCNEXT+K ) -
$ DDOT( K-1, WORK, 1, AP( KCNEXT ), 1 )
END IF
KSTEP = 2
KCNEXT = KCNEXT + K + 1
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the leading
* submatrix A(1:k+1,1:k+1)
*
KPC = ( KP-1 )*KP / 2 + 1
CALL DSWAP( KP-1, AP( KC ), 1, AP( KPC ), 1 )
KX = KPC + KP - 1
DO 40 J = KP + 1, K - 1
KX = KX + J - 1
TEMP = AP( KC+J-1 )
AP( KC+J-1 ) = AP( KX )
AP( KX ) = TEMP
40 CONTINUE
TEMP = AP( KC+K-1 )
AP( KC+K-1 ) = AP( KPC+KP-1 )
AP( KPC+KP-1 ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = AP( KC+K+K-1 )
AP( KC+K+K-1 ) = AP( KC+K+KP-1 )
AP( KC+K+KP-1 ) = TEMP
END IF
END IF
*
K = K + KSTEP
KC = KCNEXT
GO TO 30
50 CONTINUE
*
ELSE
*
* Compute inv(A) from the factorization A = L*D*L**T.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
NPP = N*( N+1 ) / 2
K = N
KC = NPP
60 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 80
*
KCNEXT = KC - ( N-K+2 )
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
AP( KC ) = ONE / AP( KC )
*
* Compute column K of the inverse.
*
IF( K.LT.N ) THEN
CALL DCOPY( N-K, AP( KC+1 ), 1, WORK, 1 )
CALL DSPMV( UPLO, N-K, -ONE, AP( KC+N-K+1 ), WORK, 1,
$ ZERO, AP( KC+1 ), 1 )
AP( KC ) = AP( KC ) - DDOT( N-K, WORK, 1, AP( KC+1 ),
$ 1 )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( AP( KCNEXT+1 ) )
AK = AP( KCNEXT ) / T
AKP1 = AP( KC ) / T
AKKP1 = AP( KCNEXT+1 ) / T
D = T*( AK*AKP1-ONE )
AP( KCNEXT ) = AKP1 / D
AP( KC ) = AK / D
AP( KCNEXT+1 ) = -AKKP1 / D
*
* Compute columns K-1 and K of the inverse.
*
IF( K.LT.N ) THEN
CALL DCOPY( N-K, AP( KC+1 ), 1, WORK, 1 )
CALL DSPMV( UPLO, N-K, -ONE, AP( KC+( N-K+1 ) ), WORK,
$ 1,
$ ZERO, AP( KC+1 ), 1 )
AP( KC ) = AP( KC ) - DDOT( N-K, WORK, 1, AP( KC+1 ),
$ 1 )
AP( KCNEXT+1 ) = AP( KCNEXT+1 ) -
$ DDOT( N-K, AP( KC+1 ), 1,
$ AP( KCNEXT+2 ), 1 )
CALL DCOPY( N-K, AP( KCNEXT+2 ), 1, WORK, 1 )
CALL DSPMV( UPLO, N-K, -ONE, AP( KC+( N-K+1 ) ), WORK,
$ 1,
$ ZERO, AP( KCNEXT+2 ), 1 )
AP( KCNEXT ) = AP( KCNEXT ) -
$ DDOT( N-K, WORK, 1, AP( KCNEXT+2 ), 1 )
END IF
KSTEP = 2
KCNEXT = KCNEXT - ( N-K+3 )
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the trailing
* submatrix A(k-1:n,k-1:n)
*
KPC = NPP - ( N-KP+1 )*( N-KP+2 ) / 2 + 1
IF( KP.LT.N )
$ CALL DSWAP( N-KP, AP( KC+KP-K+1 ), 1, AP( KPC+1 ), 1 )
KX = KC + KP - K
DO 70 J = K + 1, KP - 1
KX = KX + N - J + 1
TEMP = AP( KC+J-K )
AP( KC+J-K ) = AP( KX )
AP( KX ) = TEMP
70 CONTINUE
TEMP = AP( KC )
AP( KC ) = AP( KPC )
AP( KPC ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = AP( KC-N+K-1 )
AP( KC-N+K-1 ) = AP( KC-N+KP-1 )
AP( KC-N+KP-1 ) = TEMP
END IF
END IF
*
K = K - KSTEP
KC = KCNEXT
GO TO 60
80 CONTINUE
END IF
*
RETURN
*
* End of DSPTRI
*
END
*> \brief \b DSPTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPTRS solves a system of linear equations A*X = B with a real
*> symmetric matrix A stored in packed format using the factorization
*> A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by DSPTRF, stored as a
*> packed triangular matrix.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptrs
*
* =====================================================================
SUBROUTINE DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, K, KC, KP
DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Solve A*X = B, where A = U*D*U**T.
*
* First solve U*D*X = B, overwriting B with X.
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
KC = N*( N+1 ) / 2 + 1
10 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 30
*
KC = KC - K
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(U(K)), where U(K) is the transformation
* stored in column K of A.
*
CALL DGER( K-1, NRHS, -ONE, AP( KC ), 1, B( K, 1 ), LDB,
$ B( 1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
CALL DSCAL( NRHS, ONE / AP( KC+K-1 ), B( K, 1 ), LDB )
K = K - 1
ELSE
*
* 2 x 2 diagonal block
*
* Interchange rows K-1 and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K-1 )
$ CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(U(K)), where U(K) is the transformation
* stored in columns K-1 and K of A.
*
CALL DGER( K-2, NRHS, -ONE, AP( KC ), 1, B( K, 1 ), LDB,
$ B( 1, 1 ), LDB )
CALL DGER( K-2, NRHS, -ONE, AP( KC-( K-1 ) ), 1,
$ B( K-1, 1 ), LDB, B( 1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
AKM1K = AP( KC+K-2 )
AKM1 = AP( KC-1 ) / AKM1K
AK = AP( KC+K-1 ) / AKM1K
DENOM = AKM1*AK - ONE
DO 20 J = 1, NRHS
BKM1 = B( K-1, J ) / AKM1K
BK = B( K, J ) / AKM1K
B( K-1, J ) = ( AK*BKM1-BK ) / DENOM
B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM
20 CONTINUE
KC = KC - K + 1
K = K - 2
END IF
*
GO TO 10
30 CONTINUE
*
* Next solve U**T*X = B, overwriting B with X.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
KC = 1
40 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 50
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Multiply by inv(U**T(K)), where U(K) is the transformation
* stored in column K of A.
*
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
$ AP( KC ),
$ 1, ONE, B( K, 1 ), LDB )
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
KC = KC + K
K = K + 1
ELSE
*
* 2 x 2 diagonal block
*
* Multiply by inv(U**T(K+1)), where U(K+1) is the transformation
* stored in columns K and K+1 of A.
*
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
$ AP( KC ),
$ 1, ONE, B( K, 1 ), LDB )
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
$ AP( KC+K ), 1, ONE, B( K+1, 1 ), LDB )
*
* Interchange rows K and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
KC = KC + 2*K + 1
K = K + 2
END IF
*
GO TO 40
50 CONTINUE
*
ELSE
*
* Solve A*X = B, where A = L*D*L**T.
*
* First solve L*D*X = B, overwriting B with X.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
KC = 1
60 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 80
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(L(K)), where L(K) is the transformation
* stored in column K of A.
*
IF( K.LT.N )
$ CALL DGER( N-K, NRHS, -ONE, AP( KC+1 ), 1, B( K, 1 ),
$ LDB, B( K+1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
CALL DSCAL( NRHS, ONE / AP( KC ), B( K, 1 ), LDB )
KC = KC + N - K + 1
K = K + 1
ELSE
*
* 2 x 2 diagonal block
*
* Interchange rows K+1 and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K+1 )
$ CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(L(K)), where L(K) is the transformation
* stored in columns K and K+1 of A.
*
IF( K.LT.N-1 ) THEN
CALL DGER( N-K-1, NRHS, -ONE, AP( KC+2 ), 1, B( K,
$ 1 ),
$ LDB, B( K+2, 1 ), LDB )
CALL DGER( N-K-1, NRHS, -ONE, AP( KC+N-K+2 ), 1,
$ B( K+1, 1 ), LDB, B( K+2, 1 ), LDB )
END IF
*
* Multiply by the inverse of the diagonal block.
*
AKM1K = AP( KC+1 )
AKM1 = AP( KC ) / AKM1K
AK = AP( KC+N-K+1 ) / AKM1K
DENOM = AKM1*AK - ONE
DO 70 J = 1, NRHS
BKM1 = B( K, J ) / AKM1K
BK = B( K+1, J ) / AKM1K
B( K, J ) = ( AK*BKM1-BK ) / DENOM
B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
70 CONTINUE
KC = KC + 2*( N-K ) + 1
K = K + 2
END IF
*
GO TO 60
80 CONTINUE
*
* Next solve L**T*X = B, overwriting B with X.
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
KC = N*( N+1 ) / 2 + 1
90 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 100
*
KC = KC - ( N-K+1 )
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Multiply by inv(L**T(K)), where L(K) is the transformation
* stored in column K of A.
*
IF( K.LT.N )
$ CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, AP( KC+1 ), 1, ONE, B( K, 1 ), LDB )
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K - 1
ELSE
*
* 2 x 2 diagonal block
*
* Multiply by inv(L**T(K-1)), where L(K-1) is the transformation
* stored in columns K-1 and K of A.
*
IF( K.LT.N ) THEN
CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, AP( KC+1 ), 1, ONE, B( K, 1 ), LDB )
CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, AP( KC-( N-K ) ), 1, ONE, B( K-1, 1 ),
$ LDB )
END IF
*
* Interchange rows K and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
KC = KC - ( N-K+2 )
K = K - 2
END IF
*
GO TO 90
100 CONTINUE
END IF
*
RETURN
*
* End of DSPTRS
*
END
*> \brief \b DSTEBZ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEBZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstebz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstebz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstebz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E,
* M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER ORDER, RANGE
* INTEGER IL, INFO, IU, M, N, NSPLIT
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEBZ computes the eigenvalues of a symmetric tridiagonal
*> matrix T. The user may ask for all eigenvalues, all eigenvalues
*> in the half-open interval (VL, VU], or the IL-th through IU-th
*> eigenvalues.
*>
*> To avoid overflow, the matrix must be scaled so that its
*> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
*> accuracy, it should not be much smaller than that.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': ("All") all eigenvalues will be found.
*> = 'V': ("Value") all eigenvalues in the half-open interval
*> (VL, VU] will be found.
*> = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
*> entire matrix) will be found.
*> \endverbatim
*>
*> \param[in] ORDER
*> \verbatim
*> ORDER is CHARACTER*1
*> = 'B': ("By Block") the eigenvalues will be grouped by
*> split-off block (see IBLOCK, ISPLIT) and
*> ordered from smallest to largest within
*> the block.
*> = 'E': ("Entire matrix")
*> the eigenvalues for the entire matrix
*> will be ordered from smallest to
*> largest.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the tridiagonal matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*>
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. Eigenvalues less than or equal
*> to VL, or greater than VU, will not be returned. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*>
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. Eigenvalues less than or equal
*> to VL, or greater than VU, will not be returned. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*>
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*>
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute tolerance for the eigenvalues. An eigenvalue
*> (or cluster) is considered to be located if it has been
*> determined to lie in an interval whose width is ABSTOL or
*> less. If ABSTOL is less than or equal to zero, then ULP*|T|
*> will be used, where |T| means the 1-norm of T.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The actual number of eigenvalues found. 0 <= M <= N.
*> (See also the description of INFO=2,3.)
*> \endverbatim
*>
*> \param[out] NSPLIT
*> \verbatim
*> NSPLIT is INTEGER
*> The number of diagonal blocks in the matrix T.
*> 1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> On exit, the first M elements of W will contain the
*> eigenvalues. (DSTEBZ may use the remaining N-M elements as
*> workspace.)
*> \endverbatim
*>
*> \param[out] IBLOCK
*> \verbatim
*> IBLOCK is INTEGER array, dimension (N)
*> At each row/column j where E(j) is zero or small, the
*> matrix T is considered to split into a block diagonal
*> matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
*> block (from 1 to the number of blocks) the eigenvalue W(i)
*> belongs. (DSTEBZ may use the remaining N-M elements as
*> workspace.)
*> \endverbatim
*>
*> \param[out] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into submatrices.
*> The first submatrix consists of rows/columns 1 to ISPLIT(1),
*> the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*> etc., and the NSPLIT-th consists of rows/columns
*> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*> (Only the first NSPLIT elements will actually be used, but
*> since the user cannot know a priori what value NSPLIT will
*> have, N words must be reserved for ISPLIT.)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: some or all of the eigenvalues failed to converge or
*> were not computed:
*> =1 or 3: Bisection failed to converge for some
*> eigenvalues; these eigenvalues are flagged by a
*> negative block number. The effect is that the
*> eigenvalues may not be as accurate as the
*> absolute and relative tolerances. This is
*> generally caused by unexpectedly inaccurate
*> arithmetic.
*> =2 or 3: RANGE='I' only: Not all of the eigenvalues
*> IL:IU were found.
*> Effect: M < IU+1-IL
*> Cause: non-monotonic arithmetic, causing the
*> Sturm sequence to be non-monotonic.
*> Cure: recalculate, using RANGE='A', and pick
*> out eigenvalues IL:IU. In some cases,
*> increasing the PARAMETER "FUDGE" may
*> make things work.
*> = 4: RANGE='I', and the Gershgorin interval
*> initially used was too small. No eigenvalues
*> were computed.
*> Probable cause: your machine has sloppy
*> floating-point arithmetic.
*> Cure: Increase the PARAMETER "FUDGE",
*> recompile, and try again.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> RELFAC DOUBLE PRECISION, default = 2.0e0
*> The relative tolerance. An interval (a,b] lies within
*> "relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|),
*> where "ulp" is the machine precision (distance from 1 to
*> the next larger floating point number.)
*>
*> FUDGE DOUBLE PRECISION, default = 2
*> A "fudge factor" to widen the Gershgorin intervals. Ideally,
*> a value of 1 should work, but on machines with sloppy
*> arithmetic, this needs to be larger. The default for
*> publicly released versions should be large enough to handle
*> the worst machine around. Note that this has no effect
*> on accuracy of the solution.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stebz
*
* =====================================================================
SUBROUTINE DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D,
$ E,
$ M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER ORDER, RANGE
INTEGER IL, INFO, IU, M, N, NSPLIT
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, HALF
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ HALF = 1.0D0 / TWO )
DOUBLE PRECISION FUDGE, RELFAC
PARAMETER ( FUDGE = 2.1D0, RELFAC = 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL NCNVRG, TOOFEW
INTEGER IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO,
$ IM, IN, IOFF, IORDER, IOUT, IRANGE, ITMAX,
$ ITMP1, IW, IWOFF, J, JB, JDISC, JE, NB, NWL,
$ NWU
DOUBLE PRECISION ATOLI, BNORM, GL, GU, PIVMIN, RTOLI, SAFEMN,
$ TMP1, TMP2, TNORM, ULP, WKILL, WL, WLU, WU, WUL
* ..
* .. Local Arrays ..
INTEGER IDUMMA( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, ILAENV, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLAEBZ, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Decode RANGE
*
IF( LSAME( RANGE, 'A' ) ) THEN
IRANGE = 1
ELSE IF( LSAME( RANGE, 'V' ) ) THEN
IRANGE = 2
ELSE IF( LSAME( RANGE, 'I' ) ) THEN
IRANGE = 3
ELSE
IRANGE = 0
END IF
*
* Decode ORDER
*
IF( LSAME( ORDER, 'B' ) ) THEN
IORDER = 2
ELSE IF( LSAME( ORDER, 'E' ) ) THEN
IORDER = 1
ELSE
IORDER = 0
END IF
*
* Check for Errors
*
IF( IRANGE.LE.0 ) THEN
INFO = -1
ELSE IF( IORDER.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( IRANGE.EQ.2 ) THEN
IF( VL.GE.VU )
$ INFO = -5
ELSE IF( IRANGE.EQ.3 .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) )
$ THEN
INFO = -6
ELSE IF( IRANGE.EQ.3 .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) )
$ THEN
INFO = -7
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEBZ', -INFO )
RETURN
END IF
*
* Initialize error flags
*
INFO = 0
NCNVRG = .FALSE.
TOOFEW = .FALSE.
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
* Simplifications:
*
IF( IRANGE.EQ.3 .AND. IL.EQ.1 .AND. IU.EQ.N )
$ IRANGE = 1
*
* Get machine constants
* NB is the minimum vector length for vector bisection, or 0
* if only scalar is to be done.
*
SAFEMN = DLAMCH( 'S' )
ULP = DLAMCH( 'P' )
RTOLI = ULP*RELFAC
NB = ILAENV( 1, 'DSTEBZ', ' ', N, -1, -1, -1 )
IF( NB.LE.1 )
$ NB = 0
*
* Special Case when N=1
*
IF( N.EQ.1 ) THEN
NSPLIT = 1
ISPLIT( 1 ) = 1
IF( IRANGE.EQ.2 .AND. ( VL.GE.D( 1 ) .OR. VU.LT.D( 1 ) ) ) THEN
M = 0
ELSE
W( 1 ) = D( 1 )
IBLOCK( 1 ) = 1
M = 1
END IF
RETURN
END IF
*
* Compute Splitting Points
*
NSPLIT = 1
WORK( N ) = ZERO
PIVMIN = ONE
*
DO 10 J = 2, N
TMP1 = E( J-1 )**2
IF( ABS( D( J )*D( J-1 ) )*ULP**2+SAFEMN.GT.TMP1 ) THEN
ISPLIT( NSPLIT ) = J - 1
NSPLIT = NSPLIT + 1
WORK( J-1 ) = ZERO
ELSE
WORK( J-1 ) = TMP1
PIVMIN = MAX( PIVMIN, TMP1 )
END IF
10 CONTINUE
ISPLIT( NSPLIT ) = N
PIVMIN = PIVMIN*SAFEMN
*
* Compute Interval and ATOLI
*
IF( IRANGE.EQ.3 ) THEN
*
* RANGE='I': Compute the interval containing eigenvalues
* IL through IU.
*
* Compute Gershgorin interval for entire (split) matrix
* and use it as the initial interval
*
GU = D( 1 )
GL = D( 1 )
TMP1 = ZERO
*
DO 20 J = 1, N - 1
TMP2 = SQRT( WORK( J ) )
GU = MAX( GU, D( J )+TMP1+TMP2 )
GL = MIN( GL, D( J )-TMP1-TMP2 )
TMP1 = TMP2
20 CONTINUE
*
GU = MAX( GU, D( N )+TMP1 )
GL = MIN( GL, D( N )-TMP1 )
TNORM = MAX( ABS( GL ), ABS( GU ) )
GL = GL - FUDGE*TNORM*ULP*N - FUDGE*TWO*PIVMIN
GU = GU + FUDGE*TNORM*ULP*N + FUDGE*PIVMIN
*
* Compute Iteration parameters
*
ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
IF( ABSTOL.LE.ZERO ) THEN
ATOLI = ULP*TNORM
ELSE
ATOLI = ABSTOL
END IF
*
WORK( N+1 ) = GL
WORK( N+2 ) = GL
WORK( N+3 ) = GU
WORK( N+4 ) = GU
WORK( N+5 ) = GL
WORK( N+6 ) = GU
IWORK( 1 ) = -1
IWORK( 2 ) = -1
IWORK( 3 ) = N + 1
IWORK( 4 ) = N + 1
IWORK( 5 ) = IL - 1
IWORK( 6 ) = IU
*
CALL DLAEBZ( 3, ITMAX, N, 2, 2, NB, ATOLI, RTOLI, PIVMIN, D,
$ E,
$ WORK, IWORK( 5 ), WORK( N+1 ), WORK( N+5 ), IOUT,
$ IWORK, W, IBLOCK, IINFO )
*
IF( IWORK( 6 ).EQ.IU ) THEN
WL = WORK( N+1 )
WLU = WORK( N+3 )
NWL = IWORK( 1 )
WU = WORK( N+4 )
WUL = WORK( N+2 )
NWU = IWORK( 4 )
ELSE
WL = WORK( N+2 )
WLU = WORK( N+4 )
NWL = IWORK( 2 )
WU = WORK( N+3 )
WUL = WORK( N+1 )
NWU = IWORK( 3 )
END IF
*
IF( NWL.LT.0 .OR. NWL.GE.N .OR. NWU.LT.1 .OR. NWU.GT.N ) THEN
INFO = 4
RETURN
END IF
ELSE
*
* RANGE='A' or 'V' -- Set ATOLI
*
TNORM = MAX( ABS( D( 1 ) )+ABS( E( 1 ) ),
$ ABS( D( N ) )+ABS( E( N-1 ) ) )
*
DO 30 J = 2, N - 1
TNORM = MAX( TNORM, ABS( D( J ) )+ABS( E( J-1 ) )+
$ ABS( E( J ) ) )
30 CONTINUE
*
IF( ABSTOL.LE.ZERO ) THEN
ATOLI = ULP*TNORM
ELSE
ATOLI = ABSTOL
END IF
*
IF( IRANGE.EQ.2 ) THEN
WL = VL
WU = VU
ELSE
WL = ZERO
WU = ZERO
END IF
END IF
*
* Find Eigenvalues -- Loop Over Blocks and recompute NWL and NWU.
* NWL accumulates the number of eigenvalues .le. WL,
* NWU accumulates the number of eigenvalues .le. WU
*
M = 0
IEND = 0
INFO = 0
NWL = 0
NWU = 0
*
DO 70 JB = 1, NSPLIT
IOFF = IEND
IBEGIN = IOFF + 1
IEND = ISPLIT( JB )
IN = IEND - IOFF
*
IF( IN.EQ.1 ) THEN
*
* Special Case -- IN=1
*
IF( IRANGE.EQ.1 .OR. WL.GE.D( IBEGIN )-PIVMIN )
$ NWL = NWL + 1
IF( IRANGE.EQ.1 .OR. WU.GE.D( IBEGIN )-PIVMIN )
$ NWU = NWU + 1
IF( IRANGE.EQ.1 .OR. ( WL.LT.D( IBEGIN )-PIVMIN .AND. WU.GE.
$ D( IBEGIN )-PIVMIN ) ) THEN
M = M + 1
W( M ) = D( IBEGIN )
IBLOCK( M ) = JB
END IF
ELSE
*
* General Case -- IN > 1
*
* Compute Gershgorin Interval
* and use it as the initial interval
*
GU = D( IBEGIN )
GL = D( IBEGIN )
TMP1 = ZERO
*
DO 40 J = IBEGIN, IEND - 1
TMP2 = ABS( E( J ) )
GU = MAX( GU, D( J )+TMP1+TMP2 )
GL = MIN( GL, D( J )-TMP1-TMP2 )
TMP1 = TMP2
40 CONTINUE
*
GU = MAX( GU, D( IEND )+TMP1 )
GL = MIN( GL, D( IEND )-TMP1 )
BNORM = MAX( ABS( GL ), ABS( GU ) )
GL = GL - FUDGE*BNORM*ULP*IN - FUDGE*PIVMIN
GU = GU + FUDGE*BNORM*ULP*IN + FUDGE*PIVMIN
*
* Compute ATOLI for the current submatrix
*
IF( ABSTOL.LE.ZERO ) THEN
ATOLI = ULP*MAX( ABS( GL ), ABS( GU ) )
ELSE
ATOLI = ABSTOL
END IF
*
IF( IRANGE.GT.1 ) THEN
IF( GU.LT.WL ) THEN
NWL = NWL + IN
NWU = NWU + IN
GO TO 70
END IF
GL = MAX( GL, WL )
GU = MIN( GU, WU )
IF( GL.GE.GU )
$ GO TO 70
END IF
*
* Set Up Initial Interval
*
WORK( N+1 ) = GL
WORK( N+IN+1 ) = GU
CALL DLAEBZ( 1, 0, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
$ D( IBEGIN ), E( IBEGIN ), WORK( IBEGIN ),
$ IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IM,
$ IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
*
NWL = NWL + IWORK( 1 )
NWU = NWU + IWORK( IN+1 )
IWOFF = M - IWORK( 1 )
*
* Compute Eigenvalues
*
ITMAX = INT( ( LOG( GU-GL+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
CALL DLAEBZ( 2, ITMAX, IN, IN, 1, NB, ATOLI, RTOLI,
$ PIVMIN,
$ D( IBEGIN ), E( IBEGIN ), WORK( IBEGIN ),
$ IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IOUT,
$ IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
*
* Copy Eigenvalues Into W and IBLOCK
* Use -JB for block number for unconverged eigenvalues.
*
DO 60 J = 1, IOUT
TMP1 = HALF*( WORK( J+N )+WORK( J+IN+N ) )
*
* Flag non-convergence.
*
IF( J.GT.IOUT-IINFO ) THEN
NCNVRG = .TRUE.
IB = -JB
ELSE
IB = JB
END IF
DO 50 JE = IWORK( J ) + 1 + IWOFF,
$ IWORK( J+IN ) + IWOFF
W( JE ) = TMP1
IBLOCK( JE ) = IB
50 CONTINUE
60 CONTINUE
*
M = M + IM
END IF
70 CONTINUE
*
* If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
* If NWL+1 < IL or NWU > IU, discard extra eigenvalues.
*
IF( IRANGE.EQ.3 ) THEN
IM = 0
IDISCL = IL - 1 - NWL
IDISCU = NWU - IU
*
IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
DO 80 JE = 1, M
IF( W( JE ).LE.WLU .AND. IDISCL.GT.0 ) THEN
IDISCL = IDISCL - 1
ELSE IF( W( JE ).GE.WUL .AND. IDISCU.GT.0 ) THEN
IDISCU = IDISCU - 1
ELSE
IM = IM + 1
W( IM ) = W( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
80 CONTINUE
M = IM
END IF
IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
*
* Code to deal with effects of bad arithmetic:
* Some low eigenvalues to be discarded are not in (WL,WLU],
* or high eigenvalues to be discarded are not in (WUL,WU]
* so just kill off the smallest IDISCL/largest IDISCU
* eigenvalues, by simply finding the smallest/largest
* eigenvalue(s).
*
* (If N(w) is monotone non-decreasing, this should never
* happen.)
*
IF( IDISCL.GT.0 ) THEN
WKILL = WU
DO 100 JDISC = 1, IDISCL
IW = 0
DO 90 JE = 1, M
IF( IBLOCK( JE ).NE.0 .AND.
$ ( W( JE ).LT.WKILL .OR. IW.EQ.0 ) ) THEN
IW = JE
WKILL = W( JE )
END IF
90 CONTINUE
IBLOCK( IW ) = 0
100 CONTINUE
END IF
IF( IDISCU.GT.0 ) THEN
*
WKILL = WL
DO 120 JDISC = 1, IDISCU
IW = 0
DO 110 JE = 1, M
IF( IBLOCK( JE ).NE.0 .AND.
$ ( W( JE ).GT.WKILL .OR. IW.EQ.0 ) ) THEN
IW = JE
WKILL = W( JE )
END IF
110 CONTINUE
IBLOCK( IW ) = 0
120 CONTINUE
END IF
IM = 0
DO 130 JE = 1, M
IF( IBLOCK( JE ).NE.0 ) THEN
IM = IM + 1
W( IM ) = W( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
130 CONTINUE
M = IM
END IF
IF( IDISCL.LT.0 .OR. IDISCU.LT.0 ) THEN
TOOFEW = .TRUE.
END IF
END IF
*
* If ORDER='B', do nothing -- the eigenvalues are already sorted
* by block.
* If ORDER='E', sort the eigenvalues from smallest to largest
*
IF( IORDER.EQ.1 .AND. NSPLIT.GT.1 ) THEN
DO 150 JE = 1, M - 1
IE = 0
TMP1 = W( JE )
DO 140 J = JE + 1, M
IF( W( J ).LT.TMP1 ) THEN
IE = J
TMP1 = W( J )
END IF
140 CONTINUE
*
IF( IE.NE.0 ) THEN
ITMP1 = IBLOCK( IE )
W( IE ) = W( JE )
IBLOCK( IE ) = IBLOCK( JE )
W( JE ) = TMP1
IBLOCK( JE ) = ITMP1
END IF
150 CONTINUE
END IF
*
INFO = 0
IF( NCNVRG )
$ INFO = INFO + 1
IF( TOOFEW )
$ INFO = INFO + 2
RETURN
*
* End of DSTEBZ
*
END
*> \brief \b DSTEDC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEDC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstedc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstedc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstedc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> The eigenvectors of a full or band real symmetric matrix can also be
*> found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
*> matrix to tridiagonal form.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'I': Compute eigenvectors of tridiagonal matrix also.
*> = 'V': Compute eigenvectors of original dense symmetric
*> matrix also. On entry, Z contains the orthogonal
*> matrix used to reduce the original matrix to
*> tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the subdiagonal elements of the tridiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> On entry, if COMPZ = 'V', then Z contains the orthogonal
*> matrix used in the reduction to tridiagonal form.
*> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*> orthonormal eigenvectors of the original symmetric matrix,
*> and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*> of the symmetric tridiagonal matrix.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1.
*> If eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
*> If COMPZ = 'V' and N > 1 then LWORK must be at least
*> ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
*> where lg( N ) = smallest integer k such
*> that 2**k >= N.
*> If COMPZ = 'I' and N > 1 then LWORK must be at least
*> ( 1 + 4*N + N**2 ).
*> Note that for COMPZ = 'I' or 'V', then if N is less than or
*> equal to the minimum divide size, usually 25, then LWORK need
*> only be max(1,2*(N-1)).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
*> If COMPZ = 'V' and N > 1 then LIWORK must be at least
*> ( 6 + 6*N + 5*N*lg N ).
*> If COMPZ = 'I' and N > 1 then LIWORK must be at least
*> ( 3 + 5*N ).
*> Note that for COMPZ = 'I' or 'V', then if N is less than or
*> equal to the minimum divide size, usually 25, then LIWORK
*> need only be 1.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute an eigenvalue while
*> working on the submatrix lying in rows and columns
*> INFO/(N+1) through mod(INFO,N+1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stedc
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN,
$ LWMIN, M, SMLSIZ, START, STOREZ, STRTRW
DOUBLE PRECISION EPS, ORGNRM, P, TINY
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, ILAENV, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLAED0, DLASCL, DLASET,
$ DLASRT,
$ DSTEQR, DSTERF, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, MAX, MOD, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR.
$ ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
INFO = -6
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
SMLSIZ = ILAENV( 9, 'DSTEDC', ' ', 0, 0, 0, 0 )
IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE IF( N.LE.SMLSIZ ) THEN
LIWMIN = 1
LWMIN = 2*( N - 1 )
ELSE
LGN = INT( LOG( DBLE( N ) )/LOG( TWO ) )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( ICOMPZ.EQ.1 ) THEN
LWMIN = 1 + 3*N + 2*N*LGN + 4*N**2
LIWMIN = 6 + 6*N + 5*N*LGN
ELSE IF( ICOMPZ.EQ.2 ) THEN
LWMIN = 1 + 4*N + N**2
LIWMIN = 3 + 5*N
END IF
END IF
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT. LQUERY ) THEN
INFO = -8
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT. LQUERY ) THEN
INFO = -10
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEDC', -INFO )
RETURN
ELSE IF (LQUERY) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( N.EQ.1 ) THEN
IF( ICOMPZ.NE.0 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* If the following conditional clause is removed, then the routine
* will use the Divide and Conquer routine to compute only the
* eigenvalues, which requires (3N + 3N**2) real workspace and
* (2 + 5N + 2N lg(N)) integer workspace.
* Since on many architectures DSTERF is much faster than any other
* algorithm for finding eigenvalues only, it is used here
* as the default. If the conditional clause is removed, then
* information on the size of workspace needs to be changed.
*
* If COMPZ = 'N', use DSTERF to compute the eigenvalues.
*
IF( ICOMPZ.EQ.0 ) THEN
CALL DSTERF( N, D, E, INFO )
GO TO 50
END IF
*
* If N is smaller than the minimum divide size (SMLSIZ+1), then
* solve the problem with another solver.
*
IF( N.LE.SMLSIZ ) THEN
*
CALL DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
ELSE
*
* If COMPZ = 'V', the Z matrix must be stored elsewhere for later
* use.
*
IF( ICOMPZ.EQ.1 ) THEN
STOREZ = 1 + N*N
ELSE
STOREZ = 1
END IF
*
IF( ICOMPZ.EQ.2 ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
END IF
*
* Scale.
*
ORGNRM = DLANST( 'M', N, D, E )
IF( ORGNRM.EQ.ZERO )
$ GO TO 50
*
EPS = DLAMCH( 'Epsilon' )
*
START = 1
*
* while ( START <= N )
*
10 CONTINUE
IF( START.LE.N ) THEN
*
* Let FINISH be the position of the next subdiagonal entry
* such that E( FINISH ) <= TINY or FINISH = N if no such
* subdiagonal exists. The matrix identified by the elements
* between START and FINISH constitutes an independent
* sub-problem.
*
FINISH = START
20 CONTINUE
IF( FINISH.LT.N ) THEN
TINY = EPS*SQRT( ABS( D( FINISH ) ) )*
$ SQRT( ABS( D( FINISH+1 ) ) )
IF( ABS( E( FINISH ) ).GT.TINY ) THEN
FINISH = FINISH + 1
GO TO 20
END IF
END IF
*
* (Sub) Problem determined. Compute its size and solve it.
*
M = FINISH - START + 1
IF( M.EQ.1 ) THEN
START = FINISH + 1
GO TO 10
END IF
IF( M.GT.SMLSIZ ) THEN
*
* Scale.
*
ORGNRM = DLANST( 'M', M, D( START ), E( START ) )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ),
$ M,
$ INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1,
$ E( START ),
$ M-1, INFO )
*
IF( ICOMPZ.EQ.1 ) THEN
STRTRW = 1
ELSE
STRTRW = START
END IF
CALL DLAED0( ICOMPZ, N, M, D( START ), E( START ),
$ Z( STRTRW, START ), LDZ, WORK( 1 ), N,
$ WORK( STOREZ ), IWORK, INFO )
IF( INFO.NE.0 ) THEN
INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) +
$ MOD( INFO, ( M+1 ) ) + START - 1
GO TO 50
END IF
*
* Scale back.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ),
$ M,
$ INFO )
*
ELSE
IF( ICOMPZ.EQ.1 ) THEN
*
* Since QR won't update a Z matrix which is larger than
* the length of D, we must solve the sub-problem in a
* workspace and then multiply back into Z.
*
CALL DSTEQR( 'I', M, D( START ), E( START ), WORK,
$ M,
$ WORK( M*M+1 ), INFO )
CALL DLACPY( 'A', N, M, Z( 1, START ), LDZ,
$ WORK( STOREZ ), N )
CALL DGEMM( 'N', 'N', N, M, M, ONE,
$ WORK( STOREZ ), N, WORK, M, ZERO,
$ Z( 1, START ), LDZ )
ELSE IF( ICOMPZ.EQ.2 ) THEN
CALL DSTEQR( 'I', M, D( START ), E( START ),
$ Z( START, START ), LDZ, WORK, INFO )
ELSE
CALL DSTERF( M, D( START ), E( START ), INFO )
END IF
IF( INFO.NE.0 ) THEN
INFO = START*( N+1 ) + FINISH
GO TO 50
END IF
END IF
*
START = FINISH + 1
GO TO 10
END IF
*
* endwhile
*
IF( ICOMPZ.EQ.0 ) THEN
*
* Use Quick Sort
*
CALL DLASRT( 'I', N, D, INFO )
*
ELSE
*
* Use Selection Sort to minimize swaps of eigenvectors
*
DO 40 II = 2, N
I = II - 1
K = I
P = D( I )
DO 30 J = II, N
IF( D( J ).LT.P ) THEN
K = J
P = D( J )
END IF
30 CONTINUE
IF( K.NE.I ) THEN
D( K ) = D( I )
D( I ) = P
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
END IF
40 CONTINUE
END IF
END IF
*
50 CONTINUE
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSTEDC
*
END
*> \brief \b DSTEGR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEGR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstegr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstegr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstegr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
* ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE
* INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER ISUPPZ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
* DOUBLE PRECISION Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEGR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
*> a well defined set of pairwise different real eigenvalues, the corresponding
*> real eigenvectors are pairwise orthogonal.
*>
*> The spectrum may be computed either completely or partially by specifying
*> either an interval (VL,VU] or a range of indices IL:IU for the desired
*> eigenvalues.
*>
*> DSTEGR is a compatibility wrapper around the improved DSTEMR routine.
*> See DSTEMR for further details.
*>
*> One important change is that the ABSTOL parameter no longer provides any
*> benefit and hence is no longer used.
*>
*> Note : DSTEGR and DSTEMR work only on machines which follow
*> IEEE-754 floating-point standard in their handling of infinities and
*> NaNs. Normal execution may create these exceptional values and hence
*> may abort due to a floating point exception in environments which
*> do not conform to the IEEE-754 standard.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the N diagonal elements of the tridiagonal matrix
*> T. On exit, D is overwritten.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, the (N-1) subdiagonal elements of the tridiagonal
*> matrix T in elements 1 to N-1 of E. E(N) need not be set on
*> input, but is used internally as workspace.
*> On exit, E is overwritten.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*>
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*>
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*>
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*>
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> Unused. Was the absolute error tolerance for the
*> eigenvalues/eigenvectors in previous versions.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the selected eigenvalues in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*> If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix T
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> Supplying N columns is always safe.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*> The support of the eigenvectors in Z, i.e., the indices
*> indicating the nonzero elements in Z. The i-th computed eigenvector
*> is nonzero only in elements ISUPPZ( 2*i-1 ) through
*> ISUPPZ( 2*i ). This is relevant in the case when the matrix
*> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal
*> (and minimal) LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,18*N)
*> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (LIWORK)
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK. LIWORK >= max(1,10*N)
*> if the eigenvectors are desired, and LIWORK >= max(1,8*N)
*> if only the eigenvalues are to be computed.
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, INFO
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = 1X, internal error in DLARRE,
*> if INFO = 2X, internal error in DLARRV.
*> Here, the digit X = ABS( IINFO ) < 10, where IINFO is
*> the nonzero error code returned by DLARRE or
*> DLARRV, respectively.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stegr
*
*> \par Contributors:
* ==================
*>
*> Inderjit Dhillon, IBM Almaden, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, LBNL/NERSC, USA \n
*
* =====================================================================
SUBROUTINE DSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
DOUBLE PRECISION Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL TRYRAC
* ..
* .. External Subroutines ..
EXTERNAL DSTEMR
* ..
* .. Executable Statements ..
INFO = 0
TRYRAC = .FALSE.
CALL DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ M, W, Z, LDZ, N, ISUPPZ, TRYRAC, WORK, LWORK,
$ IWORK, LIWORK, INFO )
*
* End of DSTEGR
*
END
*> \brief \b DSTEIN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstein.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
* IWORK, IFAIL, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDZ, M, N
* ..
* .. Array Arguments ..
* INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
* $ IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEIN computes the eigenvectors of a real symmetric tridiagonal
*> matrix T corresponding to specified eigenvalues, using inverse
*> iteration.
*>
*> The maximum number of iterations allowed for each eigenvector is
*> specified by an internal parameter MAXITS (currently set to 5).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the tridiagonal matrix
*> T, in elements 1 to N-1.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of eigenvectors to be found. 0 <= M <= N.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements of W contain the eigenvalues for
*> which eigenvectors are to be computed. The eigenvalues
*> should be grouped by split-off block and ordered from
*> smallest to largest within the block. ( The output array
*> W from DSTEBZ with ORDER = 'B' is expected here. )
*> \endverbatim
*>
*> \param[in] IBLOCK
*> \verbatim
*> IBLOCK is INTEGER array, dimension (N)
*> The submatrix indices associated with the corresponding
*> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
*> the first submatrix from the top, =2 if W(i) belongs to
*> the second submatrix, etc. ( The output array IBLOCK
*> from DSTEBZ is expected here. )
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into submatrices.
*> The first submatrix consists of rows/columns 1 to
*> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
*> through ISPLIT( 2 ), etc.
*> ( The output array ISPLIT from DSTEBZ is expected here. )
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, M)
*> The computed eigenvectors. The eigenvector associated
*> with the eigenvalue W(i) is stored in the i-th column of
*> Z. Any vector which fails to converge is set to its current
*> iterate after MAXITS iterations.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (M)
*> On normal exit, all elements of IFAIL are zero.
*> If one or more eigenvectors fail to converge after
*> MAXITS iterations, then their indices are stored in
*> array IFAIL.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, then i eigenvectors failed to converge
*> in MAXITS iterations. Their indices are stored in
*> array IFAIL.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> MAXITS INTEGER, default = 5
*> The maximum number of iterations performed.
*>
*> EXTRA INTEGER, default = 2
*> The number of iterations performed after norm growth
*> criterion is satisfied, should be at least 1.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stein
*
* =====================================================================
SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
$ IWORK, IFAIL, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDZ, M, N
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
$ IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TEN, ODM3, ODM1
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1,
$ ODM3 = 1.0D-3, ODM1 = 1.0D-1 )
INTEGER MAXITS, EXTRA
PARAMETER ( MAXITS = 5, EXTRA = 2 )
* ..
* .. Local Scalars ..
INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
$ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
$ JBLK, JMAX, NBLK, NRMCHK
DOUBLE PRECISION DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
$ SCL, SEP, TOL, XJ, XJM, ZTR
* ..
* .. Local Arrays ..
INTEGER ISEED( 4 )
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DDOT, DLAMCH, DNRM2
EXTERNAL IDAMAX, DDOT, DLAMCH, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLAGTF, DLAGTS, DLARNV,
$ DSCAL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
DO 10 I = 1, M
IFAIL( I ) = 0
10 CONTINUE
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
INFO = -4
ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE
DO 20 J = 2, M
IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
INFO = -6
GO TO 30
END IF
IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
$ THEN
INFO = -5
GO TO 30
END IF
20 CONTINUE
30 CONTINUE
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEIN', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
EPS = DLAMCH( 'Precision' )
*
* Initialize seed for random number generator DLARNV.
*
DO 40 I = 1, 4
ISEED( I ) = 1
40 CONTINUE
*
* Initialize pointers.
*
INDRV1 = 0
INDRV2 = INDRV1 + N
INDRV3 = INDRV2 + N
INDRV4 = INDRV3 + N
INDRV5 = INDRV4 + N
*
* Compute eigenvectors of matrix blocks.
*
J1 = 1
DO 160 NBLK = 1, IBLOCK( M )
*
* Find starting and ending indices of block nblk.
*
IF( NBLK.EQ.1 ) THEN
B1 = 1
ELSE
B1 = ISPLIT( NBLK-1 ) + 1
END IF
BN = ISPLIT( NBLK )
BLKSIZ = BN - B1 + 1
IF( BLKSIZ.EQ.1 )
$ GO TO 60
GPIND = J1
*
* Compute reorthogonalization criterion and stopping criterion.
*
ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
DO 50 I = B1 + 1, BN - 1
ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
$ ABS( E( I ) ) )
50 CONTINUE
ORTOL = ODM3*ONENRM
*
DTPCRT = SQRT( ODM1 / BLKSIZ )
*
* Loop through eigenvalues of block nblk.
*
60 CONTINUE
JBLK = 0
DO 150 J = J1, M
IF( IBLOCK( J ).NE.NBLK ) THEN
J1 = J
GO TO 160
END IF
JBLK = JBLK + 1
XJ = W( J )
*
* Skip all the work if the block size is one.
*
IF( BLKSIZ.EQ.1 ) THEN
WORK( INDRV1+1 ) = ONE
GO TO 120
END IF
*
* If eigenvalues j and j-1 are too close, add a relatively
* small perturbation.
*
IF( JBLK.GT.1 ) THEN
EPS1 = ABS( EPS*XJ )
PERTOL = TEN*EPS1
SEP = XJ - XJM
IF( SEP.LT.PERTOL )
$ XJ = XJM + PERTOL
END IF
*
ITS = 0
NRMCHK = 0
*
* Get random starting vector.
*
CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
*
* Copy the matrix T so it won't be destroyed in factorization.
*
CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
*
* Compute LU factors with partial pivoting ( PT = LU )
*
TOL = ZERO
CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ,
$ WORK( INDRV2+2 ),
$ WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
$ IINFO )
*
* Update iteration count.
*
70 CONTINUE
ITS = ITS + 1
IF( ITS.GT.MAXITS )
$ GO TO 100
*
* Normalize and scale the righthand side vector Pb.
*
JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
SCL = BLKSIZ*ONENRM*MAX( EPS,
$ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
$ ABS( WORK( INDRV1+JMAX ) )
CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
*
* Solve the system LU = Pb.
*
CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ),
$ WORK( INDRV2+2 ),
$ WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
$ WORK( INDRV1+1 ), TOL, IINFO )
*
* Reorthogonalize by modified Gram-Schmidt if eigenvalues are
* close enough.
*
IF( JBLK.EQ.1 )
$ GO TO 90
IF( ABS( XJ-XJM ).GT.ORTOL )
$ GPIND = J
IF( GPIND.NE.J ) THEN
DO 80 I = GPIND, J - 1
ZTR = -DDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1,
$ I ),
$ 1 )
CALL DAXPY( BLKSIZ, ZTR, Z( B1, I ), 1,
$ WORK( INDRV1+1 ), 1 )
80 CONTINUE
END IF
*
* Check the infinity norm of the iterate.
*
90 CONTINUE
JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
NRM = ABS( WORK( INDRV1+JMAX ) )
*
* Continue for additional iterations after norm reaches
* stopping criterion.
*
IF( NRM.LT.DTPCRT )
$ GO TO 70
NRMCHK = NRMCHK + 1
IF( NRMCHK.LT.EXTRA+1 )
$ GO TO 70
*
GO TO 110
*
* If stopping criterion was not satisfied, update info and
* store eigenvector number in array ifail.
*
100 CONTINUE
INFO = INFO + 1
IFAIL( INFO ) = J
*
* Accept iterate as jth eigenvector.
*
110 CONTINUE
SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
IF( WORK( INDRV1+JMAX ).LT.ZERO )
$ SCL = -SCL
CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
120 CONTINUE
DO 130 I = 1, N
Z( I, J ) = ZERO
130 CONTINUE
DO 140 I = 1, BLKSIZ
Z( B1+I-1, J ) = WORK( INDRV1+I )
140 CONTINUE
*
* Save the shift to check eigenvalue spacing at next
* iteration.
*
XJM = XJ
*
150 CONTINUE
160 CONTINUE
*
RETURN
*
* End of DSTEIN
*
END
*> \brief \b DSTEMR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEMR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstemr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstemr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstemr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
* M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
* IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE
* LOGICAL TRYRAC
* INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
* DOUBLE PRECISION VL, VU
* ..
* .. Array Arguments ..
* INTEGER ISUPPZ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
* DOUBLE PRECISION Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEMR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
*> a well defined set of pairwise different real eigenvalues, the corresponding
*> real eigenvectors are pairwise orthogonal.
*>
*> The spectrum may be computed either completely or partially by specifying
*> either an interval (VL,VU] or a range of indices IL:IU for the desired
*> eigenvalues.
*>
*> Depending on the number of desired eigenvalues, these are computed either
*> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
*> computed by the use of various suitable L D L^T factorizations near clusters
*> of close eigenvalues (referred to as RRRs, Relatively Robust
*> Representations). An informal sketch of the algorithm follows.
*>
*> For each unreduced block (submatrix) of T,
*> (a) Compute T - sigma I = L D L^T, so that L and D
*> define all the wanted eigenvalues to high relative accuracy.
*> This means that small relative changes in the entries of D and L
*> cause only small relative changes in the eigenvalues and
*> eigenvectors. The standard (unfactored) representation of the
*> tridiagonal matrix T does not have this property in general.
*> (b) Compute the eigenvalues to suitable accuracy.
*> If the eigenvectors are desired, the algorithm attains full
*> accuracy of the computed eigenvalues only right before
*> the corresponding vectors have to be computed, see steps c) and d).
*> (c) For each cluster of close eigenvalues, select a new
*> shift close to the cluster, find a new factorization, and refine
*> the shifted eigenvalues to suitable accuracy.
*> (d) For each eigenvalue with a large enough relative separation compute
*> the corresponding eigenvector by forming a rank revealing twisted
*> factorization. Go back to (c) for any clusters that remain.
*>
*> For more details, see:
*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
*> 2004. Also LAPACK Working Note 154.
*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
*> tridiagonal eigenvalue/eigenvector problem",
*> Computer Science Division Technical Report No. UCB/CSD-97-971,
*> UC Berkeley, May 1997.
*>
*> Further Details
*> 1.DSTEMR works only on machines which follow IEEE-754
*> floating-point standard in their handling of infinities and NaNs.
*> This permits the use of efficient inner loops avoiding a check for
*> zero divisors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the N diagonal elements of the tridiagonal matrix
*> T. On exit, D is overwritten.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, the (N-1) subdiagonal elements of the tridiagonal
*> matrix T in elements 1 to N-1 of E. E(N) need not be set on
*> input, but is used internally as workspace.
*> On exit, E is overwritten.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*>
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*>
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*>
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*>
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the selected eigenvalues in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*> If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix T
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and can be computed with a workspace
*> query by setting NZC = -1, see below.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[in] NZC
*> \verbatim
*> NZC is INTEGER
*> The number of eigenvectors to be held in the array Z.
*> If RANGE = 'A', then NZC >= max(1,N).
*> If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
*> If RANGE = 'I', then NZC >= IU-IL+1.
*> If NZC = -1, then a workspace query is assumed; the
*> routine calculates the number of columns of the array Z that
*> are needed to hold the eigenvectors.
*> This value is returned as the first entry of the Z array, and
*> no error message related to NZC is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*> The support of the eigenvectors in Z, i.e., the indices
*> indicating the nonzero elements in Z. The i-th computed eigenvector
*> is nonzero only in elements ISUPPZ( 2*i-1 ) through
*> ISUPPZ( 2*i ). This is relevant in the case when the matrix
*> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
*> \endverbatim
*>
*> \param[in,out] TRYRAC
*> \verbatim
*> TRYRAC is LOGICAL
*> If TRYRAC = .TRUE., indicates that the code should check whether
*> the tridiagonal matrix defines its eigenvalues to high relative
*> accuracy. If so, the code uses relative-accuracy preserving
*> algorithms that might be (a bit) slower depending on the matrix.
*> If the matrix does not define its eigenvalues to high relative
*> accuracy, the code can uses possibly faster algorithms.
*> If TRYRAC = .FALSE., the code is not required to guarantee
*> relatively accurate eigenvalues and can use the fastest possible
*> techniques.
*> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
*> does not define its eigenvalues to high relative accuracy.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal
*> (and minimal) LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,18*N)
*> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (LIWORK)
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK. LIWORK >= max(1,10*N)
*> if the eigenvectors are desired, and LIWORK >= max(1,8*N)
*> if only the eigenvalues are to be computed.
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, INFO
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = 1X, internal error in DLARRE,
*> if INFO = 2X, internal error in DLARRV.
*> Here, the digit X = ABS( IINFO ) < 10, where IINFO is
*> the nonzero error code returned by DLARRE or
*> DLARRV, respectively.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stemr
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA \n
*> Aravindh Krishnamoorthy, FAU, Erlangen, Germany \n
*
* =====================================================================
SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
$ IWORK, LIWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
LOGICAL TRYRAC
INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
DOUBLE PRECISION VL, VU
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
DOUBLE PRECISION Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
$ FOUR = 4.0D0,
$ MINRGP = 1.0D-3 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY,
$ LAESWAP
INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
$ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
$ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
$ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
$ NZCMIN, OFFSET, WBEGIN, WEND
DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
$ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
$ THRESH, TMP, TNRM, WL, WU
* ..
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE,
$ DLARRJ,
$ DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
ZQUERY = ( NZC.EQ.-1 )
LAESWAP = .FALSE.
* DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
* Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N.
IF( WANTZ ) THEN
LWMIN = 18*N
LIWMIN = 10*N
ELSE
* need less workspace if only the eigenvalues are wanted
LWMIN = 12*N
LIWMIN = 8*N
ENDIF
WL = ZERO
WU = ZERO
IIL = 0
IIU = 0
NSPLIT = 0
IF( VALEIG ) THEN
* We do not reference VL, VU in the cases RANGE = 'I','A'
* The interval (WL, WU] contains all the wanted eigenvalues.
* It is either given by the user or computed in DLARRE.
WL = VL
WU = VU
ELSEIF( INDEIG ) THEN
* We do not reference IL, IU in the cases RANGE = 'V','A'
IIL = IL
IIU = IU
ENDIF
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
INFO = -7
ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
INFO = -8
ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
INFO = -9
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( WANTZ .AND. ALLEIG ) THEN
NZCMIN = N
ELSE IF( WANTZ .AND. VALEIG ) THEN
CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
$ NZCMIN, ITMP, ITMP2, INFO )
ELSE IF( WANTZ .AND. INDEIG ) THEN
NZCMIN = IIU-IIL+1
ELSE
* WANTZ .EQ. FALSE.
NZCMIN = 0
ENDIF
IF( ZQUERY .AND. INFO.EQ.0 ) THEN
Z( 1,1 ) = NZCMIN
ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
INFO = -14
END IF
END IF
IF( INFO.NE.0 ) THEN
*
CALL XERBLA( 'DSTEMR', -INFO )
*
RETURN
ELSE IF( LQUERY .OR. ZQUERY ) THEN
RETURN
END IF
*
* Handle N = 0, 1, and 2 cases immediately
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = D( 1 )
ELSE
IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
M = 1
W( 1 ) = D( 1 )
END IF
END IF
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
Z( 1, 1 ) = ONE
ISUPPZ(1) = 1
ISUPPZ(2) = 1
END IF
RETURN
END IF
*
IF( N.EQ.2 ) THEN
IF( .NOT.WANTZ ) THEN
CALL DLAE2( D(1), E(1), D(2), R1, R2 )
ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
END IF
* D/S/LAE2 and D/S/LAEV2 outputs satisfy |R1| >= |R2|. However,
* the following code requires R1 >= R2. Hence, we correct
* the order of R1, R2, CS, SN if R1 < R2 before further processing.
IF( R1.LT.R2 ) THEN
E(2) = R1
R1 = R2
R2 = E(2)
LAESWAP = .TRUE.
ENDIF
IF( ALLEIG.OR.
$ (VALEIG.AND.(R2.GT.WL).AND.
$ (R2.LE.WU)).OR.
$ (INDEIG.AND.(IIL.EQ.1)) ) THEN
M = M+1
W( M ) = R2
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
IF( LAESWAP ) THEN
Z( 1, M ) = CS
Z( 2, M ) = SN
ELSE
Z( 1, M ) = -SN
Z( 2, M ) = CS
ENDIF
* Note: At most one of SN and CS can be zero.
IF (SN.NE.ZERO) THEN
IF (CS.NE.ZERO) THEN
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M) = 2
ELSE
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M) = 1
END IF
ELSE
ISUPPZ(2*M-1) = 2
ISUPPZ(2*M) = 2
END IF
ENDIF
ENDIF
IF( ALLEIG.OR.
$ (VALEIG.AND.(R1.GT.WL).AND.
$ (R1.LE.WU)).OR.
$ (INDEIG.AND.(IIU.EQ.2)) ) THEN
M = M+1
W( M ) = R1
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
IF( LAESWAP ) THEN
Z( 1, M ) = -SN
Z( 2, M ) = CS
ELSE
Z( 1, M ) = CS
Z( 2, M ) = SN
ENDIF
* Note: At most one of SN and CS can be zero.
IF (SN.NE.ZERO) THEN
IF (CS.NE.ZERO) THEN
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M) = 2
ELSE
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M) = 1
END IF
ELSE
ISUPPZ(2*M-1) = 2
ISUPPZ(2*M) = 2
END IF
ENDIF
ENDIF
ELSE
* Continue with general N
INDGRS = 1
INDERR = 2*N + 1
INDGP = 3*N + 1
INDD = 4*N + 1
INDE2 = 5*N + 1
INDWRK = 6*N + 1
*
IINSPL = 1
IINDBL = N + 1
IINDW = 2*N + 1
IINDWK = 3*N + 1
*
* Scale matrix to allowable range, if necessary.
* The allowable range is related to the PIVMIN parameter; see the
* comments in DLARRD. The preference for scaling small values
* up is heuristic; we expect users' matrices not to be close to the
* RMAX threshold.
*
SCALE = ONE
TNRM = DLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
SCALE = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
SCALE = RMAX / TNRM
END IF
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N, SCALE, D, 1 )
CALL DSCAL( N-1, SCALE, E, 1 )
TNRM = TNRM*SCALE
IF( VALEIG ) THEN
* If eigenvalues in interval have to be found,
* scale (WL, WU] accordingly
WL = WL*SCALE
WU = WU*SCALE
ENDIF
END IF
*
* Compute the desired eigenvalues of the tridiagonal after splitting
* into smaller subblocks if the corresponding off-diagonal elements
* are small
* THRESH is the splitting parameter for DLARRE
* A negative THRESH forces the old splitting criterion based on the
* size of the off-diagonal. A positive THRESH switches to splitting
* which preserves relative accuracy.
*
IF( TRYRAC ) THEN
* Test whether the matrix warrants the more expensive relative approach.
CALL DLARRR( N, D, E, IINFO )
ELSE
* The user does not care about relative accurately eigenvalues
IINFO = -1
ENDIF
* Set the splitting criterion
IF (IINFO.EQ.0) THEN
THRESH = EPS
ELSE
THRESH = -EPS
* relative accuracy is desired but T does not guarantee it
TRYRAC = .FALSE.
ENDIF
*
IF( TRYRAC ) THEN
* Copy original diagonal, needed to guarantee relative accuracy
CALL DCOPY(N,D,1,WORK(INDD),1)
ENDIF
* Store the squares of the offdiagonal values of T
DO 5 J = 1, N-1
WORK( INDE2+J-1 ) = E(J)**2
5 CONTINUE
* Set the tolerance parameters for bisection
IF( .NOT.WANTZ ) THEN
* DLARRE computes the eigenvalues to full precision.
RTOL1 = FOUR * EPS
RTOL2 = FOUR * EPS
ELSE
* DLARRE computes the eigenvalues to less than full precision.
* DLARRV will refine the eigenvalue approximations, and we can
* need less accurate initial bisection in DLARRE.
* Note: these settings do only affect the subset case and DLARRE
RTOL1 = SQRT(EPS)
RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
ENDIF
CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
$ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
$ IWORK( IINSPL ), M, W, WORK( INDERR ),
$ WORK( INDGP ), IWORK( IINDBL ),
$ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
$ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 10 + ABS( IINFO )
RETURN
END IF
* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
* part of the spectrum. All desired eigenvalues are contained in
* (WL,WU]
IF( WANTZ ) THEN
*
* Compute the desired eigenvectors corresponding to the computed
* eigenvalues
*
CALL DLARRV( N, WL, WU, D, E,
$ PIVMIN, IWORK( IINSPL ), M,
$ 1, M, MINRGP, RTOL1, RTOL2,
$ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
$ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
$ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 20 + ABS( IINFO )
RETURN
END IF
ELSE
* DLARRE computes eigenvalues of the (shifted) root representation
* DLARRV returns the eigenvalues of the unshifted matrix.
* However, if the eigenvectors are not desired by the user, we need
* to apply the corresponding shifts from DLARRE to obtain the
* eigenvalues of the original matrix.
DO 20 J = 1, M
ITMP = IWORK( IINDBL+J-1 )
W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
20 CONTINUE
END IF
*
IF ( TRYRAC ) THEN
* Refine computed eigenvalues so that they are relatively accurate
* with respect to the original matrix T.
IBEGIN = 1
WBEGIN = 1
DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
IEND = IWORK( IINSPL+JBLK-1 )
IN = IEND - IBEGIN + 1
WEND = WBEGIN - 1
* check if any eigenvalues have to be refined in this block
36 CONTINUE
IF( WEND.LT.M ) THEN
IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
WEND = WEND + 1
GO TO 36
END IF
END IF
IF( WEND.LT.WBEGIN ) THEN
IBEGIN = IEND + 1
GO TO 39
END IF
OFFSET = IWORK(IINDW+WBEGIN-1)-1
IFIRST = IWORK(IINDW+WBEGIN-1)
ILAST = IWORK(IINDW+WEND-1)
RTOL2 = FOUR * EPS
CALL DLARRJ( IN,
$ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
$ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
$ WORK( INDERR+WBEGIN-1 ),
$ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
$ TNRM, IINFO )
IBEGIN = IEND + 1
WBEGIN = WEND + 1
39 CONTINUE
ENDIF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( M, ONE / SCALE, W, 1 )
END IF
END IF
*
* If eigenvalues are not in increasing order, then sort them,
* possibly along with eigenvectors.
*
IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN
IF( .NOT. WANTZ ) THEN
CALL DLASRT( 'I', M, W, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 3
RETURN
END IF
ELSE
DO 60 J = 1, M - 1
I = 0
TMP = W( J )
DO 50 JJ = J + 1, M
IF( W( JJ ).LT.TMP ) THEN
I = JJ
TMP = W( JJ )
END IF
50 CONTINUE
IF( I.NE.0 ) THEN
W( I ) = W( J )
W( J ) = TMP
IF( WANTZ ) THEN
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
ITMP = ISUPPZ( 2*I-1 )
ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
ISUPPZ( 2*J-1 ) = ITMP
ITMP = ISUPPZ( 2*I )
ISUPPZ( 2*I ) = ISUPPZ( 2*J )
ISUPPZ( 2*J ) = ITMP
END IF
END IF
60 CONTINUE
END IF
ENDIF
*
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
RETURN
*
* End of DSTEMR
*
END
*> \brief \b DSTEQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsteqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsteqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsteqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the implicit QL or QR method.
*> The eigenvectors of a full or band symmetric matrix can also be found
*> if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to
*> tridiagonal form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'V': Compute eigenvalues and eigenvectors of the original
*> symmetric matrix. On entry, Z must contain the
*> orthogonal matrix used to reduce the original matrix
*> to tridiagonal form.
*> = 'I': Compute eigenvalues and eigenvectors of the
*> tridiagonal matrix. Z is initialized to the identity
*> matrix.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', then Z contains the orthogonal
*> matrix used in the reduction to tridiagonal form.
*> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*> orthonormal eigenvectors of the original symmetric matrix,
*> and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*> of the symmetric tridiagonal matrix.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
*> If COMPZ = 'N', then WORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm has failed to find all the eigenvalues in
*> a total of 30*N iterations; if INFO = i, then i
*> elements of E have not converged to zero; on exit, D
*> and E contain the elements of a symmetric tridiagonal
*> matrix which is orthogonally similar to the original
*> matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup steqr
*
* =====================================================================
SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0 )
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
* ..
* .. Local Scalars ..
INTEGER I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND,
$ LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1,
$ NM1, NMAXIT
DOUBLE PRECISION ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2,
$ S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST, DLAPY2
EXTERNAL LSAME, DLAMCH, DLANST, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DLAE2, DLAEV2, DLARTG, DLASCL, DLASET,
$ DLASR,
$ DLASRT, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
$ N ) ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEQR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ICOMPZ.EQ.2 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Determine the unit roundoff and over/underflow thresholds.
*
EPS = DLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
*
* Compute the eigenvalues and eigenvectors of the tridiagonal
* matrix.
*
IF( ICOMPZ.EQ.2 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
NMAXIT = N*MAXIT
JTOT = 0
*
* Determine where the matrix splits and choose QL or QR iteration
* for each block, according to whether top or bottom diagonal
* element is smaller.
*
L1 = 1
NM1 = N - 1
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 160
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
IF( L1.LE.NM1 ) THEN
DO 20 M = L1, NM1
TST = ABS( E( M ) )
IF( TST.EQ.ZERO )
$ GO TO 30
IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
END IF
M = N
*
30 CONTINUE
L = L1
LSV = L
LEND = M
LENDSV = LEND
L1 = M + 1
IF( LEND.EQ.L )
$ GO TO 10
*
* Scale submatrix in rows and columns L to LEND
*
ANORM = DLANST( 'M', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
IF( ANORM.EQ.ZERO )
$ GO TO 10
IF( ANORM.GT.SSFMAX ) THEN
ISCALE = 1
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ),
$ N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
$ INFO )
ELSE IF( ANORM.LT.SSFMIN ) THEN
ISCALE = 2
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ),
$ N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
$ INFO )
END IF
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GT.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
40 CONTINUE
IF( L.NE.LEND ) THEN
LENDM1 = LEND - 1
DO 50 M = L, LENDM1
TST = ABS( E( M ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M+1 ) )+
$ SAFMIN )GO TO 60
50 CONTINUE
END IF
*
M = LEND
*
60 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 80
*
* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L+1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL DLAEV2( D( L ), E( L ), D( L+1 ), RT1, RT2, C,
$ S )
WORK( L ) = C
WORK( N-1+L ) = S
CALL DLASR( 'R', 'V', 'B', N, 2, WORK( L ),
$ WORK( N-1+L ), Z( 1, L ), LDZ )
ELSE
CALL DLAE2( D( L ), E( L ), D( L+1 ), RT1, RT2 )
END IF
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L+1 )-P ) / ( TWO*E( L ) )
R = DLAPY2( G, ONE )
G = D( M ) - P + ( E( L ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
MM1 = M - 1
DO 70 I = MM1, L, -1
F = S*E( I )
B = C*E( I )
CALL DLARTG( G, F, C, S, R )
IF( I.NE.M-1 )
$ E( I+1 ) = R
G = D( I+1 ) - P
R = ( D( I )-G )*S + TWO*C*B
P = S*R
D( I+1 ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = -S
END IF
*
70 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = M - L + 1
CALL DLASR( 'R', 'V', 'B', N, MM, WORK( L ),
$ WORK( N-1+L ),
$ Z( 1, L ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( L ) = G
GO TO 40
*
* Eigenvalue found.
*
80 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
90 CONTINUE
IF( L.NE.LEND ) THEN
LENDP1 = LEND + 1
DO 100 M = L, LENDP1, -1
TST = ABS( E( M-1 ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M-1 ) )+
$ SAFMIN )GO TO 110
100 CONTINUE
END IF
*
M = LEND
*
110 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 130
*
* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L-1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL DLAEV2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2, C,
$ S )
WORK( M ) = C
WORK( N-1+M ) = S
CALL DLASR( 'R', 'V', 'F', N, 2, WORK( M ),
$ WORK( N-1+M ), Z( 1, L-1 ), LDZ )
ELSE
CALL DLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 )
END IF
D( L-1 ) = RT1
D( L ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) )
R = DLAPY2( G, ONE )
G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
LM1 = L - 1
DO 120 I = M, LM1
F = S*E( I )
B = C*E( I )
CALL DLARTG( G, F, C, S, R )
IF( I.NE.M )
$ E( I-1 ) = R
G = D( I ) - P
R = ( D( I+1 )-G )*S + TWO*C*B
P = S*R
D( I ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = S
END IF
*
120 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = L - M + 1
CALL DLASR( 'R', 'V', 'F', N, MM, WORK( M ),
$ WORK( N-1+M ),
$ Z( 1, M ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( LM1 ) = G
GO TO 90
*
* Eigenvalue found.
*
130 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
*
END IF
*
* Undo scaling if necessary
*
140 CONTINUE
IF( ISCALE.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1,
$ E( LSV ),
$ N, INFO )
ELSE IF( ISCALE.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1,
$ E( LSV ),
$ N, INFO )
END IF
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.LT.NMAXIT )
$ GO TO 10
DO 150 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
150 CONTINUE
GO TO 190
*
* Order eigenvalues and eigenvectors.
*
160 CONTINUE
IF( ICOMPZ.EQ.0 ) THEN
*
* Use Quick Sort
*
CALL DLASRT( 'I', N, D, INFO )
*
ELSE
*
* Use Selection Sort to minimize swaps of eigenvectors
*
DO 180 II = 2, N
I = II - 1
K = I
P = D( I )
DO 170 J = II, N
IF( D( J ).LT.P ) THEN
K = J
P = D( J )
END IF
170 CONTINUE
IF( K.NE.I ) THEN
D( K ) = D( I )
D( I ) = P
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
END IF
180 CONTINUE
END IF
*
190 CONTINUE
RETURN
*
* End of DSTEQR
*
END
*> \brief \b DSTERF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTERF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsterf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsterf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsterf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTERF( N, D, E, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
*> using the Pal-Walker-Kahan variant of the QL or QR algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm failed to find all of the eigenvalues in
*> a total of 30*N iterations; if INFO = i, then i
*> elements of E have not converged to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup sterf
*
* =====================================================================
SUBROUTINE DSTERF( N, D, E, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0 )
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
* ..
* .. Local Scalars ..
INTEGER I, ISCALE, JTOT, L, L1, LEND, LENDSV, LSV, M,
$ NMAXIT
DOUBLE PRECISION ALPHA, ANORM, BB, C, EPS, EPS2, GAMMA, OLDC,
$ OLDGAM, P, R, RT1, RT2, RTE, S, SAFMAX, SAFMIN,
$ SIGMA, SSFMAX, SSFMIN, RMAX
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANST, DLAPY2
EXTERNAL DLAMCH, DLANST, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DLAE2, DLASCL, DLASRT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
* Quick return if possible
*
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DSTERF', -INFO )
RETURN
END IF
IF( N.LE.1 )
$ RETURN
*
* Determine the unit roundoff for this environment.
*
EPS = DLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
RMAX = DLAMCH( 'O' )
*
* Compute the eigenvalues of the tridiagonal matrix.
*
NMAXIT = N*MAXIT
SIGMA = ZERO
JTOT = 0
*
* Determine where the matrix splits and choose QL or QR iteration
* for each block, according to whether top or bottom diagonal
* element is smaller.
*
L1 = 1
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 170
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
DO 20 M = L1, N - 1
IF( ABS( E( M ) ).LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
M = N
*
30 CONTINUE
L = L1
LSV = L
LEND = M
LENDSV = LEND
L1 = M + 1
IF( LEND.EQ.L )
$ GO TO 10
*
* Scale submatrix in rows and columns L to LEND
*
ANORM = DLANST( 'M', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
IF( ANORM.EQ.ZERO )
$ GO TO 10
IF( (ANORM.GT.SSFMAX) ) THEN
ISCALE = 1
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ),
$ N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
$ INFO )
ELSE IF( ANORM.LT.SSFMIN ) THEN
ISCALE = 2
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ),
$ N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
$ INFO )
END IF
*
DO 40 I = L, LEND - 1
E( I ) = E( I )**2
40 CONTINUE
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GE.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
50 CONTINUE
IF( L.NE.LEND ) THEN
DO 60 M = L, LEND - 1
IF( ABS( E( M ) ).LE.EPS2*ABS( D( M )*D( M+1 ) ) )
$ GO TO 70
60 CONTINUE
END IF
M = LEND
*
70 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 90
*
* If remaining matrix is 2 by 2, use DLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L+1 ) THEN
RTE = SQRT( E( L ) )
CALL DLAE2( D( L ), RTE, D( L+1 ), RT1, RT2 )
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L ) )
SIGMA = ( D( L+1 )-P ) / ( TWO*RTE )
R = DLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
DO 80 I = M - 1, L, -1
BB = E( I )
R = P + BB
IF( I.NE.M-1 )
$ E( I+1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I+1 ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
80 CONTINUE
*
E( L ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 50
*
* Eigenvalue found.
*
90 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
100 CONTINUE
DO 110 M = L, LEND + 1, -1
IF( ABS( E( M-1 ) ).LE.EPS2*ABS( D( M )*D( M-1 ) ) )
$ GO TO 120
110 CONTINUE
M = LEND
*
120 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 140
*
* If remaining matrix is 2 by 2, use DLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L-1 ) THEN
RTE = SQRT( E( L-1 ) )
CALL DLAE2( D( L ), RTE, D( L-1 ), RT1, RT2 )
D( L ) = RT1
D( L-1 ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L-1 ) )
SIGMA = ( D( L-1 )-P ) / ( TWO*RTE )
R = DLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
DO 130 I = M, L - 1
BB = E( I )
R = P + BB
IF( I.NE.M )
$ E( I-1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I+1 )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
130 CONTINUE
*
E( L-1 ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 100
*
* Eigenvalue found.
*
140 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
*
END IF
*
* Undo scaling if necessary
*
150 CONTINUE
IF( ISCALE.EQ.1 )
$ CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
IF( ISCALE.EQ.2 )
$ CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.LT.NMAXIT )
$ GO TO 10
DO 160 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
160 CONTINUE
GO TO 180
*
* Sort eigenvalues in increasing order.
*
170 CONTINUE
CALL DLASRT( 'I', N, D, INFO )
*
180 CONTINUE
RETURN
*
* End of DSTERF
*
END
*> \brief <b> DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstev.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEV( JOBZ, N, D, E, Z, LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
* INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEV computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric tridiagonal matrix A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A, stored in elements 1 to N-1 of E.
*> On exit, the contents of E are destroyed.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*> eigenvectors of the matrix A, with the i-th column of Z
*> holding the eigenvector associated with D(i).
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
*> If JOBZ = 'N', WORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the algorithm failed to converge; i
*> off-diagonal elements of E did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stev
*
* =====================================================================
SUBROUTINE DSTEV( JOBZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL WANTZ
INTEGER IMAX, ISCALE
DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
$ TNRM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSTEQR, DSTERF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -6
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEV ', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
TNRM = DLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / TNRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL DSCAL( N, SIGMA, D, 1 )
CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
END IF
*
* For eigenvalues only, call DSTERF. For eigenvalues and
* eigenvectors, call DSTEQR.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, D, E, INFO )
ELSE
CALL DSTEQR( 'I', N, D, E, Z, LDZ, WORK, INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, D, 1 )
END IF
*
RETURN
*
* End of DSTEV
*
END
*> \brief <b> DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEVD( JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
* INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEVD computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric tridiagonal matrix. If eigenvectors are desired, it
*> uses a divide and conquer algorithm.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A, stored in elements 1 to N-1 of E.
*> On exit, the contents of E are destroyed.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*> eigenvectors of the matrix A, with the i-th column of Z
*> holding the eigenvector associated with D(i).
*> If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If JOBZ = 'N' or N <= 1 then LWORK must be at least 1.
*> If JOBZ = 'V' and N > 1 then LWORK must be at least
*> ( 1 + 4*N + N**2 ).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If JOBZ = 'N' or N <= 1 then LIWORK must be at least 1.
*> If JOBZ = 'V' and N > 1 then LIWORK must be at least 3+5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the algorithm failed to converge; i
*> off-diagonal elements of E did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stevd
*
* =====================================================================
SUBROUTINE DSTEVD( JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WANTZ
INTEGER ISCALE, LIWMIN, LWMIN
DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
$ TNRM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSTEDC, DSTERF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
LIWMIN = 1
LWMIN = 1
IF( N.GT.1 .AND. WANTZ ) THEN
LWMIN = 1 + 4*N + N**2
LIWMIN = 3 + 5*N
END IF
*
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -6
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -8
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
TNRM = DLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / TNRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL DSCAL( N, SIGMA, D, 1 )
CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
END IF
*
* For eigenvalues only, call DSTERF. For eigenvalues and
* eigenvectors, call DSTEDC.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, D, E, INFO )
ELSE
CALL DSTEDC( 'I', N, D, E, Z, LDZ, WORK, LWORK, IWORK,
$ LIWORK,
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 )
$ CALL DSCAL( N, ONE / SIGMA, D, 1 )
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSTEVD
*
END
*> \brief <b> DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEVR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
* M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE
* INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER ISUPPZ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEVR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix T. Eigenvalues and
*> eigenvectors can be selected by specifying either a range of values
*> or a range of indices for the desired eigenvalues.
*>
*> Whenever possible, DSTEVR calls DSTEMR to compute the
*> eigenspectrum using Relatively Robust Representations. DSTEMR
*> computes eigenvalues by the dqds algorithm, while orthogonal
*> eigenvectors are computed from various "good" L D L^T representations
*> (also known as Relatively Robust Representations). Gram-Schmidt
*> orthogonalization is avoided as far as possible. More specifically,
*> the various steps of the algorithm are as follows. For the i-th
*> unreduced block of T,
*> (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
*> is a relatively robust representation,
*> (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
*> relative accuracy by the dqds algorithm,
*> (c) If there is a cluster of close eigenvalues, "choose" sigma_i
*> close to the cluster, and go to step (a),
*> (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
*> compute the corresponding eigenvector by forming a
*> rank-revealing twisted factorization.
*> The desired accuracy of the output can be specified by the input
*> parameter ABSTOL.
*>
*> For more details, see "A new O(n^2) algorithm for the symmetric
*> tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
*> Computer Science Division Technical Report No. UCB//CSD-97-971,
*> UC Berkeley, May 1997.
*>
*>
*> Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
*> on machines which conform to the ieee-754 floating point standard.
*> DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
*> when partial spectrum requests are made.
*>
*> Normal execution of DSTEMR may create NaNs and infinities and
*> hence may abort due to a floating point exception in environments
*> which do not handle NaNs and infinities in the ieee standard default
*> manner.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
*> DSTEIN are called
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A.
*> On exit, D may be multiplied by a constant factor chosen
*> to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (max(1,N-1))
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A in elements 1 to N-1 of E.
*> On exit, E may be multiplied by a constant factor chosen
*> to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing A to tridiagonal form.
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices
*> with Guaranteed High Relative Accuracy," by Demmel and
*> Kahan, LAPACK Working Note #3.
*>
*> If high relative accuracy is important, set ABSTOL to
*> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
*> eigenvalues are computed to high relative accuracy when
*> possible in future releases. The current code does not
*> make any guarantees about high relative accuracy, but
*> future releases will. See J. Barlow and J. Demmel,
*> "Computing Accurate Eigensystems of Scaled Diagonally
*> Dominant Matrices", LAPACK Working Note #7, for a discussion
*> of which matrices define their eigenvalues to high relative
*> accuracy.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the selected eigenvalues in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*> The support of the eigenvectors in Z, i.e., the indices
*> indicating the nonzero elements in Z. The i-th eigenvector
*> is nonzero only in elements ISUPPZ( 2*i-1 ) through
*> ISUPPZ( 2*i ).
*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal (and
*> minimal) LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,20*N).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal (and
*> minimal) LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK. LIWORK >= max(1,10*N).
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: Internal error
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stevr
*
*> \par Contributors:
* ==================
*>
*> Inderjit Dhillon, IBM Almaden, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Ken Stanley, Computer Science Division, University of
*> California at Berkeley, USA \n
*>
* =====================================================================
SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ ABSTOL,
$ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
$ TRYRAC
CHARACTER ORDER
INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
$ INDIWO, ISCALE, ITMP1, J, JJ, LIWMIN, LWMIN,
$ NSPLIT
DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
$ TMP1, TNRM, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, ILAENV, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEMR, DSTEIN,
$ DSTERF,
$ DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
*
* Test the input parameters.
*
IEEEOK = ILAENV( 10, 'DSTEVR', 'N', 1, 2, 3, 4 )
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
LWMIN = MAX( 1, 20*N )
LIWMIN = MAX( 1, 10*N )
*
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -7
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -9
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -14
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEVR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = D( 1 )
ELSE
IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
M = 1
W( 1 ) = D( 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
IF( VALEIG ) THEN
VLL = VL
VUU = VU
END IF
*
TNRM = DLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / TNRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL DSCAL( N, SIGMA, D, 1 )
CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
* Initialize indices into workspaces. Note: These indices are used only
* if DSTERF or DSTEMR fail.
* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
* stores the block indices of each of the M<=N eigenvalues.
INDIBL = 1
* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
* stores the starting and finishing indices of each block.
INDISP = INDIBL + N
* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
* that corresponding to eigenvectors that fail to converge in
* DSTEIN. This information is discarded; if any fail, the driver
* returns INFO > 0.
INDIFL = INDISP + N
* INDIWO is the offset of the remaining integer workspace.
INDIWO = INDISP + N
*
* If all eigenvalues are desired, then
* call DSTERF or DSTEMR. If this fails for some eigenvalue, then
* try DSTEBZ.
*
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
IF( .NOT.WANTZ ) THEN
CALL DCOPY( N, D, 1, W, 1 )
CALL DSTERF( N, W, WORK, INFO )
ELSE
CALL DCOPY( N, D, 1, WORK( N+1 ), 1 )
IF (ABSTOL .LE. TWO*N*EPS) THEN
TRYRAC = .TRUE.
ELSE
TRYRAC = .FALSE.
END IF
CALL DSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
$ IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
$ WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
*
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 10
END IF
INFO = 0
END IF
*
* Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E,
$ M,
$ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ),
$ IWORK( INDISP ),
$ Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
10 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 30 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 20 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
20 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( I )
W( I ) = W( J )
IWORK( I ) = IWORK( J )
W( J ) = TMP1
IWORK( J ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
END IF
30 CONTINUE
END IF
*
* Causes problems with tests 19 & 20:
* IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
*
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
RETURN
*
* End of DSTEVR
*
END
*> \brief <b> DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
* M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE
* INTEGER IL, INFO, IU, LDZ, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix A. Eigenvalues and
*> eigenvectors can be selected by specifying either a range of values
*> or a range of indices for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A.
*> On exit, D may be multiplied by a constant factor chosen
*> to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (max(1,N-1))
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A in elements 1 to N-1 of E.
*> On exit, E may be multiplied by a constant factor chosen
*> to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less
*> than or equal to zero, then EPS*|T| will be used in
*> its place, where |T| is the 1-norm of the tridiagonal
*> matrix.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*DLAMCH('S').
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices
*> with Guaranteed High Relative Accuracy," by Demmel and
*> Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the selected eigenvalues in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If an eigenvector fails to converge (INFO > 0), then that
*> column of Z contains the latest approximation to the
*> eigenvector, and the index of the eigenvector is returned
*> in IFAIL. If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (N)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvectors that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, then i eigenvectors failed to converge.
*> Their indices are stored in array IFAIL.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stevx
*
* =====================================================================
SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ ABSTOL,
$ M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
INTEGER IL, INFO, IU, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
CHARACTER ORDER
INTEGER I, IMAX, INDISP, INDIWO, INDWRK,
$ ISCALE, ITMP1, J, JJ, NSPLIT
DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
$ TMP1, TNRM, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEIN, DSTEQR,
$ DSTERF,
$ DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -7
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -9
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
$ INFO = -14
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = D( 1 )
ELSE
IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
M = 1
W( 1 ) = D( 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
IF( VALEIG ) THEN
VLL = VL
VUU = VU
ELSE
VLL = ZERO
VUU = ZERO
END IF
TNRM = DLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / TNRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL DSCAL( N, SIGMA, D, 1 )
CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
*
* If all eigenvalues are desired and ABSTOL is less than zero, then
* call DSTERF or SSTEQR. If this fails for some eigenvalue, then
* try DSTEBZ.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
CALL DCOPY( N, D, 1, W, 1 )
CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
INDWRK = N + 1
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK, INFO )
ELSE
CALL DSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ),
$ INFO )
IF( INFO.EQ.0 ) THEN
DO 10 I = 1, N
IFAIL( I ) = 0
10 CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 20
END IF
INFO = 0
END IF
*
* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDWRK = 1
INDISP = 1 + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E,
$ M,
$ NSPLIT, W, IWORK( 1 ), IWORK( INDISP ),
$ WORK( INDWRK ), IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, D, E, M, W, IWORK( 1 ), IWORK( INDISP ),
$ Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL,
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
20 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 40 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 30 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
30 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( 1 + I-1 )
W( I ) = W( J )
IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
W( J ) = TMP1
IWORK( 1 + J-1 ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
40 CONTINUE
END IF
*
RETURN
*
* End of DSTEVX
*
END
*> \brief \b DSYCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsycon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsycon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsycon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYCON( UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric matrix A using the factorization
*> A = U*D*U**T or A = L*D*L**T computed by DSYTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*> estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hecon
*
* =====================================================================
SUBROUTINE DSYCON( UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, KASE
DOUBLE PRECISION AINVNM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DSYTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.LE.ZERO ) THEN
RETURN
END IF
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
DO 10 I = N, 1, -1
IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
$ RETURN
10 CONTINUE
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
DO 20 I = 1, N
IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
$ RETURN
20 CONTINUE
END IF
*
* Estimate the 1-norm of the inverse.
*
KASE = 0
30 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
*
* Multiply by inv(L*D*L**T) or inv(U*D*U**T).
*
CALL DSYTRS( UPLO, N, 1, A, LDA, IPIV, WORK, N, INFO )
GO TO 30
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
RETURN
*
* End of DSYCON
*
END
*> \brief \b DSYCONV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYCONV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyconv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyconv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyconv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYCONV( UPLO, WAY, N, A, LDA, IPIV, E, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO, WAY
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYCONV convert A given by TRF into L and D and vice-versa.
*> Get Non-diag elements of D (returned in workspace) and
*> apply or reverse permutation done in TRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] WAY
*> \verbatim
*> WAY is CHARACTER*1
*> = 'C': Convert
*> = 'R': Revert
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> E stores the supdiagonal/subdiagonal of the symmetric 1-by-1
*> or 2-by-2 block diagonal matrix D in LDLT.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup syconv
*
* =====================================================================
SUBROUTINE DSYCONV( UPLO, WAY, N, A, LDA, IPIV, E, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO, WAY
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Local Scalars ..
LOGICAL UPPER, CONVERT
INTEGER I, IP, J
DOUBLE PRECISION TEMP
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
CONVERT = LSAME( WAY, 'C' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.CONVERT .AND. .NOT.LSAME( WAY, 'R' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYCONV', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* A is UPPER
*
* Convert A (A is upper)
*
* Convert VALUE
*
IF ( CONVERT ) THEN
I=N
E(1)=ZERO
DO WHILE ( I .GT. 1 )
IF( IPIV(I) .LT. 0 ) THEN
E(I)=A(I-1,I)
E(I-1)=ZERO
A(I-1,I)=ZERO
I=I-1
ELSE
E(I)=ZERO
ENDIF
I=I-1
END DO
*
* Convert PERMUTATIONS
*
I=N
DO WHILE ( I .GE. 1 )
IF( IPIV(I) .GT. 0) THEN
IP=IPIV(I)
IF( I .LT. N) THEN
DO 12 J= I+1,N
TEMP=A(IP,J)
A(IP,J)=A(I,J)
A(I,J)=TEMP
12 CONTINUE
ENDIF
ELSE
IP=-IPIV(I)
IF( I .LT. N) THEN
DO 13 J= I+1,N
TEMP=A(IP,J)
A(IP,J)=A(I-1,J)
A(I-1,J)=TEMP
13 CONTINUE
ENDIF
I=I-1
ENDIF
I=I-1
END DO
ELSE
*
* Revert A (A is upper)
*
*
* Revert PERMUTATIONS
*
I=1
DO WHILE ( I .LE. N )
IF( IPIV(I) .GT. 0 ) THEN
IP=IPIV(I)
IF( I .LT. N) THEN
DO J= I+1,N
TEMP=A(IP,J)
A(IP,J)=A(I,J)
A(I,J)=TEMP
END DO
ENDIF
ELSE
IP=-IPIV(I)
I=I+1
IF( I .LT. N) THEN
DO J= I+1,N
TEMP=A(IP,J)
A(IP,J)=A(I-1,J)
A(I-1,J)=TEMP
END DO
ENDIF
ENDIF
I=I+1
END DO
*
* Revert VALUE
*
I=N
DO WHILE ( I .GT. 1 )
IF( IPIV(I) .LT. 0 ) THEN
A(I-1,I)=E(I)
I=I-1
ENDIF
I=I-1
END DO
END IF
ELSE
*
* A is LOWER
*
IF ( CONVERT ) THEN
*
* Convert A (A is lower)
*
*
* Convert VALUE
*
I=1
E(N)=ZERO
DO WHILE ( I .LE. N )
IF( I.LT.N .AND. IPIV(I) .LT. 0 ) THEN
E(I)=A(I+1,I)
E(I+1)=ZERO
A(I+1,I)=ZERO
I=I+1
ELSE
E(I)=ZERO
ENDIF
I=I+1
END DO
*
* Convert PERMUTATIONS
*
I=1
DO WHILE ( I .LE. N )
IF( IPIV(I) .GT. 0 ) THEN
IP=IPIV(I)
IF (I .GT. 1) THEN
DO 22 J= 1,I-1
TEMP=A(IP,J)
A(IP,J)=A(I,J)
A(I,J)=TEMP
22 CONTINUE
ENDIF
ELSE
IP=-IPIV(I)
IF (I .GT. 1) THEN
DO 23 J= 1,I-1
TEMP=A(IP,J)
A(IP,J)=A(I+1,J)
A(I+1,J)=TEMP
23 CONTINUE
ENDIF
I=I+1
ENDIF
I=I+1
END DO
ELSE
*
* Revert A (A is lower)
*
*
* Revert PERMUTATIONS
*
I=N
DO WHILE ( I .GE. 1 )
IF( IPIV(I) .GT. 0 ) THEN
IP=IPIV(I)
IF (I .GT. 1) THEN
DO J= 1,I-1
TEMP=A(I,J)
A(I,J)=A(IP,J)
A(IP,J)=TEMP
END DO
ENDIF
ELSE
IP=-IPIV(I)
I=I-1
IF (I .GT. 1) THEN
DO J= 1,I-1
TEMP=A(I+1,J)
A(I+1,J)=A(IP,J)
A(IP,J)=TEMP
END DO
ENDIF
ENDIF
I=I-1
END DO
*
* Revert VALUE
*
I=1
DO WHILE ( I .LE. N-1 )
IF( IPIV(I) .LT. 0 ) THEN
A(I+1,I)=E(I)
I=I+1
ENDIF
I=I+1
END DO
END IF
END IF
RETURN
*
* End of DSYCONV
*
END
*> \brief \b DSYEQUB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, N
* DOUBLE PRECISION AMAX, SCOND
* CHARACTER UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYEQUB computes row and column scalings intended to equilibrate a
*> symmetric matrix A (with respect to the Euclidean norm) and reduce
*> its condition number. The scale factors S are computed by the BIN
*> algorithm (see references) so that the scaled matrix B with elements
*> B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
*> the smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The N-by-N symmetric matrix whose scaling factors are to be
*> computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*> SCOND is DOUBLE PRECISION
*> If INFO = 0, S contains the ratio of the smallest S(i) to
*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
*> large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is DOUBLE PRECISION
*> Largest absolute value of any matrix element. If AMAX is
*> very close to overflow or very close to underflow, the
*> matrix should be scaled.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heequb
*
*> \par References:
* ================
*>
*> Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n
*> Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n
*> DOI 10.1023/B:NUMA.0000016606.32820.69 \n
*> Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
*>
* =====================================================================
SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION AMAX, SCOND
CHARACTER UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
INTEGER MAX_ITER
PARAMETER ( MAX_ITER = 100 )
* ..
* .. Local Scalars ..
INTEGER I, J, ITER
DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
$ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
LOGICAL UP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
LOGICAL LSAME
EXTERNAL DLAMCH, LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR.
$ LSAME( UPLO, 'L' ) ) ) THEN
INFO = -1
ELSE IF ( N .LT. 0 ) THEN
INFO = -2
ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
INFO = -4
END IF
IF ( INFO .NE. 0 ) THEN
CALL XERBLA( 'DSYEQUB', -INFO )
RETURN
END IF
UP = LSAME( UPLO, 'U' )
AMAX = ZERO
*
* Quick return if possible.
*
IF ( N .EQ. 0 ) THEN
SCOND = ONE
RETURN
END IF
DO I = 1, N
S( I ) = ZERO
END DO
AMAX = ZERO
IF ( UP ) THEN
DO J = 1, N
DO I = 1, J-1
S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
AMAX = MAX( AMAX, ABS( A( I, J ) ) )
END DO
S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
AMAX = MAX( AMAX, ABS( A( J, J ) ) )
END DO
ELSE
DO J = 1, N
S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
AMAX = MAX( AMAX, ABS( A( J, J ) ) )
DO I = J+1, N
S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
AMAX = MAX( AMAX, ABS( A( I, J ) ) )
END DO
END DO
END IF
DO J = 1, N
S( J ) = 1.0D0 / S( J )
END DO
TOL = ONE / SQRT( 2.0D0 * N )
DO ITER = 1, MAX_ITER
SCALE = 0.0D0
SUMSQ = 0.0D0
* beta = |A|s
DO I = 1, N
WORK( I ) = ZERO
END DO
IF ( UP ) THEN
DO J = 1, N
DO I = 1, J-1
WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
END DO
WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
END DO
ELSE
DO J = 1, N
WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
DO I = J+1, N
WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
END DO
END DO
END IF
* avg = s^T beta / n
AVG = 0.0D0
DO I = 1, N
AVG = AVG + S( I )*WORK( I )
END DO
AVG = AVG / N
STD = 0.0D0
DO I = N+1, 2*N
WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
END DO
CALL DLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
STD = SCALE * SQRT( SUMSQ / N )
IF ( STD .LT. TOL * AVG ) GOTO 999
DO I = 1, N
T = ABS( A( I, I ) )
SI = S( I )
C2 = ( N-1 ) * T
C1 = ( N-2 ) * ( WORK( I ) - T*SI )
C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
D = C1*C1 - 4*C0*C2
IF ( D .LE. 0 ) THEN
INFO = -1
RETURN
END IF
SI = -2*C0 / ( C1 + SQRT( D ) )
D = SI - S( I )
U = ZERO
IF ( UP ) THEN
DO J = 1, I
T = ABS( A( J, I ) )
U = U + S( J )*T
WORK( J ) = WORK( J ) + D*T
END DO
DO J = I+1,N
T = ABS( A( I, J ) )
U = U + S( J )*T
WORK( J ) = WORK( J ) + D*T
END DO
ELSE
DO J = 1, I
T = ABS( A( I, J ) )
U = U + S( J )*T
WORK( J ) = WORK( J ) + D*T
END DO
DO J = I+1,N
T = ABS( A( J, I ) )
U = U + S( J )*T
WORK( J ) = WORK( J ) + D*T
END DO
END IF
AVG = AVG + ( U + WORK( I ) ) * D / N
S( I ) = SI
END DO
END DO
999 CONTINUE
SMLNUM = DLAMCH( 'SAFEMIN' )
BIGNUM = ONE / SMLNUM
SMIN = BIGNUM
SMAX = ZERO
T = ONE / SQRT( AVG )
BASE = DLAMCH( 'B' )
U = ONE / LOG( BASE )
DO I = 1, N
S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
SMIN = MIN( SMIN, S( I ) )
SMAX = MAX( SMAX, S( I ) )
END DO
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
*
END
*> \brief <b> DSYEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyev.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYEV computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*> orthonormal eigenvectors of the matrix A.
*> If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
*> or the upper triangle (if UPLO='U') of A, including the
*> diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= max(1,3*N-1).
*> For optimal efficiency, LWORK >= (NB+2)*N,
*> where NB is the blocksize for DSYTRD returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the algorithm failed to converge; i
*> off-diagonal elements of an intermediate tridiagonal
*> form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heev
*
* =====================================================================
SUBROUTINE DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, LQUERY, WANTZ
INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE,
$ LLWORK, LWKOPT, NB
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DLASCL, DORGTR, DSCAL, DSTEQR, DSTERF,
$ DSYTRD,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
LQUERY = ( LWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( 1, ( NB+2 )*N )
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.MAX( 1, 3*N-1 ) .AND. .NOT.LQUERY )
$ INFO = -8
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
RETURN
END IF
*
IF( N.EQ.1 ) THEN
W( 1 ) = A( 1, 1 )
WORK( 1 ) = 2
IF( WANTZ )
$ A( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 )
$ CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
* Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
INDWRK = INDTAU + N
LLWORK = LWORK - INDWRK + 1
CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ),
$ WORK( INDWRK ), LLWORK, IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, first call
* DORGTR to generate the orthogonal matrix, then call DSTEQR.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DORGTR( UPLO, N, A, LDA, WORK( INDTAU ),
$ WORK( INDWRK ),
$ LLWORK, IINFO )
CALL DSTEQR( JOBZ, N, W, WORK( INDE ), A, LDA,
$ WORK( INDTAU ),
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DSYEV
*
END
*> \brief <b> DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, LDA, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYEVD computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A. If eigenvectors are desired, it uses a
*> divide and conquer algorithm.
*>
*> Because of large use of BLAS of level 3, DSYEVD needs N**2 more
*> workspace than DSYEVX.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*> orthonormal eigenvectors of the matrix A.
*> If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
*> or the upper triangle (if UPLO='U') of A, including the
*> diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK must be at least 1.
*> If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
*> If JOBZ = 'V' and N > 1, LWORK must be at least
*> 1 + 6*N + 2*N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If N <= 1, LIWORK must be at least 1.
*> If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
*> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i and JOBZ = 'N', then the algorithm failed
*> to converge; i off-diagonal elements of an intermediate
*> tridiagonal form did not converge to zero;
*> if INFO = i and JOBZ = 'V', then the algorithm failed
*> to compute an eigenvalue while working on the submatrix
*> lying in rows and columns INFO/(N+1) through
*> mod(INFO,N+1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heevd
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee \n
*> Modified description of INFO. Sven, 16 Feb 05. \n
*>
* =====================================================================
SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK,
$ IWORK,
$ LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDA, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
*
LOGICAL LOWER, LQUERY, WANTZ
INTEGER IINFO, INDE, INDTAU, INDWK2, INDWRK, ISCALE,
$ LIOPT, LIWMIN, LLWORK, LLWRK2, LOPT, LWMIN
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, DLAMCH, DLANSY, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLASCL, DORMTR, DSCAL, DSTEDC,
$ DSTERF,
$ DSYTRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
LOPT = LWMIN
LIOPT = LIWMIN
ELSE
IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 6*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N + 1
END IF
LOPT = MAX( LWMIN, 2*N +
$ N*ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1,
$ -1 ) )
LIOPT = LIWMIN
END IF
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -8
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYEVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
W( 1 ) = A( 1, 1 )
IF( WANTZ )
$ A( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 )
$ CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
* Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
INDWRK = INDTAU + N
LLWORK = LWORK - INDWRK + 1
INDWK2 = INDWRK + N*N
LLWRK2 = LWORK - INDWK2 + 1
*
CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ),
$ WORK( INDWRK ), LLWORK, IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, first call
* DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
* tridiagonal matrix, then call DORMTR to multiply it by the
* Householder transformations stored in A.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
$ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
CALL DORMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ),
$ WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO )
CALL DLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 )
$ CALL DSCAL( N, ONE / SIGMA, W, 1 )
*
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
RETURN
*
* End of DSYEVD
*
END
*> \brief <b> DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEVR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
* ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
* IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER ISUPPZ( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYEVR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
*> selected by specifying either a range of values or a range of indices
*> for the desired eigenvalues. Invocations with different choices for
*> these parameters may result in the computation of slightly different
*> eigenvalues and/or eigenvectors for the same matrix. The reason for
*> this behavior is that there exists a variety of algorithms (each
*> performing best for a particular set of options) with DSYEVR
*> attempting to select the best based on the various parameters. In all
*> cases, the computed values are accurate within the limits of finite
*> precision arithmetic.
*>
*> DSYEVR first reduces the matrix A to tridiagonal form T with a call
*> to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute
*> the eigenspectrum using Relatively Robust Representations. DSTEMR
*> computes eigenvalues by the dqds algorithm, while orthogonal
*> eigenvectors are computed from various "good" L D L^T representations
*> (also known as Relatively Robust Representations). Gram-Schmidt
*> orthogonalization is avoided as far as possible. More specifically,
*> the various steps of the algorithm are as follows.
*>
*> For each unreduced block (submatrix) of T,
*> (a) Compute T - sigma I = L D L^T, so that L and D
*> define all the wanted eigenvalues to high relative accuracy.
*> This means that small relative changes in the entries of D and L
*> cause only small relative changes in the eigenvalues and
*> eigenvectors. The standard (unfactored) representation of the
*> tridiagonal matrix T does not have this property in general.
*> (b) Compute the eigenvalues to suitable accuracy.
*> If the eigenvectors are desired, the algorithm attains full
*> accuracy of the computed eigenvalues only right before
*> the corresponding vectors have to be computed, see steps c) and d).
*> (c) For each cluster of close eigenvalues, select a new
*> shift close to the cluster, find a new factorization, and refine
*> the shifted eigenvalues to suitable accuracy.
*> (d) For each eigenvalue with a large enough relative separation compute
*> the corresponding eigenvector by forming a rank revealing twisted
*> factorization. Go back to (c) for any clusters that remain.
*>
*> The desired accuracy of the output can be specified by the input
*> parameter ABSTOL.
*>
*> For more details, see DSTEMR's documentation and:
*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
*> 2004. Also LAPACK Working Note 154.
*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
*> tridiagonal eigenvalue/eigenvector problem",
*> Computer Science Division Technical Report No. UCB/CSD-97-971,
*> UC Berkeley, May 1997.
*>
*>
*> Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
*> on machines which conform to the ieee-754 floating point standard.
*> DSYEVR calls DSTEBZ and DSTEIN on non-ieee machines and
*> when partial spectrum requests are made.
*>
*> Normal execution of DSTEMR may create NaNs and infinities and
*> hence may abort due to a floating point exception in environments
*> which do not handle NaNs and infinities in the ieee standard default
*> manner.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*>
*> This parameter influences the choice of the algorithm and
*> may alter the computed values.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
*> DSTEIN are called
*>
*> This parameter influences the choice of the algorithm and
*> may alter the computed values.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*> On exit, the lower triangle (if UPLO='L') or the upper
*> triangle (if UPLO='U') of A, including the diagonal, is
*> destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing A to tridiagonal form.
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices
*> with Guaranteed High Relative Accuracy," by Demmel and
*> Kahan, LAPACK Working Note #3.
*>
*> If high relative accuracy is important, set ABSTOL to
*> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
*> eigenvalues are computed to high relative accuracy when
*> possible in future releases. The current code does not
*> make any guarantees about high relative accuracy, but
*> future releases will. See J. Barlow and J. Demmel,
*> "Computing Accurate Eigensystems of Scaled Diagonally
*> Dominant Matrices", LAPACK Working Note #7, for a discussion
*> of which matrices define their eigenvalues to high relative
*> accuracy.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements contain the selected eigenvalues in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> Supplying N columns is always safe.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*> The support of the eigenvectors in Z, i.e., the indices
*> indicating the nonzero elements in Z. The i-th eigenvector
*> is nonzero only in elements ISUPPZ( 2*i-1 ) through
*> ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal
*> matrix). The support of the eigenvectors of A is typically
*> 1:N because of the orthogonal transformations applied by DORMTR.
*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK >= 1, else LWORK >= 26*N.
*> For optimal efficiency, LWORK >= (NB+6)*N,
*> where NB is the max of the blocksize for DSYTRD and DORMTR
*> returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: Internal error
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heevr
*
*> \par Contributors:
* ==================
*>
*> Inderjit Dhillon, IBM Almaden, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Ken Stanley, Computer Science Division, University of
*> California at Berkeley, USA \n
*> Jason Riedy, Computer Science Division, University of
*> California at Berkeley, USA \n
*>
* =====================================================================
SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL,
$ IU,
$ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
$ IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
$ TRYRAC
CHARACTER ORDER
INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
$ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
$ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
$ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT
DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
$ SIGMA, SMLNUM, TMP1, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DORMTR, DSCAL, DSTEBZ, DSTEMR,
$ DSTEIN,
$ DSTERF, DSWAP, DSYTRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
IEEEOK = ILAENV( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
*
LOWER = LSAME( UPLO, 'L' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
*
IF( N.LE.1 ) THEN
LWMIN = 1
LIWMIN = 1
ELSE
LWMIN = 26*N
LIWMIN = 10*N
END IF
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -8
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -10
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -15
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -18
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -20
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
LWKOPT = MAX( ( NB+1 )*N, LWMIN )
WORK( 1 ) = LWKOPT
IWORK( 1 ) = LIWMIN
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYEVR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( N.EQ.1 ) THEN
WORK( 1 ) = 1
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = A( 1, 1 )
ELSE
IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
M = 1
W( 1 ) = A( 1, 1 )
END IF
END IF
IF( WANTZ ) THEN
Z( 1, 1 ) = ONE
ISUPPZ( 1 ) = 1
ISUPPZ( 2 ) = 1
END IF
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
ABSTLL = ABSTOL
IF (VALEIG) THEN
VLL = VL
VUU = VU
END IF
ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
IF( LOWER ) THEN
DO 10 J = 1, N
CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
10 CONTINUE
ELSE
DO 20 J = 1, N
CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
20 CONTINUE
END IF
IF( ABSTOL.GT.0 )
$ ABSTLL = ABSTOL*SIGMA
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
* Initialize indices into workspaces. Note: The IWORK indices are
* used only if DSTERF or DSTEMR fail.
* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
* elementary reflectors used in DSYTRD.
INDTAU = 1
* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
INDD = INDTAU + N
* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
* tridiagonal matrix from DSYTRD.
INDE = INDD + N
* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
* -written by DSTEMR (the DSTERF path copies the diagonal to W).
INDDD = INDE + N
* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
* -written while computing the eigenvalues in DSTERF and DSTEMR.
INDEE = INDDD + N
* INDWK is the starting offset of the left-over workspace, and
* LLWORK is the remaining workspace size.
INDWK = INDEE + N
LLWORK = LWORK - INDWK + 1
* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
* stores the block indices of each of the M<=N eigenvalues.
INDIBL = 1
* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
* stores the starting and finishing indices of each block.
INDISP = INDIBL + N
* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
* that corresponding to eigenvectors that fail to converge in
* DSTEIN. This information is discarded; if any fail, the driver
* returns INFO > 0.
INDIFL = INDISP + N
* INDIWO is the offset of the remaining integer workspace.
INDIWO = INDIFL + N
*
* Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
$ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
*
* If all eigenvalues are desired
* then call DSTERF or DSTEMR and DORMTR.
*
IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND.
$ IEEEOK.EQ.1 ) THEN
IF( .NOT.WANTZ ) THEN
CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 )
*
IF (ABSTOL .LE. TWO*N*EPS) THEN
TRYRAC = .TRUE.
ELSE
TRYRAC = .FALSE.
END IF
CALL DSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ),
$ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ,
$ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK,
$ INFO )
*
*
*
* Apply orthogonal matrix used in reduction to tridiagonal
* form to eigenvectors returned by DSTEMR.
*
IF( WANTZ .AND. INFO.EQ.0 ) THEN
INDWKN = INDE
LLWRKN = LWORK - INDWKN + 1
CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA,
$ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
$ LLWRKN, IINFO )
END IF
END IF
*
*
IF( INFO.EQ.0 ) THEN
* Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are
* undefined.
M = N
GO TO 30
END IF
INFO = 0
END IF
*
* Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
* Also call DSTEBZ and DSTEIN if DSTEMR fails.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
$ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ),
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
$ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ),
$ INFO )
*
* Apply orthogonal matrix used in reduction to tridiagonal
* form to eigenvectors returned by DSTEIN.
*
INDWKN = INDE
LLWRKN = LWORK - INDWKN + 1
CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ),
$ Z,
$ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
* Jump here if DSTEMR/DSTEIN succeeded.
30 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
* It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
* not return this detailed information to the user.
*
IF( WANTZ ) THEN
DO 50 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 40 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
40 CONTINUE
*
IF( I.NE.0 ) THEN
W( I ) = W( J )
W( J ) = TMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
END IF
50 CONTINUE
END IF
*
* Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = LWKOPT
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSYEVR
*
END
*> \brief <b> DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
* ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
* IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
*> selected by specifying either a range of values or a range of indices
*> for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*> On exit, the lower triangle (if UPLO='L') or the upper
*> triangle (if UPLO='U') of A, including the diagonal, is
*> destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing A to tridiagonal form.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*DLAMCH('S').
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices
*> with Guaranteed High Relative Accuracy," by Demmel and
*> Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> On normal exit, the first M elements contain the selected
*> eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If an eigenvector fails to converge, then that column of Z
*> contains the latest approximation to the eigenvector, and the
*> index of the eigenvector is returned in IFAIL.
*> If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= 1, when N <= 1;
*> otherwise 8*N.
*> For optimal efficiency, LWORK >= (NB+3)*N,
*> where NB is the max of the blocksize for DSYTRD and DORMTR
*> returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (N)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvectors that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, then i eigenvectors failed to converge.
*> Their indices are stored in array IFAIL.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heevx
*
* =====================================================================
SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL,
$ IU,
$ ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
$ IFAIL, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
$ WANTZ
CHARACTER ORDER
INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
$ INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
$ ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN,
$ LWKOPT, NB, NSPLIT
DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
$ SIGMA, SMLNUM, TMP1, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DORGTR, DORMTR, DSCAL,
$ DSTEBZ,
$ DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
LOWER = LSAME( UPLO, 'L' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
LQUERY = ( LWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -8
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -10
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -15
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
LWKMIN = 8*N
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1,
$ -1 ) )
LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
$ INFO = -17
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYEVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 ) THEN
RETURN
END IF
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = A( 1, 1 )
ELSE
IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
M = 1
W( 1 ) = A( 1, 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
ABSTLL = ABSTOL
IF( VALEIG ) THEN
VLL = VL
VUU = VU
END IF
ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
IF( LOWER ) THEN
DO 10 J = 1, N
CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
10 CONTINUE
ELSE
DO 20 J = 1, N
CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
20 CONTINUE
END IF
IF( ABSTOL.GT.0 )
$ ABSTLL = ABSTOL*SIGMA
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
*
* Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
INDTAU = 1
INDE = INDTAU + N
INDD = INDE + N
INDWRK = INDD + N
LLWORK = LWORK - INDWRK + 1
CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
$ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
*
* If all eigenvalues are desired and ABSTOL is less than or equal to
* zero, then call DSTERF or DORGTR and SSTEQR. If this fails for
* some eigenvalue, then try DSTEBZ.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
INDEE = INDWRK + 2*N
IF( .NOT.WANTZ ) THEN
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ )
CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
$ WORK( INDWRK ), LLWORK, IINFO )
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
$ WORK( INDWRK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 30 I = 1, N
IFAIL( I ) = 0
30 CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 40
END IF
INFO = 0
END IF
*
* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDIBL = 1
INDISP = INDIBL + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
$ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
$ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
* Apply orthogonal matrix used in reduction to tridiagonal
* form to eigenvectors returned by DSTEIN.
*
INDWKN = INDE
LLWRKN = LWORK - INDWKN + 1
CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ),
$ Z,
$ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
40 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 60 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 50 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
50 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( INDIBL+I-1 )
W( I ) = W( J )
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
W( J ) = TMP1
IWORK( INDIBL+J-1 ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
60 CONTINUE
END IF
*
* Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DSYEVX
*
END
*> \brief \b DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygs2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygs2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygs2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGS2 reduces a real symmetric-definite generalized eigenproblem
*> to standard form.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.
*>
*> B must have been previously factorized as U**T *U or L*L**T by DPOTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*> = 2 or 3: compute U*A*U**T or L**T *A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored, and how B has been factorized.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the transformed matrix, stored in the
*> same format as A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The triangular factor from the Cholesky factorization of B,
*> as returned by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegs2
*
* =====================================================================
SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, HALF
PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER K
DOUBLE PRECISION AKK, BKK, CT
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DSYR2, DTRMV, DTRSV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGS2', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
DO 10 K = 1, N
*
* Update the upper triangle of A(k:n,k:n)
*
AKK = A( K, K )
BKK = B( K, K )
AKK = AKK / BKK**2
A( K, K ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, A( K, K+1 ), LDA )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
$ LDA )
CALL DSYR2( UPLO, N-K, -ONE, A( K, K+1 ), LDA,
$ B( K, K+1 ), LDB, A( K+1, K+1 ), LDA )
CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
$ LDA )
CALL DTRSV( UPLO, 'Transpose', 'Non-unit', N-K,
$ B( K+1, K+1 ), LDB, A( K, K+1 ), LDA )
END IF
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
DO 20 K = 1, N
*
* Update the lower triangle of A(k:n,k:n)
*
AKK = A( K, K )
BKK = B( K, K )
AKK = AKK / BKK**2
A( K, K ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, A( K+1, K ), 1 )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ),
$ 1 )
CALL DSYR2( UPLO, N-K, -ONE, A( K+1, K ), 1,
$ B( K+1, K ), 1, A( K+1, K+1 ), LDA )
CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ),
$ 1 )
CALL DTRSV( UPLO, 'No transpose', 'Non-unit', N-K,
$ B( K+1, K+1 ), LDB, A( K+1, K ), 1 )
END IF
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
DO 30 K = 1, N
*
* Update the upper triangle of A(1:k,1:k)
*
AKK = A( K, K )
BKK = B( K, K )
CALL DTRMV( UPLO, 'No transpose', 'Non-unit', K-1, B,
$ LDB, A( 1, K ), 1 )
CT = HALF*AKK
CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
CALL DSYR2( UPLO, K-1, ONE, A( 1, K ), 1, B( 1, K ),
$ 1,
$ A, LDA )
CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
CALL DSCAL( K-1, BKK, A( 1, K ), 1 )
A( K, K ) = AKK*BKK**2
30 CONTINUE
ELSE
*
* Compute L**T *A*L
*
DO 40 K = 1, N
*
* Update the lower triangle of A(1:k,1:k)
*
AKK = A( K, K )
BKK = B( K, K )
CALL DTRMV( UPLO, 'Transpose', 'Non-unit', K-1, B,
$ LDB,
$ A( K, 1 ), LDA )
CT = HALF*AKK
CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
CALL DSYR2( UPLO, K-1, ONE, A( K, 1 ), LDA, B( K, 1 ),
$ LDB, A, LDA )
CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
CALL DSCAL( K-1, BKK, A( K, 1 ), LDA )
A( K, K ) = AKK*BKK**2
40 CONTINUE
END IF
END IF
RETURN
*
* End of DSYGS2
*
END
*> \brief \b DSYGST
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygst.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGST reduces a real symmetric-definite generalized eigenproblem
*> to standard form.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*>
*> B must have been previously factorized as U**T*U or L*L**T by DPOTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*> = 2 or 3: compute U*A*U**T or L**T*A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored and B is factored as
*> U**T*U;
*> = 'L': Lower triangle of A is stored and B is factored as
*> L*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the transformed matrix, stored in the
*> same format as A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The triangular factor from the Cholesky factorization of B,
*> as returned by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegst
*
* =====================================================================
SUBROUTINE DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, HALF
PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER K, KB, NB
* ..
* .. External Subroutines ..
EXTERNAL DSYGS2, DSYMM, DSYR2K, DTRMM, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGST', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DSYGST', UPLO, N, -1, -1, -1 )
*
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code
*
CALL DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
ELSE
*
* Use blocked code
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
DO 10 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the upper triangle of A(k:n,k:n)
*
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
IF( K+KB.LE.N ) THEN
CALL DTRSM( 'Left', UPLO, 'Transpose',
$ 'Non-unit',
$ KB, N-K-KB+1, ONE, B( K, K ), LDB,
$ A( K, K+KB ), LDA )
CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF,
$ A( K, K ), LDA, B( K, K+KB ), LDB, ONE,
$ A( K, K+KB ), LDA )
CALL DSYR2K( UPLO, 'Transpose', N-K-KB+1, KB,
$ -ONE,
$ A( K, K+KB ), LDA, B( K, K+KB ), LDB,
$ ONE, A( K+KB, K+KB ), LDA )
CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF,
$ A( K, K ), LDA, B( K, K+KB ), LDB, ONE,
$ A( K, K+KB ), LDA )
CALL DTRSM( 'Right', UPLO, 'No transpose',
$ 'Non-unit', KB, N-K-KB+1, ONE,
$ B( K+KB, K+KB ), LDB, A( K, K+KB ),
$ LDA )
END IF
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
DO 20 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the lower triangle of A(k:n,k:n)
*
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
IF( K+KB.LE.N ) THEN
CALL DTRSM( 'Right', UPLO, 'Transpose',
$ 'Non-unit',
$ N-K-KB+1, KB, ONE, B( K, K ), LDB,
$ A( K+KB, K ), LDA )
CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF,
$ A( K, K ), LDA, B( K+KB, K ), LDB, ONE,
$ A( K+KB, K ), LDA )
CALL DSYR2K( UPLO, 'No transpose', N-K-KB+1, KB,
$ -ONE, A( K+KB, K ), LDA, B( K+KB, K ),
$ LDB, ONE, A( K+KB, K+KB ), LDA )
CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF,
$ A( K, K ), LDA, B( K+KB, K ), LDB, ONE,
$ A( K+KB, K ), LDA )
CALL DTRSM( 'Left', UPLO, 'No transpose',
$ 'Non-unit', N-K-KB+1, KB, ONE,
$ B( K+KB, K+KB ), LDB, A( K+KB, K ),
$ LDA )
END IF
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
DO 30 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the upper triangle of A(1:k+kb-1,1:k+kb-1)
*
CALL DTRMM( 'Left', UPLO, 'No transpose',
$ 'Non-unit',
$ K-1, KB, ONE, B, LDB, A( 1, K ), LDA )
CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K,
$ K ),
$ LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA )
CALL DSYR2K( UPLO, 'No transpose', K-1, KB, ONE,
$ A( 1, K ), LDA, B( 1, K ), LDB, ONE, A,
$ LDA )
CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K,
$ K ),
$ LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA )
CALL DTRMM( 'Right', UPLO, 'Transpose', 'Non-unit',
$ K-1, KB, ONE, B( K, K ), LDB, A( 1, K ),
$ LDA )
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
30 CONTINUE
ELSE
*
* Compute L**T*A*L
*
DO 40 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the lower triangle of A(1:k+kb-1,1:k+kb-1)
*
CALL DTRMM( 'Right', UPLO, 'No transpose',
$ 'Non-unit',
$ KB, K-1, ONE, B, LDB, A( K, 1 ), LDA )
CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ),
$ LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA )
CALL DSYR2K( UPLO, 'Transpose', K-1, KB, ONE,
$ A( K, 1 ), LDA, B( K, 1 ), LDB, ONE, A,
$ LDA )
CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ),
$ LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA )
CALL DTRMM( 'Left', UPLO, 'Transpose', 'Non-unit',
$ KB,
$ K-1, ONE, B( K, K ), LDB, A( K, 1 ), LDA )
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
40 CONTINUE
END IF
END IF
END IF
RETURN
*
* End of DSYGST
*
END
*> \brief \b DSYGV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, ITYPE, LDA, LDB, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGV computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.
*> Here A and B are assumed to be symmetric and B is also
*> positive definite.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> Specifies the problem type to be solved:
*> = 1: A*x = (lambda)*B*x
*> = 2: A*B*x = (lambda)*x
*> = 3: B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*>
*> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*> matrix Z of eigenvectors. The eigenvectors are normalized
*> as follows:
*> if ITYPE = 1 or 2, Z**T*B*Z = I;
*> if ITYPE = 3, Z**T*inv(B)*Z = I.
*> If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
*> or the lower triangle (if UPLO='L') of A, including the
*> diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the symmetric positive definite matrix B.
*> If UPLO = 'U', the leading N-by-N upper triangular part of B
*> contains the upper triangular part of the matrix B.
*> If UPLO = 'L', the leading N-by-N lower triangular part of B
*> contains the lower triangular part of the matrix B.
*>
*> On exit, if INFO <= N, the part of B containing the matrix is
*> overwritten by the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= max(1,3*N-1).
*> For optimal efficiency, LWORK >= (NB+2)*N,
*> where NB is the blocksize for DSYTRD returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: DPOTRF or DSYEV returned an error code:
*> <= N: if INFO = i, DSYEV failed to converge;
*> i off-diagonal elements of an intermediate
*> tridiagonal form did not converge to zero;
*> > N: if INFO = N + i, for 1 <= i <= N, then the leading
*> principal minor of order i of B is not positive.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegv
*
* =====================================================================
SUBROUTINE DSYGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W,
$ WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDA, LDB, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER, WANTZ
CHARACTER TRANS
INTEGER LWKMIN, LWKOPT, NB, NEIG
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DPOTRF, DSYEV, DSYGST, DTRMM, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
*
INFO = 0
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
*
IF( INFO.EQ.0 ) THEN
LWKMIN = MAX( 1, 3*N - 1 )
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( LWKMIN, ( NB + 2 )*N )
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form a Cholesky factorization of B.
*
CALL DPOTRF( UPLO, N, B, LDB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
CALL DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
IF( WANTZ ) THEN
*
* Backtransform eigenvectors to the original problem.
*
NEIG = N
IF( INFO.GT.0 )
$ NEIG = INFO - 1
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG,
$ ONE,
$ B, LDB, A, LDA )
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* For B*A*x=(lambda)*x;
* backtransform eigenvectors: x = L*y or U**T*y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG,
$ ONE,
$ B, LDB, A, LDA )
END IF
END IF
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of DSYGV
*
END
*> \brief \b DSYGVD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
* LWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
*> B are assumed to be symmetric and B is also positive definite.
*> If eigenvectors are desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> Specifies the problem type to be solved:
*> = 1: A*x = (lambda)*B*x
*> = 2: A*B*x = (lambda)*x
*> = 3: B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*>
*> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*> matrix Z of eigenvectors. The eigenvectors are normalized
*> as follows:
*> if ITYPE = 1 or 2, Z**T*B*Z = I;
*> if ITYPE = 3, Z**T*inv(B)*Z = I.
*> If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
*> or the lower triangle (if UPLO='L') of A, including the
*> diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the symmetric matrix B. If UPLO = 'U', the
*> leading N-by-N upper triangular part of B contains the
*> upper triangular part of the matrix B. If UPLO = 'L',
*> the leading N-by-N lower triangular part of B contains
*> the lower triangular part of the matrix B.
*>
*> On exit, if INFO <= N, the part of B containing the matrix is
*> overwritten by the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK >= 1.
*> If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
*> If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If N <= 1, LIWORK >= 1.
*> If JOBZ = 'N' and N > 1, LIWORK >= 1.
*> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: DPOTRF or DSYEVD returned an error code:
*> <= N: if INFO = i and JOBZ = 'N', then the algorithm
*> failed to converge; i off-diagonal elements of an
*> intermediate tridiagonal form did not converge to
*> zero;
*> if INFO = i and JOBZ = 'V', then the algorithm
*> failed to compute an eigenvalue while working on
*> the submatrix lying in rows and columns INFO/(N+1)
*> through mod(INFO,N+1);
*> > N: if INFO = N + i, for 1 <= i <= N, then the leading
*> principal minor of order i of B is not positive.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegvd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified so that no backsubstitution is performed if DSYEVD fails to
*> converge (NEIG in old code could be greater than N causing out of
*> bounds reference to A - reported by Ralf Meyer). Also corrected the
*> description of INFO and the test on ITYPE. Sven, 16 Feb 05.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*>
* =====================================================================
SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W,
$ WORK,
$ LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER, WANTZ
CHARACTER TRANS
INTEGER LIOPT, LIWMIN, LOPT, LWMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPOTRF, DSYEVD, DSYGST, DTRMM, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 6*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N + 1
END IF
LOPT = LWMIN
LIOPT = LIWMIN
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form a Cholesky factorization of B.
*
CALL DPOTRF( UPLO, N, B, LDB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
CALL DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
$ LIWORK,
$ INFO )
LOPT = INT( MAX( DBLE( LOPT ), DBLE( WORK( 1 ) ) ) )
LIOPT = INT( MAX( DBLE( LIOPT ), DBLE( IWORK( 1 ) ) ) )
*
IF( WANTZ .AND. INFO.EQ.0 ) THEN
*
* Backtransform eigenvectors to the original problem.
*
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, N, ONE,
$ B, LDB, A, LDA )
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* For B*A*x=(lambda)*x;
* backtransform eigenvectors: x = L*y or U**T*y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, N, ONE,
$ B, LDB, A, LDA )
END IF
END IF
*
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
RETURN
*
* End of DSYGVD
*
END
*> \brief \b DSYGVX
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
* VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
* LWORK, IWORK, IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
*> and B are assumed to be symmetric and B is also positive definite.
*> Eigenvalues and eigenvectors can be selected by specifying either a
*> range of values or a range of indices for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> Specifies the problem type to be solved:
*> = 1: A*x = (lambda)*B*x
*> = 2: A*B*x = (lambda)*x
*> = 3: B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A and B are stored;
*> = 'L': Lower triangle of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix pencil (A,B). N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*>
*> On exit, the lower triangle (if UPLO='L') or the upper
*> triangle (if UPLO='U') of A, including the diagonal, is
*> destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the symmetric matrix B. If UPLO = 'U', the
*> leading N-by-N upper triangular part of B contains the
*> upper triangular part of the matrix B. If UPLO = 'L',
*> the leading N-by-N lower triangular part of B contains
*> the lower triangular part of the matrix B.
*>
*> On exit, if INFO <= N, the part of B containing the matrix is
*> overwritten by the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is DOUBLE PRECISION
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less than
*> or equal to zero, then EPS*|T| will be used in its place,
*> where |T| is the 1-norm of the tridiagonal matrix obtained
*> by reducing C to tridiagonal form, where C is the symmetric
*> matrix of the standard symmetric problem to which the
*> generalized problem is transformed.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*DLAMCH('S').
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> On normal exit, the first M elements contain the selected
*> eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*> If JOBZ = 'N', then Z is not referenced.
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> The eigenvectors are normalized as follows:
*> if ITYPE = 1 or 2, Z**T*B*Z = I;
*> if ITYPE = 3, Z**T*inv(B)*Z = I.
*>
*> If an eigenvector fails to converge, then that column of Z
*> contains the latest approximation to the eigenvector, and the
*> index of the eigenvector is returned in IFAIL.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= max(1,8*N).
*> For optimal efficiency, LWORK >= (NB+3)*N,
*> where NB is the blocksize for DSYTRD returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (N)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvectors that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: DPOTRF or DSYEVX returned an error code:
*> <= N: if INFO = i, DSYEVX failed to converge;
*> i eigenvectors failed to converge. Their indices
*> are stored in array IFAIL.
*> > N: if INFO = N + i, for 1 <= i <= N, then the leading
*> principal minor of order i of B is not positive.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegvx
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
$ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
$ LWORK, IWORK, IFAIL, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
CHARACTER TRANS
INTEGER LWKMIN, LWKOPT, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DPOTRF, DSYEVX, DSYGST, DTRMM, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
UPPER = LSAME( UPLO, 'U' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
LQUERY = ( LWORK.EQ.-1 )
*
INFO = 0
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -3
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -11
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -13
END IF
END IF
END IF
IF (INFO.EQ.0) THEN
IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
INFO = -18
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
LWKMIN = MAX( 1, 8*N )
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -20
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 ) THEN
RETURN
END IF
*
* Form a Cholesky factorization of B.
*
CALL DPOTRF( UPLO, N, B, LDB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
CALL DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
$ ABSTOL,
$ M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
*
IF( WANTZ ) THEN
*
* Backtransform eigenvectors to the original problem.
*
IF( INFO.GT.0 )
$ M = INFO - 1
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE,
$ B,
$ LDB, Z, LDZ )
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* For B*A*x=(lambda)*x;
* backtransform eigenvectors: x = L*y or U**T*y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE,
$ B,
$ LDB, Z, LDZ )
END IF
END IF
*
* Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DSYGVX
*
END
*> \brief \b DSYRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
* X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric indefinite, and
*> provides error bounds and backward error estimates for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
*> upper triangular part of A contains the upper triangular part
*> of the matrix A, and the strictly lower triangular part of A
*> is not referenced. If UPLO = 'L', the leading N-by-N lower
*> triangular part of A contains the lower triangular part of
*> the matrix A, and the strictly upper triangular part of A is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> The factored form of the matrix A. AF contains the block
*> diagonal matrix D and the multipliers used to obtain the
*> factor U or L from the factorization A = U*D*U**T or
*> A = L*D*L**T as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by DSYTRS.
*> On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup herfs
*
* =====================================================================
SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
$ LDB,
$ X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER COUNT, I, J, K, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DSYMV, DSYTRS,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
$ WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
DO 40 I = 1, K - 1
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
40 CONTINUE
WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
DO 60 I = K + 1, N
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
60 CONTINUE
WORK( K ) = WORK( K ) + S
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
$ INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(A))*
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(A) is the inverse of A
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(A)*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(A) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(A**T).
*
CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
$ N,
$ INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
110 CONTINUE
ELSE IF( KASE.EQ.2 ) THEN
*
* Multiply by inv(A)*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
120 CONTINUE
CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
$ N,
$ INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DSYRFS
*
END
*> \brief <b> DSYSV computes the solution to system of linear equations A * X = B for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsysv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsysv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsysv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYSV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDB, LWORK, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYSV computes the solution to a real system of linear equations
*> A * X = B,
*> where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
*> matrices.
*>
*> The diagonal pivoting method is used to factor A as
*> A = U * D * U**T, if UPLO = 'U', or
*> A = L * D * L**T, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then
*> used to solve the system of equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the block diagonal matrix D and the
*> multipliers used to obtain the factor U or L from the
*> factorization A = U*D*U**T or A = L*D*L**T as computed by
*> DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D, as
*> determined by DSYTRF. If IPIV(k) > 0, then rows and columns
*> k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
*> diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
*> then rows and columns k-1 and -IPIV(k) were interchanged and
*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
*> IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
*> -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
*> diagonal block.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of WORK. LWORK >= 1, and for best performance
*> LWORK >= max(1,N*NB), where NB is the optimal blocksize for
*> DSYTRF.
*> for LWORK < N, TRS will be done with Level BLAS 2
*> for LWORK >= N, TRS will be done with Level BLAS 3
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, so the solution could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hesv
*
* =====================================================================
SUBROUTINE DSYSV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LWORK, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LWKOPT
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DSYTRF, DSYTRS, DSYTRS2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
LWKOPT = 1
ELSE
CALL DSYTRF( UPLO, N, A, LDA, IPIV, WORK, -1, INFO )
LWKOPT = INT( WORK( 1 ) )
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYSV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
CALL DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
IF ( LWORK.LT.N ) THEN
*
* Solve with TRS ( Use Level BLAS 2)
*
CALL DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
ELSE
*
* Solve with TRS2 ( Use Level BLAS 3)
*
CALL DSYTRS2( UPLO,N,NRHS,A,LDA,IPIV,B,LDB,WORK,INFO )
*
END IF
*
END IF
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DSYSV
*
END
*> \brief <b> DSYSVX computes the solution to system of linear equations A * X = B for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsysvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsysvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsysvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
* LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER FACT, UPLO
* INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYSVX uses the diagonal pivoting factorization to compute the
*> solution to a real system of linear equations A * X = B,
*> where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
*> matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
*> The form of the factorization is
*> A = U * D * U**T, if UPLO = 'U', or
*> A = L * D * L**T, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
*> returns with INFO = i. Otherwise, the factored form of A is used
*> to estimate the condition number of the matrix A. If the
*> reciprocal of the condition number is less than machine precision,
*> INFO = N+1 is returned as a warning, but the routine still goes on
*> to solve for X and compute error bounds as described below.
*>
*> 3. The system of equations is solved for X using the factored form
*> of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of A has been
*> supplied on entry.
*> = 'F': On entry, AF and IPIV contain the factored form of
*> A. AF and IPIV will not be modified.
*> = 'N': The matrix A will be copied to AF and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
*> upper triangular part of A contains the upper triangular part
*> of the matrix A, and the strictly lower triangular part of A
*> is not referenced. If UPLO = 'L', the leading N-by-N lower
*> triangular part of A contains the lower triangular part of
*> the matrix A, and the strictly upper triangular part of A is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> If FACT = 'F', then AF is an input argument and on entry
*> contains the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L from the factorization
*> A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
*>
*> If FACT = 'N', then AF is an output argument and on exit
*> returns the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L from the factorization
*> A = U*D*U**T or A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains details of the interchanges and the block structure
*> of D, as determined by DSYTRF.
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*>
*> If FACT = 'N', then IPIV is an output argument and on exit
*> contains details of the interchanges and the block structure
*> of D, as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A. If RCOND is less than the machine precision (in
*> particular, if RCOND = 0), the matrix is singular to working
*> precision. This condition is indicated by a return code of
*> INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of WORK. LWORK >= max(1,3*N), and for best
*> performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
*> NB is the optimal blocksize for DSYTRF.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: D(i,i) is exactly zero. The factorization
*> has been completed but the factor D is exactly
*> singular, so the solution and error bounds could
*> not be computed. RCOND = 0 is returned.
*> = N+1: D is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hesvx
*
* =====================================================================
SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
$ B,
$ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
$ IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, NOFACT
INTEGER LWKMIN, LWKOPT, NB
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DSYCON, DSYRFS, DSYTRF, DSYTRS,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
LQUERY = ( LWORK.EQ.-1 )
LWKMIN = MAX( 1, 3*N )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -11
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
*
IF( INFO.EQ.0 ) THEN
LWKOPT = LWKMIN
IF( NOFACT ) THEN
NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( LWKOPT, N*NB )
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYSVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANSY( 'I', UPLO, N, A, LDA, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK,
$ IWORK,
$ INFO )
*
* Compute the solution vectors X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
$ LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DSYSVX
*
END
*> \brief \b DSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYSWAPR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyswapr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyswapr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyswapr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYSWAPR( UPLO, N, A, LDA, I1, I2)
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER I1, I2, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, N )
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYSWAPR applies an elementary permutation on the rows and the columns of
*> a symmetric matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,*)
*> On entry, the N-by-N matrix A. On exit, the permuted matrix
*> where the rows I1 and I2 and columns I1 and I2 are interchanged.
*> If UPLO = 'U', the interchanges are applied to the upper
*> triangular part and the strictly lower triangular part of A is
*> not referenced; if UPLO = 'L', the interchanges are applied to
*> the lower triangular part and the part of A above the diagonal
*> is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] I1
*> \verbatim
*> I1 is INTEGER
*> Index of the first row to swap
*> \endverbatim
*>
*> \param[in] I2
*> \verbatim
*> I2 is INTEGER
*> Index of the second row to swap
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heswapr
*
* =====================================================================
SUBROUTINE DSYSWAPR( UPLO, N, A, LDA, I1, I2)
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER I1, I2, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
*
* =====================================================================
*
* ..
* .. Local Scalars ..
LOGICAL UPPER
DOUBLE PRECISION TMP
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSWAP
* ..
* .. Executable Statements ..
*
UPPER = LSAME( UPLO, 'U' )
IF (UPPER) THEN
*
* UPPER
* first swap
* - swap column I1 and I2 from I1 to I1-1
CALL DSWAP( I1-1, A(1,I1), 1, A(1,I2), 1 )
*
* second swap :
* - swap A(I1,I1) and A(I2,I2)
* - swap row I1 from I1+1 to I2-1 with col I2 from I1+1 to I2-1
TMP=A(I1,I1)
A(I1,I1)=A(I2,I2)
A(I2,I2)=TMP
*
CALL DSWAP( I2-I1-1, A(I1,I1+1), LDA, A(I1+1,I2), 1 )
*
* third swap
* - swap row I1 and I2 from I2+1 to N
IF ( I2.LT.N )
$ CALL DSWAP( N-I2, A(I1,I2+1), LDA, A(I2,I2+1), LDA )
*
ELSE
*
* LOWER
* first swap
* - swap row I1 and I2 from I1 to I1-1
CALL DSWAP( I1-1, A(I1,1), LDA, A(I2,1), LDA )
*
* second swap :
* - swap A(I1,I1) and A(I2,I2)
* - swap col I1 from I1+1 to I2-1 with row I2 from I1+1 to I2-1
TMP=A(I1,I1)
A(I1,I1)=A(I2,I2)
A(I2,I2)=TMP
*
CALL DSWAP( I2-I1-1, A(I1+1,I1), 1, A(I2,I1+1), LDA )
*
* third swap
* - swap col I1 and I2 from I2+1 to N
IF ( I2.LT.N )
$ CALL DSWAP( N-I2, A(I2+1,I1), 1, A(I2+1,I2), 1 )
*
ENDIF
END SUBROUTINE DSYSWAPR
*> \brief \b DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
*> form T by an orthogonal similarity transformation: Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit, if UPLO = 'U', the diagonal and first superdiagonal
*> of A are overwritten by the corresponding elements of the
*> tridiagonal matrix T, and the elements above the first
*> superdiagonal, with the array TAU, represent the orthogonal
*> matrix Q as a product of elementary reflectors; if UPLO
*> = 'L', the diagonal and first subdiagonal of A are over-
*> written by the corresponding elements of the tridiagonal
*> matrix T, and the elements below the first subdiagonal, with
*> the array TAU, represent the orthogonal matrix Q as a product
*> of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetd2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n-1) . . . H(2) H(1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*> A(1:i-1,i+1), and tau in TAU(i).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(n-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*> and tau in TAU(i).
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( d e v2 v3 v4 ) ( d )
*> ( d e v3 v4 ) ( e d )
*> ( d e v4 ) ( v1 e d )
*> ( d e ) ( v1 v2 e d )
*> ( d ) ( v1 v2 v3 e d )
*>
*> where d and e denote diagonal and off-diagonal elements of T, and vi
*> denotes an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, HALF
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
$ HALF = 1.0D0 / 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I
DOUBLE PRECISION ALPHA, TAUI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTD2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A
*
DO 10 I = N - 1, 1, -1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(1:i-1,i+1)
*
CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
E( I ) = A( I, I+1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(1:i,1:i)
*
A( I, I+1 ) = ONE
*
* Compute x := tau * A * v storing x in TAU(1:i)
*
CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1,
$ ZERO,
$ TAU, 1 )
*
* Compute w := x - 1/2 * tau * (x**T * v) * v
*
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
$ LDA )
*
A( I, I+1 ) = E( I )
END IF
D( I+1 ) = A( I+1, I+1 )
TAU( I ) = TAUI
10 CONTINUE
D( 1 ) = A( 1, 1 )
ELSE
*
* Reduce the lower triangle of A
*
DO 20 I = 1, N - 1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(i+2:n,i)
*
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAUI )
E( I ) = A( I+1, I )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(i+1:n,i+1:n)
*
A( I+1, I ) = ONE
*
* Compute x := tau * A * v storing y in TAU(i:n-1)
*
CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
*
* Compute w := x - 1/2 * tau * (x**T * v) * v
*
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1,
$ I ),
$ 1 )
CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ),
$ 1,
$ A( I+1, I+1 ), LDA )
*
A( I+1, I ) = E( I )
END IF
D( I ) = A( I, I )
TAU( I ) = TAUI
20 CONTINUE
D( N ) = A( N, N )
END IF
*
RETURN
*
* End of DSYTD2
*
END
*> \brief \b DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTF2 computes the factorization of a real symmetric matrix A using
*> the Bunch-Kaufman diagonal pivoting method:
*>
*> A = U*D*U**T or A = L*D*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, U**T is the transpose of U, and D is symmetric and
*> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L (see below for further details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D.
*>
*> If UPLO = 'U':
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
*> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*> is a 2-by-2 diagonal block.
*>
*> If UPLO = 'L':
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
*> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
*> is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, and division by zero will occur if it
*> is used to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetf2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', then A = U*D*U**T, where
*> U = P(n)*U(n)* ... *P(k)U(k)* ...,
*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I v 0 ) k-s
*> U(k) = ( 0 I 0 ) s
*> ( 0 0 I ) n-k
*> k-s s n-k
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*> and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*> If UPLO = 'L', then A = L*D*L**T, where
*> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I 0 0 ) k-1
*> L(k) = ( 0 I 0 ) s
*> ( 0 v I ) n-k-s+1
*> k-1 s n-k-s+1
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> 09-29-06 - patch from
*> Bobby Cheng, MathWorks
*>
*> Replace l.204 and l.372
*> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*> by
*> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
*>
*> 01-01-96 - Based on modifications by
*> J. Lewis, Boeing Computer Services Company
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*> 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
*> Company
*> \endverbatim
*
* =====================================================================
SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION EIGHT, SEVTEN
PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
$ ROWMAX, T, WK, WKM1, WKP1
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
INTEGER IDAMAX
EXTERNAL LSAME, IDAMAX, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTF2', -INFO )
RETURN
END IF
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
IF( UPPER ) THEN
*
* Factorize A as U*D*U**T using the upper triangle of A
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2
*
K = N
10 CONTINUE
*
* If K < 1, exit from loop
*
IF( K.LT.1 )
$ GO TO 70
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( A( K, K ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
*
IF( K.GT.1 ) THEN
IMAX = IDAMAX( K-1, A( 1, K ), 1 )
COLMAX = ABS( A( IMAX, K ) )
ELSE
COLMAX = ZERO
END IF
*
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR.
$ DISNAN(ABSAKK) ) THEN
*
* Column K is zero or underflow, or contains a NaN:
* set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
ROWMAX = ABS( A( IMAX, JMAX ) )
IF( IMAX.GT.1 ) THEN
JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K-1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
KK = K - KSTEP + 1
IF( KP.NE.KK ) THEN
*
* Interchange rows and columns KK and KP in the leading
* submatrix A(1:k,1:k)
*
CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
$ LDA )
T = A( KK, KK )
A( KK, KK ) = A( KP, KP )
A( KP, KP ) = T
IF( KSTEP.EQ.2 ) THEN
T = A( K-1, K )
A( K-1, K ) = A( KP, K )
A( KP, K ) = T
END IF
END IF
*
* Update the leading submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = U(k)*D(k)
*
* where U(k) is the k-th column of U
*
* Perform a rank-1 update of A(1:k-1,1:k-1) as
*
* A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
*
R1 = ONE / A( K, K )
CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
*
* Store U(k) in column k
*
CALL DSCAL( K-1, R1, A( 1, K ), 1 )
ELSE
*
* 2-by-2 pivot block D(k): columns k and k-1 now hold
*
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
* where U(k) and U(k-1) are the k-th and (k-1)-th columns
* of U
*
* Perform a rank-2 update of A(1:k-2,1:k-2) as
*
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
*
IF( K.GT.2 ) THEN
*
D12 = A( K-1, K )
D22 = A( K-1, K-1 ) / D12
D11 = A( K, K ) / D12
T = ONE / ( D11*D22-ONE )
D12 = T / D12
*
DO 30 J = K - 2, 1, -1
WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
WK = D12*( D22*A( J, K )-A( J, K-1 ) )
DO 20 I = J, 1, -1
A( I, J ) = A( I, J ) - A( I, K )*WK -
$ A( I, K-1 )*WKM1
20 CONTINUE
A( J, K ) = WK
A( J, K-1 ) = WKM1
30 CONTINUE
*
END IF
*
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K-1 ) = -KP
END IF
*
* Decrease K and return to the start of the main loop
*
K = K - KSTEP
GO TO 10
*
ELSE
*
* Factorize A as L*D*L**T using the lower triangle of A
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2
*
K = 1
40 CONTINUE
*
* If K > N, exit from loop
*
IF( K.GT.N )
$ GO TO 70
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( A( K, K ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
*
IF( K.LT.N ) THEN
IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
COLMAX = ABS( A( IMAX, K ) )
ELSE
COLMAX = ZERO
END IF
*
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR.
$ DISNAN(ABSAKK) ) THEN
*
* Column K is zero or underflow, or contains a NaN:
* set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
ROWMAX = ABS( A( IMAX, JMAX ) )
IF( IMAX.LT.N ) THEN
JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
$ 1 )
ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K+1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
KK = K + KSTEP - 1
IF( KP.NE.KK ) THEN
*
* Interchange rows and columns KK and KP in the trailing
* submatrix A(k:n,k:n)
*
IF( KP.LT.N )
$ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ),
$ 1 )
CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
$ LDA )
T = A( KK, KK )
A( KK, KK ) = A( KP, KP )
A( KP, KP ) = T
IF( KSTEP.EQ.2 ) THEN
T = A( K+1, K )
A( K+1, K ) = A( KP, K )
A( KP, K ) = T
END IF
END IF
*
* Update the trailing submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = L(k)*D(k)
*
* where L(k) is the k-th column of L
*
IF( K.LT.N ) THEN
*
* Perform a rank-1 update of A(k+1:n,k+1:n) as
*
* A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
*
D11 = ONE / A( K, K )
CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
$ A( K+1, K+1 ), LDA )
*
* Store L(k) in column K
*
CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
END IF
ELSE
*
* 2-by-2 pivot block D(k)
*
IF( K.LT.N-1 ) THEN
*
* Perform a rank-2 update of A(k+2:n,k+2:n) as
*
* A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))**T
*
* where L(k) and L(k+1) are the k-th and (k+1)-th
* columns of L
*
D21 = A( K+1, K )
D11 = A( K+1, K+1 ) / D21
D22 = A( K, K ) / D21
T = ONE / ( D11*D22-ONE )
D21 = T / D21
*
DO 60 J = K + 2, N
*
WK = D21*( D11*A( J, K )-A( J, K+1 ) )
WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
*
DO 50 I = J, N
A( I, J ) = A( I, J ) - A( I, K )*WK -
$ A( I, K+1 )*WKP1
50 CONTINUE
*
A( J, K ) = WK
A( J, K+1 ) = WKP1
*
60 CONTINUE
END IF
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K+1 ) = -KP
END IF
*
* Increase K and return to the start of the main loop
*
K = K + KSTEP
GO TO 40
*
END IF
*
70 CONTINUE
*
RETURN
*
* End of DSYTF2
*
END
*> \brief \b DSYTRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRD reduces a real symmetric matrix A to real symmetric
*> tridiagonal form T by an orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit, if UPLO = 'U', the diagonal and first superdiagonal
*> of A are overwritten by the corresponding elements of the
*> tridiagonal matrix T, and the elements above the first
*> superdiagonal, with the array TAU, represent the orthogonal
*> matrix Q as a product of elementary reflectors; if UPLO
*> = 'L', the diagonal and first subdiagonal of A are over-
*> written by the corresponding elements of the tridiagonal
*> matrix T, and the elements below the first subdiagonal, with
*> the array TAU, represent the orthogonal matrix Q as a product
*> of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrd
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n-1) . . . H(2) H(1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*> A(1:i-1,i+1), and tau in TAU(i).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(n-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*> and tau in TAU(i).
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( d e v2 v3 v4 ) ( d )
*> ( d e v3 v4 ) ( e d )
*> ( d e v4 ) ( v1 e d )
*> ( d e ) ( v1 v2 e d )
*> ( d ) ( v1 v2 v3 e d )
*>
*> where d and e denote diagonal and off-diagonal elements of T, and vi
*> denotes an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size.
*
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( 1, N*NB )
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NX = N
IWS = 1
IF( NB.GT.1 .AND. NB.LT.N ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code).
*
NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
IF( NX.LT.N ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code by setting NX = N.
*
NB = MAX( LWORK / LDWORK, 1 )
NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
IF( NB.LT.NBMIN )
$ NX = N
END IF
ELSE
NX = N
END IF
ELSE
NB = 1
END IF
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* Columns 1:kk are handled by the unblocked method.
*
KK = N - ( ( N-NX+NB-1 ) / NB )*NB
DO 20 I = N - NB + 1, KK + 1, -NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
$ LDWORK )
*
* Update the unreduced submatrix A(1:i-1,1:i-1), using an
* update of the form: A := A - V*W**T - W*V**T
*
CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1,
$ I ),
$ LDA, WORK, LDWORK, ONE, A, LDA )
*
* Copy superdiagonal elements back into A, and diagonal
* elements into D
*
DO 10 J = I, I + NB - 1
A( J-1, J ) = E( J-1 )
D( J ) = A( J, J )
10 CONTINUE
20 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
ELSE
*
* Reduce the lower triangle of A
*
DO 40 I = 1, N - NX, NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
$ TAU( I ), WORK, LDWORK )
*
* Update the unreduced submatrix A(i+ib:n,i+ib:n), using
* an update of the form: A := A - V*W**T - W*V**T
*
CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
$ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
$ A( I+NB, I+NB ), LDA )
*
* Copy subdiagonal elements back into A, and diagonal
* elements into D
*
DO 30 J = I, I + NB - 1
A( J+1, J ) = E( J )
D( J ) = A( J, J )
30 CONTINUE
40 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAU( I ), IINFO )
END IF
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of DSYTRD
*
END
*> \brief \b DSYTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRF computes the factorization of a real symmetric matrix A using
*> the Bunch-Kaufman diagonal pivoting method. The form of the
*> factorization is
*>
*> A = U**T*D*U or A = L*D*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L (see below for further details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D.
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of WORK. LWORK >= 1. For best performance
*> LWORK >= N*NB, where NB is the block size returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, and division by zero will occur if it
*> is used to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', then A = U**T*D*U, where
*> U = P(n)*U(n)* ... *P(k)U(k)* ...,
*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I v 0 ) k-s
*> U(k) = ( 0 I 0 ) s
*> ( 0 0 I ) n-k
*> k-s s n-k
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*> and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*> If UPLO = 'L', then A = L*D*L**T, where
*> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I 0 0 ) k-1
*> L(k) = ( 0 I 0 ) s
*> ( 0 v I ) n-k-s+1
*> k-1 s n-k-s+1
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLASYF, DSYTF2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size
*
NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( 1, N*NB )
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
NBMIN = 2
LDWORK = N
IF( NB.GT.1 .AND. NB.LT.N ) THEN
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
NB = MAX( LWORK / LDWORK, 1 )
NBMIN = MAX( 2, ILAENV( 2, 'DSYTRF', UPLO, N, -1, -1,
$ -1 ) )
END IF
ELSE
IWS = 1
END IF
IF( NB.LT.NBMIN )
$ NB = N
*
IF( UPPER ) THEN
*
* Factorize A as U**T*D*U using the upper triangle of A
*
* K is the main loop index, decreasing from N to 1 in steps of
* KB, where KB is the number of columns factorized by DLASYF;
* KB is either NB or NB-1, or K for the last block
*
K = N
10 CONTINUE
*
* If K < 1, exit from loop
*
IF( K.LT.1 )
$ GO TO 40
*
IF( K.GT.NB ) THEN
*
* Factorize columns k-kb+1:k of A and use blocked code to
* update columns 1:k-kb
*
CALL DLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK,
$ IINFO )
ELSE
*
* Use unblocked code to factorize columns 1:k of A
*
CALL DSYTF2( UPLO, K, A, LDA, IPIV, IINFO )
KB = K
END IF
*
* Set INFO on the first occurrence of a zero pivot
*
IF( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO
*
* Decrease K and return to the start of the main loop
*
K = K - KB
GO TO 10
*
ELSE
*
* Factorize A as L*D*L**T using the lower triangle of A
*
* K is the main loop index, increasing from 1 to N in steps of
* KB, where KB is the number of columns factorized by DLASYF;
* KB is either NB or NB-1, or N-K+1 for the last block
*
K = 1
20 CONTINUE
*
* If K > N, exit from loop
*
IF( K.GT.N )
$ GO TO 40
*
IF( K.LE.N-NB ) THEN
*
* Factorize columns k:k+kb-1 of A and use blocked code to
* update columns k+kb:n
*
CALL DLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA,
$ IPIV( K ),
$ WORK, LDWORK, IINFO )
ELSE
*
* Use unblocked code to factorize columns k:n of A
*
CALL DSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ),
$ IINFO )
KB = N - K + 1
END IF
*
* Set INFO on the first occurrence of a zero pivot
*
IF( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO + K - 1
*
* Adjust IPIV
*
DO 30 J = K, K + KB - 1
IF( IPIV( J ).GT.0 ) THEN
IPIV( J ) = IPIV( J ) + K - 1
ELSE
IPIV( J ) = IPIV( J ) - K + 1
END IF
30 CONTINUE
*
* Increase K and return to the start of the main loop
*
K = K + KB
GO TO 20
*
END IF
*
40 CONTINUE
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of DSYTRF
*
END
*> \brief \b DSYTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRI computes the inverse of a real symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> DSYTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the block diagonal matrix D and the multipliers
*> used to obtain the factor U or L as computed by DSYTRF.
*>
*> On exit, if INFO = 0, the (symmetric) inverse of the original
*> matrix. If UPLO = 'U', the upper triangular part of the
*> inverse is formed and the part of A below the diagonal is not
*> referenced; if UPLO = 'L' the lower triangular part of the
*> inverse is formed and the part of A above the diagonal is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*> inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetri
*
* =====================================================================
SUBROUTINE DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER K, KP, KSTEP
DOUBLE PRECISION AK, AKKP1, AKP1, D, T, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSWAP, DSYMV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
DO 10 INFO = N, 1, -1
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
$ RETURN
10 CONTINUE
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
DO 20 INFO = 1, N
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
$ RETURN
20 CONTINUE
END IF
INFO = 0
*
IF( UPPER ) THEN
*
* Compute inv(A) from the factorization A = U*D*U**T.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
30 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 40
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
A( K, K ) = ONE / A( K, K )
*
* Compute column K of the inverse.
*
IF( K.GT.1 ) THEN
CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
$ A( 1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ),
$ 1 )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( A( K, K+1 ) )
AK = A( K, K ) / T
AKP1 = A( K+1, K+1 ) / T
AKKP1 = A( K, K+1 ) / T
D = T*( AK*AKP1-ONE )
A( K, K ) = AKP1 / D
A( K+1, K+1 ) = AK / D
A( K, K+1 ) = -AKKP1 / D
*
* Compute columns K and K+1 of the inverse.
*
IF( K.GT.1 ) THEN
CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
$ A( 1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ),
$ 1 )
A( K, K+1 ) = A( K, K+1 ) -
$ DDOT( K-1, A( 1, K ), 1, A( 1, K+1 ),
$ 1 )
CALL DCOPY( K-1, A( 1, K+1 ), 1, WORK, 1 )
CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
$ A( 1, K+1 ), 1 )
A( K+1, K+1 ) = A( K+1, K+1 ) -
$ DDOT( K-1, WORK, 1, A( 1, K+1 ), 1 )
END IF
KSTEP = 2
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the leading
* submatrix A(1:k+1,1:k+1)
*
CALL DSWAP( KP-1, A( 1, K ), 1, A( 1, KP ), 1 )
CALL DSWAP( K-KP-1, A( KP+1, K ), 1, A( KP, KP+1 ), LDA )
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = A( K, K+1 )
A( K, K+1 ) = A( KP, K+1 )
A( KP, K+1 ) = TEMP
END IF
END IF
*
K = K + KSTEP
GO TO 30
40 CONTINUE
*
ELSE
*
* Compute inv(A) from the factorization A = L*D*L**T.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
50 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 60
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
A( K, K ) = ONE / A( K, K )
*
* Compute column K of the inverse.
*
IF( K.LT.N ) THEN
CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK,
$ 1,
$ ZERO, A( K+1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1,
$ K ),
$ 1 )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( A( K, K-1 ) )
AK = A( K-1, K-1 ) / T
AKP1 = A( K, K ) / T
AKKP1 = A( K, K-1 ) / T
D = T*( AK*AKP1-ONE )
A( K-1, K-1 ) = AKP1 / D
A( K, K ) = AK / D
A( K, K-1 ) = -AKKP1 / D
*
* Compute columns K-1 and K of the inverse.
*
IF( K.LT.N ) THEN
CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK,
$ 1,
$ ZERO, A( K+1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1,
$ K ),
$ 1 )
A( K, K-1 ) = A( K, K-1 ) -
$ DDOT( N-K, A( K+1, K ), 1, A( K+1,
$ K-1 ),
$ 1 )
CALL DCOPY( N-K, A( K+1, K-1 ), 1, WORK, 1 )
CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK,
$ 1,
$ ZERO, A( K+1, K-1 ), 1 )
A( K-1, K-1 ) = A( K-1, K-1 ) -
$ DDOT( N-K, WORK, 1, A( K+1, K-1 ), 1 )
END IF
KSTEP = 2
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the trailing
* submatrix A(k-1:n,k-1:n)
*
IF( KP.LT.N )
$ CALL DSWAP( N-KP, A( KP+1, K ), 1, A( KP+1, KP ), 1 )
CALL DSWAP( KP-K-1, A( K+1, K ), 1, A( KP, K+1 ), LDA )
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = A( K, K-1 )
A( K, K-1 ) = A( KP, K-1 )
A( KP, K-1 ) = TEMP
END IF
END IF
*
K = K - KSTEP
GO TO 50
60 CONTINUE
END IF
*
RETURN
*
* End of DSYTRI
*
END
*> \brief \b DSYTRI2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRI2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytri2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytri2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytri2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRI2( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRI2 computes the inverse of a DOUBLE PRECISION symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> DSYTRF. DSYTRI2 sets the LEADING DIMENSION of the workspace
*> before calling DSYTRI2X that actually computes the inverse.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the block diagonal matrix D and the multipliers
*> used to obtain the factor U or L as computed by DSYTRF.
*>
*> On exit, if INFO = 0, the (symmetric) inverse of the original
*> matrix. If UPLO = 'U', the upper triangular part of the
*> inverse is formed and the part of A below the diagonal is not
*> referenced; if UPLO = 'L' the lower triangular part of the
*> inverse is formed and the part of A above the diagonal is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N = 0, LWORK >= 1, else LWORK >= (N+NB+1)*(NB+3).
*> If LWORK = -1, then a workspace query is assumed; the routine
*> calculates:
*> - the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array,
*> - and no error message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*> inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetri2
*
* =====================================================================
SUBROUTINE DSYTRI2( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL UPPER, LQUERY
INTEGER MINSIZE, NBMAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DSYTRI, DSYTRI2X, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
*
* Get blocksize
*
NBMAX = ILAENV( 1, 'DSYTRI2', UPLO, N, -1, -1, -1 )
IF( N.EQ.0 ) THEN
MINSIZE = 1
ELSE IF( NBMAX.GE.N ) THEN
MINSIZE = N
ELSE
MINSIZE = (N+NBMAX+1)*(NBMAX+3)
END IF
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.MINSIZE .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRI2', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
WORK( 1 ) = MINSIZE
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( NBMAX.GE.N ) THEN
CALL DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )
ELSE
CALL DSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NBMAX, INFO )
END IF
*
RETURN
*
* End of DSYTRI2
*
END
*> \brief \b DSYTRI2X
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRI2X + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytri2x.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytri2x.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytri2x.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N, NB
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), WORK( N+NB+1,* )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRI2X computes the inverse of a real symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> DSYTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the NNB diagonal matrix D and the multipliers
*> used to obtain the factor U or L as computed by DSYTRF.
*>
*> On exit, if INFO = 0, the (symmetric) inverse of the original
*> matrix. If UPLO = 'U', the upper triangular part of the
*> inverse is formed and the part of A below the diagonal is not
*> referenced; if UPLO = 'L' the lower triangular part of the
*> inverse is formed and the part of A above the diagonal is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the NNB structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3)
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> Block size
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*> inverse could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetri2x
*
* =====================================================================
SUBROUTINE DSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N, NB
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), WORK( N+NB+1,* )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IINFO, IP, K, CUT, NNB
INTEGER COUNT
INTEGER J, U11, INVD
DOUBLE PRECISION AK, AKKP1, AKP1, D, T
DOUBLE PRECISION U01_I_J, U01_IP1_J
DOUBLE PRECISION U11_I_J, U11_IP1_J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSYCONV, XERBLA, DTRTRI
EXTERNAL DGEMM, DTRMM, DSYSWAPR
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
*
* Quick return if possible
*
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRI2X', -INFO )
RETURN
END IF
IF( N.EQ.0 )
$ RETURN
*
* Convert A
* Workspace got Non-diag elements of D
*
CALL DSYCONV( UPLO, 'C', N, A, LDA, IPIV, WORK, IINFO )
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
DO INFO = N, 1, -1
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
$ RETURN
END DO
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
DO INFO = 1, N
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
$ RETURN
END DO
END IF
INFO = 0
*
* Splitting Workspace
* U01 is a block (N,NB+1)
* The first element of U01 is in WORK(1,1)
* U11 is a block (NB+1,NB+1)
* The first element of U11 is in WORK(N+1,1)
U11 = N
* INVD is a block (N,2)
* The first element of INVD is in WORK(1,INVD)
INVD = NB+2
IF( UPPER ) THEN
*
* invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
*
CALL DTRTRI( UPLO, 'U', N, A, LDA, INFO )
*
* inv(D) and inv(D)*inv(U)
*
K=1
DO WHILE ( K .LE. N )
IF( IPIV( K ).GT.0 ) THEN
* 1 x 1 diagonal NNB
WORK(K,INVD) = ONE / A( K, K )
WORK(K,INVD+1) = 0
K=K+1
ELSE
* 2 x 2 diagonal NNB
T = WORK(K+1,1)
AK = A( K, K ) / T
AKP1 = A( K+1, K+1 ) / T
AKKP1 = WORK(K+1,1) / T
D = T*( AK*AKP1-ONE )
WORK(K,INVD) = AKP1 / D
WORK(K+1,INVD+1) = AK / D
WORK(K,INVD+1) = -AKKP1 / D
WORK(K+1,INVD) = -AKKP1 / D
K=K+2
END IF
END DO
*
* inv(U**T) = (inv(U))**T
*
* inv(U**T)*inv(D)*inv(U)
*
CUT=N
DO WHILE (CUT .GT. 0)
NNB=NB
IF (CUT .LE. NNB) THEN
NNB=CUT
ELSE
COUNT = 0
* count negative elements,
DO I=CUT+1-NNB,CUT
IF (IPIV(I) .LT. 0) COUNT=COUNT+1
END DO
* need a even number for a clear cut
IF (MOD(COUNT,2) .EQ. 1) NNB=NNB+1
END IF
CUT=CUT-NNB
*
* U01 Block
*
DO I=1,CUT
DO J=1,NNB
WORK(I,J)=A(I,CUT+J)
END DO
END DO
*
* U11 Block
*
DO I=1,NNB
WORK(U11+I,I)=ONE
DO J=1,I-1
WORK(U11+I,J)=ZERO
END DO
DO J=I+1,NNB
WORK(U11+I,J)=A(CUT+I,CUT+J)
END DO
END DO
*
* invD*U01
*
I=1
DO WHILE (I .LE. CUT)
IF (IPIV(I) > 0) THEN
DO J=1,NNB
WORK(I,J)=WORK(I,INVD)*WORK(I,J)
END DO
I=I+1
ELSE
DO J=1,NNB
U01_I_J = WORK(I,J)
U01_IP1_J = WORK(I+1,J)
WORK(I,J)=WORK(I,INVD)*U01_I_J+
$ WORK(I,INVD+1)*U01_IP1_J
WORK(I+1,J)=WORK(I+1,INVD)*U01_I_J+
$ WORK(I+1,INVD+1)*U01_IP1_J
END DO
I=I+2
END IF
END DO
*
* invD1*U11
*
I=1
DO WHILE (I .LE. NNB)
IF (IPIV(CUT+I) > 0) THEN
DO J=I,NNB
WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J)
END DO
I=I+1
ELSE
DO J=I,NNB
U11_I_J = WORK(U11+I,J)
U11_IP1_J = WORK(U11+I+1,J)
WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J) +
$ WORK(CUT+I,INVD+1)*WORK(U11+I+1,J)
WORK(U11+I+1,J)=WORK(CUT+I+1,INVD)*U11_I_J+
$ WORK(CUT+I+1,INVD+1)*U11_IP1_J
END DO
I=I+2
END IF
END DO
*
* U11**T*invD1*U11->U11
*
CALL DTRMM('L','U','T','U',NNB, NNB,
$ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1)
*
DO I=1,NNB
DO J=I,NNB
A(CUT+I,CUT+J)=WORK(U11+I,J)
END DO
END DO
*
* U01**T*invD*U01->A(CUT+I,CUT+J)
*
CALL DGEMM('T','N',NNB,NNB,CUT,ONE,A(1,CUT+1),LDA,
$ WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1)
*
* U11 = U11**T*invD1*U11 + U01**T*invD*U01
*
DO I=1,NNB
DO J=I,NNB
A(CUT+I,CUT+J)=A(CUT+I,CUT+J)+WORK(U11+I,J)
END DO
END DO
*
* U01 = U00**T*invD0*U01
*
CALL DTRMM('L',UPLO,'T','U',CUT, NNB,
$ ONE,A,LDA,WORK,N+NB+1)
*
* Update U01
*
DO I=1,CUT
DO J=1,NNB
A(I,CUT+J)=WORK(I,J)
END DO
END DO
*
* Next Block
*
END DO
*
* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
*
I=1
DO WHILE ( I .LE. N )
IF( IPIV(I) .GT. 0 ) THEN
IP=IPIV(I)
IF (I .LT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, I ,
$ IP )
IF (I .GT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, IP ,
$ I )
ELSE
IP=-IPIV(I)
I=I+1
IF ( (I-1) .LT. IP)
$ CALL DSYSWAPR( UPLO, N, A, LDA, I-1 ,IP )
IF ( (I-1) .GT. IP)
$ CALL DSYSWAPR( UPLO, N, A, LDA, IP ,I-1 )
ENDIF
I=I+1
END DO
ELSE
*
* LOWER...
*
* invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
*
CALL DTRTRI( UPLO, 'U', N, A, LDA, INFO )
*
* inv(D) and inv(D)*inv(U)
*
K=N
DO WHILE ( K .GE. 1 )
IF( IPIV( K ).GT.0 ) THEN
* 1 x 1 diagonal NNB
WORK(K,INVD) = ONE / A( K, K )
WORK(K,INVD+1) = 0
K=K-1
ELSE
* 2 x 2 diagonal NNB
T = WORK(K-1,1)
AK = A( K-1, K-1 ) / T
AKP1 = A( K, K ) / T
AKKP1 = WORK(K-1,1) / T
D = T*( AK*AKP1-ONE )
WORK(K-1,INVD) = AKP1 / D
WORK(K,INVD) = AK / D
WORK(K,INVD+1) = -AKKP1 / D
WORK(K-1,INVD+1) = -AKKP1 / D
K=K-2
END IF
END DO
*
* inv(U**T) = (inv(U))**T
*
* inv(U**T)*inv(D)*inv(U)
*
CUT=0
DO WHILE (CUT .LT. N)
NNB=NB
IF (CUT + NNB .GT. N) THEN
NNB=N-CUT
ELSE
COUNT = 0
* count negative elements,
DO I=CUT+1,CUT+NNB
IF (IPIV(I) .LT. 0) COUNT=COUNT+1
END DO
* need a even number for a clear cut
IF (MOD(COUNT,2) .EQ. 1) NNB=NNB+1
END IF
* L21 Block
DO I=1,N-CUT-NNB
DO J=1,NNB
WORK(I,J)=A(CUT+NNB+I,CUT+J)
END DO
END DO
* L11 Block
DO I=1,NNB
WORK(U11+I,I)=ONE
DO J=I+1,NNB
WORK(U11+I,J)=ZERO
END DO
DO J=1,I-1
WORK(U11+I,J)=A(CUT+I,CUT+J)
END DO
END DO
*
* invD*L21
*
I=N-CUT-NNB
DO WHILE (I .GE. 1)
IF (IPIV(CUT+NNB+I) > 0) THEN
DO J=1,NNB
WORK(I,J)=WORK(CUT+NNB+I,INVD)*WORK(I,J)
END DO
I=I-1
ELSE
DO J=1,NNB
U01_I_J = WORK(I,J)
U01_IP1_J = WORK(I-1,J)
WORK(I,J)=WORK(CUT+NNB+I,INVD)*U01_I_J+
$ WORK(CUT+NNB+I,INVD+1)*U01_IP1_J
WORK(I-1,J)=WORK(CUT+NNB+I-1,INVD+1)*U01_I_J+
$ WORK(CUT+NNB+I-1,INVD)*U01_IP1_J
END DO
I=I-2
END IF
END DO
*
* invD1*L11
*
I=NNB
DO WHILE (I .GE. 1)
IF (IPIV(CUT+I) > 0) THEN
DO J=1,NNB
WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J)
END DO
I=I-1
ELSE
DO J=1,NNB
U11_I_J = WORK(U11+I,J)
U11_IP1_J = WORK(U11+I-1,J)
WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J) +
$ WORK(CUT+I,INVD+1)*U11_IP1_J
WORK(U11+I-1,J)=WORK(CUT+I-1,INVD+1)*U11_I_J+
$ WORK(CUT+I-1,INVD)*U11_IP1_J
END DO
I=I-2
END IF
END DO
*
* L11**T*invD1*L11->L11
*
CALL DTRMM('L',UPLO,'T','U',NNB, NNB,
$ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1)
*
DO I=1,NNB
DO J=1,I
A(CUT+I,CUT+J)=WORK(U11+I,J)
END DO
END DO
*
IF ( (CUT+NNB) .LT. N ) THEN
*
* L21**T*invD2*L21->A(CUT+I,CUT+J)
*
CALL DGEMM('T','N',NNB,NNB,N-NNB-CUT,ONE,A(CUT+NNB+1,CUT+1)
$ ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1)
*
* L11 = L11**T*invD1*L11 + U01**T*invD*U01
*
DO I=1,NNB
DO J=1,I
A(CUT+I,CUT+J)=A(CUT+I,CUT+J)+WORK(U11+I,J)
END DO
END DO
*
* L01 = L22**T*invD2*L21
*
CALL DTRMM('L',UPLO,'T','U', N-NNB-CUT, NNB,
$ ONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1)
*
* Update L21
*
DO I=1,N-CUT-NNB
DO J=1,NNB
A(CUT+NNB+I,CUT+J)=WORK(I,J)
END DO
END DO
ELSE
*
* L11 = L11**T*invD1*L11
*
DO I=1,NNB
DO J=1,I
A(CUT+I,CUT+J)=WORK(U11+I,J)
END DO
END DO
END IF
*
* Next Block
*
CUT=CUT+NNB
END DO
*
* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
*
I=N
DO WHILE ( I .GE. 1 )
IF( IPIV(I) .GT. 0 ) THEN
IP=IPIV(I)
IF (I .LT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, I ,
$ IP )
IF (I .GT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, IP ,
$ I )
ELSE
IP=-IPIV(I)
IF ( I .LT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, I ,
$ IP )
IF ( I .GT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, IP,
$ I )
I=I-1
ENDIF
I=I-1
END DO
END IF
*
RETURN
*
* End of DSYTRI2X
*
END
*> \brief \b DSYTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRS solves a system of linear equations A*X = B with a real
*> symmetric matrix A using the factorization A = U*D*U**T or
*> A = L*D*L**T computed by DSYTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrs
*
* =====================================================================
SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, K, KP
DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Solve A*X = B, where A = U*D*U**T.
*
* First solve U*D*X = B, overwriting B with X.
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
10 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 30
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(U(K)), where U(K) is the transformation
* stored in column K of A.
*
CALL DGER( K-1, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
$ B( 1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
K = K - 1
ELSE
*
* 2 x 2 diagonal block
*
* Interchange rows K-1 and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K-1 )
$ CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(U(K)), where U(K) is the transformation
* stored in columns K-1 and K of A.
*
CALL DGER( K-2, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
$ B( 1, 1 ), LDB )
CALL DGER( K-2, NRHS, -ONE, A( 1, K-1 ), 1, B( K-1, 1 ),
$ LDB, B( 1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
AKM1K = A( K-1, K )
AKM1 = A( K-1, K-1 ) / AKM1K
AK = A( K, K ) / AKM1K
DENOM = AKM1*AK - ONE
DO 20 J = 1, NRHS
BKM1 = B( K-1, J ) / AKM1K
BK = B( K, J ) / AKM1K
B( K-1, J ) = ( AK*BKM1-BK ) / DENOM
B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM
20 CONTINUE
K = K - 2
END IF
*
GO TO 10
30 CONTINUE
*
* Next solve U**T *X = B, overwriting B with X.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
40 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 50
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Multiply by inv(U**T(K)), where U(K) is the transformation
* stored in column K of A.
*
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1,
$ K ),
$ 1, ONE, B( K, 1 ), LDB )
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K + 1
ELSE
*
* 2 x 2 diagonal block
*
* Multiply by inv(U**T(K+1)), where U(K+1) is the transformation
* stored in columns K and K+1 of A.
*
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1,
$ K ),
$ 1, ONE, B( K, 1 ), LDB )
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
$ A( 1, K+1 ), 1, ONE, B( K+1, 1 ), LDB )
*
* Interchange rows K and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K + 2
END IF
*
GO TO 40
50 CONTINUE
*
ELSE
*
* Solve A*X = B, where A = L*D*L**T.
*
* First solve L*D*X = B, overwriting B with X.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
60 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 80
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(L(K)), where L(K) is the transformation
* stored in column K of A.
*
IF( K.LT.N )
$ CALL DGER( N-K, NRHS, -ONE, A( K+1, K ), 1, B( K, 1 ),
$ LDB, B( K+1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
K = K + 1
ELSE
*
* 2 x 2 diagonal block
*
* Interchange rows K+1 and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K+1 )
$ CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(L(K)), where L(K) is the transformation
* stored in columns K and K+1 of A.
*
IF( K.LT.N-1 ) THEN
CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K ), 1, B( K,
$ 1 ),
$ LDB, B( K+2, 1 ), LDB )
CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K+1 ), 1,
$ B( K+1, 1 ), LDB, B( K+2, 1 ), LDB )
END IF
*
* Multiply by the inverse of the diagonal block.
*
AKM1K = A( K+1, K )
AKM1 = A( K, K ) / AKM1K
AK = A( K+1, K+1 ) / AKM1K
DENOM = AKM1*AK - ONE
DO 70 J = 1, NRHS
BKM1 = B( K, J ) / AKM1K
BK = B( K+1, J ) / AKM1K
B( K, J ) = ( AK*BKM1-BK ) / DENOM
B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
70 CONTINUE
K = K + 2
END IF
*
GO TO 60
80 CONTINUE
*
* Next solve L**T *X = B, overwriting B with X.
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
90 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 100
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Multiply by inv(L**T(K)), where L(K) is the transformation
* stored in column K of A.
*
IF( K.LT.N )
$ CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K - 1
ELSE
*
* 2 x 2 diagonal block
*
* Multiply by inv(L**T(K-1)), where L(K-1) is the transformation
* stored in columns K-1 and K of A.
*
IF( K.LT.N ) THEN
CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, A( K+1, K-1 ), 1, ONE, B( K-1, 1 ),
$ LDB )
END IF
*
* Interchange rows K and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K - 2
END IF
*
GO TO 90
100 CONTINUE
END IF
*
RETURN
*
* End of DSYTRS
*
END
*> \brief \b DSYTRS2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrs2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrs2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrs2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRS2( UPLO, N, NRHS, A, LDA, IPIV, B, LDB,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRS2 solves a system of linear equations A*X = B with a real
*> symmetric matrix A using the factorization A = U*D*U**T or
*> A = L*D*L**T computed by DSYTRF and converted by DSYCONV.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*D*U**T;
*> = 'L': Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by DSYTRF.
*> Note that A is input / output. This might be counter-intuitive,
*> and one may think that A is input only. A is input / output. This
*> is because, at the start of the subroutine, we permute A in a
*> "better" form and then we permute A back to its original form at
*> the end.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrs2
*
* =====================================================================
SUBROUTINE DSYTRS2( UPLO, N, NRHS, A, LDA, IPIV, B, LDB,
$ WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IINFO, J, K, KP
DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSYCONV, DSWAP, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRS2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
* Convert A
*
CALL DSYCONV( UPLO, 'C', N, A, LDA, IPIV, WORK, IINFO )
*
IF( UPPER ) THEN
*
* Solve A*X = B, where A = U*D*U**T.
*
* P**T * B
K=N
DO WHILE ( K .GE. 1 )
IF( IPIV( K ).GT.0 ) THEN
* 1 x 1 diagonal block
* Interchange rows K and IPIV(K).
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K=K-1
ELSE
* 2 x 2 diagonal block
* Interchange rows K-1 and -IPIV(K).
KP = -IPIV( K )
IF( KP.EQ.-IPIV( K-1 ) )
$ CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
K=K-2
END IF
END DO
*
* Compute (U \P**T * B) -> B [ (U \P**T * B) ]
*
CALL DTRSM('L','U','N','U',N,NRHS,ONE,A,LDA,B,LDB)
*
* Compute D \ B -> B [ D \ (U \P**T * B) ]
*
I=N
DO WHILE ( I .GE. 1 )
IF( IPIV(I) .GT. 0 ) THEN
CALL DSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB )
ELSEIF ( I .GT. 1) THEN
IF ( IPIV(I-1) .EQ. IPIV(I) ) THEN
AKM1K = WORK(I)
AKM1 = A( I-1, I-1 ) / AKM1K
AK = A( I, I ) / AKM1K
DENOM = AKM1*AK - ONE
DO 15 J = 1, NRHS
BKM1 = B( I-1, J ) / AKM1K
BK = B( I, J ) / AKM1K
B( I-1, J ) = ( AK*BKM1-BK ) / DENOM
B( I, J ) = ( AKM1*BK-BKM1 ) / DENOM
15 CONTINUE
I = I - 1
ENDIF
ENDIF
I = I - 1
END DO
*
* Compute (U**T \ B) -> B [ U**T \ (D \ (U \P**T * B) ) ]
*
CALL DTRSM('L','U','T','U',N,NRHS,ONE,A,LDA,B,LDB)
*
* P * B [ P * (U**T \ (D \ (U \P**T * B) )) ]
*
K=1
DO WHILE ( K .LE. N )
IF( IPIV( K ).GT.0 ) THEN
* 1 x 1 diagonal block
* Interchange rows K and IPIV(K).
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K=K+1
ELSE
* 2 x 2 diagonal block
* Interchange rows K-1 and -IPIV(K).
KP = -IPIV( K )
IF( K .LT. N .AND. KP.EQ.-IPIV( K+1 ) )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K=K+2
ENDIF
END DO
*
ELSE
*
* Solve A*X = B, where A = L*D*L**T.
*
* P**T * B
K=1
DO WHILE ( K .LE. N )
IF( IPIV( K ).GT.0 ) THEN
* 1 x 1 diagonal block
* Interchange rows K and IPIV(K).
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K=K+1
ELSE
* 2 x 2 diagonal block
* Interchange rows K and -IPIV(K+1).
KP = -IPIV( K+1 )
IF( KP.EQ.-IPIV( K ) )
$ CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
K=K+2
ENDIF
END DO
*
* Compute (L \P**T * B) -> B [ (L \P**T * B) ]
*
CALL DTRSM('L','L','N','U',N,NRHS,ONE,A,LDA,B,LDB)
*
* Compute D \ B -> B [ D \ (L \P**T * B) ]
*
I=1
DO WHILE ( I .LE. N )
IF( IPIV(I) .GT. 0 ) THEN
CALL DSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB )
ELSE
AKM1K = WORK(I)
AKM1 = A( I, I ) / AKM1K
AK = A( I+1, I+1 ) / AKM1K
DENOM = AKM1*AK - ONE
DO 25 J = 1, NRHS
BKM1 = B( I, J ) / AKM1K
BK = B( I+1, J ) / AKM1K
B( I, J ) = ( AK*BKM1-BK ) / DENOM
B( I+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
25 CONTINUE
I = I + 1
ENDIF
I = I + 1
END DO
*
* Compute (L**T \ B) -> B [ L**T \ (D \ (L \P**T * B) ) ]
*
CALL DTRSM('L','L','T','U',N,NRHS,ONE,A,LDA,B,LDB)
*
* P * B [ P * (L**T \ (D \ (L \P**T * B) )) ]
*
K=N
DO WHILE ( K .GE. 1 )
IF( IPIV( K ).GT.0 ) THEN
* 1 x 1 diagonal block
* Interchange rows K and IPIV(K).
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K=K-1
ELSE
* 2 x 2 diagonal block
* Interchange rows K-1 and -IPIV(K).
KP = -IPIV( K )
IF( K.GT.1 .AND. KP.EQ.-IPIV( K-1 ) )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K=K-2
ENDIF
END DO
*
END IF
*
* Revert A
*
CALL DSYCONV( UPLO, 'R', N, A, LDA, IPIV, WORK, IINFO )
*
RETURN
*
* End of DSYTRS2
*
END
*> \brief \b DTBCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTBCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtbcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtbcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtbcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTBCON( NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORM, UPLO
* INTEGER INFO, KD, LDAB, N
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTBCON estimates the reciprocal of the condition number of a
*> triangular band matrix A, in either the 1-norm or the infinity-norm.
*>
*> The norm of A is computed and an estimate is obtained for
*> norm(inv(A)), then the reciprocal of the condition number is
*> computed as
*> RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies whether the 1-norm condition number or the
*> infinity-norm condition number is required:
*> = '1' or 'O': 1-norm;
*> = 'I': Infinity-norm.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals or subdiagonals of the
*> triangular band matrix A. KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangular band matrix A, stored in the
*> first kd+1 rows of the array. The j-th column of A is stored
*> in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> If DIAG = 'U', the diagonal elements of A are not referenced
*> and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tbcon
*
* =====================================================================
SUBROUTINE DTBCON( NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND,
$ WORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, ONENRM, UPPER
CHARACTER NORMIN
INTEGER IX, KASE, KASE1
DOUBLE PRECISION AINVNM, ANORM, SCALE, SMLNUM, XNORM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLANTB
EXTERNAL LSAME, IDAMAX, DLAMCH, DLANTB
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLATBS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KD.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTBCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
*
RCOND = ZERO
SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) )
*
* Compute the norm of the triangular matrix A.
*
ANORM = DLANTB( NORM, UPLO, DIAG, N, KD, AB, LDAB, WORK )
*
* Continue only if ANORM > 0.
*
IF( ANORM.GT.ZERO ) THEN
*
* Estimate the norm of the inverse of A.
*
AINVNM = ZERO
NORMIN = 'N'
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KASE = 0
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE,
$ ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(A).
*
CALL DLATBS( UPLO, 'No transpose', DIAG, NORMIN, N,
$ KD,
$ AB, LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO )
ELSE
*
* Multiply by inv(A**T).
*
CALL DLATBS( UPLO, 'Transpose', DIAG, NORMIN, N, KD,
$ AB,
$ LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO )
END IF
NORMIN = 'Y'
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
XNORM = ABS( WORK( IX ) )
IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / ANORM ) / AINVNM
END IF
*
20 CONTINUE
RETURN
*
* End of DTBCON
*
END
*> \brief \b DTBRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTBRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtbrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtbrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtbrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
* LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTBRFS provides error bounds and backward error estimates for the
*> solution to a system of linear equations with a triangular band
*> coefficient matrix.
*>
*> The solution matrix X must be computed by DTBTRS or some other
*> means before entering this routine. DTBRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals or subdiagonals of the
*> triangular band matrix A. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangular band matrix A, stored in the
*> first kd+1 rows of the array. The j-th column of A is stored
*> in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> If DIAG = 'U', the diagonal elements of A are not referenced
*> and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> The solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tbrfs
*
* =====================================================================
SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
$ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
CHARACTER TRANST
INTEGER I, J, K, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DTBMV, DTBSV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KD.LT.0 ) THEN
INFO = -5
ELSE IF( NRHS.LT.0 ) THEN
INFO = -6
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTBRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = KD + 2
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 250 J = 1, NRHS
*
* Compute residual R = B - op(A) * X,
* where op(A) = A or A**T, depending on TRANS.
*
CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ),
$ 1 )
CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 20 I = 1, N
WORK( I ) = ABS( B( I, J ) )
20 CONTINUE
*
IF( NOTRAN ) THEN
*
* Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
IF( NOUNIT ) THEN
DO 40 K = 1, N
XK = ABS( X( K, J ) )
DO 30 I = MAX( 1, K-KD ), K
WORK( I ) = WORK( I ) +
$ ABS( AB( KD+1+I-K, K ) )*XK
30 CONTINUE
40 CONTINUE
ELSE
DO 60 K = 1, N
XK = ABS( X( K, J ) )
DO 50 I = MAX( 1, K-KD ), K - 1
WORK( I ) = WORK( I ) +
$ ABS( AB( KD+1+I-K, K ) )*XK
50 CONTINUE
WORK( K ) = WORK( K ) + XK
60 CONTINUE
END IF
ELSE
IF( NOUNIT ) THEN
DO 80 K = 1, N
XK = ABS( X( K, J ) )
DO 70 I = K, MIN( N, K+KD )
WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
70 CONTINUE
80 CONTINUE
ELSE
DO 100 K = 1, N
XK = ABS( X( K, J ) )
DO 90 I = K + 1, MIN( N, K+KD )
WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
90 CONTINUE
WORK( K ) = WORK( K ) + XK
100 CONTINUE
END IF
END IF
ELSE
*
* Compute abs(A**T)*abs(X) + abs(B).
*
IF( UPPER ) THEN
IF( NOUNIT ) THEN
DO 120 K = 1, N
S = ZERO
DO 110 I = MAX( 1, K-KD ), K
S = S + ABS( AB( KD+1+I-K, K ) )*
$ ABS( X( I, J ) )
110 CONTINUE
WORK( K ) = WORK( K ) + S
120 CONTINUE
ELSE
DO 140 K = 1, N
S = ABS( X( K, J ) )
DO 130 I = MAX( 1, K-KD ), K - 1
S = S + ABS( AB( KD+1+I-K, K ) )*
$ ABS( X( I, J ) )
130 CONTINUE
WORK( K ) = WORK( K ) + S
140 CONTINUE
END IF
ELSE
IF( NOUNIT ) THEN
DO 160 K = 1, N
S = ZERO
DO 150 I = K, MIN( N, K+KD )
S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
150 CONTINUE
WORK( K ) = WORK( K ) + S
160 CONTINUE
ELSE
DO 180 K = 1, N
S = ABS( X( K, J ) )
DO 170 I = K + 1, MIN( N, K+KD )
S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
170 CONTINUE
WORK( K ) = WORK( K ) + S
180 CONTINUE
END IF
END IF
END IF
S = ZERO
DO 190 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
190 CONTINUE
BERR( J ) = S
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(op(A)))*
* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(A)) is the inverse of op(A)
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(op(A)) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 200 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
200 CONTINUE
*
KASE = 0
210 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**T).
*
CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB,
$ WORK( N+1 ), 1 )
DO 220 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
220 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 230 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
230 CONTINUE
CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB,
$ WORK( N+1 ), 1 )
END IF
GO TO 210
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 240 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
240 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
250 CONTINUE
*
RETURN
*
* End of DTBRFS
*
END
*> \brief \b DTBTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTBTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtbtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtbtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtbtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTBTRS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
* LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, KD, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTBTRS solves a triangular system of the form
*>
*> A * X = B or A**T * X = B,
*>
*> where A is a triangular band matrix of order N, and B is an N-by-NRHS matrix.
*>
*> This subroutine verifies that A is nonsingular, but callers should note that only exact
*> singularity is detected. It is conceivable for one or more diagonal elements of A to be
*> subnormally tiny numbers without this subroutine signalling an error.
*>
*> If a possible loss of numerical precision due to near-singular matrices is a concern, the
*> caller should verify that A is nonsingular within some tolerance before calling this subroutine.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals or subdiagonals of the
*> triangular band matrix A. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> The upper or lower triangular band matrix A, stored in the
*> first kd+1 rows of AB. The j-th column of A is stored
*> in the j-th column of the array AB as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> If DIAG = 'U', the diagonal elements of A are not referenced
*> and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, if INFO = 0, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element of A is exactly zero,
*> indicating that the matrix is singular and the
*> solutions X have not been computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tbtrs
*
* =====================================================================
SUBROUTINE DTBTRS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
$ LDB, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, KD, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, UPPER
INTEGER J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DTBSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOUNIT = LSAME( DIAG, 'N' )
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.
$ LSAME( TRANS, 'T' ) .AND.
$ .NOT.LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KD.LT.0 ) THEN
INFO = -5
ELSE IF( NRHS.LT.0 ) THEN
INFO = -6
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTBTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check for singularity.
*
IF( NOUNIT ) THEN
IF( UPPER ) THEN
DO 10 INFO = 1, N
IF( AB( KD+1, INFO ).EQ.ZERO )
$ RETURN
10 CONTINUE
ELSE
DO 20 INFO = 1, N
IF( AB( 1, INFO ).EQ.ZERO )
$ RETURN
20 CONTINUE
END IF
END IF
INFO = 0
*
* Solve A * X = B or A**T * X = B.
*
DO 30 J = 1, NRHS
CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, B( 1, J ),
$ 1 )
30 CONTINUE
*
RETURN
*
* End of DTBTRS
*
END
*> \brief \b DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFSM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtfsm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtfsm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtfsm.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A,
* B, LDB )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, DIAG, SIDE, TRANS, UPLO
* INTEGER LDB, M, N
* DOUBLE PRECISION ALPHA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: * ), B( 0: LDB-1, 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Level 3 BLAS like routine for A in RFP Format.
*>
*> DTFSM solves the matrix equation
*>
*> op( A )*X = alpha*B or X*op( A ) = alpha*B
*>
*> where alpha is a scalar, X and B are m by n matrices, A is a unit, or
*> non-unit, upper or lower triangular matrix and op( A ) is one of
*>
*> op( A ) = A or op( A ) = A**T.
*>
*> A is in Rectangular Full Packed (RFP) Format.
*>
*> The matrix X is overwritten on B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': The Normal Form of RFP A is stored;
*> = 'T': The Transpose Form of RFP A is stored.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> On entry, SIDE specifies whether op( A ) appears on the left
*> or right of X as follows:
*>
*> SIDE = 'L' or 'l' op( A )*X = alpha*B.
*>
*> SIDE = 'R' or 'r' X*op( A ) = alpha*B.
*>
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the RFP matrix A came from
*> an upper or lower triangular matrix as follows:
*> UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
*> UPLO = 'L' or 'l' RFP A came from a lower triangular matrix
*>
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> On entry, TRANS specifies the form of op( A ) to be used
*> in the matrix multiplication as follows:
*>
*> TRANS = 'N' or 'n' op( A ) = A.
*>
*> TRANS = 'T' or 't' op( A ) = A'.
*>
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> On entry, DIAG specifies whether or not RFP A is unit
*> triangular as follows:
*>
*> DIAG = 'U' or 'u' A is assumed to be unit triangular.
*>
*> DIAG = 'N' or 'n' A is not assumed to be unit
*> triangular.
*>
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> On entry, M specifies the number of rows of B. M must be at
*> least zero.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the number of columns of B. N must be
*> at least zero.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> On entry, ALPHA specifies the scalar alpha. When alpha is
*> zero then A is not referenced and B need not be set before
*> entry.
*> Unchanged on exit.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (NT)
*> NT = N*(N+1)/2 if SIDE='R' and NT = M*(M+1)/2 otherwise.
*> On entry, the matrix A in RFP Format.
*> RFP Format is described by TRANSR, UPLO and N as follows:
*> If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
*> K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
*> TRANSR = 'T' then RFP is the transpose of RFP A as
*> defined when TRANSR = 'N'. The contents of RFP A are defined
*> by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
*> elements of upper packed A either in normal or
*> transpose Format. If UPLO = 'L' the RFP A contains
*> the NT elements of lower packed A either in normal or
*> transpose Format. The LDA of RFP A is (N+1)/2 when
*> TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
*> even and is N when is odd.
*> See the Note below for more details. Unchanged on exit.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> Before entry, the leading m by n part of the array B must
*> contain the right-hand side matrix B, and on exit is
*> overwritten by the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> On entry, LDB specifies the first dimension of B as declared
*> in the calling (sub) program. LDB must be at least
*> max( 1, m ).
*> Unchanged on exit.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tfsm
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*
* =====================================================================
SUBROUTINE DTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA,
$ A,
$ B, LDB )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, DIAG, SIDE, TRANS, UPLO
INTEGER LDB, M, N
DOUBLE PRECISION ALPHA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( 0: * ), B( 0: LDB-1, 0: * )
* ..
*
* =====================================================================
*
* ..
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, LSIDE, MISODD, NISODD, NORMALTRANSR,
$ NOTRANS
INTEGER M1, M2, N1, N2, K, INFO, I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DGEMM, DTRSM
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LSIDE = LSAME( SIDE, 'L' )
LOWER = LSAME( UPLO, 'L' )
NOTRANS = LSAME( TRANS, 'N' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSIDE .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -2
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -3
ELSE IF( .NOT.NOTRANS .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -4
ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND.
$ .NOT.LSAME( DIAG, 'U' ) )
$ THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( N.LT.0 ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTFSM ', -INFO )
RETURN
END IF
*
* Quick return when ( (N.EQ.0).OR.(M.EQ.0) )
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )
$ RETURN
*
* Quick return when ALPHA.EQ.(0D+0)
*
IF( ALPHA.EQ.ZERO ) THEN
DO 20 J = 0, N - 1
DO 10 I = 0, M - 1
B( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
RETURN
END IF
*
IF( LSIDE ) THEN
*
* SIDE = 'L'
*
* A is M-by-M.
* If M is odd, set NISODD = .TRUE., and M1 and M2.
* If M is even, NISODD = .FALSE., and M.
*
IF( MOD( M, 2 ).EQ.0 ) THEN
MISODD = .FALSE.
K = M / 2
ELSE
MISODD = .TRUE.
IF( LOWER ) THEN
M2 = M / 2
M1 = M - M2
ELSE
M1 = M / 2
M2 = M - M1
END IF
END IF
*
*
IF( MISODD ) THEN
*
* SIDE = 'L' and N is odd
*
IF( NORMALTRANSR ) THEN
*
* SIDE = 'L', N is odd, and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SIDE ='L', N is odd, TRANSR = 'N', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='L', N is odd, TRANSR = 'N', UPLO = 'L', and
* TRANS = 'N'
*
IF( M.EQ.1 ) THEN
CALL DTRSM( 'L', 'L', 'N', DIAG, M1, N,
$ ALPHA,
$ A, M, B, LDB )
ELSE
CALL DTRSM( 'L', 'L', 'N', DIAG, M1, N,
$ ALPHA,
$ A( 0 ), M, B, LDB )
CALL DGEMM( 'N', 'N', M2, N, M1, -ONE,
$ A( M1 ),
$ M, B, LDB, ALPHA, B( M1, 0 ), LDB )
CALL DTRSM( 'L', 'U', 'T', DIAG, M2, N, ONE,
$ A( M ), M, B( M1, 0 ), LDB )
END IF
*
ELSE
*
* SIDE ='L', N is odd, TRANSR = 'N', UPLO = 'L', and
* TRANS = 'T'
*
IF( M.EQ.1 ) THEN
CALL DTRSM( 'L', 'L', 'T', DIAG, M1, N,
$ ALPHA,
$ A( 0 ), M, B, LDB )
ELSE
CALL DTRSM( 'L', 'U', 'N', DIAG, M2, N,
$ ALPHA,
$ A( M ), M, B( M1, 0 ), LDB )
CALL DGEMM( 'T', 'N', M1, N, M2, -ONE,
$ A( M1 ),
$ M, B( M1, 0 ), LDB, ALPHA, B, LDB )
CALL DTRSM( 'L', 'L', 'T', DIAG, M1, N, ONE,
$ A( 0 ), M, B, LDB )
END IF
*
END IF
*
ELSE
*
* SIDE ='L', N is odd, TRANSR = 'N', and UPLO = 'U'
*
IF( .NOT.NOTRANS ) THEN
*
* SIDE ='L', N is odd, TRANSR = 'N', UPLO = 'U', and
* TRANS = 'N'
*
CALL DTRSM( 'L', 'L', 'N', DIAG, M1, N, ALPHA,
$ A( M2 ), M, B, LDB )
CALL DGEMM( 'T', 'N', M2, N, M1, -ONE, A( 0 ),
$ M,
$ B, LDB, ALPHA, B( M1, 0 ), LDB )
CALL DTRSM( 'L', 'U', 'T', DIAG, M2, N, ONE,
$ A( M1 ), M, B( M1, 0 ), LDB )
*
ELSE
*
* SIDE ='L', N is odd, TRANSR = 'N', UPLO = 'U', and
* TRANS = 'T'
*
CALL DTRSM( 'L', 'U', 'N', DIAG, M2, N, ALPHA,
$ A( M1 ), M, B( M1, 0 ), LDB )
CALL DGEMM( 'N', 'N', M1, N, M2, -ONE, A( 0 ),
$ M,
$ B( M1, 0 ), LDB, ALPHA, B, LDB )
CALL DTRSM( 'L', 'L', 'T', DIAG, M1, N, ONE,
$ A( M2 ), M, B, LDB )
*
END IF
*
END IF
*
ELSE
*
* SIDE = 'L', N is odd, and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SIDE ='L', N is odd, TRANSR = 'T', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='L', N is odd, TRANSR = 'T', UPLO = 'L', and
* TRANS = 'N'
*
IF( M.EQ.1 ) THEN
CALL DTRSM( 'L', 'U', 'T', DIAG, M1, N,
$ ALPHA,
$ A( 0 ), M1, B, LDB )
ELSE
CALL DTRSM( 'L', 'U', 'T', DIAG, M1, N,
$ ALPHA,
$ A( 0 ), M1, B, LDB )
CALL DGEMM( 'T', 'N', M2, N, M1, -ONE,
$ A( M1*M1 ), M1, B, LDB, ALPHA,
$ B( M1, 0 ), LDB )
CALL DTRSM( 'L', 'L', 'N', DIAG, M2, N, ONE,
$ A( 1 ), M1, B( M1, 0 ), LDB )
END IF
*
ELSE
*
* SIDE ='L', N is odd, TRANSR = 'T', UPLO = 'L', and
* TRANS = 'T'
*
IF( M.EQ.1 ) THEN
CALL DTRSM( 'L', 'U', 'N', DIAG, M1, N,
$ ALPHA,
$ A( 0 ), M1, B, LDB )
ELSE
CALL DTRSM( 'L', 'L', 'T', DIAG, M2, N,
$ ALPHA,
$ A( 1 ), M1, B( M1, 0 ), LDB )
CALL DGEMM( 'N', 'N', M1, N, M2, -ONE,
$ A( M1*M1 ), M1, B( M1, 0 ), LDB,
$ ALPHA, B, LDB )
CALL DTRSM( 'L', 'U', 'N', DIAG, M1, N, ONE,
$ A( 0 ), M1, B, LDB )
END IF
*
END IF
*
ELSE
*
* SIDE ='L', N is odd, TRANSR = 'T', and UPLO = 'U'
*
IF( .NOT.NOTRANS ) THEN
*
* SIDE ='L', N is odd, TRANSR = 'T', UPLO = 'U', and
* TRANS = 'N'
*
CALL DTRSM( 'L', 'U', 'T', DIAG, M1, N, ALPHA,
$ A( M2*M2 ), M2, B, LDB )
CALL DGEMM( 'N', 'N', M2, N, M1, -ONE, A( 0 ),
$ M2,
$ B, LDB, ALPHA, B( M1, 0 ), LDB )
CALL DTRSM( 'L', 'L', 'N', DIAG, M2, N, ONE,
$ A( M1*M2 ), M2, B( M1, 0 ), LDB )
*
ELSE
*
* SIDE ='L', N is odd, TRANSR = 'T', UPLO = 'U', and
* TRANS = 'T'
*
CALL DTRSM( 'L', 'L', 'T', DIAG, M2, N, ALPHA,
$ A( M1*M2 ), M2, B( M1, 0 ), LDB )
CALL DGEMM( 'T', 'N', M1, N, M2, -ONE, A( 0 ),
$ M2,
$ B( M1, 0 ), LDB, ALPHA, B, LDB )
CALL DTRSM( 'L', 'U', 'N', DIAG, M1, N, ONE,
$ A( M2*M2 ), M2, B, LDB )
*
END IF
*
END IF
*
END IF
*
ELSE
*
* SIDE = 'L' and N is even
*
IF( NORMALTRANSR ) THEN
*
* SIDE = 'L', N is even, and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SIDE ='L', N is even, TRANSR = 'N', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='L', N is even, TRANSR = 'N', UPLO = 'L',
* and TRANS = 'N'
*
CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ALPHA,
$ A( 1 ), M+1, B, LDB )
CALL DGEMM( 'N', 'N', K, N, K, -ONE, A( K+1 ),
$ M+1, B, LDB, ALPHA, B( K, 0 ), LDB )
CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ONE,
$ A( 0 ), M+1, B( K, 0 ), LDB )
*
ELSE
*
* SIDE ='L', N is even, TRANSR = 'N', UPLO = 'L',
* and TRANS = 'T'
*
CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ALPHA,
$ A( 0 ), M+1, B( K, 0 ), LDB )
CALL DGEMM( 'T', 'N', K, N, K, -ONE, A( K+1 ),
$ M+1, B( K, 0 ), LDB, ALPHA, B, LDB )
CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ONE,
$ A( 1 ), M+1, B, LDB )
*
END IF
*
ELSE
*
* SIDE ='L', N is even, TRANSR = 'N', and UPLO = 'U'
*
IF( .NOT.NOTRANS ) THEN
*
* SIDE ='L', N is even, TRANSR = 'N', UPLO = 'U',
* and TRANS = 'N'
*
CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ALPHA,
$ A( K+1 ), M+1, B, LDB )
CALL DGEMM( 'T', 'N', K, N, K, -ONE, A( 0 ),
$ M+1,
$ B, LDB, ALPHA, B( K, 0 ), LDB )
CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ONE,
$ A( K ), M+1, B( K, 0 ), LDB )
*
ELSE
*
* SIDE ='L', N is even, TRANSR = 'N', UPLO = 'U',
* and TRANS = 'T'
CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ALPHA,
$ A( K ), M+1, B( K, 0 ), LDB )
CALL DGEMM( 'N', 'N', K, N, K, -ONE, A( 0 ),
$ M+1,
$ B( K, 0 ), LDB, ALPHA, B, LDB )
CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ONE,
$ A( K+1 ), M+1, B, LDB )
*
END IF
*
END IF
*
ELSE
*
* SIDE = 'L', N is even, and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SIDE ='L', N is even, TRANSR = 'T', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='L', N is even, TRANSR = 'T', UPLO = 'L',
* and TRANS = 'N'
*
CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ALPHA,
$ A( K ), K, B, LDB )
CALL DGEMM( 'T', 'N', K, N, K, -ONE,
$ A( K*( K+1 ) ), K, B, LDB, ALPHA,
$ B( K, 0 ), LDB )
CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ONE,
$ A( 0 ), K, B( K, 0 ), LDB )
*
ELSE
*
* SIDE ='L', N is even, TRANSR = 'T', UPLO = 'L',
* and TRANS = 'T'
*
CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ALPHA,
$ A( 0 ), K, B( K, 0 ), LDB )
CALL DGEMM( 'N', 'N', K, N, K, -ONE,
$ A( K*( K+1 ) ), K, B( K, 0 ), LDB,
$ ALPHA, B, LDB )
CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ONE,
$ A( K ), K, B, LDB )
*
END IF
*
ELSE
*
* SIDE ='L', N is even, TRANSR = 'T', and UPLO = 'U'
*
IF( .NOT.NOTRANS ) THEN
*
* SIDE ='L', N is even, TRANSR = 'T', UPLO = 'U',
* and TRANS = 'N'
*
CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ALPHA,
$ A( K*( K+1 ) ), K, B, LDB )
CALL DGEMM( 'N', 'N', K, N, K, -ONE, A( 0 ), K,
$ B,
$ LDB, ALPHA, B( K, 0 ), LDB )
CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ONE,
$ A( K*K ), K, B( K, 0 ), LDB )
*
ELSE
*
* SIDE ='L', N is even, TRANSR = 'T', UPLO = 'U',
* and TRANS = 'T'
*
CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ALPHA,
$ A( K*K ), K, B( K, 0 ), LDB )
CALL DGEMM( 'T', 'N', K, N, K, -ONE, A( 0 ), K,
$ B( K, 0 ), LDB, ALPHA, B, LDB )
CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ONE,
$ A( K*( K+1 ) ), K, B, LDB )
*
END IF
*
END IF
*
END IF
*
END IF
*
ELSE
*
* SIDE = 'R'
*
* A is N-by-N.
* If N is odd, set NISODD = .TRUE., and N1 and N2.
* If N is even, NISODD = .FALSE., and K.
*
IF( MOD( N, 2 ).EQ.0 ) THEN
NISODD = .FALSE.
K = N / 2
ELSE
NISODD = .TRUE.
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
END IF
*
IF( NISODD ) THEN
*
* SIDE = 'R' and N is odd
*
IF( NORMALTRANSR ) THEN
*
* SIDE = 'R', N is odd, and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SIDE ='R', N is odd, TRANSR = 'N', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is odd, TRANSR = 'N', UPLO = 'L', and
* TRANS = 'N'
*
CALL DTRSM( 'R', 'U', 'T', DIAG, M, N2, ALPHA,
$ A( N ), N, B( 0, N1 ), LDB )
CALL DGEMM( 'N', 'N', M, N1, N2, -ONE, B( 0,
$ N1 ),
$ LDB, A( N1 ), N, ALPHA, B( 0, 0 ),
$ LDB )
CALL DTRSM( 'R', 'L', 'N', DIAG, M, N1, ONE,
$ A( 0 ), N, B( 0, 0 ), LDB )
*
ELSE
*
* SIDE ='R', N is odd, TRANSR = 'N', UPLO = 'L', and
* TRANS = 'T'
*
CALL DTRSM( 'R', 'L', 'T', DIAG, M, N1, ALPHA,
$ A( 0 ), N, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'T', M, N2, N1, -ONE, B( 0,
$ 0 ),
$ LDB, A( N1 ), N, ALPHA, B( 0, N1 ),
$ LDB )
CALL DTRSM( 'R', 'U', 'N', DIAG, M, N2, ONE,
$ A( N ), N, B( 0, N1 ), LDB )
*
END IF
*
ELSE
*
* SIDE ='R', N is odd, TRANSR = 'N', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is odd, TRANSR = 'N', UPLO = 'U', and
* TRANS = 'N'
*
CALL DTRSM( 'R', 'L', 'T', DIAG, M, N1, ALPHA,
$ A( N2 ), N, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'N', M, N2, N1, -ONE, B( 0,
$ 0 ),
$ LDB, A( 0 ), N, ALPHA, B( 0, N1 ),
$ LDB )
CALL DTRSM( 'R', 'U', 'N', DIAG, M, N2, ONE,
$ A( N1 ), N, B( 0, N1 ), LDB )
*
ELSE
*
* SIDE ='R', N is odd, TRANSR = 'N', UPLO = 'U', and
* TRANS = 'T'
*
CALL DTRSM( 'R', 'U', 'T', DIAG, M, N2, ALPHA,
$ A( N1 ), N, B( 0, N1 ), LDB )
CALL DGEMM( 'N', 'T', M, N1, N2, -ONE, B( 0,
$ N1 ),
$ LDB, A( 0 ), N, ALPHA, B( 0, 0 ), LDB )
CALL DTRSM( 'R', 'L', 'N', DIAG, M, N1, ONE,
$ A( N2 ), N, B( 0, 0 ), LDB )
*
END IF
*
END IF
*
ELSE
*
* SIDE = 'R', N is odd, and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SIDE ='R', N is odd, TRANSR = 'T', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is odd, TRANSR = 'T', UPLO = 'L', and
* TRANS = 'N'
*
CALL DTRSM( 'R', 'L', 'N', DIAG, M, N2, ALPHA,
$ A( 1 ), N1, B( 0, N1 ), LDB )
CALL DGEMM( 'N', 'T', M, N1, N2, -ONE, B( 0,
$ N1 ),
$ LDB, A( N1*N1 ), N1, ALPHA, B( 0, 0 ),
$ LDB )
CALL DTRSM( 'R', 'U', 'T', DIAG, M, N1, ONE,
$ A( 0 ), N1, B( 0, 0 ), LDB )
*
ELSE
*
* SIDE ='R', N is odd, TRANSR = 'T', UPLO = 'L', and
* TRANS = 'T'
*
CALL DTRSM( 'R', 'U', 'N', DIAG, M, N1, ALPHA,
$ A( 0 ), N1, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'N', M, N2, N1, -ONE, B( 0,
$ 0 ),
$ LDB, A( N1*N1 ), N1, ALPHA, B( 0, N1 ),
$ LDB )
CALL DTRSM( 'R', 'L', 'T', DIAG, M, N2, ONE,
$ A( 1 ), N1, B( 0, N1 ), LDB )
*
END IF
*
ELSE
*
* SIDE ='R', N is odd, TRANSR = 'T', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is odd, TRANSR = 'T', UPLO = 'U', and
* TRANS = 'N'
*
CALL DTRSM( 'R', 'U', 'N', DIAG, M, N1, ALPHA,
$ A( N2*N2 ), N2, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'T', M, N2, N1, -ONE, B( 0,
$ 0 ),
$ LDB, A( 0 ), N2, ALPHA, B( 0, N1 ),
$ LDB )
CALL DTRSM( 'R', 'L', 'T', DIAG, M, N2, ONE,
$ A( N1*N2 ), N2, B( 0, N1 ), LDB )
*
ELSE
*
* SIDE ='R', N is odd, TRANSR = 'T', UPLO = 'U', and
* TRANS = 'T'
*
CALL DTRSM( 'R', 'L', 'N', DIAG, M, N2, ALPHA,
$ A( N1*N2 ), N2, B( 0, N1 ), LDB )
CALL DGEMM( 'N', 'N', M, N1, N2, -ONE, B( 0,
$ N1 ),
$ LDB, A( 0 ), N2, ALPHA, B( 0, 0 ),
$ LDB )
CALL DTRSM( 'R', 'U', 'T', DIAG, M, N1, ONE,
$ A( N2*N2 ), N2, B( 0, 0 ), LDB )
*
END IF
*
END IF
*
END IF
*
ELSE
*
* SIDE = 'R' and N is even
*
IF( NORMALTRANSR ) THEN
*
* SIDE = 'R', N is even, and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SIDE ='R', N is even, TRANSR = 'N', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is even, TRANSR = 'N', UPLO = 'L',
* and TRANS = 'N'
*
CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ALPHA,
$ A( 0 ), N+1, B( 0, K ), LDB )
CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, K ),
$ LDB, A( K+1 ), N+1, ALPHA, B( 0, 0 ),
$ LDB )
CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ONE,
$ A( 1 ), N+1, B( 0, 0 ), LDB )
*
ELSE
*
* SIDE ='R', N is even, TRANSR = 'N', UPLO = 'L',
* and TRANS = 'T'
*
CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ALPHA,
$ A( 1 ), N+1, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, 0 ),
$ LDB, A( K+1 ), N+1, ALPHA, B( 0, K ),
$ LDB )
CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ONE,
$ A( 0 ), N+1, B( 0, K ), LDB )
*
END IF
*
ELSE
*
* SIDE ='R', N is even, TRANSR = 'N', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is even, TRANSR = 'N', UPLO = 'U',
* and TRANS = 'N'
*
CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ALPHA,
$ A( K+1 ), N+1, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, 0 ),
$ LDB, A( 0 ), N+1, ALPHA, B( 0, K ),
$ LDB )
CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ONE,
$ A( K ), N+1, B( 0, K ), LDB )
*
ELSE
*
* SIDE ='R', N is even, TRANSR = 'N', UPLO = 'U',
* and TRANS = 'T'
*
CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ALPHA,
$ A( K ), N+1, B( 0, K ), LDB )
CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, K ),
$ LDB, A( 0 ), N+1, ALPHA, B( 0, 0 ),
$ LDB )
CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ONE,
$ A( K+1 ), N+1, B( 0, 0 ), LDB )
*
END IF
*
END IF
*
ELSE
*
* SIDE = 'R', N is even, and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SIDE ='R', N is even, TRANSR = 'T', and UPLO = 'L'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is even, TRANSR = 'T', UPLO = 'L',
* and TRANS = 'N'
*
CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ALPHA,
$ A( 0 ), K, B( 0, K ), LDB )
CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, K ),
$ LDB, A( ( K+1 )*K ), K, ALPHA,
$ B( 0, 0 ), LDB )
CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ONE,
$ A( K ), K, B( 0, 0 ), LDB )
*
ELSE
*
* SIDE ='R', N is even, TRANSR = 'T', UPLO = 'L',
* and TRANS = 'T'
*
CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ALPHA,
$ A( K ), K, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, 0 ),
$ LDB, A( ( K+1 )*K ), K, ALPHA,
$ B( 0, K ), LDB )
CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ONE,
$ A( 0 ), K, B( 0, K ), LDB )
*
END IF
*
ELSE
*
* SIDE ='R', N is even, TRANSR = 'T', and UPLO = 'U'
*
IF( NOTRANS ) THEN
*
* SIDE ='R', N is even, TRANSR = 'T', UPLO = 'U',
* and TRANS = 'N'
*
CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ALPHA,
$ A( ( K+1 )*K ), K, B( 0, 0 ), LDB )
CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, 0 ),
$ LDB, A( 0 ), K, ALPHA, B( 0, K ), LDB )
CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ONE,
$ A( K*K ), K, B( 0, K ), LDB )
*
ELSE
*
* SIDE ='R', N is even, TRANSR = 'T', UPLO = 'U',
* and TRANS = 'T'
*
CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ALPHA,
$ A( K*K ), K, B( 0, K ), LDB )
CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, K ),
$ LDB, A( 0 ), K, ALPHA, B( 0, 0 ), LDB )
CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ONE,
$ A( ( K+1 )*K ), K, B( 0, 0 ), LDB )
*
END IF
*
END IF
*
END IF
*
END IF
END IF
*
RETURN
*
* End of DTFSM
*
END
*> \brief \b DTFTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtftri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtftri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtftri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO, DIAG
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTFTRI computes the inverse of a triangular matrix A stored in RFP
*> format.
*>
*> This is a Level 3 BLAS version of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': The Normal TRANSR of RFP A is stored;
*> = 'T': The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (0:nt-1);
*> nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
*> Positive Definite matrix A in RFP format. RFP format is
*> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
*> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
*> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
*> the transpose of RFP A as defined when
*> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
*> follows: If UPLO = 'U' the RFP A contains the nt elements of
*> upper packed A; If UPLO = 'L' the RFP A contains the nt
*> elements of lower packed A. The LDA of RFP A is (N+1)/2 when
*> TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
*> even and N is odd. See the Note below for more details.
*>
*> On exit, the (triangular) inverse of the original matrix, in
*> the same storage format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, A(i,i) is exactly zero. The triangular
*> matrix is singular and its inverse can not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tftri
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO, DIAG
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( 0: * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, NISODD, NORMALTRANSR
INTEGER N1, N2, K
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DTRMM, DTRTRI
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND.
$ .NOT.LSAME( DIAG, 'U' ) )
$ THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTFTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* If N is odd, set NISODD = .TRUE.
* If N is even, set K = N/2 and NISODD = .FALSE.
*
IF( MOD( N, 2 ).EQ.0 ) THEN
K = N / 2
NISODD = .FALSE.
ELSE
NISODD = .TRUE.
END IF
*
* Set N1 and N2 depending on LOWER
*
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
*
*
* start execution: there are eight cases
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
* T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
* T1 -> a(0), T2 -> a(n), S -> a(n1)
*
CALL DTRTRI( 'L', DIAG, N1, A( 0 ), N, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'L', 'N', DIAG, N2, N1, -ONE, A( 0 ),
$ N, A( N1 ), N )
CALL DTRTRI( 'U', DIAG, N2, A( N ), N, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'U', 'T', DIAG, N2, N1, ONE, A( N ),
$ N,
$ A( N1 ), N )
*
ELSE
*
* SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
* T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
* T1 -> a(n2), T2 -> a(n1), S -> a(0)
*
CALL DTRTRI( 'L', DIAG, N1, A( N2 ), N, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'L', 'T', DIAG, N1, N2, -ONE,
$ A( N2 ),
$ N, A( 0 ), N )
CALL DTRTRI( 'U', DIAG, N2, A( N1 ), N, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'U', 'N', DIAG, N1, N2, ONE, A( N1 ),
$ N, A( 0 ), N )
*
END IF
*
ELSE
*
* N is odd and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE and N is odd
* T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1)
*
CALL DTRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'U', 'N', DIAG, N1, N2, -ONE, A( 0 ),
$ N1, A( N1*N1 ), N1 )
CALL DTRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'L', 'T', DIAG, N1, N2, ONE, A( 1 ),
$ N1, A( N1*N1 ), N1 )
*
ELSE
*
* SRPA for UPPER, TRANSPOSE and N is odd
* T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0)
*
CALL DTRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'U', 'T', DIAG, N2, N1, -ONE,
$ A( N2*N2 ), N2, A( 0 ), N2 )
CALL DTRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + N1
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'L', 'N', DIAG, N2, N1, ONE,
$ A( N1*N2 ), N2, A( 0 ), N2 )
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
* T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
CALL DTRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'L', 'N', DIAG, K, K, -ONE, A( 1 ),
$ N+1, A( K+1 ), N+1 )
CALL DTRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'U', 'T', DIAG, K, K, ONE, A( 0 ),
$ N+1,
$ A( K+1 ), N+1 )
*
ELSE
*
* SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
* T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
CALL DTRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'L', 'T', DIAG, K, K, -ONE, A( K+1 ),
$ N+1, A( 0 ), N+1 )
CALL DTRTRI( 'U', DIAG, K, A( K ), N+1, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'U', 'N', DIAG, K, K, ONE, A( K ),
$ N+1,
$ A( 0 ), N+1 )
END IF
ELSE
*
* N is even and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE and N is even (see paper)
* T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
* T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
CALL DTRTRI( 'U', DIAG, K, A( K ), K, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'U', 'N', DIAG, K, K, -ONE, A( K ),
$ K,
$ A( K*( K+1 ) ), K )
CALL DTRTRI( 'L', DIAG, K, A( 0 ), K, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'L', 'T', DIAG, K, K, ONE, A( 0 ), K,
$ A( K*( K+1 ) ), K )
ELSE
*
* SRPA for UPPER, TRANSPOSE and N is even (see paper)
* T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
* T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
CALL DTRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO )
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'R', 'U', 'T', DIAG, K, K, -ONE,
$ A( K*( K+1 ) ), K, A( 0 ), K )
CALL DTRTRI( 'L', DIAG, K, A( K*K ), K, INFO )
IF( INFO.GT.0 )
$ INFO = INFO + K
IF( INFO.GT.0 )
$ RETURN
CALL DTRMM( 'L', 'L', 'N', DIAG, K, K, ONE, A( K*K ),
$ K,
$ A( 0 ), K )
END IF
END IF
END IF
*
RETURN
*
* End of DTFTRI
*
END
*> \brief \b DTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFTTP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtfttp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtfttp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtfttp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTFTTP( TRANSR, UPLO, N, ARF, AP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( 0: * ), ARF( 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTFTTP copies a triangular matrix A from rectangular full packed
*> format (TF) to standard packed format (TP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': ARF is in Normal format;
*> = 'T': ARF is in Transpose format;
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ARF
*> \verbatim
*> ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*> On entry, the upper or lower triangular matrix A stored in
*> RFP format. For a further discussion see Notes below.
*> \endverbatim
*>
*> \param[out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*> On exit, the upper or lower triangular matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tfttp
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTFTTP( TRANSR, UPLO, N, ARF, AP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( 0: * ), ARF( 0: * )
* ..
*
* =====================================================================
*
* .. Parameters ..
* ..
* .. Local Scalars ..
LOGICAL LOWER, NISODD, NORMALTRANSR
INTEGER N1, N2, K, NT
INTEGER I, J, IJ
INTEGER IJP, JP, LDA, JS
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTFTTP', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( NORMALTRANSR ) THEN
AP( 0 ) = ARF( 0 )
ELSE
AP( 0 ) = ARF( 0 )
END IF
RETURN
END IF
*
* Size of array ARF(0:NT-1)
*
NT = N*( N+1 ) / 2
*
* Set N1 and N2 depending on LOWER
*
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
*
* If N is odd, set NISODD = .TRUE.
* If N is even, set K = N/2 and NISODD = .FALSE.
*
* set lda of ARF^C; ARF^C is (0:(N+1)/2-1,0:N-noe)
* where noe = 0 if n is even, noe = 1 if n is odd
*
IF( MOD( N, 2 ).EQ.0 ) THEN
K = N / 2
NISODD = .FALSE.
LDA = N + 1
ELSE
NISODD = .TRUE.
LDA = N
END IF
*
* ARF^C has lda rows and n+1-noe cols
*
IF( .NOT.NORMALTRANSR )
$ LDA = ( N+1 ) / 2
*
* start execution: there are eight cases
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
* T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
* T1 -> a(0), T2 -> a(n), S -> a(n1); lda = n
*
IJP = 0
JP = 0
DO J = 0, N2
DO I = J, N - 1
IJ = I + JP
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JP = JP + LDA
END DO
DO I = 0, N2 - 1
DO J = 1 + I, N2
IJ = I + J*LDA
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
END DO
*
ELSE
*
* SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
* T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
* T1 -> a(n2), T2 -> a(n1), S -> a(0)
*
IJP = 0
DO J = 0, N1 - 1
IJ = N2 + J
DO I = 0, J
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
IJ = IJ + LDA
END DO
END DO
JS = 0
DO J = N1, N - 1
IJ = JS
DO IJ = JS, JS + J
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
*
END IF
*
ELSE
*
* N is odd and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE and N is odd
* T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
* T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
*
IJP = 0
DO I = 0, N2
DO IJ = I*( LDA+1 ), N*LDA - 1, LDA
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
END DO
JS = 1
DO J = 0, N2 - 1
DO IJ = JS, JS + N2 - J - 1
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JS = JS + LDA + 1
END DO
*
ELSE
*
* SRPA for UPPER, TRANSPOSE and N is odd
* T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
* T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
*
IJP = 0
JS = N2*LDA
DO J = 0, N1 - 1
DO IJ = JS, JS + J
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
DO I = 0, N1
DO IJ = I, I + ( N1+I )*LDA, LDA
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
END DO
*
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
* T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
IJP = 0
JP = 0
DO J = 0, K - 1
DO I = J, N - 1
IJ = 1 + I + JP
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JP = JP + LDA
END DO
DO I = 0, K - 1
DO J = I, K - 1
IJ = I + J*LDA
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
END DO
*
ELSE
*
* SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
* T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
* T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
IJP = 0
DO J = 0, K - 1
IJ = K + 1 + J
DO I = 0, J
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
IJ = IJ + LDA
END DO
END DO
JS = 0
DO J = K, N - 1
IJ = JS
DO IJ = JS, JS + J
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
*
END IF
*
ELSE
*
* N is even and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* SRPA for LOWER, TRANSPOSE and N is even (see paper)
* T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
* T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
IJP = 0
DO I = 0, K - 1
DO IJ = I + ( I+1 )*LDA, ( N+1 )*LDA - 1, LDA
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
END DO
JS = 0
DO J = 0, K - 1
DO IJ = JS, JS + K - J - 1
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JS = JS + LDA + 1
END DO
*
ELSE
*
* SRPA for UPPER, TRANSPOSE and N is even (see paper)
* T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
* T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
IJP = 0
JS = ( K+1 )*LDA
DO J = 0, K - 1
DO IJ = JS, JS + J
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
DO I = 0, K - 1
DO IJ = I, I + ( K+I )*LDA, LDA
AP( IJP ) = ARF( IJ )
IJP = IJP + 1
END DO
END DO
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of DTFTTP
*
END
*> \brief \b DTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFTTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtfttr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtfttr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtfttr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTFTTR( TRANSR, UPLO, N, ARF, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: LDA-1, 0: * ), ARF( 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTFTTR copies a triangular matrix A from rectangular full packed
*> format (TF) to standard full format (TR).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': ARF is in Normal format;
*> = 'T': ARF is in Transpose format.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices ARF and A. N >= 0.
*> \endverbatim
*>
*> \param[in] ARF
*> \verbatim
*> ARF is DOUBLE PRECISION array, dimension (N*(N+1)/2).
*> On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
*> matrix A in RFP format. See the "Notes" below for more
*> details.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On exit, the triangular matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of the array A contains
*> the upper triangular matrix, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of the array A contains
*> the lower triangular matrix, and the strictly upper
*> triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tfttr
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*
* =====================================================================
SUBROUTINE DTFTTR( TRANSR, UPLO, N, ARF, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO
INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( 0: LDA-1, 0: * ), ARF( 0: * )
* ..
*
* =====================================================================
*
* ..
* .. Local Scalars ..
LOGICAL LOWER, NISODD, NORMALTRANSR
INTEGER N1, N2, K, NT, NX2, NP1X2
INTEGER I, J, L, IJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTFTTR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 ) THEN
IF( N.EQ.1 ) THEN
A( 0, 0 ) = ARF( 0 )
END IF
RETURN
END IF
*
* Size of array ARF(0:nt-1)
*
NT = N*( N+1 ) / 2
*
* set N1 and N2 depending on LOWER: for N even N1=N2=K
*
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
*
* If N is odd, set NISODD = .TRUE., LDA=N+1 and A is (N+1)--by--K2.
* If N is even, set K = N/2 and NISODD = .FALSE., LDA=N and A is
* N--by--(N+1)/2.
*
IF( MOD( N, 2 ).EQ.0 ) THEN
K = N / 2
NISODD = .FALSE.
IF( .NOT.LOWER )
$ NP1X2 = N + N + 2
ELSE
NISODD = .TRUE.
IF( .NOT.LOWER )
$ NX2 = N + N
END IF
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'N', and UPLO = 'L'
*
IJ = 0
DO J = 0, N2
DO I = N1, N2 + J
A( N2+J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
DO I = J, N - 1
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is odd, TRANSR = 'N', and UPLO = 'U'
*
IJ = NT - N
DO J = N - 1, N1, -1
DO I = 0, J
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
DO L = J - N1, N1 - 1
A( J-N1, L ) = ARF( IJ )
IJ = IJ + 1
END DO
IJ = IJ - NX2
END DO
*
END IF
*
ELSE
*
* N is odd and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'T', and UPLO = 'L'
*
IJ = 0
DO J = 0, N2 - 1
DO I = 0, J
A( J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
DO I = N1 + J, N - 1
A( I, N1+J ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
DO J = N2, N - 1
DO I = 0, N1 - 1
A( J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is odd, TRANSR = 'T', and UPLO = 'U'
*
IJ = 0
DO J = 0, N1
DO I = N1, N - 1
A( J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
DO J = 0, N1 - 1
DO I = 0, J
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
DO L = N2 + J, N - 1
A( N2+J, L ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
*
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'N', and UPLO = 'L'
*
IJ = 0
DO J = 0, K - 1
DO I = K, K + J
A( K+J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
DO I = J, N - 1
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is even, TRANSR = 'N', and UPLO = 'U'
*
IJ = NT - N - 1
DO J = N - 1, K, -1
DO I = 0, J
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
DO L = J - K, K - 1
A( J-K, L ) = ARF( IJ )
IJ = IJ + 1
END DO
IJ = IJ - NP1X2
END DO
*
END IF
*
ELSE
*
* N is even and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'T', and UPLO = 'L'
*
IJ = 0
J = K
DO I = K, N - 1
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
DO J = 0, K - 2
DO I = 0, J
A( J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
DO I = K + 1 + J, N - 1
A( I, K+1+J ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
DO J = K - 1, N - 1
DO I = 0, K - 1
A( J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is even, TRANSR = 'T', and UPLO = 'U'
*
IJ = 0
DO J = 0, K
DO I = K, N - 1
A( J, I ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
DO J = 0, K - 2
DO I = 0, J
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
DO L = K + 1 + J, N - 1
A( K+1+J, L ) = ARF( IJ )
IJ = IJ + 1
END DO
END DO
* Note that here, on exit of the loop, J = K-1
DO I = 0, J
A( I, J ) = ARF( IJ )
IJ = IJ + 1
END DO
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of DTFTTR
*
END
*> \brief \b DTGEVC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGEVC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgevc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgevc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgevc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
* LDVL, VR, LDVR, MM, M, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, SIDE
* INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* DOUBLE PRECISION P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGEVC computes some or all of the right and/or left eigenvectors of
*> a pair of real matrices (S,P), where S is a quasi-triangular matrix
*> and P is upper triangular. Matrix pairs of this type are produced by
*> the generalized Schur factorization of a matrix pair (A,B):
*>
*> A = Q*S*Z**T, B = Q*P*Z**T
*>
*> as computed by DGGHRD + DHGEQZ.
*>
*> The right eigenvector x and the left eigenvector y of (S,P)
*> corresponding to an eigenvalue w are defined by:
*>
*> S*x = w*P*x, (y**H)*S = w*(y**H)*P,
*>
*> where y**H denotes the conjugate transpose of y.
*> The eigenvalues are not input to this routine, but are computed
*> directly from the diagonal blocks of S and P.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of (S,P), or the products Z*X and/or Q*Y,
*> where Z and Q are input matrices.
*> If Q and Z are the orthogonal factors from the generalized Schur
*> factorization of a matrix pair (A,B), then Z*X and Q*Y
*> are the matrices of right and left eigenvectors of (A,B).
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': compute right eigenvectors only;
*> = 'L': compute left eigenvectors only;
*> = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute all right and/or left eigenvectors;
*> = 'B': compute all right and/or left eigenvectors,
*> backtransformed by the matrices in VR and/or VL;
*> = 'S': compute selected right and/or left eigenvectors,
*> specified by the logical array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY='S', SELECT specifies the eigenvectors to be
*> computed. If w(j) is a real eigenvalue, the corresponding
*> real eigenvector is computed if SELECT(j) is .TRUE..
*> If w(j) and w(j+1) are the real and imaginary parts of a
*> complex eigenvalue, the corresponding complex eigenvector
*> is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
*> and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
*> set to .FALSE..
*> Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices S and P. N >= 0.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (LDS,N)
*> The upper quasi-triangular matrix S from a generalized Schur
*> factorization, as computed by DHGEQZ.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*> LDS is INTEGER
*> The leading dimension of array S. LDS >= max(1,N).
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is DOUBLE PRECISION array, dimension (LDP,N)
*> The upper triangular matrix P from a generalized Schur
*> factorization, as computed by DHGEQZ.
*> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
*> of S must be in positive diagonal form.
*> \endverbatim
*>
*> \param[in] LDP
*> \verbatim
*> LDP is INTEGER
*> The leading dimension of array P. LDP >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*> contain an N-by-N matrix Q (usually the orthogonal matrix Q
*> of left Schur vectors returned by DHGEQZ).
*> On exit, if SIDE = 'L' or 'B', VL contains:
*> if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
*> if HOWMNY = 'B', the matrix Q*Y;
*> if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
*> SELECT, stored consecutively in the columns of
*> VL, in the same order as their eigenvalues.
*>
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part, and the second the imaginary part.
*>
*> Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of array VL. LDVL >= 1, and if
*> SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*> contain an N-by-N matrix Z (usually the orthogonal matrix Z
*> of right Schur vectors returned by DHGEQZ).
*>
*> On exit, if SIDE = 'R' or 'B', VR contains:
*> if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
*> if HOWMNY = 'B' or 'b', the matrix Z*X;
*> if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
*> specified by SELECT, stored consecutively in the
*> columns of VR, in the same order as their
*> eigenvalues.
*>
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part and the second the imaginary part.
*>
*> Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1, and if
*> SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of columns in the arrays VL and/or VR actually
*> used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
*> is set to N. Each selected real eigenvector occupies one
*> column and each selected complex eigenvector occupies two
*> columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (6*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
*> eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgevc
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Allocation of workspace:
*> ---------- -- ---------
*>
*> WORK( j ) = 1-norm of j-th column of A, above the diagonal
*> WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
*> WORK( 2*N+1:3*N ) = real part of eigenvector
*> WORK( 3*N+1:4*N ) = imaginary part of eigenvector
*> WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
*> WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
*>
*> Rowwise vs. columnwise solution methods:
*> ------- -- ---------- -------- -------
*>
*> Finding a generalized eigenvector consists basically of solving the
*> singular triangular system
*>
*> (A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
*>
*> Consider finding the i-th right eigenvector (assume all eigenvalues
*> are real). The equation to be solved is:
*> n i
*> 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
*> k=j k=j
*>
*> where C = (A - w B) (The components v(i+1:n) are 0.)
*>
*> The "rowwise" method is:
*>
*> (1) v(i) := 1
*> for j = i-1,. . .,1:
*> i
*> (2) compute s = - sum C(j,k) v(k) and
*> k=j+1
*>
*> (3) v(j) := s / C(j,j)
*>
*> Step 2 is sometimes called the "dot product" step, since it is an
*> inner product between the j-th row and the portion of the eigenvector
*> that has been computed so far.
*>
*> The "columnwise" method consists basically in doing the sums
*> for all the rows in parallel. As each v(j) is computed, the
*> contribution of v(j) times the j-th column of C is added to the
*> partial sums. Since FORTRAN arrays are stored columnwise, this has
*> the advantage that at each step, the elements of C that are accessed
*> are adjacent to one another, whereas with the rowwise method, the
*> elements accessed at a step are spaced LDS (and LDP) words apart.
*>
*> When finding left eigenvectors, the matrix in question is the
*> transpose of the one in storage, so the rowwise method then
*> actually accesses columns of A and B at each step, and so is the
*> preferred method.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
$ LDVL, VR, LDVR, MM, M, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
DOUBLE PRECISION P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, SAFETY
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0,
$ SAFETY = 1.0D+2 )
* ..
* .. Local Scalars ..
LOGICAL COMPL, COMPR, IL2BY2, ILABAD, ILALL, ILBACK,
$ ILBBAD, ILCOMP, ILCPLX, LSA, LSB
INTEGER I, IBEG, IEIG, IEND, IHWMNY, IINFO, IM, ISIDE,
$ J, JA, JC, JE, JR, JW, NA, NW
DOUBLE PRECISION ACOEF, ACOEFA, ANORM, ASCALE, BCOEFA, BCOEFI,
$ BCOEFR, BIG, BIGNUM, BNORM, BSCALE, CIM2A,
$ CIM2B, CIMAGA, CIMAGB, CRE2A, CRE2B, CREALA,
$ CREALB, DMIN, SAFMIN, SALFAR, SBETA, SCALE,
$ SMALL, TEMP, TEMP2, TEMP2I, TEMP2R, ULP, XMAX,
$ XSCALE
* ..
* .. Local Arrays ..
DOUBLE PRECISION BDIAG( 2 ), SUM( 2, 2 ), SUMS( 2, 2 ),
$ SUMP( 2, 2 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DLACPY, DLAG2, DLALN2,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Decode and Test the input parameters
*
IF( LSAME( HOWMNY, 'A' ) ) THEN
IHWMNY = 1
ILALL = .TRUE.
ILBACK = .FALSE.
ELSE IF( LSAME( HOWMNY, 'S' ) ) THEN
IHWMNY = 2
ILALL = .FALSE.
ILBACK = .FALSE.
ELSE IF( LSAME( HOWMNY, 'B' ) ) THEN
IHWMNY = 3
ILALL = .TRUE.
ILBACK = .TRUE.
ELSE
IHWMNY = -1
ILALL = .TRUE.
END IF
*
IF( LSAME( SIDE, 'R' ) ) THEN
ISIDE = 1
COMPL = .FALSE.
COMPR = .TRUE.
ELSE IF( LSAME( SIDE, 'L' ) ) THEN
ISIDE = 2
COMPL = .TRUE.
COMPR = .FALSE.
ELSE IF( LSAME( SIDE, 'B' ) ) THEN
ISIDE = 3
COMPL = .TRUE.
COMPR = .TRUE.
ELSE
ISIDE = -1
END IF
*
INFO = 0
IF( ISIDE.LT.0 ) THEN
INFO = -1
ELSE IF( IHWMNY.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDS.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDP.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGEVC', -INFO )
RETURN
END IF
*
* Count the number of eigenvectors to be computed
*
IF( .NOT.ILALL ) THEN
IM = 0
ILCPLX = .FALSE.
DO 10 J = 1, N
IF( ILCPLX ) THEN
ILCPLX = .FALSE.
GO TO 10
END IF
IF( J.LT.N ) THEN
IF( S( J+1, J ).NE.ZERO )
$ ILCPLX = .TRUE.
END IF
IF( ILCPLX ) THEN
IF( SELECT( J ) .OR. SELECT( J+1 ) )
$ IM = IM + 2
ELSE
IF( SELECT( J ) )
$ IM = IM + 1
END IF
10 CONTINUE
ELSE
IM = N
END IF
*
* Check 2-by-2 diagonal blocks of A, B
*
ILABAD = .FALSE.
ILBBAD = .FALSE.
DO 20 J = 1, N - 1
IF( S( J+1, J ).NE.ZERO ) THEN
IF( P( J, J ).EQ.ZERO .OR. P( J+1, J+1 ).EQ.ZERO .OR.
$ P( J, J+1 ).NE.ZERO )ILBBAD = .TRUE.
IF( J.LT.N-1 ) THEN
IF( S( J+2, J+1 ).NE.ZERO )
$ ILABAD = .TRUE.
END IF
END IF
20 CONTINUE
*
IF( ILABAD ) THEN
INFO = -5
ELSE IF( ILBBAD ) THEN
INFO = -7
ELSE IF( COMPL .AND. LDVL.LT.N .OR. LDVL.LT.1 ) THEN
INFO = -10
ELSE IF( COMPR .AND. LDVR.LT.N .OR. LDVR.LT.1 ) THEN
INFO = -12
ELSE IF( MM.LT.IM ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGEVC', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = IM
IF( N.EQ.0 )
$ RETURN
*
* Machine Constants
*
SAFMIN = DLAMCH( 'Safe minimum' )
BIG = ONE / SAFMIN
ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
SMALL = SAFMIN*N / ULP
BIG = ONE / SMALL
BIGNUM = ONE / ( SAFMIN*N )
*
* Compute the 1-norm of each column of the strictly upper triangular
* part (i.e., excluding all elements belonging to the diagonal
* blocks) of A and B to check for possible overflow in the
* triangular solver.
*
ANORM = ABS( S( 1, 1 ) )
IF( N.GT.1 )
$ ANORM = ANORM + ABS( S( 2, 1 ) )
BNORM = ABS( P( 1, 1 ) )
WORK( 1 ) = ZERO
WORK( N+1 ) = ZERO
*
DO 50 J = 2, N
TEMP = ZERO
TEMP2 = ZERO
IF( S( J, J-1 ).EQ.ZERO ) THEN
IEND = J - 1
ELSE
IEND = J - 2
END IF
DO 30 I = 1, IEND
TEMP = TEMP + ABS( S( I, J ) )
TEMP2 = TEMP2 + ABS( P( I, J ) )
30 CONTINUE
WORK( J ) = TEMP
WORK( N+J ) = TEMP2
DO 40 I = IEND + 1, MIN( J+1, N )
TEMP = TEMP + ABS( S( I, J ) )
TEMP2 = TEMP2 + ABS( P( I, J ) )
40 CONTINUE
ANORM = MAX( ANORM, TEMP )
BNORM = MAX( BNORM, TEMP2 )
50 CONTINUE
*
ASCALE = ONE / MAX( ANORM, SAFMIN )
BSCALE = ONE / MAX( BNORM, SAFMIN )
*
* Left eigenvectors
*
IF( COMPL ) THEN
IEIG = 0
*
* Main loop over eigenvalues
*
ILCPLX = .FALSE.
DO 220 JE = 1, N
*
* Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
* (b) this would be the second of a complex pair.
* Check for complex eigenvalue, so as to be sure of which
* entry(-ies) of SELECT to look at.
*
IF( ILCPLX ) THEN
ILCPLX = .FALSE.
GO TO 220
END IF
NW = 1
IF( JE.LT.N ) THEN
IF( S( JE+1, JE ).NE.ZERO ) THEN
ILCPLX = .TRUE.
NW = 2
END IF
END IF
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE IF( ILCPLX ) THEN
ILCOMP = SELECT( JE ) .OR. SELECT( JE+1 )
ELSE
ILCOMP = SELECT( JE )
END IF
IF( .NOT.ILCOMP )
$ GO TO 220
*
* Decide if (a) singular pencil, (b) real eigenvalue, or
* (c) complex eigenvalue.
*
IF( .NOT.ILCPLX ) THEN
IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- return unit eigenvector
*
IEIG = IEIG + 1
DO 60 JR = 1, N
VL( JR, IEIG ) = ZERO
60 CONTINUE
VL( IEIG, IEIG ) = ONE
GO TO 220
END IF
END IF
*
* Clear vector
*
DO 70 JR = 1, NW*N
WORK( 2*N+JR ) = ZERO
70 CONTINUE
* T
* Compute coefficients in ( a A - b B ) y = 0
* a is ACOEF
* b is BCOEFR + i*BCOEFI
*
IF( .NOT.ILCPLX ) THEN
*
* Real eigenvalue
*
TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
$ ABS( P( JE, JE ) )*BSCALE, SAFMIN )
SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*P( JE, JE ) )*BSCALE
ACOEF = SBETA*ASCALE
BCOEFR = SALFAR*BSCALE
BCOEFI = ZERO
*
* Scale to avoid underflow
*
SCALE = ONE
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
$ SMALL
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEF ),
$ ABS( BCOEFR ) ) ) )
IF( LSA ) THEN
ACOEF = ASCALE*( SCALE*SBETA )
ELSE
ACOEF = SCALE*ACOEF
END IF
IF( LSB ) THEN
BCOEFR = BSCALE*( SCALE*SALFAR )
ELSE
BCOEFR = SCALE*BCOEFR
END IF
END IF
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR )
*
* First component is 1
*
WORK( 2*N+JE ) = ONE
XMAX = ONE
ELSE
*
* Complex eigenvalue
*
CALL DLAG2( S( JE, JE ), LDS, P( JE, JE ), LDP,
$ SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
$ BCOEFI )
BCOEFI = -BCOEFI
IF( BCOEFI.EQ.ZERO ) THEN
INFO = JE
RETURN
END IF
*
* Scale to avoid over/underflow
*
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
SCALE = ONE
IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
$ SCALE = ( SAFMIN / ULP ) / ACOEFA
IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
$ SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
IF( SAFMIN*ACOEFA.GT.ASCALE )
$ SCALE = ASCALE / ( SAFMIN*ACOEFA )
IF( SAFMIN*BCOEFA.GT.BSCALE )
$ SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
IF( SCALE.NE.ONE ) THEN
ACOEF = SCALE*ACOEF
ACOEFA = ABS( ACOEF )
BCOEFR = SCALE*BCOEFR
BCOEFI = SCALE*BCOEFI
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
END IF
*
* Compute first two components of eigenvector
*
TEMP = ACOEF*S( JE+1, JE )
TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
TEMP2I = -BCOEFI*P( JE, JE )
IF( ABS( TEMP ).GT.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
WORK( 2*N+JE ) = ONE
WORK( 3*N+JE ) = ZERO
WORK( 2*N+JE+1 ) = -TEMP2R / TEMP
WORK( 3*N+JE+1 ) = -TEMP2I / TEMP
ELSE
WORK( 2*N+JE+1 ) = ONE
WORK( 3*N+JE+1 ) = ZERO
TEMP = ACOEF*S( JE, JE+1 )
WORK( 2*N+JE ) = ( BCOEFR*P( JE+1, JE+1 )-ACOEF*
$ S( JE+1, JE+1 ) ) / TEMP
WORK( 3*N+JE ) = BCOEFI*P( JE+1, JE+1 ) / TEMP
END IF
XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
$ ABS( WORK( 2*N+JE+1 ) )+ABS( WORK( 3*N+JE+1 ) ) )
END IF
*
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* T
* Triangular solve of (a A - b B) y = 0
*
* T
* (rowwise in (a A - b B) , or columnwise in (a A - b B) )
*
IL2BY2 = .FALSE.
*
DO 160 J = JE + NW, N
IF( IL2BY2 ) THEN
IL2BY2 = .FALSE.
GO TO 160
END IF
*
NA = 1
BDIAG( 1 ) = P( J, J )
IF( J.LT.N ) THEN
IF( S( J+1, J ).NE.ZERO ) THEN
IL2BY2 = .TRUE.
BDIAG( 2 ) = P( J+1, J+1 )
NA = 2
END IF
END IF
*
* Check whether scaling is necessary for dot products
*
XSCALE = ONE / MAX( ONE, XMAX )
TEMP = MAX( WORK( J ), WORK( N+J ),
$ ACOEFA*WORK( J )+BCOEFA*WORK( N+J ) )
IF( IL2BY2 )
$ TEMP = MAX( TEMP, WORK( J+1 ), WORK( N+J+1 ),
$ ACOEFA*WORK( J+1 )+BCOEFA*WORK( N+J+1 ) )
IF( TEMP.GT.BIGNUM*XSCALE ) THEN
DO 90 JW = 0, NW - 1
DO 80 JR = JE, J - 1
WORK( ( JW+2 )*N+JR ) = XSCALE*
$ WORK( ( JW+2 )*N+JR )
80 CONTINUE
90 CONTINUE
XMAX = XMAX*XSCALE
END IF
*
* Compute dot products
*
* j-1
* SUM = sum conjg( a*S(k,j) - b*P(k,j) )*x(k)
* k=je
*
* To reduce the op count, this is done as
*
* _ j-1 _ j-1
* a*conjg( sum S(k,j)*x(k) ) - b*conjg( sum P(k,j)*x(k) )
* k=je k=je
*
* which may cause underflow problems if A or B are close
* to underflow. (E.g., less than SMALL.)
*
*
DO 120 JW = 1, NW
DO 110 JA = 1, NA
SUMS( JA, JW ) = ZERO
SUMP( JA, JW ) = ZERO
*
DO 100 JR = JE, J - 1
SUMS( JA, JW ) = SUMS( JA, JW ) +
$ S( JR, J+JA-1 )*
$ WORK( ( JW+1 )*N+JR )
SUMP( JA, JW ) = SUMP( JA, JW ) +
$ P( JR, J+JA-1 )*
$ WORK( ( JW+1 )*N+JR )
100 CONTINUE
110 CONTINUE
120 CONTINUE
*
DO 130 JA = 1, NA
IF( ILCPLX ) THEN
SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
$ BCOEFR*SUMP( JA, 1 ) -
$ BCOEFI*SUMP( JA, 2 )
SUM( JA, 2 ) = -ACOEF*SUMS( JA, 2 ) +
$ BCOEFR*SUMP( JA, 2 ) +
$ BCOEFI*SUMP( JA, 1 )
ELSE
SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
$ BCOEFR*SUMP( JA, 1 )
END IF
130 CONTINUE
*
* T
* Solve ( a A - b B ) y = SUM(,)
* with scaling and perturbation of the denominator
*
CALL DLALN2( .TRUE., NA, NW, DMIN, ACOEF, S( J, J ),
$ LDS,
$ BDIAG( 1 ), BDIAG( 2 ), SUM, 2, BCOEFR,
$ BCOEFI, WORK( 2*N+J ), N, SCALE, TEMP,
$ IINFO )
IF( SCALE.LT.ONE ) THEN
DO 150 JW = 0, NW - 1
DO 140 JR = JE, J - 1
WORK( ( JW+2 )*N+JR ) = SCALE*
$ WORK( ( JW+2 )*N+JR )
140 CONTINUE
150 CONTINUE
XMAX = SCALE*XMAX
END IF
XMAX = MAX( XMAX, TEMP )
160 CONTINUE
*
* Copy eigenvector to VL, back transforming if
* HOWMNY='B'.
*
IEIG = IEIG + 1
IF( ILBACK ) THEN
DO 170 JW = 0, NW - 1
CALL DGEMV( 'N', N, N+1-JE, ONE, VL( 1, JE ), LDVL,
$ WORK( ( JW+2 )*N+JE ), 1, ZERO,
$ WORK( ( JW+4 )*N+1 ), 1 )
170 CONTINUE
CALL DLACPY( ' ', N, NW, WORK( 4*N+1 ), N, VL( 1,
$ JE ),
$ LDVL )
IBEG = 1
ELSE
CALL DLACPY( ' ', N, NW, WORK( 2*N+1 ), N, VL( 1,
$ IEIG ),
$ LDVL )
IBEG = JE
END IF
*
* Scale eigenvector
*
XMAX = ZERO
IF( ILCPLX ) THEN
DO 180 J = IBEG, N
XMAX = MAX( XMAX, ABS( VL( J, IEIG ) )+
$ ABS( VL( J, IEIG+1 ) ) )
180 CONTINUE
ELSE
DO 190 J = IBEG, N
XMAX = MAX( XMAX, ABS( VL( J, IEIG ) ) )
190 CONTINUE
END IF
*
IF( XMAX.GT.SAFMIN ) THEN
XSCALE = ONE / XMAX
*
DO 210 JW = 0, NW - 1
DO 200 JR = IBEG, N
VL( JR, IEIG+JW ) = XSCALE*VL( JR, IEIG+JW )
200 CONTINUE
210 CONTINUE
END IF
IEIG = IEIG + NW - 1
*
220 CONTINUE
END IF
*
* Right eigenvectors
*
IF( COMPR ) THEN
IEIG = IM + 1
*
* Main loop over eigenvalues
*
ILCPLX = .FALSE.
DO 500 JE = N, 1, -1
*
* Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
* (b) this would be the second of a complex pair.
* Check for complex eigenvalue, so as to be sure of which
* entry(-ies) of SELECT to look at -- if complex, SELECT(JE)
* or SELECT(JE-1).
* If this is a complex pair, the 2-by-2 diagonal block
* corresponding to the eigenvalue is in rows/columns JE-1:JE
*
IF( ILCPLX ) THEN
ILCPLX = .FALSE.
GO TO 500
END IF
NW = 1
IF( JE.GT.1 ) THEN
IF( S( JE, JE-1 ).NE.ZERO ) THEN
ILCPLX = .TRUE.
NW = 2
END IF
END IF
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE IF( ILCPLX ) THEN
ILCOMP = SELECT( JE ) .OR. SELECT( JE-1 )
ELSE
ILCOMP = SELECT( JE )
END IF
IF( .NOT.ILCOMP )
$ GO TO 500
*
* Decide if (a) singular pencil, (b) real eigenvalue, or
* (c) complex eigenvalue.
*
IF( .NOT.ILCPLX ) THEN
IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- unit eigenvector
*
IEIG = IEIG - 1
DO 230 JR = 1, N
VR( JR, IEIG ) = ZERO
230 CONTINUE
VR( IEIG, IEIG ) = ONE
GO TO 500
END IF
END IF
*
* Clear vector
*
DO 250 JW = 0, NW - 1
DO 240 JR = 1, N
WORK( ( JW+2 )*N+JR ) = ZERO
240 CONTINUE
250 CONTINUE
*
* Compute coefficients in ( a A - b B ) x = 0
* a is ACOEF
* b is BCOEFR + i*BCOEFI
*
IF( .NOT.ILCPLX ) THEN
*
* Real eigenvalue
*
TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
$ ABS( P( JE, JE ) )*BSCALE, SAFMIN )
SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*P( JE, JE ) )*BSCALE
ACOEF = SBETA*ASCALE
BCOEFR = SALFAR*BSCALE
BCOEFI = ZERO
*
* Scale to avoid underflow
*
SCALE = ONE
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
$ SMALL
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEF ),
$ ABS( BCOEFR ) ) ) )
IF( LSA ) THEN
ACOEF = ASCALE*( SCALE*SBETA )
ELSE
ACOEF = SCALE*ACOEF
END IF
IF( LSB ) THEN
BCOEFR = BSCALE*( SCALE*SALFAR )
ELSE
BCOEFR = SCALE*BCOEFR
END IF
END IF
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR )
*
* First component is 1
*
WORK( 2*N+JE ) = ONE
XMAX = ONE
*
* Compute contribution from column JE of A and B to sum
* (See "Further Details", above.)
*
DO 260 JR = 1, JE - 1
WORK( 2*N+JR ) = BCOEFR*P( JR, JE ) -
$ ACOEF*S( JR, JE )
260 CONTINUE
ELSE
*
* Complex eigenvalue
*
CALL DLAG2( S( JE-1, JE-1 ), LDS, P( JE-1, JE-1 ),
$ LDP,
$ SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
$ BCOEFI )
IF( BCOEFI.EQ.ZERO ) THEN
INFO = JE - 1
RETURN
END IF
*
* Scale to avoid over/underflow
*
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
SCALE = ONE
IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
$ SCALE = ( SAFMIN / ULP ) / ACOEFA
IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
$ SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
IF( SAFMIN*ACOEFA.GT.ASCALE )
$ SCALE = ASCALE / ( SAFMIN*ACOEFA )
IF( SAFMIN*BCOEFA.GT.BSCALE )
$ SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
IF( SCALE.NE.ONE ) THEN
ACOEF = SCALE*ACOEF
ACOEFA = ABS( ACOEF )
BCOEFR = SCALE*BCOEFR
BCOEFI = SCALE*BCOEFI
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
END IF
*
* Compute first two components of eigenvector
* and contribution to sums
*
TEMP = ACOEF*S( JE, JE-1 )
TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
TEMP2I = -BCOEFI*P( JE, JE )
IF( ABS( TEMP ).GE.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
WORK( 2*N+JE ) = ONE
WORK( 3*N+JE ) = ZERO
WORK( 2*N+JE-1 ) = -TEMP2R / TEMP
WORK( 3*N+JE-1 ) = -TEMP2I / TEMP
ELSE
WORK( 2*N+JE-1 ) = ONE
WORK( 3*N+JE-1 ) = ZERO
TEMP = ACOEF*S( JE-1, JE )
WORK( 2*N+JE ) = ( BCOEFR*P( JE-1, JE-1 )-ACOEF*
$ S( JE-1, JE-1 ) ) / TEMP
WORK( 3*N+JE ) = BCOEFI*P( JE-1, JE-1 ) / TEMP
END IF
*
XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
$ ABS( WORK( 2*N+JE-1 ) )+ABS( WORK( 3*N+JE-1 ) ) )
*
* Compute contribution from columns JE and JE-1
* of A and B to the sums.
*
CREALA = ACOEF*WORK( 2*N+JE-1 )
CIMAGA = ACOEF*WORK( 3*N+JE-1 )
CREALB = BCOEFR*WORK( 2*N+JE-1 ) -
$ BCOEFI*WORK( 3*N+JE-1 )
CIMAGB = BCOEFI*WORK( 2*N+JE-1 ) +
$ BCOEFR*WORK( 3*N+JE-1 )
CRE2A = ACOEF*WORK( 2*N+JE )
CIM2A = ACOEF*WORK( 3*N+JE )
CRE2B = BCOEFR*WORK( 2*N+JE ) - BCOEFI*WORK( 3*N+JE )
CIM2B = BCOEFI*WORK( 2*N+JE ) + BCOEFR*WORK( 3*N+JE )
DO 270 JR = 1, JE - 2
WORK( 2*N+JR ) = -CREALA*S( JR, JE-1 ) +
$ CREALB*P( JR, JE-1 ) -
$ CRE2A*S( JR, JE ) + CRE2B*P( JR, JE )
WORK( 3*N+JR ) = -CIMAGA*S( JR, JE-1 ) +
$ CIMAGB*P( JR, JE-1 ) -
$ CIM2A*S( JR, JE ) + CIM2B*P( JR, JE )
270 CONTINUE
END IF
*
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* Columnwise triangular solve of (a A - b B) x = 0
*
IL2BY2 = .FALSE.
DO 370 J = JE - NW, 1, -1
*
* If a 2-by-2 block, is in position j-1:j, wait until
* next iteration to process it (when it will be j:j+1)
*
IF( .NOT.IL2BY2 .AND. J.GT.1 ) THEN
IF( S( J, J-1 ).NE.ZERO ) THEN
IL2BY2 = .TRUE.
GO TO 370
END IF
END IF
BDIAG( 1 ) = P( J, J )
IF( IL2BY2 ) THEN
NA = 2
BDIAG( 2 ) = P( J+1, J+1 )
ELSE
NA = 1
END IF
*
* Compute x(j) (and x(j+1), if 2-by-2 block)
*
CALL DLALN2( .FALSE., NA, NW, DMIN, ACOEF, S( J, J ),
$ LDS, BDIAG( 1 ), BDIAG( 2 ), WORK( 2*N+J ),
$ N, BCOEFR, BCOEFI, SUM, 2, SCALE, TEMP,
$ IINFO )
IF( SCALE.LT.ONE ) THEN
*
DO 290 JW = 0, NW - 1
DO 280 JR = 1, JE
WORK( ( JW+2 )*N+JR ) = SCALE*
$ WORK( ( JW+2 )*N+JR )
280 CONTINUE
290 CONTINUE
END IF
XMAX = MAX( SCALE*XMAX, TEMP )
*
DO 310 JW = 1, NW
DO 300 JA = 1, NA
WORK( ( JW+1 )*N+J+JA-1 ) = SUM( JA, JW )
300 CONTINUE
310 CONTINUE
*
* w = w + x(j)*(a S(*,j) - b P(*,j) ) with scaling
*
IF( J.GT.1 ) THEN
*
* Check whether scaling is necessary for sum.
*
XSCALE = ONE / MAX( ONE, XMAX )
TEMP = ACOEFA*WORK( J ) + BCOEFA*WORK( N+J )
IF( IL2BY2 )
$ TEMP = MAX( TEMP, ACOEFA*WORK( J+1 )+BCOEFA*
$ WORK( N+J+1 ) )
TEMP = MAX( TEMP, ACOEFA, BCOEFA )
IF( TEMP.GT.BIGNUM*XSCALE ) THEN
*
DO 330 JW = 0, NW - 1
DO 320 JR = 1, JE
WORK( ( JW+2 )*N+JR ) = XSCALE*
$ WORK( ( JW+2 )*N+JR )
320 CONTINUE
330 CONTINUE
XMAX = XMAX*XSCALE
END IF
*
* Compute the contributions of the off-diagonals of
* column j (and j+1, if 2-by-2 block) of A and B to the
* sums.
*
*
DO 360 JA = 1, NA
IF( ILCPLX ) THEN
CREALA = ACOEF*WORK( 2*N+J+JA-1 )
CIMAGA = ACOEF*WORK( 3*N+J+JA-1 )
CREALB = BCOEFR*WORK( 2*N+J+JA-1 ) -
$ BCOEFI*WORK( 3*N+J+JA-1 )
CIMAGB = BCOEFI*WORK( 2*N+J+JA-1 ) +
$ BCOEFR*WORK( 3*N+J+JA-1 )
DO 340 JR = 1, J - 1
WORK( 2*N+JR ) = WORK( 2*N+JR ) -
$ CREALA*S( JR, J+JA-1 ) +
$ CREALB*P( JR, J+JA-1 )
WORK( 3*N+JR ) = WORK( 3*N+JR ) -
$ CIMAGA*S( JR, J+JA-1 ) +
$ CIMAGB*P( JR, J+JA-1 )
340 CONTINUE
ELSE
CREALA = ACOEF*WORK( 2*N+J+JA-1 )
CREALB = BCOEFR*WORK( 2*N+J+JA-1 )
DO 350 JR = 1, J - 1
WORK( 2*N+JR ) = WORK( 2*N+JR ) -
$ CREALA*S( JR, J+JA-1 ) +
$ CREALB*P( JR, J+JA-1 )
350 CONTINUE
END IF
360 CONTINUE
END IF
*
IL2BY2 = .FALSE.
370 CONTINUE
*
* Copy eigenvector to VR, back transforming if
* HOWMNY='B'.
*
IEIG = IEIG - NW
IF( ILBACK ) THEN
*
DO 410 JW = 0, NW - 1
DO 380 JR = 1, N
WORK( ( JW+4 )*N+JR ) = WORK( ( JW+2 )*N+1 )*
$ VR( JR, 1 )
380 CONTINUE
*
* A series of compiler directives to defeat
* vectorization for the next loop
*
*
DO 400 JC = 2, JE
DO 390 JR = 1, N
WORK( ( JW+4 )*N+JR ) = WORK( ( JW+4 )*N+JR ) +
$ WORK( ( JW+2 )*N+JC )*VR( JR, JC )
390 CONTINUE
400 CONTINUE
410 CONTINUE
*
DO 430 JW = 0, NW - 1
DO 420 JR = 1, N
VR( JR, IEIG+JW ) = WORK( ( JW+4 )*N+JR )
420 CONTINUE
430 CONTINUE
*
IEND = N
ELSE
DO 450 JW = 0, NW - 1
DO 440 JR = 1, N
VR( JR, IEIG+JW ) = WORK( ( JW+2 )*N+JR )
440 CONTINUE
450 CONTINUE
*
IEND = JE
END IF
*
* Scale eigenvector
*
XMAX = ZERO
IF( ILCPLX ) THEN
DO 460 J = 1, IEND
XMAX = MAX( XMAX, ABS( VR( J, IEIG ) )+
$ ABS( VR( J, IEIG+1 ) ) )
460 CONTINUE
ELSE
DO 470 J = 1, IEND
XMAX = MAX( XMAX, ABS( VR( J, IEIG ) ) )
470 CONTINUE
END IF
*
IF( XMAX.GT.SAFMIN ) THEN
XSCALE = ONE / XMAX
DO 490 JW = 0, NW - 1
DO 480 JR = 1, IEND
VR( JR, IEIG+JW ) = XSCALE*VR( JR, IEIG+JW )
480 CONTINUE
490 CONTINUE
END IF
500 CONTINUE
END IF
*
RETURN
*
* End of DTGEVC
*
END
*> \brief \b DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGEX2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgex2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgex2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgex2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
* LDZ, J1, N1, N2, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* LOGICAL WANTQ, WANTZ
* INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
*> of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
*> (A, B) by an orthogonal equivalence transformation.
*>
*> (A, B) must be in generalized real Schur canonical form (as returned
*> by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
*> diagonal blocks. B is upper triangular.
*>
*> Optionally, the matrices Q and Z of generalized Schur vectors are
*> updated.
*>
*> Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
*> Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTQ
*> \verbatim
*> WANTQ is LOGICAL
*> .TRUE. : update the left transformation matrix Q;
*> .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> .TRUE. : update the right transformation matrix Z;
*> .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimensions (LDA,N)
*> On entry, the matrix A in the pair (A, B).
*> On exit, the updated matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimensions (LDB,N)
*> On entry, the matrix B in the pair (A, B).
*> On exit, the updated matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
*> On exit, the updated matrix Q.
*> Not referenced if WANTQ = .FALSE..
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= 1.
*> If WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
*> On exit, the updated matrix Z.
*> Not referenced if WANTZ = .FALSE..
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1.
*> If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*> J1 is INTEGER
*> The index to the first block (A11, B11). 1 <= J1 <= N.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The order of the first block (A11, B11). N1 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> The order of the second block (A22, B22). N2 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: Successful exit
*> >0: If INFO = 1, the transformed matrix (A, B) would be
*> too far from generalized Schur form; the blocks are
*> not swapped and (A, B) and (Q, Z) are unchanged.
*> The problem of swapping is too ill-conditioned.
*> <0: If INFO = -16: LWORK is too small. Appropriate value
*> for LWORK is returned in WORK(1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgex2
*
*> \par Further Details:
* =====================
*>
*> In the current code both weak and strong stability tests are
*> performed. The user can omit the strong stability test by changing
*> the internal logical parameter WANDS to .FALSE.. See ref. [2] for
*> details.
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*>
*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*> Estimation: Theory, Algorithms and Software,
*> Report UMINF - 94.04, Department of Computing Science, Umea
*> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
*> Note 87. To appear in Numerical Algorithms, 1996.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, J1, N1, N2, WORK, LWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
* loops. Sven Hammarling, 1/5/02.
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TWENTY
PARAMETER ( TWENTY = 2.0D+01 )
INTEGER LDST
PARAMETER ( LDST = 4 )
LOGICAL WANDS
PARAMETER ( WANDS = .TRUE. )
* ..
* .. Local Scalars ..
LOGICAL STRONG, WEAK
INTEGER I, IDUM, LINFO, M
DOUBLE PRECISION BQRA21, BRQA21, DDUM, DNORMA, DNORMB, DSCALE,
$ DSUM, EPS, F, G, SA, SB, SCALE, SMLNUM,
$ THRESHA, THRESHB
* ..
* .. Local Arrays ..
INTEGER IWORK( LDST + 2 )
DOUBLE PRECISION AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ),
$ IRCOP( LDST, LDST ), LI( LDST, LDST ),
$ LICOP( LDST, LDST ), S( LDST, LDST ),
$ SCPY( LDST, LDST ), T( LDST, LDST ),
$ TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2,
$ DLARTG,
$ DLASET, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2,
$ DROT, DSCAL, DTGSY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 )
$ RETURN
IF( N1.GT.N .OR. ( J1+N1 ).GT.N )
$ RETURN
M = N1 + N2
IF( LWORK.LT.MAX( 1, N*M, M*M*2 ) ) THEN
INFO = -16
WORK( 1 ) = MAX( 1, N*M, M*M*2 )
RETURN
END IF
*
WEAK = .FALSE.
STRONG = .FALSE.
*
* Make a local copy of selected block
*
CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, LI, LDST )
CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, IR, LDST )
CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
*
* Compute threshold for testing acceptance of swapping.
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
DSCALE = ZERO
DSUM = ONE
CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
DNORMA = DSCALE*SQRT( DSUM )
DSCALE = ZERO
DSUM = ONE
CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
DNORMB = DSCALE*SQRT( DSUM )
*
* THRES has been changed from
* THRESH = MAX( TEN*EPS*SA, SMLNUM )
* to
* THRESH = MAX( TWENTY*EPS*SA, SMLNUM )
* on 04/01/10.
* "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by
* Jim Demmel and Guillaume Revy. See forum post 1783.
*
THRESHA = MAX( TWENTY*EPS*DNORMA, SMLNUM )
THRESHB = MAX( TWENTY*EPS*DNORMB, SMLNUM )
*
IF( M.EQ.2 ) THEN
*
* CASE 1: Swap 1-by-1 and 1-by-1 blocks.
*
* Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
* using Givens rotations and perform the swap tentatively.
*
F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
SA = ABS( S( 2, 2 ) ) * ABS( T( 1, 1 ) )
SB = ABS( S( 1, 1 ) ) * ABS( T( 2, 2 ) )
CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
IR( 2, 1 ) = -IR( 1, 2 )
IR( 2, 2 ) = IR( 1, 1 )
CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
$ IR( 2, 1 ) )
CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
$ IR( 2, 1 ) )
IF( SA.GE.SB ) THEN
CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2,
$ 1 ),
$ DDUM )
ELSE
CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2,
$ 1 ),
$ DDUM )
END IF
CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
$ LI( 2, 1 ) )
CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
$ LI( 2, 1 ) )
LI( 2, 2 ) = LI( 1, 1 )
LI( 1, 2 ) = -LI( 2, 1 )
*
* Weak stability test: |S21| <= O(EPS F-norm((A)))
* and |T21| <= O(EPS F-norm((B)))
*
WEAK = ABS( S( 2, 1 ) ) .LE. THRESHA .AND.
$ ABS( T( 2, 1 ) ) .LE. THRESHB
IF( .NOT.WEAK )
$ GO TO 70
*
IF( WANDS ) THEN
*
* Strong stability test:
* F-norm((A-QL**H*S*QR)) <= O(EPS*F-norm((A)))
* and
* F-norm((B-QL**H*T*QR)) <= O(EPS*F-norm((B)))
*
CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA,
$ WORK( M*M+1 ),
$ M )
CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST,
$ ZERO,
$ WORK, M )
CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST,
$ ONE,
$ WORK( M*M+1 ), M )
DSCALE = ZERO
DSUM = ONE
CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
SA = DSCALE*SQRT( DSUM )
*
CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB,
$ WORK( M*M+1 ),
$ M )
CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST,
$ ZERO,
$ WORK, M )
CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST,
$ ONE,
$ WORK( M*M+1 ), M )
DSCALE = ZERO
DSUM = ONE
CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
SB = DSCALE*SQRT( DSUM )
STRONG = SA.LE.THRESHA .AND. SB.LE.THRESHB
IF( .NOT.STRONG )
$ GO TO 70
END IF
*
* Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
* (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
*
CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
$ IR( 2, 1 ) )
CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
$ IR( 2, 1 ) )
CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
$ LI( 1, 1 ), LI( 2, 1 ) )
CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
$ LI( 1, 1 ), LI( 2, 1 ) )
*
* Set N1-by-N2 (2,1) - blocks to ZERO.
*
A( J1+1, J1 ) = ZERO
B( J1+1, J1 ) = ZERO
*
* Accumulate transformations into Q and Z if requested.
*
IF( WANTZ )
$ CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ),
$ IR( 2, 1 ) )
IF( WANTQ )
$ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ),
$ LI( 2, 1 ) )
*
* Exit with INFO = 0 if swap was successfully performed.
*
RETURN
*
ELSE
*
* CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
* and 2-by-2 blocks.
*
* Solve the generalized Sylvester equation
* S11 * R - L * S22 = SCALE * S12
* T11 * R - L * T22 = SCALE * T12
* for R and L. Solutions in LI and IR.
*
CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST )
CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST,
$ IR( N2+1, N1+1 ), LDST )
CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST,
$ IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
$ LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
$ LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
*
* Compute orthogonal matrix QL:
*
* QL**T * LI = [ TL ]
* [ 0 ]
* where
* LI = [ -L ]
* [ SCALE * identity(N2) ]
*
DO 10 I = 1, N2
CALL DSCAL( N1, -ONE, LI( 1, I ), 1 )
LI( N1+I, I ) = SCALE
10 CONTINUE
CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
*
* Compute orthogonal matrix RQ:
*
* IR * RQ**T = [ 0 TR],
*
* where IR = [ SCALE * identity(N1), R ]
*
DO 20 I = 1, N1
IR( N2+I, I ) = SCALE
20 CONTINUE
CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
*
* Perform the swapping tentatively:
*
CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
$ WORK, M )
CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO,
$ S,
$ LDST )
CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
$ WORK, M )
CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO,
$ T,
$ LDST )
CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST )
CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST )
CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST )
CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST )
*
* Triangularize the B-part by an RQ factorization.
* Apply transformation (from left) to A-part, giving S.
*
CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST,
$ WORK,
$ LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST,
$ WORK,
$ LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
*
* Compute F-norm(S21) in BRQA21. (T21 is 0.)
*
DSCALE = ZERO
DSUM = ONE
DO 30 I = 1, N2
CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM )
30 CONTINUE
BRQA21 = DSCALE*SQRT( DSUM )
*
* Triangularize the B-part by a QR factorization.
* Apply transformation (from right) to A-part, giving S.
*
CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO )
IF( LINFO.NE.0 )
$ GO TO 70
CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY,
$ LDST,
$ WORK, INFO )
CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP,
$ LDST,
$ WORK, INFO )
IF( LINFO.NE.0 )
$ GO TO 70
*
* Compute F-norm(S21) in BQRA21. (T21 is 0.)
*
DSCALE = ZERO
DSUM = ONE
DO 40 I = 1, N2
CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM )
40 CONTINUE
BQRA21 = DSCALE*SQRT( DSUM )
*
* Decide which method to use.
* Weak stability test:
* F-norm(S21) <= O(EPS * F-norm((S)))
*
IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESHA ) THEN
CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
ELSE IF( BRQA21.GE.THRESHA ) THEN
GO TO 70
END IF
*
* Set lower triangle of B-part to zero
*
CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, T(2,1), LDST )
*
IF( WANDS ) THEN
*
* Strong stability test:
* F-norm((A-QL**H*S*QR)) <= O(EPS*F-norm((A)))
* and
* F-norm((B-QL**H*T*QR)) <= O(EPS*F-norm((B)))
*
CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA,
$ WORK( M*M+1 ),
$ M )
CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST,
$ ZERO,
$ WORK, M )
CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST,
$ ONE,
$ WORK( M*M+1 ), M )
DSCALE = ZERO
DSUM = ONE
CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
SA = DSCALE*SQRT( DSUM )
*
CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB,
$ WORK( M*M+1 ),
$ M )
CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST,
$ ZERO,
$ WORK, M )
CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST,
$ ONE,
$ WORK( M*M+1 ), M )
DSCALE = ZERO
DSUM = ONE
CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
SB = DSCALE*SQRT( DSUM )
STRONG = SA.LE.THRESHA .AND. SB.LE.THRESHB
IF( .NOT.STRONG )
$ GO TO 70
*
END IF
*
* If the swap is accepted ("weakly" and "strongly"), apply the
* transformations and set N1-by-N2 (2,1)-block to zero.
*
CALL DLASET( 'Full', N1, N2, ZERO, ZERO, S(N2+1,1), LDST )
*
* copy back M-by-M diagonal block starting at index J1 of (A, B)
*
CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA )
CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB )
CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, T, LDST )
*
* Standardize existing 2-by-2 blocks.
*
CALL DLASET( 'Full', M, M, ZERO, ZERO, WORK, M )
WORK( 1 ) = ONE
T( 1, 1 ) = ONE
IDUM = LWORK - M*M - 2
IF( N2.GT.1 ) THEN
CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI,
$ BE,
$ WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) )
WORK( M+1 ) = -WORK( 2 )
WORK( M+2 ) = WORK( 1 )
T( N2, N2 ) = T( 1, 1 )
T( 1, 2 ) = -T( 2, 1 )
END IF
WORK( M*M ) = ONE
T( M, M ) = ONE
*
IF( N1.GT.1 ) THEN
CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ),
$ LDB,
$ TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ),
$ WORK( N2*M+N2+2 ), T( N2+1, N2+1 ),
$ T( M, M-1 ) )
WORK( M*M ) = WORK( N2*M+N2+1 )
WORK( M*M-1 ) = -WORK( N2*M+N2+2 )
T( M, M ) = T( N2+1, N2+1 )
T( M-1, M ) = -T( M, M-1 )
END IF
CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1,
$ J1+N2 ),
$ LDA, ZERO, WORK( M*M+1 ), N2 )
CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1,
$ J1+N2 ),
$ LDA )
CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1,
$ J1+N2 ),
$ LDB, ZERO, WORK( M*M+1 ), N2 )
CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1,
$ J1+N2 ),
$ LDB )
CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO,
$ WORK( M*M+1 ), M )
CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST )
CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA,
$ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA )
CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDB,
$ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB )
CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO,
$ WORK, M )
CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST )
*
* Accumulate transformations into Q and Z if requested.
*
IF( WANTQ ) THEN
CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI,
$ LDST, ZERO, WORK, N )
CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ )
*
END IF
*
IF( WANTZ ) THEN
CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR,
$ LDST, ZERO, WORK, N )
CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ )
*
END IF
*
* Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
* (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
*
I = J1 + M
IF( I.LE.N ) THEN
CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
$ A( J1, I ), LDA, ZERO, WORK, M )
CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
$ B( J1, I ), LDB, ZERO, WORK, M )
CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
END IF
I = J1 - 1
IF( I.GT.0 ) THEN
CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR,
$ LDST, ZERO, WORK, I )
CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA )
CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR,
$ LDST, ZERO, WORK, I )
CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB )
END IF
*
* Exit with INFO = 0 if swap was successfully performed.
*
RETURN
*
END IF
*
* Exit with INFO = 1 if swap was rejected.
*
70 CONTINUE
*
INFO = 1
RETURN
*
* End of DTGEX2
*
END
*> \brief \b DTGEXC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGEXC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgexc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgexc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgexc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
* LDZ, IFST, ILST, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* LOGICAL WANTQ, WANTZ
* INTEGER IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGEXC reorders the generalized real Schur decomposition of a real
*> matrix pair (A,B) using an orthogonal equivalence transformation
*>
*> (A, B) = Q * (A, B) * Z**T,
*>
*> so that the diagonal block of (A, B) with row index IFST is moved
*> to row ILST.
*>
*> (A, B) must be in generalized real Schur canonical form (as returned
*> by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
*> diagonal blocks. B is upper triangular.
*>
*> Optionally, the matrices Q and Z of generalized Schur vectors are
*> updated.
*>
*> Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
*> Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTQ
*> \verbatim
*> WANTQ is LOGICAL
*> .TRUE. : update the left transformation matrix Q;
*> .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> .TRUE. : update the right transformation matrix Z;
*> .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the matrix A in generalized real Schur canonical
*> form.
*> On exit, the updated matrix A, again in generalized
*> real Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the matrix B in generalized real Schur canonical
*> form (A,B).
*> On exit, the updated matrix B, again in generalized
*> real Schur canonical form (A,B).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
*> On exit, the updated matrix Q.
*> If WANTQ = .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= 1.
*> If WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
*> On exit, the updated matrix Z.
*> If WANTZ = .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1.
*> If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[in,out] IFST
*> \verbatim
*> IFST is INTEGER
*> \endverbatim
*>
*> \param[in,out] ILST
*> \verbatim
*> ILST is INTEGER
*> Specify the reordering of the diagonal blocks of (A, B).
*> The block with row index IFST is moved to row ILST, by a
*> sequence of swapping between adjacent blocks.
*> On exit, if IFST pointed on entry to the second row of
*> a 2-by-2 block, it is changed to point to the first row;
*> ILST always points to the first row of the block in its
*> final position (which may differ from its input value by
*> +1 or -1). 1 <= IFST, ILST <= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: successful exit.
*> <0: if INFO = -i, the i-th argument had an illegal value.
*> =1: The transformed matrix pair (A, B) would be too far
*> from generalized Schur form; the problem is ill-
*> conditioned. (A, B) may have been partially reordered,
*> and ILST points to the first row of the current
*> position of the block being moved.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgexc
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, IFST, ILST, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER HERE, LWMIN, NBF, NBL, NBNEXT
* ..
* .. External Subroutines ..
EXTERNAL DTGEX2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode and test input arguments.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. ( LDQ.LT.MAX( 1, N ) ) ) THEN
INFO = -9
ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. ( LDZ.LT.MAX( 1, N ) ) ) THEN
INFO = -11
ELSE IF( IFST.LT.1 .OR. IFST.GT.N ) THEN
INFO = -12
ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) THEN
INFO = -13
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LWMIN = 1
ELSE
LWMIN = 4*N + 16
END IF
WORK(1) = LWMIN
*
IF (LWORK.LT.LWMIN .AND. .NOT.LQUERY) THEN
INFO = -15
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGEXC', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
* Determine the first row of the specified block and find out
* if it is 1-by-1 or 2-by-2.
*
IF( IFST.GT.1 ) THEN
IF( A( IFST, IFST-1 ).NE.ZERO )
$ IFST = IFST - 1
END IF
NBF = 1
IF( IFST.LT.N ) THEN
IF( A( IFST+1, IFST ).NE.ZERO )
$ NBF = 2
END IF
*
* Determine the first row of the final block
* and find out if it is 1-by-1 or 2-by-2.
*
IF( ILST.GT.1 ) THEN
IF( A( ILST, ILST-1 ).NE.ZERO )
$ ILST = ILST - 1
END IF
NBL = 1
IF( ILST.LT.N ) THEN
IF( A( ILST+1, ILST ).NE.ZERO )
$ NBL = 2
END IF
IF( IFST.EQ.ILST )
$ RETURN
*
IF( IFST.LT.ILST ) THEN
*
* Update ILST.
*
IF( NBF.EQ.2 .AND. NBL.EQ.1 )
$ ILST = ILST - 1
IF( NBF.EQ.1 .AND. NBL.EQ.2 )
$ ILST = ILST + 1
*
HERE = IFST
*
10 CONTINUE
*
* Swap with next one below.
*
IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
* Current block either 1-by-1 or 2-by-2.
*
NBNEXT = 1
IF( HERE+NBF+1.LE.N ) THEN
IF( A( HERE+NBF+1, HERE+NBF ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, HERE, NBF, NBNEXT, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + NBNEXT
*
* Test if 2-by-2 block breaks into two 1-by-1 blocks.
*
IF( NBF.EQ.2 ) THEN
IF( A( HERE+1, HERE ).EQ.ZERO )
$ NBF = 3
END IF
*
ELSE
*
* Current block consists of two 1-by-1 blocks, each of which
* must be swapped individually.
*
NBNEXT = 1
IF( HERE+3.LE.N ) THEN
IF( A( HERE+3, HERE+2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, HERE+1, 1, NBNEXT, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
IF( NBNEXT.EQ.1 ) THEN
*
* Swap two 1-by-1 blocks.
*
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ,
$ Z,
$ LDZ, HERE, 1, 1, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + 1
*
ELSE
*
* Recompute NBNEXT in case of 2-by-2 split.
*
IF( A( HERE+2, HERE+1 ).EQ.ZERO )
$ NBNEXT = 1
IF( NBNEXT.EQ.2 ) THEN
*
* 2-by-2 block did not split.
*
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
$ LDQ,
$ Z, LDZ, HERE, 1, NBNEXT, WORK, LWORK,
$ INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + 2
ELSE
*
* 2-by-2 block did split.
*
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
$ LDQ,
$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + 1
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
$ LDQ,
$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + 1
END IF
*
END IF
END IF
IF( HERE.LT.ILST )
$ GO TO 10
ELSE
HERE = IFST
*
20 CONTINUE
*
* Swap with next one below.
*
IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
* Current block either 1-by-1 or 2-by-2.
*
NBNEXT = 1
IF( HERE.GE.3 ) THEN
IF( A( HERE-1, HERE-2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, HERE-NBNEXT, NBNEXT, NBF, WORK, LWORK,
$ INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - NBNEXT
*
* Test if 2-by-2 block breaks into two 1-by-1 blocks.
*
IF( NBF.EQ.2 ) THEN
IF( A( HERE+1, HERE ).EQ.ZERO )
$ NBF = 3
END IF
*
ELSE
*
* Current block consists of two 1-by-1 blocks, each of which
* must be swapped individually.
*
NBNEXT = 1
IF( HERE.GE.3 ) THEN
IF( A( HERE-1, HERE-2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, HERE-NBNEXT, NBNEXT, 1, WORK, LWORK,
$ INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
IF( NBNEXT.EQ.1 ) THEN
*
* Swap two 1-by-1 blocks.
*
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ,
$ Z,
$ LDZ, HERE, NBNEXT, 1, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - 1
ELSE
*
* Recompute NBNEXT in case of 2-by-2 split.
*
IF( A( HERE, HERE-1 ).EQ.ZERO )
$ NBNEXT = 1
IF( NBNEXT.EQ.2 ) THEN
*
* 2-by-2 block did not split.
*
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
$ LDQ,
$ Z, LDZ, HERE-1, 2, 1, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - 2
ELSE
*
* 2-by-2 block did split.
*
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
$ LDQ,
$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - 1
CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
$ LDQ,
$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - 1
END IF
END IF
END IF
IF( HERE.GT.ILST )
$ GO TO 20
END IF
ILST = HERE
WORK( 1 ) = LWMIN
RETURN
*
* End of DTGEXC
*
END
*> \brief \b DTGSEN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSEN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsen.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsen.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsen.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL,
* PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* LOGICAL WANTQ, WANTZ
* INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
* $ M, N
* DOUBLE PRECISION PL, PR
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ),
* $ WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGSEN reorders the generalized real Schur decomposition of a real
*> matrix pair (A, B) (in terms of an orthonormal equivalence trans-
*> formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
*> appears in the leading diagonal blocks of the upper quasi-triangular
*> matrix A and the upper triangular B. The leading columns of Q and
*> Z form orthonormal bases of the corresponding left and right eigen-
*> spaces (deflating subspaces). (A, B) must be in generalized real
*> Schur canonical form (as returned by DGGES), i.e. A is block upper
*> triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
*> triangular.
*>
*> DTGSEN also computes the generalized eigenvalues
*>
*> w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
*>
*> of the reordered matrix pair (A, B).
*>
*> Optionally, DTGSEN computes the estimates of reciprocal condition
*> numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
*> (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
*> between the matrix pairs (A11, B11) and (A22,B22) that correspond to
*> the selected cluster and the eigenvalues outside the cluster, resp.,
*> and norms of "projections" onto left and right eigenspaces w.r.t.
*> the selected cluster in the (1,1)-block.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] IJOB
*> \verbatim
*> IJOB is INTEGER
*> Specifies whether condition numbers are required for the
*> cluster of eigenvalues (PL and PR) or the deflating subspaces
*> (Difu and Difl):
*> =0: Only reorder w.r.t. SELECT. No extras.
*> =1: Reciprocal of norms of "projections" onto left and right
*> eigenspaces w.r.t. the selected cluster (PL and PR).
*> =2: Upper bounds on Difu and Difl. F-norm-based estimate
*> (DIF(1:2)).
*> =3: Estimate of Difu and Difl. 1-norm-based estimate
*> (DIF(1:2)).
*> About 5 times as expensive as IJOB = 2.
*> =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
*> version to get it all.
*> =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
*> \endverbatim
*>
*> \param[in] WANTQ
*> \verbatim
*> WANTQ is LOGICAL
*> .TRUE. : update the left transformation matrix Q;
*> .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> .TRUE. : update the right transformation matrix Z;
*> .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> SELECT specifies the eigenvalues in the selected cluster.
*> To select a real eigenvalue w(j), SELECT(j) must be set to
*> .TRUE.. To select a complex conjugate pair of eigenvalues
*> w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
*> either SELECT(j) or SELECT(j+1) or both must be set to
*> .TRUE.; a complex conjugate pair of eigenvalues must be
*> either both included in the cluster or both excluded.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension(LDA,N)
*> On entry, the upper quasi-triangular matrix A, with (A, B) in
*> generalized real Schur canonical form.
*> On exit, A is overwritten by the reordered matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension(LDB,N)
*> On entry, the upper triangular matrix B, with (A, B) in
*> generalized real Schur canonical form.
*> On exit, B is overwritten by the reordered matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*>
*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
*> and BETA(j),j=1,...,N are the diagonals of the complex Schur
*> form (S,T) that would result if the 2-by-2 diagonal blocks of
*> the real generalized Schur form of (A,B) were further reduced
*> to triangular form using complex unitary transformations.
*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*> positive, then the j-th and (j+1)-st eigenvalues are a
*> complex conjugate pair, with ALPHAI(j+1) negative.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
*> On exit, Q has been postmultiplied by the left orthogonal
*> transformation matrix which reorder (A, B); The leading M
*> columns of Q form orthonormal bases for the specified pair of
*> left eigenspaces (deflating subspaces).
*> If WANTQ = .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= 1;
*> and if WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
*> On exit, Z has been postmultiplied by the left orthogonal
*> transformation matrix which reorder (A, B); The leading M
*> columns of Z form orthonormal bases for the specified pair of
*> left eigenspaces (deflating subspaces).
*> If WANTZ = .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1;
*> If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The dimension of the specified pair of left and right eigen-
*> spaces (deflating subspaces). 0 <= M <= N.
*> \endverbatim
*>
*> \param[out] PL
*> \verbatim
*> PL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] PR
*> \verbatim
*> PR is DOUBLE PRECISION
*>
*> If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
*> reciprocal of the norm of "projections" onto left and right
*> eigenspaces with respect to the selected cluster.
*> 0 < PL, PR <= 1.
*> If M = 0 or M = N, PL = PR = 1.
*> If IJOB = 0, 2 or 3, PL and PR are not referenced.
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*> DIF is DOUBLE PRECISION array, dimension (2).
*> If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
*> If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
*> Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
*> estimates of Difu and Difl.
*> If M = 0 or N, DIF(1:2) = F-norm([A, B]).
*> If IJOB = 0 or 1, DIF is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 4*N+16.
*> If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
*> If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK. LIWORK >= 1.
*> If IJOB = 1, 2 or 4, LIWORK >= N+6.
*> If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: Successful exit.
*> <0: If INFO = -i, the i-th argument had an illegal value.
*> =1: Reordering of (A, B) failed because the transformed
*> matrix pair (A, B) would be too far from generalized
*> Schur form; the problem is very ill-conditioned.
*> (A, B) may have been partially reordered.
*> If requested, 0 is returned in DIF(*), PL and PR.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> DTGSEN first collects the selected eigenvalues by computing
*> orthogonal U and W that move them to the top left corner of (A, B).
*> In other words, the selected eigenvalues are the eigenvalues of
*> (A11, B11) in:
*>
*> U**T*(A, B)*W = (A11 A12) (B11 B12) n1
*> ( 0 A22),( 0 B22) n2
*> n1 n2 n1 n2
*>
*> where N = n1+n2 and U**T means the transpose of U. The first n1 columns
*> of U and W span the specified pair of left and right eigenspaces
*> (deflating subspaces) of (A, B).
*>
*> If (A, B) has been obtained from the generalized real Schur
*> decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
*> reordered generalized real Schur form of (C, D) is given by
*>
*> (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
*>
*> and the first n1 columns of Q*U and Z*W span the corresponding
*> deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
*>
*> Note that if the selected eigenvalue is sufficiently ill-conditioned,
*> then its value may differ significantly from its value before
*> reordering.
*>
*> The reciprocal condition numbers of the left and right eigenspaces
*> spanned by the first n1 columns of U and W (or Q*U and Z*W) may
*> be returned in DIF(1:2), corresponding to Difu and Difl, resp.
*>
*> The Difu and Difl are defined as:
*>
*> Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
*> and
*> Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
*>
*> where sigma-min(Zu) is the smallest singular value of the
*> (2*n1*n2)-by-(2*n1*n2) matrix
*>
*> Zu = [ kron(In2, A11) -kron(A22**T, In1) ]
*> [ kron(In2, B11) -kron(B22**T, In1) ].
*>
*> Here, Inx is the identity matrix of size nx and A22**T is the
*> transpose of A22. kron(X, Y) is the Kronecker product between
*> the matrices X and Y.
*>
*> When DIF(2) is small, small changes in (A, B) can cause large changes
*> in the deflating subspace. An approximate (asymptotic) bound on the
*> maximum angular error in the computed deflating subspaces is
*>
*> EPS * norm((A, B)) / DIF(2),
*>
*> where EPS is the machine precision.
*>
*> The reciprocal norm of the projectors on the left and right
*> eigenspaces associated with (A11, B11) may be returned in PL and PR.
*> They are computed as follows. First we compute L and R so that
*> P*(A, B)*Q is block diagonal, where
*>
*> P = ( I -L ) n1 Q = ( I R ) n1
*> ( 0 I ) n2 and ( 0 I ) n2
*> n1 n2 n1 n2
*>
*> and (L, R) is the solution to the generalized Sylvester equation
*>
*> A11*R - L*A22 = -A12
*> B11*R - L*B22 = -B12
*>
*> Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
*> An approximate (asymptotic) bound on the average absolute error of
*> the selected eigenvalues is
*>
*> EPS * norm((A, B)) / PL.
*>
*> There are also global error bounds which valid for perturbations up
*> to a certain restriction: A lower bound (x) on the smallest
*> F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
*> coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
*> (i.e. (A + E, B + F), is
*>
*> x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
*>
*> An approximate bound on x can be computed from DIF(1:2), PL and PR.
*>
*> If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
*> (L', R') and unperturbed (L, R) left and right deflating subspaces
*> associated with the selected cluster in the (1,1)-blocks can be
*> bounded as
*>
*> max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
*> max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
*>
*> See LAPACK User's Guide section 4.11 or the following references
*> for more information.
*>
*> Note that if the default method for computing the Frobenius-norm-
*> based estimate DIF is not wanted (see DLATDF), then the parameter
*> IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
*> (IJOB = 2 will be used)). See DTGSYL for more details.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*>
*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*> Estimation: Theory, Algorithms and Software,
*> Report UMINF - 94.04, Department of Computing Science, Umea
*> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
*> Note 87. To appear in Numerical Algorithms, 1996.
*>
*> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*> for Solving the Generalized Sylvester Equation and Estimating the
*> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*> Department of Computing Science, Umea University, S-901 87 Umea,
*> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
*> Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
*> 1996.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B,
$ LDB,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL,
$ PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
$ M, N
DOUBLE PRECISION PL, PR
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER IDIFJB
PARAMETER ( IDIFJB = 3 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, PAIR, SWAP, WANTD, WANTD1, WANTD2,
$ WANTP
INTEGER I, IERR, IJB, K, KASE, KK, KS, LIWMIN, LWMIN,
$ MN2, N1, N2
DOUBLE PRECISION DSCALE, DSUM, EPS, RDSCAL, SMLNUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLACPY, DLAG2, DLASSQ, DTGEXC,
$ DTGSYL,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -14
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -16
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSEN', -INFO )
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
IERR = 0
*
WANTP = IJOB.EQ.1 .OR. IJOB.GE.4
WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4
WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5
WANTD = WANTD1 .OR. WANTD2
*
* Set M to the dimension of the specified pair of deflating
* subspaces.
*
M = 0
PAIR = .FALSE.
IF( .NOT.LQUERY .OR. IJOB.NE.0 ) THEN
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( A( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
$ M = M + 2
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
END IF
*
IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
LWMIN = MAX( 1, 4*N+16, 2*M*( N-M ) )
LIWMIN = MAX( 1, N+6 )
ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN
LWMIN = MAX( 1, 4*N+16, 4*M*( N-M ) )
LIWMIN = MAX( 1, 2*M*( N-M ), N+6 )
ELSE
LWMIN = MAX( 1, 4*N+16 )
LIWMIN = 1
END IF
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -22
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -24
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSEN', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible.
*
IF( M.EQ.N .OR. M.EQ.0 ) THEN
IF( WANTP ) THEN
PL = ONE
PR = ONE
END IF
IF( WANTD ) THEN
DSCALE = ZERO
DSUM = ONE
DO 20 I = 1, N
CALL DLASSQ( N, A( 1, I ), 1, DSCALE, DSUM )
CALL DLASSQ( N, B( 1, I ), 1, DSCALE, DSUM )
20 CONTINUE
DIF( 1 ) = DSCALE*SQRT( DSUM )
DIF( 2 ) = DIF( 1 )
END IF
GO TO 60
END IF
*
* Collect the selected blocks at the top-left corner of (A, B).
*
KS = 0
PAIR = .FALSE.
DO 30 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
*
SWAP = SELECT( K )
IF( K.LT.N ) THEN
IF( A( K+1, K ).NE.ZERO ) THEN
PAIR = .TRUE.
SWAP = SWAP .OR. SELECT( K+1 )
END IF
END IF
*
IF( SWAP ) THEN
KS = KS + 1
*
* Swap the K-th block to position KS.
* Perform the reordering of diagonal blocks in (A, B)
* by orthogonal transformation matrices and update
* Q and Z accordingly (if requested):
*
KK = K
IF( K.NE.KS )
$ CALL DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
$ LDQ,
$ Z, LDZ, KK, KS, WORK, LWORK, IERR )
*
IF( IERR.GT.0 ) THEN
*
* Swap is rejected: exit.
*
INFO = 1
IF( WANTP ) THEN
PL = ZERO
PR = ZERO
END IF
IF( WANTD ) THEN
DIF( 1 ) = ZERO
DIF( 2 ) = ZERO
END IF
GO TO 60
END IF
*
IF( PAIR )
$ KS = KS + 1
END IF
END IF
30 CONTINUE
IF( WANTP ) THEN
*
* Solve generalized Sylvester equation for R and L
* and compute PL and PR.
*
N1 = M
N2 = N - M
I = N1 + 1
IJB = 0
CALL DLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 )
CALL DLACPY( 'Full', N1, N2, B( 1, I ), LDB,
$ WORK( N1*N2+1 ),
$ N1 )
CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
$ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1,
$ DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
$ LWORK-2*N1*N2, IWORK, IERR )
*
* Estimate the reciprocal of norms of "projections" onto left
* and right eigenspaces.
*
RDSCAL = ZERO
DSUM = ONE
CALL DLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM )
PL = RDSCAL*SQRT( DSUM )
IF( PL.EQ.ZERO ) THEN
PL = ONE
ELSE
PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) )
END IF
RDSCAL = ZERO
DSUM = ONE
CALL DLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM )
PR = RDSCAL*SQRT( DSUM )
IF( PR.EQ.ZERO ) THEN
PR = ONE
ELSE
PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) )
END IF
END IF
*
IF( WANTD ) THEN
*
* Compute estimates of Difu and Difl.
*
IF( WANTD1 ) THEN
N1 = M
N2 = N - M
I = N1 + 1
IJB = IDIFJB
*
* Frobenius norm-based Difu-estimate.
*
CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA,
$ WORK,
$ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ),
$ N1, DSCALE, DIF( 1 ), WORK( 2*N1*N2+1 ),
$ LWORK-2*N1*N2, IWORK, IERR )
*
* Frobenius norm-based Difl-estimate.
*
CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA,
$ WORK,
$ N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ),
$ N2, DSCALE, DIF( 2 ), WORK( 2*N1*N2+1 ),
$ LWORK-2*N1*N2, IWORK, IERR )
ELSE
*
*
* Compute 1-norm-based estimates of Difu and Difl using
* reversed communication with DLACN2. In each step a
* generalized Sylvester equation or a transposed variant
* is solved.
*
KASE = 0
N1 = M
N2 = N - M
I = N1 + 1
IJB = 0
MN2 = 2*N1*N2
*
* 1-norm-based estimate of Difu.
*
40 CONTINUE
CALL DLACN2( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 1 ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve generalized Sylvester equation.
*
CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ),
$ LDA,
$ WORK, N1, B, LDB, B( I, I ), LDB,
$ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
$ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
ELSE
*
* Solve the transposed variant.
*
CALL DTGSYL( 'T', IJB, N1, N2, A, LDA, A( I, I ),
$ LDA,
$ WORK, N1, B, LDB, B( I, I ), LDB,
$ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
$ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
END IF
GO TO 40
END IF
DIF( 1 ) = DSCALE / DIF( 1 )
*
* 1-norm-based estimate of Difl.
*
50 CONTINUE
CALL DLACN2( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 2 ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve generalized Sylvester equation.
*
CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A,
$ LDA,
$ WORK, N2, B( I, I ), LDB, B, LDB,
$ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
$ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
ELSE
*
* Solve the transposed variant.
*
CALL DTGSYL( 'T', IJB, N2, N1, A( I, I ), LDA, A,
$ LDA,
$ WORK, N2, B( I, I ), LDB, B, LDB,
$ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
$ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
END IF
GO TO 50
END IF
DIF( 2 ) = DSCALE / DIF( 2 )
*
END IF
END IF
*
60 CONTINUE
*
* Compute generalized eigenvalues of reordered pair (A, B) and
* normalize the generalized Schur form.
*
PAIR = .FALSE.
DO 80 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
*
IF( K.LT.N ) THEN
IF( A( K+1, K ).NE.ZERO ) THEN
PAIR = .TRUE.
END IF
END IF
*
IF( PAIR ) THEN
*
* Compute the eigenvalue(s) at position K.
*
WORK( 1 ) = A( K, K )
WORK( 2 ) = A( K+1, K )
WORK( 3 ) = A( K, K+1 )
WORK( 4 ) = A( K+1, K+1 )
WORK( 5 ) = B( K, K )
WORK( 6 ) = B( K+1, K )
WORK( 7 ) = B( K, K+1 )
WORK( 8 ) = B( K+1, K+1 )
CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS,
$ BETA( K ),
$ BETA( K+1 ), ALPHAR( K ), ALPHAR( K+1 ),
$ ALPHAI( K ) )
ALPHAI( K+1 ) = -ALPHAI( K )
*
ELSE
*
IF( SIGN( ONE, B( K, K ) ).LT.ZERO ) THEN
*
* If B(K,K) is negative, make it positive
*
DO 70 I = 1, N
A( K, I ) = -A( K, I )
B( K, I ) = -B( K, I )
IF( WANTQ ) Q( I, K ) = -Q( I, K )
70 CONTINUE
END IF
*
ALPHAR( K ) = A( K, K )
ALPHAI( K ) = ZERO
BETA( K ) = B( K, K )
*
END IF
END IF
80 CONTINUE
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DTGSEN
*
END
*> \brief \b DTGSJA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSJA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsja.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsja.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsja.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
* LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
* Q, LDQ, WORK, NCYCLE, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBQ, JOBU, JOBV
* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
* $ NCYCLE, P
* DOUBLE PRECISION TOLA, TOLB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
* $ V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGSJA computes the generalized singular value decomposition (GSVD)
*> of two real upper triangular (or trapezoidal) matrices A and B.
*>
*> On entry, it is assumed that matrices A and B have the following
*> forms, which may be obtained by the preprocessing subroutine DGGSVP
*> from a general M-by-N matrix A and P-by-N matrix B:
*>
*> N-K-L K L
*> A = K ( 0 A12 A13 ) if M-K-L >= 0;
*> L ( 0 0 A23 )
*> M-K-L ( 0 0 0 )
*>
*> N-K-L K L
*> A = K ( 0 A12 A13 ) if M-K-L < 0;
*> M-K ( 0 0 A23 )
*>
*> N-K-L K L
*> B = L ( 0 0 B13 )
*> P-L ( 0 0 0 )
*>
*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
*> otherwise A23 is (M-K)-by-L upper trapezoidal.
*>
*> On exit,
*>
*> U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
*>
*> where U, V and Q are orthogonal matrices.
*> R is a nonsingular upper triangular matrix, and D1 and D2 are
*> ``diagonal'' matrices, which are of the following structures:
*>
*> If M-K-L >= 0,
*>
*> K L
*> D1 = K ( I 0 )
*> L ( 0 C )
*> M-K-L ( 0 0 )
*>
*> K L
*> D2 = L ( 0 S )
*> P-L ( 0 0 )
*>
*> N-K-L K L
*> ( 0 R ) = K ( 0 R11 R12 ) K
*> L ( 0 0 R22 ) L
*>
*> where
*>
*> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
*> S = diag( BETA(K+1), ... , BETA(K+L) ),
*> C**2 + S**2 = I.
*>
*> R is stored in A(1:K+L,N-K-L+1:N) on exit.
*>
*> If M-K-L < 0,
*>
*> K M-K K+L-M
*> D1 = K ( I 0 0 )
*> M-K ( 0 C 0 )
*>
*> K M-K K+L-M
*> D2 = M-K ( 0 S 0 )
*> K+L-M ( 0 0 I )
*> P-L ( 0 0 0 )
*>
*> N-K-L K M-K K+L-M
*> ( 0 R ) = K ( 0 R11 R12 R13 )
*> M-K ( 0 0 R22 R23 )
*> K+L-M ( 0 0 0 R33 )
*>
*> where
*> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
*> S = diag( BETA(K+1), ... , BETA(M) ),
*> C**2 + S**2 = I.
*>
*> R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
*> ( 0 R22 R23 )
*> in B(M-K+1:L,N+M-K-L+1:N) on exit.
*>
*> The computation of the orthogonal transformation matrices U, V or Q
*> is optional. These matrices may either be formed explicitly, or they
*> may be postmultiplied into input matrices U1, V1, or Q1.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> = 'U': U must contain an orthogonal matrix U1 on entry, and
*> the product U1*U is returned;
*> = 'I': U is initialized to the unit matrix, and the
*> orthogonal matrix U is returned;
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> = 'V': V must contain an orthogonal matrix V1 on entry, and
*> the product V1*V is returned;
*> = 'I': V is initialized to the unit matrix, and the
*> orthogonal matrix V is returned;
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*> JOBQ is CHARACTER*1
*> = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
*> the product Q1*Q is returned;
*> = 'I': Q is initialized to the unit matrix, and the
*> orthogonal matrix Q is returned;
*> = 'N': Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*>
*> K and L specify the subblocks in the input matrices A and B:
*> A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
*> of A and B, whose GSVD is going to be computed by DTGSJA.
*> See Further Details.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
*> matrix R or part of R. See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
*> a part of R. See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in] TOLA
*> \verbatim
*> TOLA is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] TOLB
*> \verbatim
*> TOLB is DOUBLE PRECISION
*>
*> TOLA and TOLB are the convergence criteria for the Jacobi-
*> Kogbetliantz iteration procedure. Generally, they are the
*> same as used in the preprocessing step, say
*> TOLA = max(M,N)*norm(A)*MAZHEPS,
*> TOLB = max(P,N)*norm(B)*MAZHEPS.
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*>
*> On exit, ALPHA and BETA contain the generalized singular
*> value pairs of A and B;
*> ALPHA(1:K) = 1,
*> BETA(1:K) = 0,
*> and if M-K-L >= 0,
*> ALPHA(K+1:K+L) = diag(C),
*> BETA(K+1:K+L) = diag(S),
*> or if M-K-L < 0,
*> ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
*> BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
*> Furthermore, if K+L < N,
*> ALPHA(K+L+1:N) = 0 and
*> BETA(K+L+1:N) = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,M)
*> On entry, if JOBU = 'U', U must contain a matrix U1 (usually
*> the orthogonal matrix returned by DGGSVP).
*> On exit,
*> if JOBU = 'I', U contains the orthogonal matrix U;
*> if JOBU = 'U', U contains the product U1*U.
*> If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,M) if
*> JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,P)
*> On entry, if JOBV = 'V', V must contain a matrix V1 (usually
*> the orthogonal matrix returned by DGGSVP).
*> On exit,
*> if JOBV = 'I', V contains the orthogonal matrix V;
*> if JOBV = 'V', V contains the product V1*V.
*> If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,P) if
*> JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
*> the orthogonal matrix returned by DGGSVP).
*> On exit,
*> if JOBQ = 'I', Q contains the orthogonal matrix Q;
*> if JOBQ = 'Q', Q contains the product Q1*Q.
*> If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N) if
*> JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] NCYCLE
*> \verbatim
*> NCYCLE is INTEGER
*> The number of cycles required for convergence.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1: the procedure does not converge after MAXIT cycles.
*> \endverbatim
*>
*> \verbatim
*> Internal Parameters
*> ===================
*>
*> MAXIT INTEGER
*> MAXIT specifies the total loops that the iterative procedure
*> may take. If after MAXIT cycles, the routine fails to
*> converge, we return INFO = 1.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsja
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
*> min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
*> matrix B13 to the form:
*>
*> U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
*>
*> where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
*> of Z. C1 and S1 are diagonal matrices satisfying
*>
*> C1**2 + S1**2 = I,
*>
*> and R1 is an L-by-L nonsingular upper triangular matrix.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
$ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
$ Q, LDQ, WORK, NCYCLE, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
$ NCYCLE, P
DOUBLE PRECISION TOLA, TOLB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), Q( LDQ, * ), U( LDU, * ),
$ V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 40 )
DOUBLE PRECISION ZERO, ONE, HUGENUM
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
*
LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
INTEGER I, J, KCYCLE
DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
$ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET,
$ DROT,
$ DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, HUGE
PARAMETER ( HUGENUM = HUGE(ZERO) )
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
INITU = LSAME( JOBU, 'I' )
WANTU = INITU .OR. LSAME( JOBU, 'U' )
*
INITV = LSAME( JOBV, 'I' )
WANTV = INITV .OR. LSAME( JOBV, 'V' )
*
INITQ = LSAME( JOBQ, 'I' )
WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
*
INFO = 0
IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( INITV .OR.
$ WANTV .OR.
$ LSAME( JOBV, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( INITQ .OR.
$ WANTQ .OR.
$ LSAME( JOBQ, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 ) THEN
INFO = -5
ELSE IF( N.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -12
ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
INFO = -18
ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
INFO = -20
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -22
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSJA', -INFO )
RETURN
END IF
*
* Initialize U, V and Q, if necessary
*
IF( INITU )
$ CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
IF( INITV )
$ CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
IF( INITQ )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
*
* Loop until convergence
*
UPPER = .FALSE.
DO 40 KCYCLE = 1, MAXIT
*
UPPER = .NOT.UPPER
*
DO 20 I = 1, L - 1
DO 10 J = I + 1, L
*
A1 = ZERO
A2 = ZERO
A3 = ZERO
IF( K+I.LE.M )
$ A1 = A( K+I, N-L+I )
IF( K+J.LE.M )
$ A3 = A( K+J, N-L+J )
*
B1 = B( I, N-L+I )
B3 = B( J, N-L+J )
*
IF( UPPER ) THEN
IF( K+I.LE.M )
$ A2 = A( K+I, N-L+J )
B2 = B( I, N-L+J )
ELSE
IF( K+J.LE.M )
$ A2 = A( K+J, N-L+I )
B2 = B( J, N-L+I )
END IF
*
CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
$ CSV, SNV, CSQ, SNQ )
*
* Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A
*
IF( K+J.LE.M )
$ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I,
$ N-L+1 ),
$ LDA, CSU, SNU )
*
* Update I-th and J-th rows of matrix B: V**T *B
*
CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
$ CSV, SNV )
*
* Update (N-L+I)-th and (N-L+J)-th columns of matrices
* A and B: A*Q and B*Q
*
CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
$ A( 1, N-L+I ), 1, CSQ, SNQ )
*
CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
$ SNQ )
*
IF( UPPER ) THEN
IF( K+I.LE.M )
$ A( K+I, N-L+J ) = ZERO
B( I, N-L+J ) = ZERO
ELSE
IF( K+J.LE.M )
$ A( K+J, N-L+I ) = ZERO
B( J, N-L+I ) = ZERO
END IF
*
* Update orthogonal matrices U, V, Q, if desired.
*
IF( WANTU .AND. K+J.LE.M )
$ CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
$ SNU )
*
IF( WANTV )
$ CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV,
$ SNV )
*
IF( WANTQ )
$ CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1,
$ CSQ,
$ SNQ )
*
10 CONTINUE
20 CONTINUE
*
IF( .NOT.UPPER ) THEN
*
* The matrices A13 and B13 were lower triangular at the start
* of the cycle, and are now upper triangular.
*
* Convergence test: test the parallelism of the corresponding
* rows of A and B.
*
ERROR = ZERO
DO 30 I = 1, MIN( L, M-K )
CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ),
$ 1 )
CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
ERROR = MAX( ERROR, SSMIN )
30 CONTINUE
*
IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
$ GO TO 50
END IF
*
* End of cycle loop
*
40 CONTINUE
*
* The algorithm has not converged after MAXIT cycles.
*
INFO = 1
GO TO 100
*
50 CONTINUE
*
* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
* Compute the generalized singular value pairs (ALPHA, BETA), and
* set the triangular matrix R to array A.
*
DO 60 I = 1, K
ALPHA( I ) = ONE
BETA( I ) = ZERO
60 CONTINUE
*
DO 70 I = 1, MIN( L, M-K )
*
A1 = A( K+I, N-L+I )
B1 = B( I, N-L+I )
GAMMA = B1 / A1
*
IF( (GAMMA.LE.HUGENUM).AND.(GAMMA.GE.-HUGENUM) ) THEN
*
* change sign if necessary
*
IF( GAMMA.LT.ZERO ) THEN
CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
IF( WANTV )
$ CALL DSCAL( P, -ONE, V( 1, I ), 1 )
END IF
*
CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ),
$ ALPHA( K+I ),
$ RWK )
*
IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I,
$ N-L+I ),
$ LDA )
ELSE
CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
$ LDB )
CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I,
$ N-L+I ),
$ LDA )
END IF
*
ELSE
*
ALPHA( K+I ) = ZERO
BETA( K+I ) = ONE
CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
$ LDA )
*
END IF
*
70 CONTINUE
*
* Post-assignment
*
DO 80 I = M + 1, K + L
ALPHA( I ) = ZERO
BETA( I ) = ONE
80 CONTINUE
*
IF( K+L.LT.N ) THEN
DO 90 I = K + L + 1, N
ALPHA( I ) = ZERO
BETA( I ) = ZERO
90 CONTINUE
END IF
*
100 CONTINUE
NCYCLE = KCYCLE
RETURN
*
* End of DTGSJA
*
END
*> \brief \b DTGSNA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsna.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
* LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, JOB
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
* $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGSNA estimates reciprocal condition numbers for specified
*> eigenvalues and/or eigenvectors of a matrix pair (A, B) in
*> generalized real Schur canonical form (or of any matrix pair
*> (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
*> Z**T denotes the transpose of Z.
*>
*> (A, B) must be in generalized real Schur form (as returned by DGGES),
*> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
*> blocks. B is upper triangular.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies whether condition numbers are required for
*> eigenvalues (S) or eigenvectors (DIF):
*> = 'E': for eigenvalues only (S);
*> = 'V': for eigenvectors only (DIF);
*> = 'B': for both eigenvalues and eigenvectors (S and DIF).
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute condition numbers for all eigenpairs;
*> = 'S': compute condition numbers for selected eigenpairs
*> specified by the array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY = 'S', SELECT specifies the eigenpairs for which
*> condition numbers are required. To select condition numbers
*> for the eigenpair corresponding to a real eigenvalue w(j),
*> SELECT(j) must be set to .TRUE.. To select condition numbers
*> corresponding to a complex conjugate pair of eigenvalues w(j)
*> and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
*> set to .TRUE..
*> If HOWMNY = 'A', SELECT is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the square matrix pair (A, B). N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The upper quasi-triangular matrix A in the pair (A,B).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The upper triangular matrix B in the pair (A,B).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,M)
*> If JOB = 'E' or 'B', VL must contain left eigenvectors of
*> (A, B), corresponding to the eigenpairs specified by HOWMNY
*> and SELECT. The eigenvectors must be stored in consecutive
*> columns of VL, as returned by DTGEVC.
*> If JOB = 'V', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1.
*> If JOB = 'E' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,M)
*> If JOB = 'E' or 'B', VR must contain right eigenvectors of
*> (A, B), corresponding to the eigenpairs specified by HOWMNY
*> and SELECT. The eigenvectors must be stored in consecutive
*> columns ov VR, as returned by DTGEVC.
*> If JOB = 'V', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1.
*> If JOB = 'E' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (MM)
*> If JOB = 'E' or 'B', the reciprocal condition numbers of the
*> selected eigenvalues, stored in consecutive elements of the
*> array. For a complex conjugate pair of eigenvalues two
*> consecutive elements of S are set to the same value. Thus
*> S(j), DIF(j), and the j-th columns of VL and VR all
*> correspond to the same eigenpair (but not in general the
*> j-th eigenpair, unless all eigenpairs are selected).
*> If JOB = 'V', S is not referenced.
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*> DIF is DOUBLE PRECISION array, dimension (MM)
*> If JOB = 'V' or 'B', the estimated reciprocal condition
*> numbers of the selected eigenvectors, stored in consecutive
*> elements of the array. For a complex eigenvector two
*> consecutive elements of DIF are set to the same value. If
*> the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
*> is set to 0; this can only occur when the true value would be
*> very small anyway.
*> If JOB = 'E', DIF is not referenced.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of elements in the arrays S and DIF. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of elements of the arrays S and DIF used to store
*> the specified condition numbers; for each selected real
*> eigenvalue one element is used, and for each selected complex
*> conjugate pair of eigenvalues, two elements are used.
*> If HOWMNY = 'A', M is set to N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N + 6)
*> If JOB = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: Successful exit
*> <0: If INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsna
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The reciprocal of the condition number of a generalized eigenvalue
*> w = (a, b) is defined as
*>
*> S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
*>
*> where u and v are the left and right eigenvectors of (A, B)
*> corresponding to w; |z| denotes the absolute value of the complex
*> number, and norm(u) denotes the 2-norm of the vector u.
*> The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
*> of the matrix pair (A, B). If both a and b equal zero, then (A B) is
*> singular and S(I) = -1 is returned.
*>
*> An approximate error bound on the chordal distance between the i-th
*> computed generalized eigenvalue w and the corresponding exact
*> eigenvalue lambda is
*>
*> chord(w, lambda) <= EPS * norm(A, B) / S(I)
*>
*> where EPS is the machine precision.
*>
*> The reciprocal of the condition number DIF(i) of right eigenvector u
*> and left eigenvector v corresponding to the generalized eigenvalue w
*> is defined as follows:
*>
*> a) If the i-th eigenvalue w = (a,b) is real
*>
*> Suppose U and V are orthogonal transformations such that
*>
*> U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
*> ( 0 S22 ),( 0 T22 ) n-1
*> 1 n-1 1 n-1
*>
*> Then the reciprocal condition number DIF(i) is
*>
*> Difl((a, b), (S22, T22)) = sigma-min( Zl ),
*>
*> where sigma-min(Zl) denotes the smallest singular value of the
*> 2(n-1)-by-2(n-1) matrix
*>
*> Zl = [ kron(a, In-1) -kron(1, S22) ]
*> [ kron(b, In-1) -kron(1, T22) ] .
*>
*> Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
*> Kronecker product between the matrices X and Y.
*>
*> Note that if the default method for computing DIF(i) is wanted
*> (see DLATDF), then the parameter DIFDRI (see below) should be
*> changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
*> See DTGSYL for more details.
*>
*> b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
*>
*> Suppose U and V are orthogonal transformations such that
*>
*> U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
*> ( 0 S22 ),( 0 T22) n-2
*> 2 n-2 2 n-2
*>
*> and (S11, T11) corresponds to the complex conjugate eigenvalue
*> pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
*> that
*>
*> U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
*> ( 0 s22 ) ( 0 t22 )
*>
*> where the generalized eigenvalues w = s11/t11 and
*> conjg(w) = s22/t22.
*>
*> Then the reciprocal condition number DIF(i) is bounded by
*>
*> min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
*>
*> where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
*> Z1 is the complex 2-by-2 matrix
*>
*> Z1 = [ s11 -s22 ]
*> [ t11 -t22 ],
*>
*> This is done by computing (using real arithmetic) the
*> roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
*> where Z1**T denotes the transpose of Z1 and det(X) denotes
*> the determinant of X.
*>
*> and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
*> upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
*>
*> Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ]
*> [ kron(T11**T, In-2) -kron(I2, T22) ]
*>
*> Note that if the default method for computing DIF is wanted (see
*> DLATDF), then the parameter DIFDRI (see below) should be changed
*> from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
*> for more details.
*>
*> For each eigenvalue/vector specified by SELECT, DIF stores a
*> Frobenius norm-based estimate of Difl.
*>
*> An approximate error bound for the i-th computed eigenvector VL(i) or
*> VR(i) is given by
*>
*> EPS * norm(A, B) / DIF(i).
*>
*> See ref. [2-3] for more details and further references.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*>
*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*> Estimation: Theory, Algorithms and Software,
*> Report UMINF - 94.04, Department of Computing Science, Umea
*> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
*> Note 87. To appear in Numerical Algorithms, 1996.
*>
*> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*> for Solving the Generalized Sylvester Equation and Estimating the
*> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*> Department of Computing Science, Umea University, S-901 87 Umea,
*> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
*> Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
*> No 1, 1996.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, JOB
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
$ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER DIFDRI
PARAMETER ( DIFDRI = 3 )
DOUBLE PRECISION ZERO, ONE, TWO, FOUR
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ FOUR = 4.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
INTEGER I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2
DOUBLE PRECISION ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND,
$ EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM,
$ TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV,
$ UHBVI
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUMMY( 1 ), DUMMY1( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2
EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DLACPY, DLAG2, DTGEXC, DTGSYL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
*
SOMCON = LSAME( HOWMNY, 'S' )
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
INFO = -10
ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
INFO = -12
ELSE
*
* Set M to the number of eigenpairs for which condition numbers
* are required, and test MM.
*
IF( SOMCON ) THEN
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( A( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
$ M = M + 2
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
ELSE
M = N
END IF
*
IF( N.EQ.0 ) THEN
LWMIN = 1
ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
LWMIN = 2*N*( N + 2 ) + 16
ELSE
LWMIN = N
END IF
WORK( 1 ) = LWMIN
*
IF( MM.LT.M ) THEN
INFO = -15
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSNA', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
KS = 0
PAIR = .FALSE.
*
DO 20 K = 1, N
*
* Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block.
*
IF( PAIR ) THEN
PAIR = .FALSE.
GO TO 20
ELSE
IF( K.LT.N )
$ PAIR = A( K+1, K ).NE.ZERO
END IF
*
* Determine whether condition numbers are required for the k-th
* eigenpair.
*
IF( SOMCON ) THEN
IF( PAIR ) THEN
IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
$ GO TO 20
ELSE
IF( .NOT.SELECT( K ) )
$ GO TO 20
END IF
END IF
*
KS = KS + 1
*
IF( WANTS ) THEN
*
* Compute the reciprocal condition number of the k-th
* eigenvalue.
*
IF( PAIR ) THEN
*
* Complex eigenvalue pair.
*
RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ),
$ DNRM2( N, VR( 1, KS+1 ), 1 ) )
LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ),
$ DNRM2( N, VL( 1, KS+1 ), 1 ) )
CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1,
$ ZERO,
$ WORK, 1 )
TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1,
$ ZERO, WORK, 1 )
TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
UHAV = TMPRR + TMPII
UHAVI = TMPIR - TMPRI
CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1,
$ ZERO,
$ WORK, 1 )
TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1,
$ ZERO, WORK, 1 )
TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
UHBV = TMPRR + TMPII
UHBVI = TMPIR - TMPRI
UHAV = DLAPY2( UHAV, UHAVI )
UHBV = DLAPY2( UHBV, UHBVI )
COND = DLAPY2( UHAV, UHBV )
S( KS ) = COND / ( RNRM*LNRM )
S( KS+1 ) = S( KS )
*
ELSE
*
* Real eigenvalue.
*
RNRM = DNRM2( N, VR( 1, KS ), 1 )
LNRM = DNRM2( N, VL( 1, KS ), 1 )
CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1,
$ ZERO,
$ WORK, 1 )
UHAV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1,
$ ZERO,
$ WORK, 1 )
UHBV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
COND = DLAPY2( UHAV, UHBV )
IF( COND.EQ.ZERO ) THEN
S( KS ) = -ONE
ELSE
S( KS ) = COND / ( RNRM*LNRM )
END IF
END IF
END IF
*
IF( WANTDF ) THEN
IF( N.EQ.1 ) THEN
DIF( KS ) = DLAPY2( A( 1, 1 ), B( 1, 1 ) )
GO TO 20
END IF
*
* Estimate the reciprocal condition number of the k-th
* eigenvectors.
IF( PAIR ) THEN
*
* Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)).
* Compute the eigenvalue(s) at position K.
*
WORK( 1 ) = A( K, K )
WORK( 2 ) = A( K+1, K )
WORK( 3 ) = A( K, K+1 )
WORK( 4 ) = A( K+1, K+1 )
WORK( 5 ) = B( K, K )
WORK( 6 ) = B( K+1, K )
WORK( 7 ) = B( K, K+1 )
WORK( 8 ) = B( K+1, K+1 )
CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA,
$ DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI )
ALPRQT = ONE
C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA )
C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI
ROOT1 = C1 + SQRT( C1*C1-4.0D0*C2 )
ROOT1 = ROOT1 / TWO
ROOT2 = C2 / ROOT1
COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) )
END IF
*
* Copy the matrix (A, B) to the array WORK and swap the
* diagonal block beginning at A(k,k) to the (1,1) position.
*
CALL DLACPY( 'Full', N, N, A, LDA, WORK, N )
CALL DLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
IFST = K
ILST = 1
*
CALL DTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ),
$ N,
$ DUMMY, 1, DUMMY1, 1, IFST, ILST,
$ WORK( N*N*2+1 ), LWORK-2*N*N, IERR )
*
IF( IERR.GT.0 ) THEN
*
* Ill-conditioned problem - swap rejected.
*
DIF( KS ) = ZERO
ELSE
*
* Reordering successful, solve generalized Sylvester
* equation for R and L,
* A22 * R - L * A11 = A12
* B22 * R - L * B11 = B12,
* and compute estimate of Difl((A11,B11), (A22, B22)).
*
N1 = 1
IF( WORK( 2 ).NE.ZERO )
$ N1 = 2
N2 = N - N1
IF( N2.EQ.0 ) THEN
DIF( KS ) = COND
ELSE
I = N*N + 1
IZ = 2*N*N + 1
CALL DTGSYL( 'N', DIFDRI, N2, N1,
$ WORK( N*N1+N1+1 ),
$ N, WORK, N, WORK( N1+1 ), N,
$ WORK( N*N1+N1+I ), N, WORK( I ), N,
$ WORK( N1+I ), N, SCALE, DIF( KS ),
$ WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR )
*
IF( PAIR )
$ DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ),
$ COND )
END IF
END IF
IF( PAIR )
$ DIF( KS+1 ) = DIF( KS )
END IF
IF( PAIR )
$ KS = KS + 1
*
20 CONTINUE
WORK( 1 ) = LWMIN
RETURN
*
* End of DTGSNA
*
END
*> \brief \b DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSY2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsy2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsy2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsy2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
* LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
* IWORK, PQ, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
* $ PQ
* DOUBLE PRECISION RDSCAL, RDSUM, SCALE
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
* $ D( LDD, * ), E( LDE, * ), F( LDF, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGSY2 solves the generalized Sylvester equation:
*>
*> A * R - L * B = scale * C (1)
*> D * R - L * E = scale * F,
*>
*> using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
*> (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
*> N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
*> must be in generalized Schur canonical form, i.e. A, B are upper
*> quasi triangular and D, E are upper triangular. The solution (R, L)
*> overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
*> chosen to avoid overflow.
*>
*> In matrix notation solving equation (1) corresponds to solve
*> Z*x = scale*b, where Z is defined as
*>
*> Z = [ kron(In, A) -kron(B**T, Im) ] (2)
*> [ kron(In, D) -kron(E**T, Im) ],
*>
*> Ik is the identity matrix of size k and X**T is the transpose of X.
*> kron(X, Y) is the Kronecker product between the matrices X and Y.
*> In the process of solving (1), we solve a number of such systems
*> where Dim(In), Dim(In) = 1 or 2.
*>
*> If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,
*> which is equivalent to solve for R and L in
*>
*> A**T * R + D**T * L = scale * C (3)
*> R * B**T + L * E**T = scale * -F
*>
*> This case is used to compute an estimate of Dif[(A, D), (B, E)] =
*> sigma_min(Z) using reverse communication with DLACON.
*>
*> DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
*> of an upper bound on the separation between to matrix pairs. Then
*> the input (A, D), (B, E) are sub-pencils of the matrix pair in
*> DTGSYL. See DTGSYL for details.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': solve the generalized Sylvester equation (1).
*> = 'T': solve the 'transposed' system (3).
*> \endverbatim
*>
*> \param[in] IJOB
*> \verbatim
*> IJOB is INTEGER
*> Specifies what kind of functionality to be performed.
*> = 0: solve (1) only.
*> = 1: A contribution from this subsystem to a Frobenius
*> norm-based estimate of the separation between two matrix
*> pairs is computed. (look ahead strategy is used).
*> = 2: A contribution from this subsystem to a Frobenius
*> norm-based estimate of the separation between two matrix
*> pairs is computed. (DGECON on sub-systems is used.)
*> Not referenced if TRANS = 'T'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> On entry, M specifies the order of A and D, and the row
*> dimension of C, F, R and L.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of B and E, and the column
*> dimension of C, F, R and L.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, M)
*> On entry, A contains an upper quasi triangular matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the matrix A. LDA >= max(1, M).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, B contains an upper quasi triangular matrix.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the matrix B. LDB >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC, N)
*> On entry, C contains the right-hand-side of the first matrix
*> equation in (1).
*> On exit, if IJOB = 0, C has been overwritten by the
*> solution R.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the matrix C. LDC >= max(1, M).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (LDD, M)
*> On entry, D contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDD
*> \verbatim
*> LDD is INTEGER
*> The leading dimension of the matrix D. LDD >= max(1, M).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (LDE, N)
*> On entry, E contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDE
*> \verbatim
*> LDE is INTEGER
*> The leading dimension of the matrix E. LDE >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] F
*> \verbatim
*> F is DOUBLE PRECISION array, dimension (LDF, N)
*> On entry, F contains the right-hand-side of the second matrix
*> equation in (1).
*> On exit, if IJOB = 0, F has been overwritten by the
*> solution L.
*> \endverbatim
*>
*> \param[in] LDF
*> \verbatim
*> LDF is INTEGER
*> The leading dimension of the matrix F. LDF >= max(1, M).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
*> R and L (C and F on entry) will hold the solutions to a
*> slightly perturbed system but the input matrices A, B, D and
*> E have not been changed. If SCALE = 0, R and L will hold the
*> solutions to the homogeneous system with C = F = 0. Normally,
*> SCALE = 1.
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*> RDSUM is DOUBLE PRECISION
*> On entry, the sum of squares of computed contributions to
*> the Dif-estimate under computation by DTGSYL, where the
*> scaling factor RDSCAL (see below) has been factored out.
*> On exit, the corresponding sum of squares updated with the
*> contributions from the current sub-system.
*> If TRANS = 'T' RDSUM is not touched.
*> NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*> RDSCAL is DOUBLE PRECISION
*> On entry, scaling factor used to prevent overflow in RDSUM.
*> On exit, RDSCAL is updated w.r.t. the current contributions
*> in RDSUM.
*> If TRANS = 'T', RDSCAL is not touched.
*> NOTE: RDSCAL only makes sense when DTGSY2 is called by
*> DTGSYL.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (M+N+2)
*> \endverbatim
*>
*> \param[out] PQ
*> \verbatim
*> PQ is INTEGER
*> On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
*> 8-by-8) solved by this routine.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, if INFO is set to
*> =0: Successful exit
*> <0: If INFO = -i, the i-th argument had an illegal value.
*> >0: The matrix pairs (A, D) and (B, E) have common or very
*> close eigenvalues.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsy2
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC,
$ D,
$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
$ IWORK, PQ, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
$ PQ
DOUBLE PRECISION RDSCAL, RDSUM, SCALE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
* ..
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET.
* Sven Hammarling, 27/5/02.
*
* .. Parameters ..
INTEGER LDZ
PARAMETER ( LDZ = 8 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER I, IE, IERR, II, IS, ISP1, J, JE, JJ, JS, JSP1,
$ K, MB, NB, P, Q, ZDIM
DOUBLE PRECISION ALPHA, SCALOC
* ..
* .. Local Arrays ..
INTEGER IPIV( LDZ ), JPIV( LDZ )
DOUBLE PRECISION RHS( LDZ ), Z( LDZ, LDZ )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DGER,
$ DGESC2,
$ DGETC2, DLASET, DLATDF, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode and test input parameters
*
INFO = 0
IERR = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -1
ELSE IF( NOTRAN ) THEN
IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
INFO = -2
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( M.LE.0 ) THEN
INFO = -3
ELSE IF( N.LE.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
INFO = -12
ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
INFO = -16
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSY2', -INFO )
RETURN
END IF
*
* Determine block structure of A
*
PQ = 0
P = 0
I = 1
10 CONTINUE
IF( I.GT.M )
$ GO TO 20
P = P + 1
IWORK( P ) = I
IF( I.EQ.M )
$ GO TO 20
IF( A( I+1, I ).NE.ZERO ) THEN
I = I + 2
ELSE
I = I + 1
END IF
GO TO 10
20 CONTINUE
IWORK( P+1 ) = M + 1
*
* Determine block structure of B
*
Q = P + 1
J = 1
30 CONTINUE
IF( J.GT.N )
$ GO TO 40
Q = Q + 1
IWORK( Q ) = J
IF( J.EQ.N )
$ GO TO 40
IF( B( J+1, J ).NE.ZERO ) THEN
J = J + 2
ELSE
J = J + 1
END IF
GO TO 30
40 CONTINUE
IWORK( Q+1 ) = N + 1
PQ = P*( Q-P-1 )
*
IF( NOTRAN ) THEN
*
* Solve (I, J) - subsystem
* A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
* D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
* for I = P, P - 1, ..., 1; J = 1, 2, ..., Q
*
SCALE = ONE
SCALOC = ONE
DO 120 J = P + 2, Q
JS = IWORK( J )
JSP1 = JS + 1
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
DO 110 I = P, 1, -1
*
IS = IWORK( I )
ISP1 = IS + 1
IE = IWORK( I+1 ) - 1
MB = IE - IS + 1
ZDIM = MB*NB*2
*
IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 2-by-2 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = D( IS, IS )
Z( 1, 2 ) = -B( JS, JS )
Z( 2, 2 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = F( IS, JS )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 50 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
50 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
F( IS, JS ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
ALPHA = -RHS( 1 )
CALL DAXPY( IS-1, ALPHA, A( 1, IS ), 1, C( 1,
$ JS ),
$ 1 )
CALL DAXPY( IS-1, ALPHA, D( 1, IS ), 1, F( 1,
$ JS ),
$ 1 )
END IF
IF( J.LT.Q ) THEN
CALL DAXPY( N-JE, RHS( 2 ), B( JS, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 2 ), E( JS, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build a 4-by-4 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = ZERO
Z( 3, 1 ) = D( IS, IS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = ZERO
Z( 2, 2 ) = A( IS, IS )
Z( 3, 2 ) = ZERO
Z( 4, 2 ) = D( IS, IS )
*
Z( 1, 3 ) = -B( JS, JS )
Z( 2, 3 ) = -B( JS, JSP1 )
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = -E( JS, JSP1 )
*
Z( 1, 4 ) = -B( JSP1, JS )
Z( 2, 4 ) = -B( JSP1, JSP1 )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( IS, JSP1 )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( IS, JSP1 )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 60 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
60 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( IS, JSP1 ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( IS, JSP1 ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGER( IS-1, NB, -ONE, A( 1, IS ), 1,
$ RHS( 1 ),
$ 1, C( 1, JS ), LDC )
CALL DGER( IS-1, NB, -ONE, D( 1, IS ), 1,
$ RHS( 1 ),
$ 1, F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
CALL DAXPY( N-JE, RHS( 3 ), B( JS, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 3 ), E( JS, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
CALL DAXPY( N-JE, RHS( 4 ), B( JSP1, JE+1 ),
$ LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 4 ), E( JSP1, JE+1 ),
$ LDE,
$ F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 4-by-4 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( ISP1, IS )
Z( 3, 1 ) = D( IS, IS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = A( IS, ISP1 )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 3, 2 ) = D( IS, ISP1 )
Z( 4, 2 ) = D( ISP1, ISP1 )
*
Z( 1, 3 ) = -B( JS, JS )
Z( 2, 3 ) = ZERO
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = -B( JS, JS )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( ISP1, JS )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( ISP1, JS )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 70 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
70 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( ISP1, JS ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( ISP1, JS ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGEMV( 'N', IS-1, MB, -ONE, A( 1, IS ),
$ LDA,
$ RHS( 1 ), 1, ONE, C( 1, JS ), 1 )
CALL DGEMV( 'N', IS-1, MB, -ONE, D( 1, IS ),
$ LDD,
$ RHS( 1 ), 1, ONE, F( 1, JS ), 1 )
END IF
IF( J.LT.Q ) THEN
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
$ B( JS, JE+1 ), LDB, C( IS, JE+1 ), LDC )
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
$ E( JS, JE+1 ), LDE, F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build an 8-by-8 system Z * x = RHS
*
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( ISP1, IS )
Z( 5, 1 ) = D( IS, IS )
*
Z( 1, 2 ) = A( IS, ISP1 )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 5, 2 ) = D( IS, ISP1 )
Z( 6, 2 ) = D( ISP1, ISP1 )
*
Z( 3, 3 ) = A( IS, IS )
Z( 4, 3 ) = A( ISP1, IS )
Z( 7, 3 ) = D( IS, IS )
*
Z( 3, 4 ) = A( IS, ISP1 )
Z( 4, 4 ) = A( ISP1, ISP1 )
Z( 7, 4 ) = D( IS, ISP1 )
Z( 8, 4 ) = D( ISP1, ISP1 )
*
Z( 1, 5 ) = -B( JS, JS )
Z( 3, 5 ) = -B( JS, JSP1 )
Z( 5, 5 ) = -E( JS, JS )
Z( 7, 5 ) = -E( JS, JSP1 )
*
Z( 2, 6 ) = -B( JS, JS )
Z( 4, 6 ) = -B( JS, JSP1 )
Z( 6, 6 ) = -E( JS, JS )
Z( 8, 6 ) = -E( JS, JSP1 )
*
Z( 1, 7 ) = -B( JSP1, JS )
Z( 3, 7 ) = -B( JSP1, JSP1 )
Z( 7, 7 ) = -E( JSP1, JSP1 )
*
Z( 2, 8 ) = -B( JSP1, JS )
Z( 4, 8 ) = -B( JSP1, JSP1 )
Z( 8, 8 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
K = 1
II = MB*NB + 1
DO 80 JJ = 0, NB - 1
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ),
$ 1 )
K = K + MB
II = II + MB
80 CONTINUE
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 90 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
90 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
K = 1
II = MB*NB + 1
DO 100 JJ = 0, NB - 1
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ),
$ 1 )
K = K + MB
II = II + MB
100 CONTINUE
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ A( 1, IS ), LDA, RHS( 1 ), MB, ONE,
$ C( 1, JS ), LDC )
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ D( 1, IS ), LDD, RHS( 1 ), MB, ONE,
$ F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
K = MB*NB + 1
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
$ RHS( K ),
$ MB, B( JS, JE+1 ), LDB, ONE,
$ C( IS, JE+1 ), LDC )
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
$ RHS( K ),
$ MB, E( JS, JE+1 ), LDE, ONE,
$ F( IS, JE+1 ), LDF )
END IF
*
END IF
*
110 CONTINUE
120 CONTINUE
ELSE
*
* Solve (I, J) - subsystem
* A(I, I)**T * R(I, J) + D(I, I)**T * L(J, J) = C(I, J)
* R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
* for I = 1, 2, ..., P, J = Q, Q - 1, ..., 1
*
SCALE = ONE
SCALOC = ONE
DO 200 I = 1, P
*
IS = IWORK( I )
ISP1 = IS + 1
IE = IWORK ( I+1 ) - 1
MB = IE - IS + 1
DO 190 J = Q, P + 2, -1
*
JS = IWORK( J )
JSP1 = JS + 1
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
ZDIM = MB*NB*2
IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 2-by-2 system Z**T * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = -B( JS, JS )
Z( 1, 2 ) = D( IS, IS )
Z( 2, 2 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = F( IS, JS )
*
* Solve Z**T * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 130 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
130 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
F( IS, JS ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
ALPHA = RHS( 1 )
CALL DAXPY( JS-1, ALPHA, B( 1, JS ), 1, F( IS,
$ 1 ),
$ LDF )
ALPHA = RHS( 2 )
CALL DAXPY( JS-1, ALPHA, E( 1, JS ), 1, F( IS,
$ 1 ),
$ LDF )
END IF
IF( I.LT.P ) THEN
ALPHA = -RHS( 1 )
CALL DAXPY( M-IE, ALPHA, A( IS, IE+1 ), LDA,
$ C( IE+1, JS ), 1 )
ALPHA = -RHS( 2 )
CALL DAXPY( M-IE, ALPHA, D( IS, IE+1 ), LDD,
$ C( IE+1, JS ), 1 )
END IF
*
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build a 4-by-4 system Z**T * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = ZERO
Z( 3, 1 ) = -B( JS, JS )
Z( 4, 1 ) = -B( JSP1, JS )
*
Z( 1, 2 ) = ZERO
Z( 2, 2 ) = A( IS, IS )
Z( 3, 2 ) = -B( JS, JSP1 )
Z( 4, 2 ) = -B( JSP1, JSP1 )
*
Z( 1, 3 ) = D( IS, IS )
Z( 2, 3 ) = ZERO
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = D( IS, IS )
Z( 3, 4 ) = -E( JS, JSP1 )
Z( 4, 4 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( IS, JSP1 )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( IS, JSP1 )
*
* Solve Z**T * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 140 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
140 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( IS, JSP1 ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( IS, JSP1 ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DAXPY( JS-1, RHS( 1 ), B( 1, JS ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 2 ), B( 1, JSP1 ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 3 ), E( 1, JS ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 4 ), E( 1, JSP1 ), 1,
$ F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGER( M-IE, NB, -ONE, A( IS, IE+1 ), LDA,
$ RHS( 1 ), 1, C( IE+1, JS ), LDC )
CALL DGER( M-IE, NB, -ONE, D( IS, IE+1 ), LDD,
$ RHS( 3 ), 1, C( IE+1, JS ), LDC )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 4-by-4 system Z**T * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( IS, ISP1 )
Z( 3, 1 ) = -B( JS, JS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = A( ISP1, IS )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 3, 2 ) = ZERO
Z( 4, 2 ) = -B( JS, JS )
*
Z( 1, 3 ) = D( IS, IS )
Z( 2, 3 ) = D( IS, ISP1 )
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = D( ISP1, ISP1 )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( ISP1, JS )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( ISP1, JS )
*
* Solve Z**T * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 150 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
150 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( ISP1, JS ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( ISP1, JS ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DGER( MB, JS-1, ONE, RHS( 1 ), 1, B( 1,
$ JS ),
$ 1, F( IS, 1 ), LDF )
CALL DGER( MB, JS-1, ONE, RHS( 3 ), 1, E( 1,
$ JS ),
$ 1, F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGEMV( 'T', MB, M-IE, -ONE, A( IS, IE+1 ),
$ LDA, RHS( 1 ), 1, ONE, C( IE+1, JS ),
$ 1 )
CALL DGEMV( 'T', MB, M-IE, -ONE, D( IS, IE+1 ),
$ LDD, RHS( 3 ), 1, ONE, C( IE+1, JS ),
$ 1 )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build an 8-by-8 system Z**T * x = RHS
*
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( IS, ISP1 )
Z( 5, 1 ) = -B( JS, JS )
Z( 7, 1 ) = -B( JSP1, JS )
*
Z( 1, 2 ) = A( ISP1, IS )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 6, 2 ) = -B( JS, JS )
Z( 8, 2 ) = -B( JSP1, JS )
*
Z( 3, 3 ) = A( IS, IS )
Z( 4, 3 ) = A( IS, ISP1 )
Z( 5, 3 ) = -B( JS, JSP1 )
Z( 7, 3 ) = -B( JSP1, JSP1 )
*
Z( 3, 4 ) = A( ISP1, IS )
Z( 4, 4 ) = A( ISP1, ISP1 )
Z( 6, 4 ) = -B( JS, JSP1 )
Z( 8, 4 ) = -B( JSP1, JSP1 )
*
Z( 1, 5 ) = D( IS, IS )
Z( 2, 5 ) = D( IS, ISP1 )
Z( 5, 5 ) = -E( JS, JS )
*
Z( 2, 6 ) = D( ISP1, ISP1 )
Z( 6, 6 ) = -E( JS, JS )
*
Z( 3, 7 ) = D( IS, IS )
Z( 4, 7 ) = D( IS, ISP1 )
Z( 5, 7 ) = -E( JS, JSP1 )
Z( 7, 7 ) = -E( JSP1, JSP1 )
*
Z( 4, 8 ) = D( ISP1, ISP1 )
Z( 6, 8 ) = -E( JS, JSP1 )
Z( 8, 8 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
K = 1
II = MB*NB + 1
DO 160 JJ = 0, NB - 1
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ),
$ 1 )
K = K + MB
II = II + MB
160 CONTINUE
*
*
* Solve Z**T * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 170 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
170 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
K = 1
II = MB*NB + 1
DO 180 JJ = 0, NB - 1
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ),
$ 1 )
K = K + MB
II = II + MB
180 CONTINUE
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
$ C( IS, JS ), LDC, B( 1, JS ), LDB, ONE,
$ F( IS, 1 ), LDF )
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
$ F( IS, JS ), LDF, E( 1, JS ), LDE, ONE,
$ F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ A( IS, IE+1 ), LDA, C( IS, JS ), LDC,
$ ONE, C( IE+1, JS ), LDC )
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ D( IS, IE+1 ), LDD, F( IS, JS ), LDF,
$ ONE, C( IE+1, JS ), LDC )
END IF
*
END IF
*
190 CONTINUE
200 CONTINUE
*
END IF
RETURN
*
* End of DTGSY2
*
END
*> \brief \b DTGSYL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSYL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsyl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsyl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsyl.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
* LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
* $ LWORK, M, N
* DOUBLE PRECISION DIF, SCALE
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
* $ D( LDD, * ), E( LDE, * ), F( LDF, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGSYL solves the generalized Sylvester equation:
*>
*> A * R - L * B = scale * C (1)
*> D * R - L * E = scale * F
*>
*> where R and L are unknown m-by-n matrices, (A, D), (B, E) and
*> (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
*> respectively, with real entries. (A, D) and (B, E) must be in
*> generalized (real) Schur canonical form, i.e. A, B are upper quasi
*> triangular and D, E are upper triangular.
*>
*> The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
*> scaling factor chosen to avoid overflow.
*>
*> In matrix notation (1) is equivalent to solve Zx = scale b, where
*> Z is defined as
*>
*> Z = [ kron(In, A) -kron(B**T, Im) ] (2)
*> [ kron(In, D) -kron(E**T, Im) ].
*>
*> Here Ik is the identity matrix of size k and X**T is the transpose of
*> X. kron(X, Y) is the Kronecker product between the matrices X and Y.
*>
*> If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
*> which is equivalent to solve for R and L in
*>
*> A**T * R + D**T * L = scale * C (3)
*> R * B**T + L * E**T = scale * -F
*>
*> This case (TRANS = 'T') is used to compute an one-norm-based estimate
*> of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
*> and (B,E), using DLACON.
*>
*> If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
*> of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
*> reciprocal of the smallest singular value of Z. See [1-2] for more
*> information.
*>
*> This is a level 3 BLAS algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': solve the generalized Sylvester equation (1).
*> = 'T': solve the 'transposed' system (3).
*> \endverbatim
*>
*> \param[in] IJOB
*> \verbatim
*> IJOB is INTEGER
*> Specifies what kind of functionality to be performed.
*> = 0: solve (1) only.
*> = 1: The functionality of 0 and 3.
*> = 2: The functionality of 0 and 4.
*> = 3: Only an estimate of Dif[(A,D), (B,E)] is computed.
*> (look ahead strategy IJOB = 1 is used).
*> = 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
*> ( DGECON on sub-systems is used ).
*> Not referenced if TRANS = 'T'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The order of the matrices A and D, and the row dimension of
*> the matrices C, F, R and L.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices B and E, and the column dimension
*> of the matrices C, F, R and L.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, M)
*> The upper quasi triangular matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1, M).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> The upper quasi triangular matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC, N)
*> On entry, C contains the right-hand-side of the first matrix
*> equation in (1) or (3).
*> On exit, if IJOB = 0, 1 or 2, C has been overwritten by
*> the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
*> the solution achieved during the computation of the
*> Dif-estimate.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1, M).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (LDD, M)
*> The upper triangular matrix D.
*> \endverbatim
*>
*> \param[in] LDD
*> \verbatim
*> LDD is INTEGER
*> The leading dimension of the array D. LDD >= max(1, M).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (LDE, N)
*> The upper triangular matrix E.
*> \endverbatim
*>
*> \param[in] LDE
*> \verbatim
*> LDE is INTEGER
*> The leading dimension of the array E. LDE >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] F
*> \verbatim
*> F is DOUBLE PRECISION array, dimension (LDF, N)
*> On entry, F contains the right-hand-side of the second matrix
*> equation in (1) or (3).
*> On exit, if IJOB = 0, 1 or 2, F has been overwritten by
*> the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
*> the solution achieved during the computation of the
*> Dif-estimate.
*> \endverbatim
*>
*> \param[in] LDF
*> \verbatim
*> LDF is INTEGER
*> The leading dimension of the array F. LDF >= max(1, M).
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*> DIF is DOUBLE PRECISION
*> On exit DIF is the reciprocal of a lower bound of the
*> reciprocal of the Dif-function, i.e. DIF is an upper bound of
*> Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
*> IF IJOB = 0 or TRANS = 'T', DIF is not touched.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> On exit SCALE is the scaling factor in (1) or (3).
*> If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
*> to a slightly perturbed system but the input matrices A, B, D
*> and E have not been changed. If SCALE = 0, C and F hold the
*> solutions R and L, respectively, to the homogeneous system
*> with C = F = 0. Normally, SCALE = 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK > = 1.
*> If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (M+N+6)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: successful exit
*> <0: If INFO = -i, the i-th argument had an illegal value.
*> >0: (A, D) and (B, E) have common or close eigenvalues.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsyl
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*> for Solving the Generalized Sylvester Equation and Estimating the
*> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*> Department of Computing Science, Umea University, S-901 87 Umea,
*> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
*> Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
*> No 1, 1996.
*>
*> [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
*> Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
*> Appl., 15(4):1045-1060, 1994
*>
*> [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
*> Condition Estimators for Solving the Generalized Sylvester
*> Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
*> July 1989, pp 745-751.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC,
$ D,
$ LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
$ LWORK, M, N
DOUBLE PRECISION DIF, SCALE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * ),
$ WORK( * )
* ..
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET.
* Sven Hammarling, 1/5/02.
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, NOTRAN
INTEGER I, IE, IFUNC, IROUND, IS, ISOLVE, J, JE, JS, K,
$ LINFO, LWMIN, MB, NB, P, PPQQ, PQ, Q
DOUBLE PRECISION DSCALE, DSUM, SCALE2, SCALOC
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, DSCAL, DTGSY2,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test input parameters
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -1
ELSE IF( NOTRAN ) THEN
IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.4 ) ) THEN
INFO = -2
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( M.LE.0 ) THEN
INFO = -3
ELSE IF( N.LE.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
INFO = -12
ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
INFO = -16
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
IF( NOTRAN ) THEN
IF( IJOB.EQ.1 .OR. IJOB.EQ.2 ) THEN
LWMIN = MAX( 1, 2*M*N )
ELSE
LWMIN = 1
END IF
ELSE
LWMIN = 1
END IF
WORK( 1 ) = LWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -20
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSYL', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
SCALE = 1
IF( NOTRAN ) THEN
IF( IJOB.NE.0 ) THEN
DIF = 0
END IF
END IF
RETURN
END IF
*
* Determine optimal block sizes MB and NB
*
MB = ILAENV( 2, 'DTGSYL', TRANS, M, N, -1, -1 )
NB = ILAENV( 5, 'DTGSYL', TRANS, M, N, -1, -1 )
*
ISOLVE = 1
IFUNC = 0
IF( NOTRAN ) THEN
IF( IJOB.GE.3 ) THEN
IFUNC = IJOB - 2
CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
ELSE IF( IJOB.GE.1 ) THEN
ISOLVE = 2
END IF
END IF
*
IF( ( MB.LE.1 .AND. NB.LE.1 ) .OR. ( MB.GE.M .AND. NB.GE.N ) )
$ THEN
*
DO 30 IROUND = 1, ISOLVE
*
* Use unblocked Level 2 solver
*
DSCALE = ZERO
DSUM = ONE
PQ = 0
CALL DTGSY2( TRANS, IFUNC, M, N, A, LDA, B, LDB, C, LDC,
$ D,
$ LDD, E, LDE, F, LDF, SCALE, DSUM, DSCALE,
$ IWORK, PQ, INFO )
IF( DSCALE.NE.ZERO ) THEN
IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
ELSE
DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
END IF
END IF
*
IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
IF( NOTRAN ) THEN
IFUNC = IJOB
END IF
SCALE2 = SCALE
CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
SCALE = SCALE2
END IF
30 CONTINUE
*
RETURN
END IF
*
* Determine block structure of A
*
P = 0
I = 1
40 CONTINUE
IF( I.GT.M )
$ GO TO 50
P = P + 1
IWORK( P ) = I
I = I + MB
IF( I.GE.M )
$ GO TO 50
IF( A( I, I-1 ).NE.ZERO )
$ I = I + 1
GO TO 40
50 CONTINUE
*
IWORK( P+1 ) = M + 1
IF( IWORK( P ).EQ.IWORK( P+1 ) )
$ P = P - 1
*
* Determine block structure of B
*
Q = P + 1
J = 1
60 CONTINUE
IF( J.GT.N )
$ GO TO 70
Q = Q + 1
IWORK( Q ) = J
J = J + NB
IF( J.GE.N )
$ GO TO 70
IF( B( J, J-1 ).NE.ZERO )
$ J = J + 1
GO TO 60
70 CONTINUE
*
IWORK( Q+1 ) = N + 1
IF( IWORK( Q ).EQ.IWORK( Q+1 ) )
$ Q = Q - 1
*
IF( NOTRAN ) THEN
*
DO 150 IROUND = 1, ISOLVE
*
* Solve (I, J)-subsystem
* A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
* D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
* for I = P, P - 1,..., 1; J = 1, 2,..., Q
*
DSCALE = ZERO
DSUM = ONE
PQ = 0
SCALE = ONE
DO 130 J = P + 2, Q
JS = IWORK( J )
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
DO 120 I = P, 1, -1
IS = IWORK( I )
IE = IWORK( I+1 ) - 1
MB = IE - IS + 1
PPQQ = 0
CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ),
$ LDA,
$ B( JS, JS ), LDB, C( IS, JS ), LDC,
$ D( IS, IS ), LDD, E( JS, JS ), LDE,
$ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
$ IWORK( Q+2 ), PPQQ, LINFO )
IF( LINFO.GT.0 )
$ INFO = LINFO
*
PQ = PQ + PPQQ
IF( SCALOC.NE.ONE ) THEN
DO 80 K = 1, JS - 1
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
80 CONTINUE
DO 90 K = JS, JE
CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
90 CONTINUE
DO 100 K = JS, JE
CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
100 CONTINUE
DO 110 K = JE + 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
110 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ A( 1, IS ), LDA, C( IS, JS ), LDC, ONE,
$ C( 1, JS ), LDC )
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ D( 1, IS ), LDD, C( IS, JS ), LDC, ONE,
$ F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
$ F( IS, JS ), LDF, B( JS, JE+1 ), LDB,
$ ONE, C( IS, JE+1 ), LDC )
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
$ F( IS, JS ), LDF, E( JS, JE+1 ), LDE,
$ ONE, F( IS, JE+1 ), LDF )
END IF
120 CONTINUE
130 CONTINUE
IF( DSCALE.NE.ZERO ) THEN
IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
ELSE
DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
END IF
END IF
IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
IF( NOTRAN ) THEN
IFUNC = IJOB
END IF
SCALE2 = SCALE
CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
SCALE = SCALE2
END IF
150 CONTINUE
*
ELSE
*
* Solve transposed (I, J)-subsystem
* A(I, I)**T * R(I, J) + D(I, I)**T * L(I, J) = C(I, J)
* R(I, J) * B(J, J)**T + L(I, J) * E(J, J)**T = -F(I, J)
* for I = 1,2,..., P; J = Q, Q-1,..., 1
*
SCALE = ONE
DO 210 I = 1, P
IS = IWORK( I )
IE = IWORK( I+1 ) - 1
MB = IE - IS + 1
DO 200 J = Q, P + 2, -1
JS = IWORK( J )
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
$ B( JS, JS ), LDB, C( IS, JS ), LDC,
$ D( IS, IS ), LDD, E( JS, JS ), LDE,
$ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
$ IWORK( Q+2 ), PPQQ, LINFO )
IF( LINFO.GT.0 )
$ INFO = LINFO
IF( SCALOC.NE.ONE ) THEN
DO 160 K = 1, JS - 1
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
160 CONTINUE
DO 170 K = JS, JE
CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
170 CONTINUE
DO 180 K = JS, JE
CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
180 CONTINUE
DO 190 K = JE + 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
190 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Substitute R(I, J) and L(I, J) into remaining equation.
*
IF( J.GT.P+2 ) THEN
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, C( IS,
$ JS ),
$ LDC, B( 1, JS ), LDB, ONE, F( IS, 1 ),
$ LDF )
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, F( IS,
$ JS ),
$ LDF, E( 1, JS ), LDE, ONE, F( IS, 1 ),
$ LDF )
END IF
IF( I.LT.P ) THEN
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ A( IS, IE+1 ), LDA, C( IS, JS ), LDC, ONE,
$ C( IE+1, JS ), LDC )
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ D( IS, IE+1 ), LDD, F( IS, JS ), LDF, ONE,
$ C( IE+1, JS ), LDC )
END IF
200 CONTINUE
210 CONTINUE
*
END IF
*
WORK( 1 ) = LWMIN
*
RETURN
*
* End of DTGSYL
*
END
*> \brief \b DTPCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPCON( NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORM, UPLO
* INTEGER INFO, N
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPCON estimates the reciprocal of the condition number of a packed
*> triangular matrix A, in either the 1-norm or the infinity-norm.
*>
*> The norm of A is computed and an estimate is obtained for
*> norm(inv(A)), then the reciprocal of the condition number is
*> computed as
*> RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies whether the 1-norm condition number or the
*> infinity-norm condition number is required:
*> = '1' or 'O': 1-norm;
*> = 'I': Infinity-norm.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangular matrix A, packed columnwise in
*> a linear array. The j-th column of A is stored in the array
*> AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> If DIAG = 'U', the diagonal elements of A are not referenced
*> and are assumed to be 1.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpcon
*
* =====================================================================
SUBROUTINE DTPCON( NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER INFO, N
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, ONENRM, UPPER
CHARACTER NORMIN
INTEGER IX, KASE, KASE1
DOUBLE PRECISION AINVNM, ANORM, SCALE, SMLNUM, XNORM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLANTP
EXTERNAL LSAME, IDAMAX, DLAMCH, DLANTP
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLATPS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
*
RCOND = ZERO
SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) )
*
* Compute the norm of the triangular matrix A.
*
ANORM = DLANTP( NORM, UPLO, DIAG, N, AP, WORK )
*
* Continue only if ANORM > 0.
*
IF( ANORM.GT.ZERO ) THEN
*
* Estimate the norm of the inverse of A.
*
AINVNM = ZERO
NORMIN = 'N'
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KASE = 0
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE,
$ ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(A).
*
CALL DLATPS( UPLO, 'No transpose', DIAG, NORMIN, N,
$ AP,
$ WORK, SCALE, WORK( 2*N+1 ), INFO )
ELSE
*
* Multiply by inv(A**T).
*
CALL DLATPS( UPLO, 'Transpose', DIAG, NORMIN, N, AP,
$ WORK, SCALE, WORK( 2*N+1 ), INFO )
END IF
NORMIN = 'Y'
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
XNORM = ABS( WORK( IX ) )
IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / ANORM ) / AINVNM
END IF
*
20 CONTINUE
RETURN
*
* End of DTPCON
*
END
*> \brief \b DTPMQRT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPMQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpmqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpmqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpmqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPMQRT( SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT,
* A, LDA, B, LDB, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDV, LDA, LDB, M, N, L, NB, LDT
* ..
* .. Array Arguments ..
* DOUBLE PRECISION V( LDV, * ), A( LDA, * ), B( LDB, * ),
* $ T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPMQRT applies a real orthogonal matrix Q obtained from a
*> "triangular-pentagonal" real block reflector H to a general
*> real matrix C, which consists of two blocks A and B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix B. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix B. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The order of the trapezoidal part of V.
*> K >= L >= 0. See Further Details.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The block size used for the storage of T. K >= NB >= 1.
*> This must be the same value of NB used to generate T
*> in CTPQRT.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> CTPQRT in B. See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If SIDE = 'L', LDV >= max(1,M);
*> if SIDE = 'R', LDV >= max(1,N).
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The upper triangular factors of the block reflectors
*> as returned by CTPQRT, stored as a NB-by-K matrix.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,N) if SIDE = 'L' or
*> (LDA,K) if SIDE = 'R'
*> On entry, the K-by-N or M-by-K matrix A.
*> On exit, A is overwritten by the corresponding block of
*> Q*C or Q**T*C or C*Q or C*Q**T. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDC >= max(1,K);
*> If SIDE = 'R', LDC >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the M-by-N matrix B.
*> On exit, B is overwritten by the corresponding block of
*> Q*C or Q**T*C or C*Q or C*Q**T. See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B.
*> LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array. The dimension of WORK is
*> N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpmqrt
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The columns of the pentagonal matrix V contain the elementary reflectors
*> H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
*> trapezoidal block V2:
*>
*> V = [V1]
*> [V2].
*>
*> The size of the trapezoidal block V2 is determined by the parameter L,
*> where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L
*> rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular;
*> if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
*>
*> If SIDE = 'L': C = [A] where A is K-by-N, B is M-by-N and V is M-by-K.
*> [B]
*>
*> If SIDE = 'R': C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K.
*>
*> The real orthogonal matrix Q is formed from V and T.
*>
*> If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
*>
*> If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C.
*>
*> If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
*>
*> If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTPMQRT( SIDE, TRANS, M, N, K, L, NB, V, LDV, T,
$ LDT,
$ A, LDA, B, LDB, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDV, LDA, LDB, M, N, L, NB, LDT
* ..
* .. Array Arguments ..
DOUBLE PRECISION V( LDV, * ), A( LDA, * ), B( LDB, * ),
$ T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* ..
* .. Local Scalars ..
LOGICAL LEFT, RIGHT, TRAN, NOTRAN
INTEGER I, IB, MB, LB, KF, LDAQ, LDVQ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DTPRFB, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* .. Test the input arguments ..
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
RIGHT = LSAME( SIDE, 'R' )
TRAN = LSAME( TRANS, 'T' )
NOTRAN = LSAME( TRANS, 'N' )
*
IF ( LEFT ) THEN
LDVQ = MAX( 1, M )
LDAQ = MAX( 1, K )
ELSE IF ( RIGHT ) THEN
LDVQ = MAX( 1, N )
LDAQ = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.RIGHT ) THEN
INFO = -1
ELSE IF( .NOT.TRAN .AND. .NOT.NOTRAN ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 ) THEN
INFO = -5
ELSE IF( L.LT.0 .OR. L.GT.K ) THEN
INFO = -6
ELSE IF( NB.LT.1 .OR. (NB.GT.K .AND. K.GT.0) ) THEN
INFO = -7
ELSE IF( LDV.LT.LDVQ ) THEN
INFO = -9
ELSE IF( LDT.LT.NB ) THEN
INFO = -11
ELSE IF( LDA.LT.LDAQ ) THEN
INFO = -13
ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
INFO = -15
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPMQRT', -INFO )
RETURN
END IF
*
* .. Quick return if possible ..
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) RETURN
*
IF( LEFT .AND. TRAN ) THEN
*
DO I = 1, K, NB
IB = MIN( NB, K-I+1 )
MB = MIN( M-L+I+IB-1, M )
IF( I.GE.L ) THEN
LB = 0
ELSE
LB = MB-M+L-I+1
END IF
CALL DTPRFB( 'L', 'T', 'F', 'C', MB, N, IB, LB,
$ V( 1, I ), LDV, T( 1, I ), LDT,
$ A( I, 1 ), LDA, B, LDB, WORK, IB )
END DO
*
ELSE IF( RIGHT .AND. NOTRAN ) THEN
*
DO I = 1, K, NB
IB = MIN( NB, K-I+1 )
MB = MIN( N-L+I+IB-1, N )
IF( I.GE.L ) THEN
LB = 0
ELSE
LB = MB-N+L-I+1
END IF
CALL DTPRFB( 'R', 'N', 'F', 'C', M, MB, IB, LB,
$ V( 1, I ), LDV, T( 1, I ), LDT,
$ A( 1, I ), LDA, B, LDB, WORK, M )
END DO
*
ELSE IF( LEFT .AND. NOTRAN ) THEN
*
KF = ((K-1)/NB)*NB+1
DO I = KF, 1, -NB
IB = MIN( NB, K-I+1 )
MB = MIN( M-L+I+IB-1, M )
IF( I.GE.L ) THEN
LB = 0
ELSE
LB = MB-M+L-I+1
END IF
CALL DTPRFB( 'L', 'N', 'F', 'C', MB, N, IB, LB,
$ V( 1, I ), LDV, T( 1, I ), LDT,
$ A( I, 1 ), LDA, B, LDB, WORK, IB )
END DO
*
ELSE IF( RIGHT .AND. TRAN ) THEN
*
KF = ((K-1)/NB)*NB+1
DO I = KF, 1, -NB
IB = MIN( NB, K-I+1 )
MB = MIN( N-L+I+IB-1, N )
IF( I.GE.L ) THEN
LB = 0
ELSE
LB = MB-N+L-I+1
END IF
CALL DTPRFB( 'R', 'T', 'F', 'C', M, MB, IB, LB,
$ V( 1, I ), LDV, T( 1, I ), LDT,
$ A( 1, I ), LDA, B, LDB, WORK, M )
END DO
*
END IF
*
RETURN
*
* End of DTPMQRT
*
END
*> \brief \b DTPQRT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPQRT( M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LDT, N, M, L, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPQRT computes a blocked QR factorization of a real
*> "triangular-pentagonal" matrix C, which is composed of a
*> triangular block A and pentagonal block B, using the compact
*> WY representation for Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix B.
*> M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix B, and the order of the
*> triangular matrix A.
*> N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of rows of the upper trapezoidal part of B.
*> MIN(M,N) >= L >= 0. See Further Details.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The block size to be used in the blocked QR. N >= NB >= 1.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the upper triangular N-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the pentagonal M-by-N matrix B. The first M-L rows
*> are rectangular, and the last L rows are upper trapezoidal.
*> On exit, B contains the pentagonal matrix V. See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The upper triangular block reflectors stored in compact form
*> as a sequence of upper triangular blocks. See Further Details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (NB*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpqrt
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The input matrix C is a (N+M)-by-N matrix
*>
*> C = [ A ]
*> [ B ]
*>
*> where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
*> matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
*> upper trapezoidal matrix B2:
*>
*> B = [ B1 ] <- (M-L)-by-N rectangular
*> [ B2 ] <- L-by-N upper trapezoidal.
*>
*> The upper trapezoidal matrix B2 consists of the first L rows of a
*> N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
*> B is rectangular M-by-N; if M=L=N, B is upper triangular.
*>
*> The matrix W stores the elementary reflectors H(i) in the i-th column
*> below the diagonal (of A) in the (N+M)-by-N input matrix C
*>
*> C = [ A ] <- upper triangular N-by-N
*> [ B ] <- M-by-N pentagonal
*>
*> so that W can be represented as
*>
*> W = [ I ] <- identity, N-by-N
*> [ V ] <- M-by-N, same form as B.
*>
*> Thus, all of information needed for W is contained on exit in B, which
*> we call V above. Note that V has the same form as B; that is,
*>
*> V = [ V1 ] <- (M-L)-by-N rectangular
*> [ V2 ] <- L-by-N upper trapezoidal.
*>
*> The columns of V represent the vectors which define the H(i)'s.
*>
*> The number of blocks is B = ceiling(N/NB), where each
*> block is of order NB except for the last block, which is of order
*> IB = N - (B-1)*NB. For each of the B blocks, a upper triangular block
*> reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
*> for the last block) T's are stored in the NB-by-N matrix T as
*>
*> T = [T1 T2 ... TB].
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTPQRT( M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDT, N, M, L, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* ..
* .. Local Scalars ..
INTEGER I, IB, LB, MB, IINFO
* ..
* .. External Subroutines ..
EXTERNAL DTPQRT2, DTPRFB, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( L.LT.0 .OR. (L.GT.MIN(M,N) .AND. MIN(M,N).GE.0)) THEN
INFO = -3
ELSE IF( NB.LT.1 .OR. (NB.GT.N .AND. N.GT.0)) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
INFO = -8
ELSE IF( LDT.LT.NB ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPQRT', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) RETURN
*
DO I = 1, N, NB
*
* Compute the QR factorization of the current block
*
IB = MIN( N-I+1, NB )
MB = MIN( M-L+I+IB-1, M )
IF( I.GE.L ) THEN
LB = 0
ELSE
LB = MB-M+L-I+1
END IF
*
CALL DTPQRT2( MB, IB, LB, A(I,I), LDA, B( 1, I ), LDB,
$ T(1, I ), LDT, IINFO )
*
* Update by applying H**T to B(:,I+IB:N) from the left
*
IF( I+IB.LE.N ) THEN
CALL DTPRFB( 'L', 'T', 'F', 'C', MB, N-I-IB+1, IB, LB,
$ B( 1, I ), LDB, T( 1, I ), LDT,
$ A( I, I+IB ), LDA, B( 1, I+IB ), LDB,
$ WORK, IB )
END IF
END DO
RETURN
*
* End of DTPQRT
*
END
*> \brief \b DTPQRT2 computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPQRT2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpqrt2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpqrt2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpqrt2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPQRT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LDT, N, M, L
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPQRT2 computes a QR factorization of a real "triangular-pentagonal"
*> matrix C, which is composed of a triangular block A and pentagonal block B,
*> using the compact WY representation for Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The total number of rows of the matrix B.
*> M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix B, and the order of
*> the triangular matrix A.
*> N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of rows of the upper trapezoidal part of B.
*> MIN(M,N) >= L >= 0. See Further Details.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the upper triangular N-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the pentagonal M-by-N matrix B. The first M-L rows
*> are rectangular, and the last L rows are upper trapezoidal.
*> On exit, B contains the pentagonal matrix V. See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The N-by-N upper triangular factor T of the block reflector.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpqrt2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The input matrix C is a (N+M)-by-N matrix
*>
*> C = [ A ]
*> [ B ]
*>
*> where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
*> matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
*> upper trapezoidal matrix B2:
*>
*> B = [ B1 ] <- (M-L)-by-N rectangular
*> [ B2 ] <- L-by-N upper trapezoidal.
*>
*> The upper trapezoidal matrix B2 consists of the first L rows of a
*> N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
*> B is rectangular M-by-N; if M=L=N, B is upper triangular.
*>
*> The matrix W stores the elementary reflectors H(i) in the i-th column
*> below the diagonal (of A) in the (N+M)-by-N input matrix C
*>
*> C = [ A ] <- upper triangular N-by-N
*> [ B ] <- M-by-N pentagonal
*>
*> so that W can be represented as
*>
*> W = [ I ] <- identity, N-by-N
*> [ V ] <- M-by-N, same form as B.
*>
*> Thus, all of information needed for W is contained on exit in B, which
*> we call V above. Note that V has the same form as B; that is,
*>
*> V = [ V1 ] <- (M-L)-by-N rectangular
*> [ V2 ] <- L-by-N upper trapezoidal.
*>
*> The columns of V represent the vectors which define the H(i)'s.
*> The (M+N)-by-(M+N) block reflector H is then given by
*>
*> H = I - W * T * W**T
*>
*> where W^H is the conjugate transpose of W and T is the upper triangular
*> factor of the block reflector.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTPQRT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDT, N, M, L
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER( ONE = 1.0, ZERO = 0.0 )
* ..
* .. Local Scalars ..
INTEGER I, J, P, MP, NP
DOUBLE PRECISION ALPHA
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DGEMV, DGER, DTRMV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( L.LT.0 .OR. L.GT.MIN(M,N) ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
INFO = -7
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPQRT2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 ) RETURN
*
DO I = 1, N
*
* Generate elementary reflector H(I) to annihilate B(:,I)
*
P = M-L+MIN( L, I )
CALL DLARFG( P+1, A( I, I ), B( 1, I ), 1, T( I, 1 ) )
IF( I.LT.N ) THEN
*
* W(1:N-I) := C(I:M,I+1:N)^H * C(I:M,I) [use W = T(:,N)]
*
DO J = 1, N-I
T( J, N ) = (A( I, I+J ))
END DO
CALL DGEMV( 'T', P, N-I, ONE, B( 1, I+1 ), LDB,
$ B( 1, I ), 1, ONE, T( 1, N ), 1 )
*
* C(I:M,I+1:N) = C(I:m,I+1:N) + alpha*C(I:M,I)*W(1:N-1)^H
*
ALPHA = -(T( I, 1 ))
DO J = 1, N-I
A( I, I+J ) = A( I, I+J ) + ALPHA*(T( J, N ))
END DO
CALL DGER( P, N-I, ALPHA, B( 1, I ), 1,
$ T( 1, N ), 1, B( 1, I+1 ), LDB )
END IF
END DO
*
DO I = 2, N
*
* T(1:I-1,I) := C(I:M,1:I-1)^H * (alpha * C(I:M,I))
*
ALPHA = -T( I, 1 )
DO J = 1, I-1
T( J, I ) = ZERO
END DO
P = MIN( I-1, L )
MP = MIN( M-L+1, M )
NP = MIN( P+1, N )
*
* Triangular part of B2
*
DO J = 1, P
T( J, I ) = ALPHA*B( M-L+J, I )
END DO
CALL DTRMV( 'U', 'T', 'N', P, B( MP, 1 ), LDB,
$ T( 1, I ), 1 )
*
* Rectangular part of B2
*
CALL DGEMV( 'T', L, I-1-P, ALPHA, B( MP, NP ), LDB,
$ B( MP, I ), 1, ZERO, T( NP, I ), 1 )
*
* B1
*
CALL DGEMV( 'T', M-L, I-1, ALPHA, B, LDB, B( 1, I ), 1,
$ ONE, T( 1, I ), 1 )
*
* T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
*
CALL DTRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 )
*
* T(I,I) = tau(I)
*
T( I, I ) = T( I, 1 )
T( I, 1 ) = ZERO
END DO
*
* End of DTPQRT2
*
END
*> \brief \b DTPRFB applies a real "triangular-pentagonal" block reflector to a real matrix, which is composed of two blocks.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPRFB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtprfb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtprfb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtprfb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPRFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L,
* V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, SIDE, STOREV, TRANS
* INTEGER K, L, LDA, LDB, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * ),
* $ V( LDV, * ), WORK( LDWORK, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPRFB applies a real "triangular-pentagonal" block reflector H or its
*> transpose H**T to a real matrix C, which is composed of two
*> blocks A and B, either from the left or right.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply H or H**T from the Left
*> = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply H (No transpose)
*> = 'T': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Indicates how H is formed from a product of elementary
*> reflectors
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Indicates how the vectors which define the elementary
*> reflectors are stored:
*> = 'C': Columns
*> = 'R': Rows
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix B.
*> M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix B.
*> N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the matrix T, i.e. the number of elementary
*> reflectors whose product defines the block reflector.
*> K >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The order of the trapezoidal part of V.
*> K >= L >= 0. See Further Details.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,M) if STOREV = 'R' and SIDE = 'L'
*> (LDV,N) if STOREV = 'R' and SIDE = 'R'
*> The pentagonal matrix V, which contains the elementary reflectors
*> H(1), H(2), ..., H(K). See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
*> if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
*> if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The triangular K-by-K matrix T in the representation of the
*> block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T.
*> LDT >= K.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'
*> On entry, the K-by-N or M-by-K matrix A.
*> On exit, A is overwritten by the corresponding block of
*> H*C or H**T*C or C*H or C*H**T. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,K);
*> If SIDE = 'R', LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the M-by-N matrix B.
*> On exit, B is overwritten by the corresponding block of
*> H*C or H**T*C or C*H or C*H**T. See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B.
*> LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (LDWORK,N) if SIDE = 'L',
*> (LDWORK,K) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*> LDWORK is INTEGER
*> The leading dimension of the array WORK.
*> If SIDE = 'L', LDWORK >= K;
*> if SIDE = 'R', LDWORK >= M.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tprfb
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix C is a composite matrix formed from blocks A and B.
*> The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
*> and if SIDE = 'L', A is of size K-by-N.
*>
*> If SIDE = 'R' and DIRECT = 'F', C = [A B].
*>
*> If SIDE = 'L' and DIRECT = 'F', C = [A]
*> [B].
*>
*> If SIDE = 'R' and DIRECT = 'B', C = [B A].
*>
*> If SIDE = 'L' and DIRECT = 'B', C = [B]
*> [A].
*>
*> The pentagonal matrix V is composed of a rectangular block V1 and a
*> trapezoidal block V2. The size of the trapezoidal block is determined by
*> the parameter L, where 0<=L<=K. If L=K, the V2 block of V is triangular;
*> if L=0, there is no trapezoidal block, thus V = V1 is rectangular.
*>
*> If DIRECT = 'F' and STOREV = 'C': V = [V1]
*> [V2]
*> - V2 is upper trapezoidal (first L rows of K-by-K upper triangular)
*>
*> If DIRECT = 'F' and STOREV = 'R': V = [V1 V2]
*>
*> - V2 is lower trapezoidal (first L columns of K-by-K lower triangular)
*>
*> If DIRECT = 'B' and STOREV = 'C': V = [V2]
*> [V1]
*> - V2 is lower trapezoidal (last L rows of K-by-K lower triangular)
*>
*> If DIRECT = 'B' and STOREV = 'R': V = [V2 V1]
*>
*> - V2 is upper trapezoidal (last L columns of K-by-K upper triangular)
*>
*> If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.
*>
*> If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.
*>
*> If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.
*>
*> If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTPRFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L,
$ V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, SIDE, STOREV, TRANS
INTEGER K, L, LDA, LDB, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * ),
$ V( LDV, * ), WORK( LDWORK, * )
* ..
*
* ==========================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0, ZERO = 0.0 )
* ..
* .. Local Scalars ..
INTEGER I, J, MP, NP, KP
LOGICAL LEFT, FORWARD, COLUMN, RIGHT, BACKWARD, ROW
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DTRMM
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 .OR. K.LE.0 .OR. L.LT.0 ) RETURN
*
IF( LSAME( STOREV, 'C' ) ) THEN
COLUMN = .TRUE.
ROW = .FALSE.
ELSE IF ( LSAME( STOREV, 'R' ) ) THEN
COLUMN = .FALSE.
ROW = .TRUE.
ELSE
COLUMN = .FALSE.
ROW = .FALSE.
END IF
*
IF( LSAME( SIDE, 'L' ) ) THEN
LEFT = .TRUE.
RIGHT = .FALSE.
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
LEFT = .FALSE.
RIGHT = .TRUE.
ELSE
LEFT = .FALSE.
RIGHT = .FALSE.
END IF
*
IF( LSAME( DIRECT, 'F' ) ) THEN
FORWARD = .TRUE.
BACKWARD = .FALSE.
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
FORWARD = .FALSE.
BACKWARD = .TRUE.
ELSE
FORWARD = .FALSE.
BACKWARD = .FALSE.
END IF
*
* ---------------------------------------------------------------------------
*
IF( COLUMN .AND. FORWARD .AND. LEFT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ I ] (K-by-K)
* [ V ] (M-by-K)
*
* Form H C or H**T C where C = [ A ] (K-by-N)
* [ B ] (M-by-N)
*
* H = I - W T W**T or H**T = I - W T**T W**T
*
* A = A - T (A + V**T B) or A = A - T**T (A + V**T B)
* B = B - V T (A + V**T B) or B = B - V T**T (A + V**T B)
*
* ---------------------------------------------------------------------------
*
MP = MIN( M-L+1, M )
KP = MIN( L+1, K )
*
DO J = 1, N
DO I = 1, L
WORK( I, J ) = B( M-L+I, J )
END DO
END DO
CALL DTRMM( 'L', 'U', 'T', 'N', L, N, ONE, V( MP, 1 ), LDV,
$ WORK, LDWORK )
CALL DGEMM( 'T', 'N', L, N, M-L, ONE, V, LDV, B, LDB,
$ ONE, WORK, LDWORK )
CALL DGEMM( 'T', 'N', K-L, N, M, ONE, V( 1, KP ), LDV,
$ B, LDB, ZERO, WORK( KP, 1 ), LDWORK )
*
DO J = 1, N
DO I = 1, K
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'L', 'U', TRANS, 'N', K, N, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, N
DO I = 1, K
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'N', 'N', M-L, N, K, -ONE, V, LDV, WORK, LDWORK,
$ ONE, B, LDB )
CALL DGEMM( 'N', 'N', L, N, K-L, -ONE, V( MP, KP ), LDV,
$ WORK( KP, 1 ), LDWORK, ONE, B( MP, 1 ), LDB )
CALL DTRMM( 'L', 'U', 'N', 'N', L, N, ONE, V( MP, 1 ), LDV,
$ WORK, LDWORK )
DO J = 1, N
DO I = 1, L
B( M-L+I, J ) = B( M-L+I, J ) - WORK( I, J )
END DO
END DO
*
* ---------------------------------------------------------------------------
*
ELSE IF( COLUMN .AND. FORWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ I ] (K-by-K)
* [ V ] (N-by-K)
*
* Form C H or C H**T where C = [ A B ] (A is M-by-K, B is M-by-N)
*
* H = I - W T W**T or H**T = I - W T**T W**T
*
* A = A - (A + B V) T or A = A - (A + B V) T**T
* B = B - (A + B V) T V**T or B = B - (A + B V) T**T V**T
*
* ---------------------------------------------------------------------------
*
NP = MIN( N-L+1, N )
KP = MIN( L+1, K )
*
DO J = 1, L
DO I = 1, M
WORK( I, J ) = B( I, N-L+J )
END DO
END DO
CALL DTRMM( 'R', 'U', 'N', 'N', M, L, ONE, V( NP, 1 ), LDV,
$ WORK, LDWORK )
CALL DGEMM( 'N', 'N', M, L, N-L, ONE, B, LDB,
$ V, LDV, ONE, WORK, LDWORK )
CALL DGEMM( 'N', 'N', M, K-L, N, ONE, B, LDB,
$ V( 1, KP ), LDV, ZERO, WORK( 1, KP ), LDWORK )
*
DO J = 1, K
DO I = 1, M
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'R', 'U', TRANS, 'N', M, K, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, K
DO I = 1, M
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'N', 'T', M, N-L, K, -ONE, WORK, LDWORK,
$ V, LDV, ONE, B, LDB )
CALL DGEMM( 'N', 'T', M, L, K-L, -ONE, WORK( 1, KP ),
$ LDWORK,
$ V( NP, KP ), LDV, ONE, B( 1, NP ), LDB )
CALL DTRMM( 'R', 'U', 'T', 'N', M, L, ONE, V( NP, 1 ), LDV,
$ WORK, LDWORK )
DO J = 1, L
DO I = 1, M
B( I, N-L+J ) = B( I, N-L+J ) - WORK( I, J )
END DO
END DO
*
* ---------------------------------------------------------------------------
*
ELSE IF( COLUMN .AND. BACKWARD .AND. LEFT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ V ] (M-by-K)
* [ I ] (K-by-K)
*
* Form H C or H**T C where C = [ B ] (M-by-N)
* [ A ] (K-by-N)
*
* H = I - W T W**T or H**T = I - W T**T W**T
*
* A = A - T (A + V**T B) or A = A - T**T (A + V**T B)
* B = B - V T (A + V**T B) or B = B - V T**T (A + V**T B)
*
* ---------------------------------------------------------------------------
*
MP = MIN( L+1, M )
KP = MIN( K-L+1, K )
*
DO J = 1, N
DO I = 1, L
WORK( K-L+I, J ) = B( I, J )
END DO
END DO
*
CALL DTRMM( 'L', 'L', 'T', 'N', L, N, ONE, V( 1, KP ), LDV,
$ WORK( KP, 1 ), LDWORK )
CALL DGEMM( 'T', 'N', L, N, M-L, ONE, V( MP, KP ), LDV,
$ B( MP, 1 ), LDB, ONE, WORK( KP, 1 ), LDWORK )
CALL DGEMM( 'T', 'N', K-L, N, M, ONE, V, LDV,
$ B, LDB, ZERO, WORK, LDWORK )
*
DO J = 1, N
DO I = 1, K
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'L', 'L', TRANS, 'N', K, N, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, N
DO I = 1, K
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'N', 'N', M-L, N, K, -ONE, V( MP, 1 ), LDV,
$ WORK, LDWORK, ONE, B( MP, 1 ), LDB )
CALL DGEMM( 'N', 'N', L, N, K-L, -ONE, V, LDV,
$ WORK, LDWORK, ONE, B, LDB )
CALL DTRMM( 'L', 'L', 'N', 'N', L, N, ONE, V( 1, KP ), LDV,
$ WORK( KP, 1 ), LDWORK )
DO J = 1, N
DO I = 1, L
B( I, J ) = B( I, J ) - WORK( K-L+I, J )
END DO
END DO
*
* ---------------------------------------------------------------------------
*
ELSE IF( COLUMN .AND. BACKWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ V ] (N-by-K)
* [ I ] (K-by-K)
*
* Form C H or C H**T where C = [ B A ] (B is M-by-N, A is M-by-K)
*
* H = I - W T W**T or H**T = I - W T**T W**T
*
* A = A - (A + B V) T or A = A - (A + B V) T**T
* B = B - (A + B V) T V**T or B = B - (A + B V) T**T V**T
*
* ---------------------------------------------------------------------------
*
NP = MIN( L+1, N )
KP = MIN( K-L+1, K )
*
DO J = 1, L
DO I = 1, M
WORK( I, K-L+J ) = B( I, J )
END DO
END DO
CALL DTRMM( 'R', 'L', 'N', 'N', M, L, ONE, V( 1, KP ), LDV,
$ WORK( 1, KP ), LDWORK )
CALL DGEMM( 'N', 'N', M, L, N-L, ONE, B( 1, NP ), LDB,
$ V( NP, KP ), LDV, ONE, WORK( 1, KP ), LDWORK )
CALL DGEMM( 'N', 'N', M, K-L, N, ONE, B, LDB,
$ V, LDV, ZERO, WORK, LDWORK )
*
DO J = 1, K
DO I = 1, M
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'R', 'L', TRANS, 'N', M, K, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, K
DO I = 1, M
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'N', 'T', M, N-L, K, -ONE, WORK, LDWORK,
$ V( NP, 1 ), LDV, ONE, B( 1, NP ), LDB )
CALL DGEMM( 'N', 'T', M, L, K-L, -ONE, WORK, LDWORK,
$ V, LDV, ONE, B, LDB )
CALL DTRMM( 'R', 'L', 'T', 'N', M, L, ONE, V( 1, KP ), LDV,
$ WORK( 1, KP ), LDWORK )
DO J = 1, L
DO I = 1, M
B( I, J ) = B( I, J ) - WORK( I, K-L+J )
END DO
END DO
*
* ---------------------------------------------------------------------------
*
ELSE IF( ROW .AND. FORWARD .AND. LEFT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ I V ] ( I is K-by-K, V is K-by-M )
*
* Form H C or H**T C where C = [ A ] (K-by-N)
* [ B ] (M-by-N)
*
* H = I - W**T T W or H**T = I - W**T T**T W
*
* A = A - T (A + V B) or A = A - T**T (A + V B)
* B = B - V**T T (A + V B) or B = B - V**T T**T (A + V B)
*
* ---------------------------------------------------------------------------
*
MP = MIN( M-L+1, M )
KP = MIN( L+1, K )
*
DO J = 1, N
DO I = 1, L
WORK( I, J ) = B( M-L+I, J )
END DO
END DO
CALL DTRMM( 'L', 'L', 'N', 'N', L, N, ONE, V( 1, MP ), LDV,
$ WORK, LDB )
CALL DGEMM( 'N', 'N', L, N, M-L, ONE, V, LDV,B, LDB,
$ ONE, WORK, LDWORK )
CALL DGEMM( 'N', 'N', K-L, N, M, ONE, V( KP, 1 ), LDV,
$ B, LDB, ZERO, WORK( KP, 1 ), LDWORK )
*
DO J = 1, N
DO I = 1, K
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'L', 'U', TRANS, 'N', K, N, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, N
DO I = 1, K
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'T', 'N', M-L, N, K, -ONE, V, LDV, WORK, LDWORK,
$ ONE, B, LDB )
CALL DGEMM( 'T', 'N', L, N, K-L, -ONE, V( KP, MP ), LDV,
$ WORK( KP, 1 ), LDWORK, ONE, B( MP, 1 ), LDB )
CALL DTRMM( 'L', 'L', 'T', 'N', L, N, ONE, V( 1, MP ), LDV,
$ WORK, LDWORK )
DO J = 1, N
DO I = 1, L
B( M-L+I, J ) = B( M-L+I, J ) - WORK( I, J )
END DO
END DO
*
* ---------------------------------------------------------------------------
*
ELSE IF( ROW .AND. FORWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ I V ] ( I is K-by-K, V is K-by-N )
*
* Form C H or C H**T where C = [ A B ] (A is M-by-K, B is M-by-N)
*
* H = I - W**T T W or H**T = I - W**T T**T W
*
* A = A - (A + B V**T) T or A = A - (A + B V**T) T**T
* B = B - (A + B V**T) T V or B = B - (A + B V**T) T**T V
*
* ---------------------------------------------------------------------------
*
NP = MIN( N-L+1, N )
KP = MIN( L+1, K )
*
DO J = 1, L
DO I = 1, M
WORK( I, J ) = B( I, N-L+J )
END DO
END DO
CALL DTRMM( 'R', 'L', 'T', 'N', M, L, ONE, V( 1, NP ), LDV,
$ WORK, LDWORK )
CALL DGEMM( 'N', 'T', M, L, N-L, ONE, B, LDB, V, LDV,
$ ONE, WORK, LDWORK )
CALL DGEMM( 'N', 'T', M, K-L, N, ONE, B, LDB,
$ V( KP, 1 ), LDV, ZERO, WORK( 1, KP ), LDWORK )
*
DO J = 1, K
DO I = 1, M
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'R', 'U', TRANS, 'N', M, K, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, K
DO I = 1, M
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'N', 'N', M, N-L, K, -ONE, WORK, LDWORK,
$ V, LDV, ONE, B, LDB )
CALL DGEMM( 'N', 'N', M, L, K-L, -ONE, WORK( 1, KP ),
$ LDWORK,
$ V( KP, NP ), LDV, ONE, B( 1, NP ), LDB )
CALL DTRMM( 'R', 'L', 'N', 'N', M, L, ONE, V( 1, NP ), LDV,
$ WORK, LDWORK )
DO J = 1, L
DO I = 1, M
B( I, N-L+J ) = B( I, N-L+J ) - WORK( I, J )
END DO
END DO
*
* ---------------------------------------------------------------------------
*
ELSE IF( ROW .AND. BACKWARD .AND. LEFT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ V I ] ( I is K-by-K, V is K-by-M )
*
* Form H C or H**T C where C = [ B ] (M-by-N)
* [ A ] (K-by-N)
*
* H = I - W**T T W or H**T = I - W**T T**T W
*
* A = A - T (A + V B) or A = A - T**T (A + V B)
* B = B - V**T T (A + V B) or B = B - V**T T**T (A + V B)
*
* ---------------------------------------------------------------------------
*
MP = MIN( L+1, M )
KP = MIN( K-L+1, K )
*
DO J = 1, N
DO I = 1, L
WORK( K-L+I, J ) = B( I, J )
END DO
END DO
CALL DTRMM( 'L', 'U', 'N', 'N', L, N, ONE, V( KP, 1 ), LDV,
$ WORK( KP, 1 ), LDWORK )
CALL DGEMM( 'N', 'N', L, N, M-L, ONE, V( KP, MP ), LDV,
$ B( MP, 1 ), LDB, ONE, WORK( KP, 1 ), LDWORK )
CALL DGEMM( 'N', 'N', K-L, N, M, ONE, V, LDV, B, LDB,
$ ZERO, WORK, LDWORK )
*
DO J = 1, N
DO I = 1, K
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'L', 'L ', TRANS, 'N', K, N, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, N
DO I = 1, K
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'T', 'N', M-L, N, K, -ONE, V( 1, MP ), LDV,
$ WORK, LDWORK, ONE, B( MP, 1 ), LDB )
CALL DGEMM( 'T', 'N', L, N, K-L, -ONE, V, LDV,
$ WORK, LDWORK, ONE, B, LDB )
CALL DTRMM( 'L', 'U', 'T', 'N', L, N, ONE, V( KP, 1 ), LDV,
$ WORK( KP, 1 ), LDWORK )
DO J = 1, N
DO I = 1, L
B( I, J ) = B( I, J ) - WORK( K-L+I, J )
END DO
END DO
*
* ---------------------------------------------------------------------------
*
ELSE IF( ROW .AND. BACKWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
* Let W = [ V I ] ( I is K-by-K, V is K-by-N )
*
* Form C H or C H**T where C = [ B A ] (A is M-by-K, B is M-by-N)
*
* H = I - W**T T W or H**T = I - W**T T**T W
*
* A = A - (A + B V**T) T or A = A - (A + B V**T) T**T
* B = B - (A + B V**T) T V or B = B - (A + B V**T) T**T V
*
* ---------------------------------------------------------------------------
*
NP = MIN( L+1, N )
KP = MIN( K-L+1, K )
*
DO J = 1, L
DO I = 1, M
WORK( I, K-L+J ) = B( I, J )
END DO
END DO
CALL DTRMM( 'R', 'U', 'T', 'N', M, L, ONE, V( KP, 1 ), LDV,
$ WORK( 1, KP ), LDWORK )
CALL DGEMM( 'N', 'T', M, L, N-L, ONE, B( 1, NP ), LDB,
$ V( KP, NP ), LDV, ONE, WORK( 1, KP ), LDWORK )
CALL DGEMM( 'N', 'T', M, K-L, N, ONE, B, LDB, V, LDV,
$ ZERO, WORK, LDWORK )
*
DO J = 1, K
DO I = 1, M
WORK( I, J ) = WORK( I, J ) + A( I, J )
END DO
END DO
*
CALL DTRMM( 'R', 'L', TRANS, 'N', M, K, ONE, T, LDT,
$ WORK, LDWORK )
*
DO J = 1, K
DO I = 1, M
A( I, J ) = A( I, J ) - WORK( I, J )
END DO
END DO
*
CALL DGEMM( 'N', 'N', M, N-L, K, -ONE, WORK, LDWORK,
$ V( 1, NP ), LDV, ONE, B( 1, NP ), LDB )
CALL DGEMM( 'N', 'N', M, L, K-L , -ONE, WORK, LDWORK,
$ V, LDV, ONE, B, LDB )
CALL DTRMM( 'R', 'U', 'N', 'N', M, L, ONE, V( KP, 1 ), LDV,
$ WORK( 1, KP ), LDWORK )
DO J = 1, L
DO I = 1, M
B( I, J ) = B( I, J ) - WORK( I, K-L+J )
END DO
END DO
*
END IF
*
RETURN
*
* End of DTPRFB
*
END
*> \brief \b DTPRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtprfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtprfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtprfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
* FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
* $ WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPRFS provides error bounds and backward error estimates for the
*> solution to a system of linear equations with a triangular packed
*> coefficient matrix.
*>
*> The solution matrix X must be computed by DTPTRS or some other
*> means before entering this routine. DTPRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangular matrix A, packed columnwise in
*> a linear array. The j-th column of A is stored in the array
*> AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> If DIAG = 'U', the diagonal elements of A are not referenced
*> and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> The solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tprfs
*
* =====================================================================
SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X,
$ LDX,
$ FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
$ WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
CHARACTER TRANST
INTEGER I, J, K, KASE, KC, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DTPMV, DTPSV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 250 J = 1, NRHS
*
* Compute residual R = B - op(A) * X,
* where op(A) = A or A**T, depending on TRANS.
*
CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 20 I = 1, N
WORK( I ) = ABS( B( I, J ) )
20 CONTINUE
*
IF( NOTRAN ) THEN
*
* Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
KC = 1
IF( NOUNIT ) THEN
DO 40 K = 1, N
XK = ABS( X( K, J ) )
DO 30 I = 1, K
WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
30 CONTINUE
KC = KC + K
40 CONTINUE
ELSE
DO 60 K = 1, N
XK = ABS( X( K, J ) )
DO 50 I = 1, K - 1
WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
50 CONTINUE
WORK( K ) = WORK( K ) + XK
KC = KC + K
60 CONTINUE
END IF
ELSE
KC = 1
IF( NOUNIT ) THEN
DO 80 K = 1, N
XK = ABS( X( K, J ) )
DO 70 I = K, N
WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
70 CONTINUE
KC = KC + N - K + 1
80 CONTINUE
ELSE
DO 100 K = 1, N
XK = ABS( X( K, J ) )
DO 90 I = K + 1, N
WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
90 CONTINUE
WORK( K ) = WORK( K ) + XK
KC = KC + N - K + 1
100 CONTINUE
END IF
END IF
ELSE
*
* Compute abs(A**T)*abs(X) + abs(B).
*
IF( UPPER ) THEN
KC = 1
IF( NOUNIT ) THEN
DO 120 K = 1, N
S = ZERO
DO 110 I = 1, K
S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
110 CONTINUE
WORK( K ) = WORK( K ) + S
KC = KC + K
120 CONTINUE
ELSE
DO 140 K = 1, N
S = ABS( X( K, J ) )
DO 130 I = 1, K - 1
S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
130 CONTINUE
WORK( K ) = WORK( K ) + S
KC = KC + K
140 CONTINUE
END IF
ELSE
KC = 1
IF( NOUNIT ) THEN
DO 160 K = 1, N
S = ZERO
DO 150 I = K, N
S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
150 CONTINUE
WORK( K ) = WORK( K ) + S
KC = KC + N - K + 1
160 CONTINUE
ELSE
DO 180 K = 1, N
S = ABS( X( K, J ) )
DO 170 I = K + 1, N
S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
170 CONTINUE
WORK( K ) = WORK( K ) + S
KC = KC + N - K + 1
180 CONTINUE
END IF
END IF
END IF
S = ZERO
DO 190 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
190 CONTINUE
BERR( J ) = S
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(op(A)))*
* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(A)) is the inverse of op(A)
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(op(A)) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 200 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
200 CONTINUE
*
KASE = 0
210 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**T).
*
CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ),
$ 1 )
DO 220 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
220 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 230 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
230 CONTINUE
CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
END IF
GO TO 210
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 240 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
240 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
250 CONTINUE
*
RETURN
*
* End of DTPRFS
*
END
*> \brief \b DTPTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtptri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtptri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtptri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPTRI( UPLO, DIAG, N, AP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPTRI computes the inverse of a real upper or lower triangular
*> matrix A stored in packed format.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangular matrix A, stored
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
*> See below for further details.
*> On exit, the (triangular) inverse of the original matrix, in
*> the same packed storage format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, A(i,i) is exactly zero. The triangular
*> matrix is singular and its inverse can not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tptri
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> A triangular matrix A can be transferred to packed storage using one
*> of the following program segments:
*>
*> UPLO = 'U': UPLO = 'L':
*>
*> JC = 1 JC = 1
*> DO 2 J = 1, N DO 2 J = 1, N
*> DO 1 I = 1, J DO 1 I = J, N
*> AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J)
*> 1 CONTINUE 1 CONTINUE
*> JC = JC + J JC = JC + N - J + 1
*> 2 CONTINUE 2 CONTINUE
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTPTRI( UPLO, DIAG, N, AP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, UPPER
INTEGER J, JC, JCLAST, JJ
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DTPMV, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOUNIT = LSAME( DIAG, 'N' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPTRI', -INFO )
RETURN
END IF
*
* Check for singularity if non-unit.
*
IF( NOUNIT ) THEN
IF( UPPER ) THEN
JJ = 0
DO 10 INFO = 1, N
JJ = JJ + INFO
IF( AP( JJ ).EQ.ZERO )
$ RETURN
10 CONTINUE
ELSE
JJ = 1
DO 20 INFO = 1, N
IF( AP( JJ ).EQ.ZERO )
$ RETURN
JJ = JJ + N - INFO + 1
20 CONTINUE
END IF
INFO = 0
END IF
*
IF( UPPER ) THEN
*
* Compute inverse of upper triangular matrix.
*
JC = 1
DO 30 J = 1, N
IF( NOUNIT ) THEN
AP( JC+J-1 ) = ONE / AP( JC+J-1 )
AJJ = -AP( JC+J-1 )
ELSE
AJJ = -ONE
END IF
*
* Compute elements 1:j-1 of j-th column.
*
CALL DTPMV( 'Upper', 'No transpose', DIAG, J-1, AP,
$ AP( JC ), 1 )
CALL DSCAL( J-1, AJJ, AP( JC ), 1 )
JC = JC + J
30 CONTINUE
*
ELSE
*
* Compute inverse of lower triangular matrix.
*
JC = N*( N+1 ) / 2
DO 40 J = N, 1, -1
IF( NOUNIT ) THEN
AP( JC ) = ONE / AP( JC )
AJJ = -AP( JC )
ELSE
AJJ = -ONE
END IF
IF( J.LT.N ) THEN
*
* Compute elements j+1:n of j-th column.
*
CALL DTPMV( 'Lower', 'No transpose', DIAG, N-J,
$ AP( JCLAST ), AP( JC+1 ), 1 )
CALL DSCAL( N-J, AJJ, AP( JC+1 ), 1 )
END IF
JCLAST = JC
JC = JC - N + J - 2
40 CONTINUE
END IF
*
RETURN
*
* End of DTPTRI
*
END
*> \brief \b DTPTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtptrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtptrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtptrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPTRS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPTRS solves a triangular system of the form
*>
*> A * X = B or A**T * X = B,
*>
*> where A is a triangular matrix of order N stored in packed format, and B is an N-by-NRHS matrix.
*>
*> This subroutine verifies that A is nonsingular, but callers should note that only exact
*> singularity is detected. It is conceivable for one or more diagonal elements of A to be
*> subnormally tiny numbers without this subroutine signalling an error.
*>
*> If a possible loss of numerical precision due to near-singular matrices is a concern, the
*> caller should verify that A is nonsingular within some tolerance before calling this subroutine.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangular matrix A, packed columnwise in
*> a linear array. The j-th column of A is stored in the array
*> AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, if INFO = 0, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element of A is exactly zero,
*> indicating that the matrix is singular and the
*> solutions X have not been computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tptrs
*
* =====================================================================
SUBROUTINE DTPTRS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, UPPER
INTEGER J, JC
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DTPSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOUNIT = LSAME( DIAG, 'N' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.
$ LSAME( TRANS, 'T' ) .AND.
$ .NOT.LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check for singularity.
*
IF( NOUNIT ) THEN
IF( UPPER ) THEN
JC = 1
DO 10 INFO = 1, N
IF( AP( JC+INFO-1 ).EQ.ZERO )
$ RETURN
JC = JC + INFO
10 CONTINUE
ELSE
JC = 1
DO 20 INFO = 1, N
IF( AP( JC ).EQ.ZERO )
$ RETURN
JC = JC + N - INFO + 1
20 CONTINUE
END IF
END IF
INFO = 0
*
* Solve A * x = b or A**T * x = b.
*
DO 30 J = 1, NRHS
CALL DTPSV( UPLO, TRANS, DIAG, N, AP, B( 1, J ), 1 )
30 CONTINUE
*
RETURN
*
* End of DTPTRS
*
END
*> \brief \b DTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpttf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpttf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpttf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPTTF( TRANSR, UPLO, N, AP, ARF, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( 0: * ), ARF( 0: * )
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPTTF copies a triangular matrix A from standard packed format (TP)
*> to rectangular full packed format (TF).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': ARF in Normal format is wanted;
*> = 'T': ARF in Conjugate-transpose format is wanted.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*> On entry, the upper or lower triangular matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] ARF
*> \verbatim
*> ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*> On exit, the upper or lower triangular matrix A stored in
*> RFP format. For a further discussion see Notes below.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpttf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTPTTF( TRANSR, UPLO, N, AP, ARF, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( 0: * ), ARF( 0: * )
*
* =====================================================================
*
* .. Parameters ..
* ..
* .. Local Scalars ..
LOGICAL LOWER, NISODD, NORMALTRANSR
INTEGER N1, N2, K, NT
INTEGER I, J, IJ
INTEGER IJP, JP, LDA, JS
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPTTF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( NORMALTRANSR ) THEN
ARF( 0 ) = AP( 0 )
ELSE
ARF( 0 ) = AP( 0 )
END IF
RETURN
END IF
*
* Size of array ARF(0:NT-1)
*
NT = N*( N+1 ) / 2
*
* Set N1 and N2 depending on LOWER
*
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
*
* If N is odd, set NISODD = .TRUE.
* If N is even, set K = N/2 and NISODD = .FALSE.
*
* set lda of ARF^C; ARF^C is (0:(N+1)/2-1,0:N-noe)
* where noe = 0 if n is even, noe = 1 if n is odd
*
IF( MOD( N, 2 ).EQ.0 ) THEN
K = N / 2
NISODD = .FALSE.
LDA = N + 1
ELSE
NISODD = .TRUE.
LDA = N
END IF
*
* ARF^C has lda rows and n+1-noe cols
*
IF( .NOT.NORMALTRANSR )
$ LDA = ( N+1 ) / 2
*
* start execution: there are eight cases
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'N', and UPLO = 'L'
*
IJP = 0
JP = 0
DO J = 0, N2
DO I = J, N - 1
IJ = I + JP
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JP = JP + LDA
END DO
DO I = 0, N2 - 1
DO J = 1 + I, N2
IJ = I + J*LDA
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
END DO
*
ELSE
*
* N is odd, TRANSR = 'N', and UPLO = 'U'
*
IJP = 0
DO J = 0, N1 - 1
IJ = N2 + J
DO I = 0, J
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
IJ = IJ + LDA
END DO
END DO
JS = 0
DO J = N1, N - 1
IJ = JS
DO IJ = JS, JS + J
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
*
END IF
*
ELSE
*
* N is odd and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'T', and UPLO = 'L'
*
IJP = 0
DO I = 0, N2
DO IJ = I*( LDA+1 ), N*LDA - 1, LDA
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
END DO
JS = 1
DO J = 0, N2 - 1
DO IJ = JS, JS + N2 - J - 1
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JS = JS + LDA + 1
END DO
*
ELSE
*
* N is odd, TRANSR = 'T', and UPLO = 'U'
*
IJP = 0
JS = N2*LDA
DO J = 0, N1 - 1
DO IJ = JS, JS + J
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
DO I = 0, N1
DO IJ = I, I + ( N1+I )*LDA, LDA
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
END DO
*
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'N', and UPLO = 'L'
*
IJP = 0
JP = 0
DO J = 0, K - 1
DO I = J, N - 1
IJ = 1 + I + JP
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JP = JP + LDA
END DO
DO I = 0, K - 1
DO J = I, K - 1
IJ = I + J*LDA
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
END DO
*
ELSE
*
* N is even, TRANSR = 'N', and UPLO = 'U'
*
IJP = 0
DO J = 0, K - 1
IJ = K + 1 + J
DO I = 0, J
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
IJ = IJ + LDA
END DO
END DO
JS = 0
DO J = K, N - 1
IJ = JS
DO IJ = JS, JS + J
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
*
END IF
*
ELSE
*
* N is even and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'T', and UPLO = 'L'
*
IJP = 0
DO I = 0, K - 1
DO IJ = I + ( I+1 )*LDA, ( N+1 )*LDA - 1, LDA
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
END DO
JS = 0
DO J = 0, K - 1
DO IJ = JS, JS + K - J - 1
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JS = JS + LDA + 1
END DO
*
ELSE
*
* N is even, TRANSR = 'T', and UPLO = 'U'
*
IJP = 0
JS = ( K+1 )*LDA
DO J = 0, K - 1
DO IJ = JS, JS + J
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
JS = JS + LDA
END DO
DO I = 0, K - 1
DO IJ = I, I + ( K+I )*LDA, LDA
ARF( IJ ) = AP( IJP )
IJP = IJP + 1
END DO
END DO
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of DTPTTF
*
END
*> \brief \b DTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpttr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpttr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpttr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTPTTR( UPLO, N, AP, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), AP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPTTR copies a triangular matrix A from standard packed format (TP)
*> to standard full format (TR).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular.
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*> On entry, the upper or lower triangular matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension ( LDA, N )
*> On exit, the triangular matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpttr
*
* =====================================================================
SUBROUTINE DTPTTR( UPLO, N, AP, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), AP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
* ..
* .. Local Scalars ..
LOGICAL LOWER
INTEGER I, J, K
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTPTTR', -INFO )
RETURN
END IF
*
IF( LOWER ) THEN
K = 0
DO J = 1, N
DO I = J, N
K = K + 1
A( I, J ) = AP( K )
END DO
END DO
ELSE
K = 0
DO J = 1, N
DO I = 1, J
K = K + 1
A( I, J ) = AP( K )
END DO
END DO
END IF
*
*
RETURN
*
* End of DTPTTR
*
END
*> \brief \b DTRCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRCON( NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, NORM, UPLO
* INTEGER INFO, LDA, N
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRCON estimates the reciprocal of the condition number of a
*> triangular matrix A, in either the 1-norm or the infinity-norm.
*>
*> The norm of A is computed and an estimate is obtained for
*> norm(inv(A)), then the reciprocal of the condition number is
*> computed as
*> RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies whether the 1-norm condition number or the
*> infinity-norm condition number is required:
*> = '1' or 'O': 1-norm;
*> = 'I': Infinity-norm.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The triangular matrix A. If UPLO = 'U', the leading N-by-N
*> upper triangular part of the array A contains the upper
*> triangular matrix, and the strictly lower triangular part of
*> A is not referenced. If UPLO = 'L', the leading N-by-N lower
*> triangular part of the array A contains the lower triangular
*> matrix, and the strictly upper triangular part of A is not
*> referenced. If DIAG = 'U', the diagonal elements of A are
*> also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trcon
*
* =====================================================================
SUBROUTINE DTRCON( NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER INFO, LDA, N
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, ONENRM, UPPER
CHARACTER NORMIN
INTEGER IX, KASE, KASE1
DOUBLE PRECISION AINVNM, ANORM, SCALE, SMLNUM, XNORM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLANTR
EXTERNAL LSAME, IDAMAX, DLAMCH, DLANTR
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLATRS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
*
RCOND = ZERO
SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) )
*
* Compute the norm of the triangular matrix A.
*
ANORM = DLANTR( NORM, UPLO, DIAG, N, N, A, LDA, WORK )
*
* Continue only if ANORM > 0.
*
IF( ANORM.GT.ZERO ) THEN
*
* Estimate the norm of the inverse of A.
*
AINVNM = ZERO
NORMIN = 'N'
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KASE = 0
10 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE,
$ ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(A).
*
CALL DLATRS( UPLO, 'No transpose', DIAG, NORMIN, N, A,
$ LDA, WORK, SCALE, WORK( 2*N+1 ), INFO )
ELSE
*
* Multiply by inv(A**T).
*
CALL DLATRS( UPLO, 'Transpose', DIAG, NORMIN, N, A,
$ LDA,
$ WORK, SCALE, WORK( 2*N+1 ), INFO )
END IF
NORMIN = 'Y'
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
IF( SCALE.NE.ONE ) THEN
IX = IDAMAX( N, WORK, 1 )
XNORM = ABS( WORK( IX ) )
IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL DRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / ANORM ) / AINVNM
END IF
*
20 CONTINUE
RETURN
*
* End of DTRCON
*
END
*> \brief \b DTREVC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREVC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrevc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrevc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrevc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
* LDVR, MM, M, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, SIDE
* INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTREVC computes some or all of the right and/or left eigenvectors of
*> a real upper quasi-triangular matrix T.
*> Matrices of this type are produced by the Schur factorization of
*> a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.
*>
*> The right eigenvector x and the left eigenvector y of T corresponding
*> to an eigenvalue w are defined by:
*>
*> T*x = w*x, (y**H)*T = w*(y**H)
*>
*> where y**H denotes the conjugate transpose of y.
*> The eigenvalues are not input to this routine, but are read directly
*> from the diagonal blocks of T.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
*> input matrix. If Q is the orthogonal factor that reduces a matrix
*> A to Schur form T, then Q*X and Q*Y are the matrices of right and
*> left eigenvectors of A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': compute right eigenvectors only;
*> = 'L': compute left eigenvectors only;
*> = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute all right and/or left eigenvectors;
*> = 'B': compute all right and/or left eigenvectors,
*> backtransformed by the matrices in VR and/or VL;
*> = 'S': compute selected right and/or left eigenvectors,
*> as indicated by the logical array SELECT.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY = 'S', SELECT specifies the eigenvectors to be
*> computed.
*> If w(j) is a real eigenvalue, the corresponding real
*> eigenvector is computed if SELECT(j) is .TRUE..
*> If w(j) and w(j+1) are the real and imaginary parts of a
*> complex eigenvalue, the corresponding complex eigenvector is
*> computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
*> on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
*> .FALSE..
*> Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The upper quasi-triangular matrix T in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*> contain an N-by-N matrix Q (usually the orthogonal matrix Q
*> of Schur vectors returned by DHSEQR).
*> On exit, if SIDE = 'L' or 'B', VL contains:
*> if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
*> if HOWMNY = 'B', the matrix Q*Y;
*> if HOWMNY = 'S', the left eigenvectors of T specified by
*> SELECT, stored consecutively in the columns
*> of VL, in the same order as their
*> eigenvalues.
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part, and the second the imaginary part.
*> Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1, and if
*> SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*> contain an N-by-N matrix Q (usually the orthogonal matrix Q
*> of Schur vectors returned by DHSEQR).
*> On exit, if SIDE = 'R' or 'B', VR contains:
*> if HOWMNY = 'A', the matrix X of right eigenvectors of T;
*> if HOWMNY = 'B', the matrix Q*X;
*> if HOWMNY = 'S', the right eigenvectors of T specified by
*> SELECT, stored consecutively in the columns
*> of VR, in the same order as their
*> eigenvalues.
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part and the second the imaginary part.
*> Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1, and if
*> SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of columns in the arrays VL and/or VR actually
*> used to store the eigenvectors.
*> If HOWMNY = 'A' or 'B', M is set to N.
*> Each selected real eigenvector occupies one column and each
*> selected complex eigenvector occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trevc
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The algorithm used in this program is basically backward (forward)
*> substitution, with scaling to make the the code robust against
*> possible overflow.
*>
*> Each eigenvector is normalized so that the element of largest
*> magnitude has magnitude 1; here the magnitude of a complex number
*> (x,y) is taken to be |x| + |y|.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
$ VR,
$ LDVR, MM, M, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLV, BOTHV, LEFTV, OVER, PAIR, RIGHTV, SOMEV
INTEGER I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, N2
DOUBLE PRECISION BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
$ SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
$ XNORM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DLALN2, DSCAL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Local Arrays ..
DOUBLE PRECISION X( 2, 2 )
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
BOTHV = LSAME( SIDE, 'B' )
RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
ALLV = LSAME( HOWMNY, 'A' )
OVER = LSAME( HOWMNY, 'B' )
SOMEV = LSAME( HOWMNY, 'S' )
*
INFO = 0
IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -1
ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
INFO = -8
ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
INFO = -10
ELSE
*
* Set M to the number of columns required to store the selected
* eigenvectors, standardize the array SELECT if necessary, and
* test MM.
*
IF( SOMEV ) THEN
M = 0
PAIR = .FALSE.
DO 10 J = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
SELECT( J ) = .FALSE.
ELSE
IF( J.LT.N ) THEN
IF( T( J+1, J ).EQ.ZERO ) THEN
IF( SELECT( J ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN
SELECT( J ) = .TRUE.
M = M + 2
END IF
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
ELSE
M = N
END IF
*
IF( MM.LT.M ) THEN
INFO = -11
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTREVC', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* Set the constants to control overflow.
*
UNFL = DLAMCH( 'Safe minimum' )
OVFL = ONE / UNFL
ULP = DLAMCH( 'Precision' )
SMLNUM = UNFL*( N / ULP )
BIGNUM = ( ONE-ULP ) / SMLNUM
*
* Compute 1-norm of each column of strictly upper triangular
* part of T to control overflow in triangular solver.
*
WORK( 1 ) = ZERO
DO 30 J = 2, N
WORK( J ) = ZERO
DO 20 I = 1, J - 1
WORK( J ) = WORK( J ) + ABS( T( I, J ) )
20 CONTINUE
30 CONTINUE
*
* Index IP is used to specify the real or complex eigenvalue:
* IP = 0, real eigenvalue,
* 1, first of conjugate complex pair: (wr,wi)
* -1, second of conjugate complex pair: (wr,wi)
*
N2 = 2*N
*
IF( RIGHTV ) THEN
*
* Compute right eigenvectors.
*
IP = 0
IS = M
DO 140 KI = N, 1, -1
*
IF( IP.EQ.1 )
$ GO TO 130
IF( KI.EQ.1 )
$ GO TO 40
IF( T( KI, KI-1 ).EQ.ZERO )
$ GO TO 40
IP = -1
*
40 CONTINUE
IF( SOMEV ) THEN
IF( IP.EQ.0 ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 130
ELSE
IF( .NOT.SELECT( KI-1 ) )
$ GO TO 130
END IF
END IF
*
* Compute the KI-th eigenvalue (WR,WI).
*
WR = T( KI, KI )
WI = ZERO
IF( IP.NE.0 )
$ WI = SQRT( ABS( T( KI, KI-1 ) ) )*
$ SQRT( ABS( T( KI-1, KI ) ) )
SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
IF( IP.EQ.0 ) THEN
*
* Real right eigenvector
*
WORK( KI+N ) = ONE
*
* Form right-hand side
*
DO 50 K = 1, KI - 1
WORK( K+N ) = -T( K, KI )
50 CONTINUE
*
* Solve the upper quasi-triangular system:
* (T(1:KI-1,1:KI-1) - WR)*X = SCALE*WORK.
*
JNXT = KI - 1
DO 60 J = KI - 1, 1, -1
IF( J.GT.JNXT )
$ GO TO 60
J1 = J
J2 = J
JNXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale X(1,1) to avoid overflow when updating
* the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
WORK( J+N ) = X( 1, 1 )
*
* Update right-hand side
*
CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
*
ELSE
*
* 2-by-2 diagonal block
*
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE,
$ T( J-1, J-1 ), LDT, ONE, ONE,
$ WORK( J-1+N ), N, WR, ZERO, X, 2,
$ SCALE, XNORM, IERR )
*
* Scale X(1,1) and X(2,1) to avoid overflow when
* updating the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
BETA = MAX( WORK( J-1 ), WORK( J ) )
IF( BETA.GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
X( 2, 1 ) = X( 2, 1 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
WORK( J-1+N ) = X( 1, 1 )
WORK( J+N ) = X( 2, 1 )
*
* Update right-hand side
*
CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
END IF
60 CONTINUE
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS ), 1 )
*
II = IDAMAX( KI, VR( 1, IS ), 1 )
REMAX = ONE / ABS( VR( II, IS ) )
CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
DO 70 K = KI + 1, N
VR( K, IS ) = ZERO
70 CONTINUE
ELSE
IF( KI.GT.1 )
$ CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR,
$ WORK( 1+N ), 1, WORK( KI+N ),
$ VR( 1, KI ), 1 )
*
II = IDAMAX( N, VR( 1, KI ), 1 )
REMAX = ONE / ABS( VR( II, KI ) )
CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
END IF
*
ELSE
*
* Complex right eigenvector.
*
* Initial solve
* [ (T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I* WI)]*X = 0.
* [ (T(KI,KI-1) T(KI,KI) ) ]
*
IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN
WORK( KI-1+N ) = ONE
WORK( KI+N2 ) = WI / T( KI-1, KI )
ELSE
WORK( KI-1+N ) = -WI / T( KI, KI-1 )
WORK( KI+N2 ) = ONE
END IF
WORK( KI+N ) = ZERO
WORK( KI-1+N2 ) = ZERO
*
* Form right-hand side
*
DO 80 K = 1, KI - 2
WORK( K+N ) = -WORK( KI-1+N )*T( K, KI-1 )
WORK( K+N2 ) = -WORK( KI+N2 )*T( K, KI )
80 CONTINUE
*
* Solve upper quasi-triangular system:
* (T(1:KI-2,1:KI-2) - (WR+i*WI))*X = SCALE*(WORK+i*WORK2)
*
JNXT = KI - 2
DO 90 J = KI - 2, 1, -1
IF( J.GT.JNXT )
$ GO TO 90
J1 = J
J2 = J
JNXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR, WI,
$ X, 2, SCALE, XNORM, IERR )
*
* Scale X(1,1) and X(1,2) to avoid overflow when
* updating the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
X( 1, 2 ) = X( 1, 2 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
END IF
WORK( J+N ) = X( 1, 1 )
WORK( J+N2 ) = X( 1, 2 )
*
* Update the right-hand side
*
CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1,
$ WORK( 1+N2 ), 1 )
*
ELSE
*
* 2-by-2 diagonal block
*
CALL DLALN2( .FALSE., 2, 2, SMIN, ONE,
$ T( J-1, J-1 ), LDT, ONE, ONE,
$ WORK( J-1+N ), N, WR, WI, X, 2, SCALE,
$ XNORM, IERR )
*
* Scale X to avoid overflow when updating
* the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
BETA = MAX( WORK( J-1 ), WORK( J ) )
IF( BETA.GT.BIGNUM / XNORM ) THEN
REC = ONE / XNORM
X( 1, 1 ) = X( 1, 1 )*REC
X( 1, 2 ) = X( 1, 2 )*REC
X( 2, 1 ) = X( 2, 1 )*REC
X( 2, 2 ) = X( 2, 2 )*REC
SCALE = SCALE*REC
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
END IF
WORK( J-1+N ) = X( 1, 1 )
WORK( J+N ) = X( 2, 1 )
WORK( J-1+N2 ) = X( 1, 2 )
WORK( J+N2 ) = X( 2, 2 )
*
* Update the right-hand side
*
CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1,
$ WORK( 1+N2 ), 1 )
CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1,
$ WORK( 1+N2 ), 1 )
END IF
90 CONTINUE
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS-1 ), 1 )
CALL DCOPY( KI, WORK( 1+N2 ), 1, VR( 1, IS ), 1 )
*
EMAX = ZERO
DO 100 K = 1, KI
EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+
$ ABS( VR( K, IS ) ) )
100 CONTINUE
*
REMAX = ONE / EMAX
CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 )
CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
DO 110 K = KI + 1, N
VR( K, IS-1 ) = ZERO
VR( K, IS ) = ZERO
110 CONTINUE
*
ELSE
*
IF( KI.GT.2 ) THEN
CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
$ WORK( 1+N ), 1, WORK( KI-1+N ),
$ VR( 1, KI-1 ), 1 )
CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
$ WORK( 1+N2 ), 1, WORK( KI+N2 ),
$ VR( 1, KI ), 1 )
ELSE
CALL DSCAL( N, WORK( KI-1+N ), VR( 1, KI-1 ),
$ 1 )
CALL DSCAL( N, WORK( KI+N2 ), VR( 1, KI ), 1 )
END IF
*
EMAX = ZERO
DO 120 K = 1, N
EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+
$ ABS( VR( K, KI ) ) )
120 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 )
CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
END IF
END IF
*
IS = IS - 1
IF( IP.NE.0 )
$ IS = IS - 1
130 CONTINUE
IF( IP.EQ.1 )
$ IP = 0
IF( IP.EQ.-1 )
$ IP = 1
140 CONTINUE
END IF
*
IF( LEFTV ) THEN
*
* Compute left eigenvectors.
*
IP = 0
IS = 1
DO 260 KI = 1, N
*
IF( IP.EQ.-1 )
$ GO TO 250
IF( KI.EQ.N )
$ GO TO 150
IF( T( KI+1, KI ).EQ.ZERO )
$ GO TO 150
IP = 1
*
150 CONTINUE
IF( SOMEV ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 250
END IF
*
* Compute the KI-th eigenvalue (WR,WI).
*
WR = T( KI, KI )
WI = ZERO
IF( IP.NE.0 )
$ WI = SQRT( ABS( T( KI, KI+1 ) ) )*
$ SQRT( ABS( T( KI+1, KI ) ) )
SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
IF( IP.EQ.0 ) THEN
*
* Real left eigenvector.
*
WORK( KI+N ) = ONE
*
* Form right-hand side
*
DO 160 K = KI + 1, N
WORK( K+N ) = -T( KI, K )
160 CONTINUE
*
* Solve the quasi-triangular system:
* (T(KI+1:N,KI+1:N) - WR)**T*X = SCALE*WORK
*
VMAX = ONE
VCRIT = BIGNUM
*
JNXT = KI + 1
DO 170 J = KI + 1, N
IF( J.LT.JNXT )
$ GO TO 170
J1 = J
J2 = J
JNXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side.
*
IF( WORK( J ).GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-1, T( KI+1, J ), 1,
$ WORK( KI+1+N ), 1 )
*
* Solve (T(J,J)-WR)**T*X = WORK
*
CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
WORK( J+N ) = X( 1, 1 )
VMAX = MAX( ABS( WORK( J+N ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
ELSE
*
* 2-by-2 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side.
*
BETA = MAX( WORK( J ), WORK( J+1 ) )
IF( BETA.GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-1, T( KI+1, J ), 1,
$ WORK( KI+1+N ), 1 )
*
WORK( J+1+N ) = WORK( J+1+N ) -
$ DDOT( J-KI-1, T( KI+1, J+1 ), 1,
$ WORK( KI+1+N ), 1 )
*
* Solve
* [T(J,J)-WR T(J,J+1) ]**T * X = SCALE*( WORK1 )
* [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
WORK( J+N ) = X( 1, 1 )
WORK( J+1+N ) = X( 2, 1 )
*
VMAX = MAX( ABS( WORK( J+N ) ),
$ ABS( WORK( J+1+N ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
END IF
170 CONTINUE
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ),
$ 1 )
*
II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
REMAX = ONE / ABS( VL( II, IS ) )
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
*
DO 180 K = 1, KI - 1
VL( K, IS ) = ZERO
180 CONTINUE
*
ELSE
*
IF( KI.LT.N )
$ CALL DGEMV( 'N', N, N-KI, ONE, VL( 1, KI+1 ),
$ LDVL,
$ WORK( KI+1+N ), 1, WORK( KI+N ),
$ VL( 1, KI ), 1 )
*
II = IDAMAX( N, VL( 1, KI ), 1 )
REMAX = ONE / ABS( VL( II, KI ) )
CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
*
END IF
*
ELSE
*
* Complex left eigenvector.
*
* Initial solve:
* ((T(KI,KI) T(KI,KI+1) )**T - (WR - I* WI))*X = 0.
* ((T(KI+1,KI) T(KI+1,KI+1)) )
*
IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN
WORK( KI+N ) = WI / T( KI, KI+1 )
WORK( KI+1+N2 ) = ONE
ELSE
WORK( KI+N ) = ONE
WORK( KI+1+N2 ) = -WI / T( KI+1, KI )
END IF
WORK( KI+1+N ) = ZERO
WORK( KI+N2 ) = ZERO
*
* Form right-hand side
*
DO 190 K = KI + 2, N
WORK( K+N ) = -WORK( KI+N )*T( KI, K )
WORK( K+N2 ) = -WORK( KI+1+N2 )*T( KI+1, K )
190 CONTINUE
*
* Solve complex quasi-triangular system:
* ( T(KI+2,N:KI+2,N) - (WR-i*WI) )*X = WORK1+i*WORK2
*
VMAX = ONE
VCRIT = BIGNUM
*
JNXT = KI + 2
DO 200 J = KI + 2, N
IF( J.LT.JNXT )
$ GO TO 200
J1 = J
J2 = J
JNXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
* Scale if necessary to avoid overflow when
* forming the right-hand side elements.
*
IF( WORK( J ).GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N ), 1 )
WORK( J+N2 ) = WORK( J+N2 ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N2 ), 1 )
*
* Solve (T(J,J)-(WR-i*WI))*(X11+i*X12)= WK+I*WK2
*
CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ -WI, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
END IF
WORK( J+N ) = X( 1, 1 )
WORK( J+N2 ) = X( 1, 2 )
VMAX = MAX( ABS( WORK( J+N ) ),
$ ABS( WORK( J+N2 ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
ELSE
*
* 2-by-2 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side elements.
*
BETA = MAX( WORK( J ), WORK( J+1 ) )
IF( BETA.GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N ), 1 )
*
WORK( J+N2 ) = WORK( J+N2 ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N2 ), 1 )
*
WORK( J+1+N ) = WORK( J+1+N ) -
$ DDOT( J-KI-2, T( KI+2, J+1 ), 1,
$ WORK( KI+2+N ), 1 )
*
WORK( J+1+N2 ) = WORK( J+1+N2 ) -
$ DDOT( J-KI-2, T( KI+2, J+1 ),
$ 1,
$ WORK( KI+2+N2 ), 1 )
*
* Solve 2-by-2 complex linear equation
* ([T(j,j) T(j,j+1) ]**T-(wr-i*wi)*I)*X = SCALE*B
* ([T(j+1,j) T(j+1,j+1)] )
*
CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ -WI, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
END IF
WORK( J+N ) = X( 1, 1 )
WORK( J+N2 ) = X( 1, 2 )
WORK( J+1+N ) = X( 2, 1 )
WORK( J+1+N2 ) = X( 2, 2 )
VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ),
$ ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
END IF
200 CONTINUE
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ),
$ 1 )
CALL DCOPY( N-KI+1, WORK( KI+N2 ), 1, VL( KI,
$ IS+1 ),
$ 1 )
*
EMAX = ZERO
DO 220 K = KI, N
EMAX = MAX( EMAX, ABS( VL( K, IS ) )+
$ ABS( VL( K, IS+1 ) ) )
220 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 )
*
DO 230 K = 1, KI - 1
VL( K, IS ) = ZERO
VL( K, IS+1 ) = ZERO
230 CONTINUE
ELSE
IF( KI.LT.N-1 ) THEN
CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
$ LDVL, WORK( KI+2+N ), 1, WORK( KI+N ),
$ VL( 1, KI ), 1 )
CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
$ LDVL, WORK( KI+2+N2 ), 1,
$ WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 )
ELSE
CALL DSCAL( N, WORK( KI+N ), VL( 1, KI ), 1 )
CALL DSCAL( N, WORK( KI+1+N2 ), VL( 1, KI+1 ),
$ 1 )
END IF
*
EMAX = ZERO
DO 240 K = 1, N
EMAX = MAX( EMAX, ABS( VL( K, KI ) )+
$ ABS( VL( K, KI+1 ) ) )
240 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 )
*
END IF
*
END IF
*
IS = IS + 1
IF( IP.NE.0 )
$ IS = IS + 1
250 CONTINUE
IF( IP.EQ.-1 )
$ IP = 0
IF( IP.EQ.1 )
$ IP = -1
*
260 CONTINUE
*
END IF
*
RETURN
*
* End of DTREVC
*
END
*> \brief \b DTREVC3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREVC3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrevc3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrevc3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrevc3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
* VR, LDVR, MM, M, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, SIDE
* INTEGER INFO, LDT, LDVL, LDVR, LWORK, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTREVC3 computes some or all of the right and/or left eigenvectors of
*> a real upper quasi-triangular matrix T.
*> Matrices of this type are produced by the Schur factorization of
*> a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.
*>
*> The right eigenvector x and the left eigenvector y of T corresponding
*> to an eigenvalue w are defined by:
*>
*> T*x = w*x, (y**T)*T = w*(y**T)
*>
*> where y**T denotes the transpose of the vector y.
*> The eigenvalues are not input to this routine, but are read directly
*> from the diagonal blocks of T.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
*> input matrix. If Q is the orthogonal factor that reduces a matrix
*> A to Schur form T, then Q*X and Q*Y are the matrices of right and
*> left eigenvectors of A.
*>
*> This uses a Level 3 BLAS version of the back transformation.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': compute right eigenvectors only;
*> = 'L': compute left eigenvectors only;
*> = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute all right and/or left eigenvectors;
*> = 'B': compute all right and/or left eigenvectors,
*> backtransformed by the matrices in VR and/or VL;
*> = 'S': compute selected right and/or left eigenvectors,
*> as indicated by the logical array SELECT.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY = 'S', SELECT specifies the eigenvectors to be
*> computed.
*> If w(j) is a real eigenvalue, the corresponding real
*> eigenvector is computed if SELECT(j) is .TRUE..
*> If w(j) and w(j+1) are the real and imaginary parts of a
*> complex eigenvalue, the corresponding complex eigenvector is
*> computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
*> on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
*> .FALSE..
*> Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The upper quasi-triangular matrix T in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*> contain an N-by-N matrix Q (usually the orthogonal matrix Q
*> of Schur vectors returned by DHSEQR).
*> On exit, if SIDE = 'L' or 'B', VL contains:
*> if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
*> if HOWMNY = 'B', the matrix Q*Y;
*> if HOWMNY = 'S', the left eigenvectors of T specified by
*> SELECT, stored consecutively in the columns
*> of VL, in the same order as their
*> eigenvalues.
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part, and the second the imaginary part.
*> Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL.
*> LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*> contain an N-by-N matrix Q (usually the orthogonal matrix Q
*> of Schur vectors returned by DHSEQR).
*> On exit, if SIDE = 'R' or 'B', VR contains:
*> if HOWMNY = 'A', the matrix X of right eigenvectors of T;
*> if HOWMNY = 'B', the matrix Q*X;
*> if HOWMNY = 'S', the right eigenvectors of T specified by
*> SELECT, stored consecutively in the columns
*> of VR, in the same order as their
*> eigenvalues.
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part and the second the imaginary part.
*> Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR.
*> LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of columns in the arrays VL and/or VR actually
*> used to store the eigenvectors.
*> If HOWMNY = 'A' or 'B', M is set to N.
*> Each selected real eigenvector occupies one column and each
*> selected complex eigenvector occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of array WORK. LWORK >= max(1,3*N).
*> For optimum performance, LWORK >= N + 2*N*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trevc3
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The algorithm used in this program is basically backward (forward)
*> substitution, with scaling to make the the code robust against
*> possible overflow.
*>
*> Each eigenvector is normalized so that the element of largest
*> magnitude has magnitude 1; here the magnitude of a complex number
*> (x,y) is taken to be |x| + |y|.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
$ VR, LDVR, MM, M, WORK, LWORK, INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDT, LDVL, LDVR, LWORK, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
INTEGER NBMIN, NBMAX
PARAMETER ( NBMIN = 8, NBMAX = 128 )
* ..
* .. Local Scalars ..
LOGICAL ALLV, BOTHV, LEFTV, LQUERY, OVER, PAIR,
$ RIGHTV, SOMEV
INTEGER I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI,
$ IV, MAXWRK, NB, KI2
DOUBLE PRECISION BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
$ SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
$ XNORM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX, ILAENV
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, ILAENV, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DLALN2, DSCAL,
$ XERBLA,
$ DGEMM, DLASET, DLACPY
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Local Arrays ..
DOUBLE PRECISION X( 2, 2 )
INTEGER ISCOMPLEX( NBMAX )
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
BOTHV = LSAME( SIDE, 'B' )
RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
ALLV = LSAME( HOWMNY, 'A' )
OVER = LSAME( HOWMNY, 'B' )
SOMEV = LSAME( HOWMNY, 'S' )
*
INFO = 0
NB = ILAENV( 1, 'DTREVC', SIDE // HOWMNY, N, -1, -1, -1 )
MAXWRK = MAX( 1, N + 2*N*NB )
WORK(1) = MAXWRK
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -1
ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
INFO = -8
ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN
INFO = -14
ELSE
*
* Set M to the number of columns required to store the selected
* eigenvectors, standardize the array SELECT if necessary, and
* test MM.
*
IF( SOMEV ) THEN
M = 0
PAIR = .FALSE.
DO 10 J = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
SELECT( J ) = .FALSE.
ELSE
IF( J.LT.N ) THEN
IF( T( J+1, J ).EQ.ZERO ) THEN
IF( SELECT( J ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN
SELECT( J ) = .TRUE.
M = M + 2
END IF
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
ELSE
M = N
END IF
*
IF( MM.LT.M ) THEN
INFO = -11
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTREVC3', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* Use blocked version of back-transformation if sufficient workspace.
* Zero-out the workspace to avoid potential NaN propagation.
*
IF( OVER .AND. LWORK .GE. N + 2*N*NBMIN ) THEN
NB = (LWORK - N) / (2*N)
NB = MIN( NB, NBMAX )
CALL DLASET( 'F', N, 1+2*NB, ZERO, ZERO, WORK, N )
ELSE
NB = 1
END IF
*
* Set the constants to control overflow.
*
UNFL = DLAMCH( 'Safe minimum' )
OVFL = ONE / UNFL
ULP = DLAMCH( 'Precision' )
SMLNUM = UNFL*( N / ULP )
BIGNUM = ( ONE-ULP ) / SMLNUM
*
* Compute 1-norm of each column of strictly upper triangular
* part of T to control overflow in triangular solver.
*
WORK( 1 ) = ZERO
DO 30 J = 2, N
WORK( J ) = ZERO
DO 20 I = 1, J - 1
WORK( J ) = WORK( J ) + ABS( T( I, J ) )
20 CONTINUE
30 CONTINUE
*
* Index IP is used to specify the real or complex eigenvalue:
* IP = 0, real eigenvalue,
* 1, first of conjugate complex pair: (wr,wi)
* -1, second of conjugate complex pair: (wr,wi)
* ISCOMPLEX array stores IP for each column in current block.
*
IF( RIGHTV ) THEN
*
* ============================================================
* Compute right eigenvectors.
*
* IV is index of column in current block.
* For complex right vector, uses IV-1 for real part and IV for complex part.
* Non-blocked version always uses IV=2;
* blocked version starts with IV=NB, goes down to 1 or 2.
* (Note the "0-th" column is used for 1-norms computed above.)
IV = 2
IF( NB.GT.2 ) THEN
IV = NB
END IF
IP = 0
IS = M
DO 140 KI = N, 1, -1
IF( IP.EQ.-1 ) THEN
* previous iteration (ki+1) was second of conjugate pair,
* so this ki is first of conjugate pair; skip to end of loop
IP = 1
GO TO 140
ELSE IF( KI.EQ.1 ) THEN
* last column, so this ki must be real eigenvalue
IP = 0
ELSE IF( T( KI, KI-1 ).EQ.ZERO ) THEN
* zero on sub-diagonal, so this ki is real eigenvalue
IP = 0
ELSE
* non-zero on sub-diagonal, so this ki is second of conjugate pair
IP = -1
END IF
IF( SOMEV ) THEN
IF( IP.EQ.0 ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 140
ELSE
IF( .NOT.SELECT( KI-1 ) )
$ GO TO 140
END IF
END IF
*
* Compute the KI-th eigenvalue (WR,WI).
*
WR = T( KI, KI )
WI = ZERO
IF( IP.NE.0 )
$ WI = SQRT( ABS( T( KI, KI-1 ) ) )*
$ SQRT( ABS( T( KI-1, KI ) ) )
SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
IF( IP.EQ.0 ) THEN
*
* --------------------------------------------------------
* Real right eigenvector
*
WORK( KI + IV*N ) = ONE
*
* Form right-hand side.
*
DO 50 K = 1, KI - 1
WORK( K + IV*N ) = -T( K, KI )
50 CONTINUE
*
* Solve upper quasi-triangular system:
* [ T(1:KI-1,1:KI-1) - WR ]*X = SCALE*WORK.
*
JNXT = KI - 1
DO 60 J = KI - 1, 1, -1
IF( J.GT.JNXT )
$ GO TO 60
J1 = J
J2 = J
JNXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale X(1,1) to avoid overflow when updating
* the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 )
WORK( J+IV*N ) = X( 1, 1 )
*
* Update right-hand side
*
CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
$ WORK( 1+IV*N ), 1 )
*
ELSE
*
* 2-by-2 diagonal block
*
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE,
$ T( J-1, J-1 ), LDT, ONE, ONE,
$ WORK( J-1+IV*N ), N, WR, ZERO, X, 2,
$ SCALE, XNORM, IERR )
*
* Scale X(1,1) and X(2,1) to avoid overflow when
* updating the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
BETA = MAX( WORK( J-1 ), WORK( J ) )
IF( BETA.GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
X( 2, 1 ) = X( 2, 1 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 )
WORK( J-1+IV*N ) = X( 1, 1 )
WORK( J +IV*N ) = X( 2, 1 )
*
* Update right-hand side
*
CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
$ WORK( 1+IV*N ), 1 )
CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
$ WORK( 1+IV*N ), 1 )
END IF
60 CONTINUE
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
* ------------------------------
* no back-transform: copy x to VR and normalize.
CALL DCOPY( KI, WORK( 1 + IV*N ), 1, VR( 1, IS ),
$ 1 )
*
II = IDAMAX( KI, VR( 1, IS ), 1 )
REMAX = ONE / ABS( VR( II, IS ) )
CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
DO 70 K = KI + 1, N
VR( K, IS ) = ZERO
70 CONTINUE
*
ELSE IF( NB.EQ.1 ) THEN
* ------------------------------
* version 1: back-transform each vector with GEMV, Q*x.
IF( KI.GT.1 )
$ CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR,
$ WORK( 1 + IV*N ), 1, WORK( KI + IV*N ),
$ VR( 1, KI ), 1 )
*
II = IDAMAX( N, VR( 1, KI ), 1 )
REMAX = ONE / ABS( VR( II, KI ) )
CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
*
ELSE
* ------------------------------
* version 2: back-transform block of vectors with GEMM
* zero out below vector
DO K = KI + 1, N
WORK( K + IV*N ) = ZERO
END DO
ISCOMPLEX( IV ) = IP
* back-transform and normalization is done below
END IF
ELSE
*
* --------------------------------------------------------
* Complex right eigenvector.
*
* Initial solve
* [ ( T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I*WI) ]*X = 0.
* [ ( T(KI, KI-1) T(KI, KI) ) ]
*
IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN
WORK( KI-1 + (IV-1)*N ) = ONE
WORK( KI + (IV )*N ) = WI / T( KI-1, KI )
ELSE
WORK( KI-1 + (IV-1)*N ) = -WI / T( KI, KI-1 )
WORK( KI + (IV )*N ) = ONE
END IF
WORK( KI + (IV-1)*N ) = ZERO
WORK( KI-1 + (IV )*N ) = ZERO
*
* Form right-hand side.
*
DO 80 K = 1, KI - 2
WORK( K+(IV-1)*N ) = -WORK( KI-1+(IV-1)*N )*T(K,KI-1)
WORK( K+(IV )*N ) = -WORK( KI +(IV )*N )*T(K,KI )
80 CONTINUE
*
* Solve upper quasi-triangular system:
* [ T(1:KI-2,1:KI-2) - (WR+i*WI) ]*X = SCALE*(WORK+i*WORK2)
*
JNXT = KI - 2
DO 90 J = KI - 2, 1, -1
IF( J.GT.JNXT )
$ GO TO 90
J1 = J
J2 = J
JNXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+(IV-1)*N ), N,
$ WR, WI, X, 2, SCALE, XNORM, IERR )
*
* Scale X(1,1) and X(1,2) to avoid overflow when
* updating the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
X( 1, 2 ) = X( 1, 2 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ),
$ 1 )
CALL DSCAL( KI, SCALE, WORK( 1+(IV )*N ),
$ 1 )
END IF
WORK( J+(IV-1)*N ) = X( 1, 1 )
WORK( J+(IV )*N ) = X( 1, 2 )
*
* Update the right-hand side
*
CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
$ WORK( 1+(IV-1)*N ), 1 )
CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1,
$ WORK( 1+(IV )*N ), 1 )
*
ELSE
*
* 2-by-2 diagonal block
*
CALL DLALN2( .FALSE., 2, 2, SMIN, ONE,
$ T( J-1, J-1 ), LDT, ONE, ONE,
$ WORK( J-1+(IV-1)*N ), N, WR, WI, X, 2,
$ SCALE, XNORM, IERR )
*
* Scale X to avoid overflow when updating
* the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
BETA = MAX( WORK( J-1 ), WORK( J ) )
IF( BETA.GT.BIGNUM / XNORM ) THEN
REC = ONE / XNORM
X( 1, 1 ) = X( 1, 1 )*REC
X( 1, 2 ) = X( 1, 2 )*REC
X( 2, 1 ) = X( 2, 1 )*REC
X( 2, 2 ) = X( 2, 2 )*REC
SCALE = SCALE*REC
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ),
$ 1 )
CALL DSCAL( KI, SCALE, WORK( 1+(IV )*N ),
$ 1 )
END IF
WORK( J-1+(IV-1)*N ) = X( 1, 1 )
WORK( J +(IV-1)*N ) = X( 2, 1 )
WORK( J-1+(IV )*N ) = X( 1, 2 )
WORK( J +(IV )*N ) = X( 2, 2 )
*
* Update the right-hand side
*
CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
$ WORK( 1+(IV-1)*N ), 1 )
CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
$ WORK( 1+(IV-1)*N ), 1 )
CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1,
$ WORK( 1+(IV )*N ), 1 )
CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1,
$ WORK( 1+(IV )*N ), 1 )
END IF
90 CONTINUE
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
* ------------------------------
* no back-transform: copy x to VR and normalize.
CALL DCOPY( KI, WORK( 1+(IV-1)*N ), 1, VR(1,IS-1),
$ 1 )
CALL DCOPY( KI, WORK( 1+(IV )*N ), 1, VR(1,IS ),
$ 1 )
*
EMAX = ZERO
DO 100 K = 1, KI
EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+
$ ABS( VR( K, IS ) ) )
100 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 )
CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
DO 110 K = KI + 1, N
VR( K, IS-1 ) = ZERO
VR( K, IS ) = ZERO
110 CONTINUE
*
ELSE IF( NB.EQ.1 ) THEN
* ------------------------------
* version 1: back-transform each vector with GEMV, Q*x.
IF( KI.GT.2 ) THEN
CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
$ WORK( 1 + (IV-1)*N ), 1,
$ WORK( KI-1 + (IV-1)*N ), VR(1,KI-1), 1)
CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
$ WORK( 1 + (IV)*N ), 1,
$ WORK( KI + (IV)*N ), VR( 1, KI ), 1 )
ELSE
CALL DSCAL( N, WORK(KI-1+(IV-1)*N), VR(1,KI-1),
$ 1)
CALL DSCAL( N, WORK(KI +(IV )*N), VR(1,KI ),
$ 1)
END IF
*
EMAX = ZERO
DO 120 K = 1, N
EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+
$ ABS( VR( K, KI ) ) )
120 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 )
CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
*
ELSE
* ------------------------------
* version 2: back-transform block of vectors with GEMM
* zero out below vector
DO K = KI + 1, N
WORK( K + (IV-1)*N ) = ZERO
WORK( K + (IV )*N ) = ZERO
END DO
ISCOMPLEX( IV-1 ) = -IP
ISCOMPLEX( IV ) = IP
IV = IV - 1
* back-transform and normalization is done below
END IF
END IF
IF( NB.GT.1 ) THEN
* --------------------------------------------------------
* Blocked version of back-transform
* For complex case, KI2 includes both vectors (KI-1 and KI)
IF( IP.EQ.0 ) THEN
KI2 = KI
ELSE
KI2 = KI - 1
END IF
* Columns IV:NB of work are valid vectors.
* When the number of vectors stored reaches NB-1 or NB,
* or if this was last vector, do the GEMM
IF( (IV.LE.2) .OR. (KI2.EQ.1) ) THEN
CALL DGEMM( 'N', 'N', N, NB-IV+1, KI2+NB-IV, ONE,
$ VR, LDVR,
$ WORK( 1 + (IV)*N ), N,
$ ZERO,
$ WORK( 1 + (NB+IV)*N ), N )
* normalize vectors
DO K = IV, NB
IF( ISCOMPLEX(K).EQ.0 ) THEN
* real eigenvector
II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 )
REMAX = ONE / ABS( WORK( II + (NB+K)*N ) )
ELSE IF( ISCOMPLEX(K).EQ.1 ) THEN
* first eigenvector of conjugate pair
EMAX = ZERO
DO II = 1, N
EMAX = MAX( EMAX,
$ ABS( WORK( II + (NB+K )*N ) )+
$ ABS( WORK( II + (NB+K+1)*N ) ) )
END DO
REMAX = ONE / EMAX
* else if ISCOMPLEX(K).EQ.-1
* second eigenvector of conjugate pair
* reuse same REMAX as previous K
END IF
CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 )
END DO
CALL DLACPY( 'F', N, NB-IV+1,
$ WORK( 1 + (NB+IV)*N ), N,
$ VR( 1, KI2 ), LDVR )
IV = NB
ELSE
IV = IV - 1
END IF
END IF ! blocked back-transform
*
IS = IS - 1
IF( IP.NE.0 )
$ IS = IS - 1
140 CONTINUE
END IF
IF( LEFTV ) THEN
*
* ============================================================
* Compute left eigenvectors.
*
* IV is index of column in current block.
* For complex left vector, uses IV for real part and IV+1 for complex part.
* Non-blocked version always uses IV=1;
* blocked version starts with IV=1, goes up to NB-1 or NB.
* (Note the "0-th" column is used for 1-norms computed above.)
IV = 1
IP = 0
IS = 1
DO 260 KI = 1, N
IF( IP.EQ.1 ) THEN
* previous iteration (ki-1) was first of conjugate pair,
* so this ki is second of conjugate pair; skip to end of loop
IP = -1
GO TO 260
ELSE IF( KI.EQ.N ) THEN
* last column, so this ki must be real eigenvalue
IP = 0
ELSE IF( T( KI+1, KI ).EQ.ZERO ) THEN
* zero on sub-diagonal, so this ki is real eigenvalue
IP = 0
ELSE
* non-zero on sub-diagonal, so this ki is first of conjugate pair
IP = 1
END IF
*
IF( SOMEV ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 260
END IF
*
* Compute the KI-th eigenvalue (WR,WI).
*
WR = T( KI, KI )
WI = ZERO
IF( IP.NE.0 )
$ WI = SQRT( ABS( T( KI, KI+1 ) ) )*
$ SQRT( ABS( T( KI+1, KI ) ) )
SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
IF( IP.EQ.0 ) THEN
*
* --------------------------------------------------------
* Real left eigenvector
*
WORK( KI + IV*N ) = ONE
*
* Form right-hand side.
*
DO 160 K = KI + 1, N
WORK( K + IV*N ) = -T( KI, K )
160 CONTINUE
*
* Solve transposed quasi-triangular system:
* [ T(KI+1:N,KI+1:N) - WR ]**T * X = SCALE*WORK
*
VMAX = ONE
VCRIT = BIGNUM
*
JNXT = KI + 1
DO 170 J = KI + 1, N
IF( J.LT.JNXT )
$ GO TO 170
J1 = J
J2 = J
JNXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side.
*
IF( WORK( J ).GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+IV*N ) = WORK( J+IV*N ) -
$ DDOT( J-KI-1, T( KI+1, J ), 1,
$ WORK( KI+1+IV*N ), 1 )
*
* Solve [ T(J,J) - WR ]**T * X = WORK
*
CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ),
$ 1 )
WORK( J+IV*N ) = X( 1, 1 )
VMAX = MAX( ABS( WORK( J+IV*N ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
ELSE
*
* 2-by-2 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side.
*
BETA = MAX( WORK( J ), WORK( J+1 ) )
IF( BETA.GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+IV*N ) = WORK( J+IV*N ) -
$ DDOT( J-KI-1, T( KI+1, J ), 1,
$ WORK( KI+1+IV*N ), 1 )
*
WORK( J+1+IV*N ) = WORK( J+1+IV*N ) -
$ DDOT( J-KI-1, T( KI+1, J+1 ),
$ 1,
$ WORK( KI+1+IV*N ), 1 )
*
* Solve
* [ T(J,J)-WR T(J,J+1) ]**T * X = SCALE*( WORK1 )
* [ T(J+1,J) T(J+1,J+1)-WR ] ( WORK2 )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ),
$ 1 )
WORK( J +IV*N ) = X( 1, 1 )
WORK( J+1+IV*N ) = X( 2, 1 )
*
VMAX = MAX( ABS( WORK( J +IV*N ) ),
$ ABS( WORK( J+1+IV*N ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
END IF
170 CONTINUE
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
* ------------------------------
* no back-transform: copy x to VL and normalize.
CALL DCOPY( N-KI+1, WORK( KI + IV*N ), 1,
$ VL( KI, IS ), 1 )
*
II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
REMAX = ONE / ABS( VL( II, IS ) )
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
*
DO 180 K = 1, KI - 1
VL( K, IS ) = ZERO
180 CONTINUE
*
ELSE IF( NB.EQ.1 ) THEN
* ------------------------------
* version 1: back-transform each vector with GEMV, Q*x.
IF( KI.LT.N )
$ CALL DGEMV( 'N', N, N-KI, ONE,
$ VL( 1, KI+1 ), LDVL,
$ WORK( KI+1 + IV*N ), 1,
$ WORK( KI + IV*N ), VL( 1, KI ), 1 )
*
II = IDAMAX( N, VL( 1, KI ), 1 )
REMAX = ONE / ABS( VL( II, KI ) )
CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
*
ELSE
* ------------------------------
* version 2: back-transform block of vectors with GEMM
* zero out above vector
* could go from KI-NV+1 to KI-1
DO K = 1, KI - 1
WORK( K + IV*N ) = ZERO
END DO
ISCOMPLEX( IV ) = IP
* back-transform and normalization is done below
END IF
ELSE
*
* --------------------------------------------------------
* Complex left eigenvector.
*
* Initial solve:
* [ ( T(KI,KI) T(KI,KI+1) )**T - (WR - I* WI) ]*X = 0.
* [ ( T(KI+1,KI) T(KI+1,KI+1) ) ]
*
IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN
WORK( KI + (IV )*N ) = WI / T( KI, KI+1 )
WORK( KI+1 + (IV+1)*N ) = ONE
ELSE
WORK( KI + (IV )*N ) = ONE
WORK( KI+1 + (IV+1)*N ) = -WI / T( KI+1, KI )
END IF
WORK( KI+1 + (IV )*N ) = ZERO
WORK( KI + (IV+1)*N ) = ZERO
*
* Form right-hand side.
*
DO 190 K = KI + 2, N
WORK( K+(IV )*N ) = -WORK( KI +(IV )*N )*T(KI, K)
WORK( K+(IV+1)*N ) = -WORK( KI+1+(IV+1)*N )*T(KI+1,K)
190 CONTINUE
*
* Solve transposed quasi-triangular system:
* [ T(KI+2:N,KI+2:N)**T - (WR-i*WI) ]*X = WORK1+i*WORK2
*
VMAX = ONE
VCRIT = BIGNUM
*
JNXT = KI + 2
DO 200 J = KI + 2, N
IF( J.LT.JNXT )
$ GO TO 200
J1 = J
J2 = J
JNXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
* Scale if necessary to avoid overflow when
* forming the right-hand side elements.
*
IF( WORK( J ).GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK(KI+(IV )*N),
$ 1 )
CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N),
$ 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+(IV )*N ) = WORK( J+(IV)*N ) -
$ DDOT( J-KI-2, T( KI+2, J ),
$ 1,
$ WORK( KI+2+(IV)*N ), 1 )
WORK( J+(IV+1)*N ) = WORK( J+(IV+1)*N ) -
$ DDOT( J-KI-2, T( KI+2, J ),
$ 1,
$ WORK( KI+2+(IV+1)*N ), 1 )
*
* Solve [ T(J,J)-(WR-i*WI) ]*(X11+i*X12)= WK+I*WK2
*
CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
$ J ),
$ LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
$ -WI, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV )*N),
$ 1)
CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N),
$ 1)
END IF
WORK( J+(IV )*N ) = X( 1, 1 )
WORK( J+(IV+1)*N ) = X( 1, 2 )
VMAX = MAX( ABS( WORK( J+(IV )*N ) ),
$ ABS( WORK( J+(IV+1)*N ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
ELSE
*
* 2-by-2 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side elements.
*
BETA = MAX( WORK( J ), WORK( J+1 ) )
IF( BETA.GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK(KI+(IV )*N),
$ 1 )
CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N),
$ 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J +(IV )*N ) = WORK( J+(IV)*N ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+(IV)*N ), 1 )
*
WORK( J +(IV+1)*N ) = WORK( J+(IV+1)*N ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+(IV+1)*N ), 1 )
*
WORK( J+1+(IV )*N ) = WORK( J+1+(IV)*N ) -
$ DDOT( J-KI-2, T( KI+2, J+1 ),
$ 1,
$ WORK( KI+2+(IV)*N ), 1 )
*
WORK( J+1+(IV+1)*N ) = WORK( J+1+(IV+1)*N ) -
$ DDOT( J-KI-2, T( KI+2, J+1 ),
$ 1,
$ WORK( KI+2+(IV+1)*N ), 1 )
*
* Solve 2-by-2 complex linear equation
* [ (T(j,j) T(j,j+1) )**T - (wr-i*wi)*I ]*X = SCALE*B
* [ (T(j+1,j) T(j+1,j+1)) ]
*
CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
$ -WI, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV )*N),
$ 1)
CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N),
$ 1)
END IF
WORK( J +(IV )*N ) = X( 1, 1 )
WORK( J +(IV+1)*N ) = X( 1, 2 )
WORK( J+1+(IV )*N ) = X( 2, 1 )
WORK( J+1+(IV+1)*N ) = X( 2, 2 )
VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ),
$ ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ),
$ VMAX )
VCRIT = BIGNUM / VMAX
*
END IF
200 CONTINUE
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
* ------------------------------
* no back-transform: copy x to VL and normalize.
CALL DCOPY( N-KI+1, WORK( KI + (IV )*N ), 1,
$ VL( KI, IS ), 1 )
CALL DCOPY( N-KI+1, WORK( KI + (IV+1)*N ), 1,
$ VL( KI, IS+1 ), 1 )
*
EMAX = ZERO
DO 220 K = KI, N
EMAX = MAX( EMAX, ABS( VL( K, IS ) )+
$ ABS( VL( K, IS+1 ) ) )
220 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 )
*
DO 230 K = 1, KI - 1
VL( K, IS ) = ZERO
VL( K, IS+1 ) = ZERO
230 CONTINUE
*
ELSE IF( NB.EQ.1 ) THEN
* ------------------------------
* version 1: back-transform each vector with GEMV, Q*x.
IF( KI.LT.N-1 ) THEN
CALL DGEMV( 'N', N, N-KI-1, ONE,
$ VL( 1, KI+2 ), LDVL,
$ WORK( KI+2 + (IV)*N ), 1,
$ WORK( KI + (IV)*N ),
$ VL( 1, KI ), 1 )
CALL DGEMV( 'N', N, N-KI-1, ONE,
$ VL( 1, KI+2 ), LDVL,
$ WORK( KI+2 + (IV+1)*N ), 1,
$ WORK( KI+1 + (IV+1)*N ),
$ VL( 1, KI+1 ), 1 )
ELSE
CALL DSCAL( N, WORK(KI+ (IV )*N), VL(1, KI ),
$ 1)
CALL DSCAL( N, WORK(KI+1+(IV+1)*N), VL(1, KI+1),
$ 1)
END IF
*
EMAX = ZERO
DO 240 K = 1, N
EMAX = MAX( EMAX, ABS( VL( K, KI ) )+
$ ABS( VL( K, KI+1 ) ) )
240 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 )
*
ELSE
* ------------------------------
* version 2: back-transform block of vectors with GEMM
* zero out above vector
* could go from KI-NV+1 to KI-1
DO K = 1, KI - 1
WORK( K + (IV )*N ) = ZERO
WORK( K + (IV+1)*N ) = ZERO
END DO
ISCOMPLEX( IV ) = IP
ISCOMPLEX( IV+1 ) = -IP
IV = IV + 1
* back-transform and normalization is done below
END IF
END IF
IF( NB.GT.1 ) THEN
* --------------------------------------------------------
* Blocked version of back-transform
* For complex case, KI2 includes both vectors (KI and KI+1)
IF( IP.EQ.0 ) THEN
KI2 = KI
ELSE
KI2 = KI + 1
END IF
* Columns 1:IV of work are valid vectors.
* When the number of vectors stored reaches NB-1 or NB,
* or if this was last vector, do the GEMM
IF( (IV.GE.NB-1) .OR. (KI2.EQ.N) ) THEN
CALL DGEMM( 'N', 'N', N, IV, N-KI2+IV, ONE,
$ VL( 1, KI2-IV+1 ), LDVL,
$ WORK( KI2-IV+1 + (1)*N ), N,
$ ZERO,
$ WORK( 1 + (NB+1)*N ), N )
* normalize vectors
DO K = 1, IV
IF( ISCOMPLEX(K).EQ.0) THEN
* real eigenvector
II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 )
REMAX = ONE / ABS( WORK( II + (NB+K)*N ) )
ELSE IF( ISCOMPLEX(K).EQ.1) THEN
* first eigenvector of conjugate pair
EMAX = ZERO
DO II = 1, N
EMAX = MAX( EMAX,
$ ABS( WORK( II + (NB+K )*N ) )+
$ ABS( WORK( II + (NB+K+1)*N ) ) )
END DO
REMAX = ONE / EMAX
* else if ISCOMPLEX(K).EQ.-1
* second eigenvector of conjugate pair
* reuse same REMAX as previous K
END IF
CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 )
END DO
CALL DLACPY( 'F', N, IV,
$ WORK( 1 + (NB+1)*N ), N,
$ VL( 1, KI2-IV+1 ), LDVL )
IV = 1
ELSE
IV = IV + 1
END IF
END IF ! blocked back-transform
*
IS = IS + 1
IF( IP.NE.0 )
$ IS = IS + 1
260 CONTINUE
END IF
*
RETURN
*
* End of DTREVC3
*
END
*> \brief \b DTREXC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREXC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrexc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrexc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrexc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPQ
* INTEGER IFST, ILST, INFO, LDQ, LDT, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTREXC reorders the real Schur factorization of a real matrix
*> A = Q*T*Q**T, so that the diagonal block of T with row index IFST is
*> moved to row ILST.
*>
*> The real Schur form T is reordered by an orthogonal similarity
*> transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
*> is updated by postmultiplying it with Z.
*>
*> T must be in Schur canonical form (as returned by DHSEQR), that is,
*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*> 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> = 'V': update the matrix Q of Schur vectors;
*> = 'N': do not update Q.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> If N == 0 arguments ILST and IFST may be any value.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> On entry, the upper quasi-triangular matrix T, in Schur
*> Schur canonical form.
*> On exit, the reordered upper quasi-triangular matrix, again
*> in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
*> On exit, if COMPQ = 'V', Q has been postmultiplied by the
*> orthogonal transformation matrix Z which reorders T.
*> If COMPQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= 1, and if
*> COMPQ = 'V', LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IFST
*> \verbatim
*> IFST is INTEGER
*> \endverbatim
*>
*> \param[in,out] ILST
*> \verbatim
*> ILST is INTEGER
*>
*> Specify the reordering of the diagonal blocks of T.
*> The block with row index IFST is moved to row ILST, by a
*> sequence of transpositions between adjacent blocks.
*> On exit, if IFST pointed on entry to the second row of a
*> 2-by-2 block, it is changed to point to the first row; ILST
*> always points to the first row of the block in its final
*> position (which may differ from its input value by +1 or -1).
*> 1 <= IFST <= N; 1 <= ILST <= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> = 1: two adjacent blocks were too close to swap (the problem
*> is very ill-conditioned); T may have been partially
*> reordered, and ILST points to the first row of the
*> current position of the block being moved.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trexc
*
* =====================================================================
SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER IFST, ILST, INFO, LDQ, LDT, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL WANTQ
INTEGER HERE, NBF, NBL, NBNEXT
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLAEXC, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode and test the input arguments.
*
INFO = 0
WANTQ = LSAME( COMPQ, 'V' )
IF( .NOT.WANTQ .AND. .NOT.LSAME( COMPQ, 'N' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.MAX( 1, N ) ) ) THEN
INFO = -6
ELSE IF(( IFST.LT.1 .OR. IFST.GT.N ).AND.( N.GT.0 )) THEN
INFO = -7
ELSE IF(( ILST.LT.1 .OR. ILST.GT.N ).AND.( N.GT.0 )) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTREXC', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
* Determine the first row of specified block
* and find out it is 1 by 1 or 2 by 2.
*
IF( IFST.GT.1 ) THEN
IF( T( IFST, IFST-1 ).NE.ZERO )
$ IFST = IFST - 1
END IF
NBF = 1
IF( IFST.LT.N ) THEN
IF( T( IFST+1, IFST ).NE.ZERO )
$ NBF = 2
END IF
*
* Determine the first row of the final block
* and find out it is 1 by 1 or 2 by 2.
*
IF( ILST.GT.1 ) THEN
IF( T( ILST, ILST-1 ).NE.ZERO )
$ ILST = ILST - 1
END IF
NBL = 1
IF( ILST.LT.N ) THEN
IF( T( ILST+1, ILST ).NE.ZERO )
$ NBL = 2
END IF
*
IF( IFST.EQ.ILST )
$ RETURN
*
IF( IFST.LT.ILST ) THEN
*
* Update ILST
*
IF( NBF.EQ.2 .AND. NBL.EQ.1 )
$ ILST = ILST - 1
IF( NBF.EQ.1 .AND. NBL.EQ.2 )
$ ILST = ILST + 1
*
HERE = IFST
*
10 CONTINUE
*
* Swap block with next one below
*
IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
* Current block either 1 by 1 or 2 by 2
*
NBNEXT = 1
IF( HERE+NBF+1.LE.N ) THEN
IF( T( HERE+NBF+1, HERE+NBF ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBF, NBNEXT,
$ WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + NBNEXT
*
* Test if 2 by 2 block breaks into two 1 by 1 blocks
*
IF( NBF.EQ.2 ) THEN
IF( T( HERE+1, HERE ).EQ.ZERO )
$ NBF = 3
END IF
*
ELSE
*
* Current block consists of two 1 by 1 blocks each of which
* must be swapped individually
*
NBNEXT = 1
IF( HERE+3.LE.N ) THEN
IF( T( HERE+3, HERE+2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1, NBNEXT,
$ WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
IF( NBNEXT.EQ.1 ) THEN
*
* Swap two 1 by 1 blocks, no problems possible
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1,
$ NBNEXT,
$ WORK, INFO )
HERE = HERE + 1
ELSE
*
* Recompute NBNEXT in case 2 by 2 split
*
IF( T( HERE+2, HERE+1 ).EQ.ZERO )
$ NBNEXT = 1
IF( NBNEXT.EQ.2 ) THEN
*
* 2 by 2 Block did not split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1,
$ NBNEXT, WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + 2
ELSE
*
* 2 by 2 Block did split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1,
$ WORK, INFO )
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1,
$ 1,
$ WORK, INFO )
HERE = HERE + 2
END IF
END IF
END IF
IF( HERE.LT.ILST )
$ GO TO 10
*
ELSE
*
HERE = IFST
20 CONTINUE
*
* Swap block with next one above
*
IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
* Current block either 1 by 1 or 2 by 2
*
NBNEXT = 1
IF( HERE.GE.3 ) THEN
IF( T( HERE-1, HERE-2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT,
$ NBNEXT,
$ NBF, WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - NBNEXT
*
* Test if 2 by 2 block breaks into two 1 by 1 blocks
*
IF( NBF.EQ.2 ) THEN
IF( T( HERE+1, HERE ).EQ.ZERO )
$ NBF = 3
END IF
*
ELSE
*
* Current block consists of two 1 by 1 blocks each of which
* must be swapped individually
*
NBNEXT = 1
IF( HERE.GE.3 ) THEN
IF( T( HERE-1, HERE-2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT,
$ NBNEXT,
$ 1, WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
IF( NBNEXT.EQ.1 ) THEN
*
* Swap two 1 by 1 blocks, no problems possible
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBNEXT,
$ 1,
$ WORK, INFO )
HERE = HERE - 1
ELSE
*
* Recompute NBNEXT in case 2 by 2 split
*
IF( T( HERE, HERE-1 ).EQ.ZERO )
$ NBNEXT = 1
IF( NBNEXT.EQ.2 ) THEN
*
* 2 by 2 Block did not split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 2,
$ 1,
$ WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - 2
ELSE
*
* 2 by 2 Block did split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1,
$ WORK, INFO )
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 1,
$ 1,
$ WORK, INFO )
HERE = HERE - 2
END IF
END IF
END IF
IF( HERE.GT.ILST )
$ GO TO 20
END IF
ILST = HERE
*
RETURN
*
* End of DTREXC
*
END
*> \brief \b DTRRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
* LDX, FERR, BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, LDA, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
* $ WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRRFS provides error bounds and backward error estimates for the
*> solution to a system of linear equations with a triangular
*> coefficient matrix.
*>
*> The solution matrix X must be computed by DTRTRS or some other
*> means before entering this routine. DTRRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The triangular matrix A. If UPLO = 'U', the leading N-by-N
*> upper triangular part of the array A contains the upper
*> triangular matrix, and the strictly lower triangular part of
*> A is not referenced. If UPLO = 'L', the leading N-by-N lower
*> triangular part of the array A contains the lower triangular
*> matrix, and the strictly upper triangular part of A is not
*> referenced. If DIAG = 'U', the diagonal elements of A are
*> also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> The solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trrfs
*
* =====================================================================
SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB,
$ X,
$ LDX, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, LDA, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
$ WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
CHARACTER TRANST
INTEGER I, J, K, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DTRMV, DTRSV,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRRFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 250 J = 1, NRHS
*
* Compute residual R = B - op(A) * X,
* where op(A) = A or A**T, depending on TRANS.
*
CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 )
CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 20 I = 1, N
WORK( I ) = ABS( B( I, J ) )
20 CONTINUE
*
IF( NOTRAN ) THEN
*
* Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
IF( NOUNIT ) THEN
DO 40 K = 1, N
XK = ABS( X( K, J ) )
DO 30 I = 1, K
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
30 CONTINUE
40 CONTINUE
ELSE
DO 60 K = 1, N
XK = ABS( X( K, J ) )
DO 50 I = 1, K - 1
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
50 CONTINUE
WORK( K ) = WORK( K ) + XK
60 CONTINUE
END IF
ELSE
IF( NOUNIT ) THEN
DO 80 K = 1, N
XK = ABS( X( K, J ) )
DO 70 I = K, N
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
70 CONTINUE
80 CONTINUE
ELSE
DO 100 K = 1, N
XK = ABS( X( K, J ) )
DO 90 I = K + 1, N
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
90 CONTINUE
WORK( K ) = WORK( K ) + XK
100 CONTINUE
END IF
END IF
ELSE
*
* Compute abs(A**T)*abs(X) + abs(B).
*
IF( UPPER ) THEN
IF( NOUNIT ) THEN
DO 120 K = 1, N
S = ZERO
DO 110 I = 1, K
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
110 CONTINUE
WORK( K ) = WORK( K ) + S
120 CONTINUE
ELSE
DO 140 K = 1, N
S = ABS( X( K, J ) )
DO 130 I = 1, K - 1
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
130 CONTINUE
WORK( K ) = WORK( K ) + S
140 CONTINUE
END IF
ELSE
IF( NOUNIT ) THEN
DO 160 K = 1, N
S = ZERO
DO 150 I = K, N
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
150 CONTINUE
WORK( K ) = WORK( K ) + S
160 CONTINUE
ELSE
DO 180 K = 1, N
S = ABS( X( K, J ) )
DO 170 I = K + 1, N
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
170 CONTINUE
WORK( K ) = WORK( K ) + S
180 CONTINUE
END IF
END IF
END IF
S = ZERO
DO 190 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
190 CONTINUE
BERR( J ) = S
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(op(A)))*
* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(A)) is the inverse of op(A)
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(op(A)) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 200 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
200 CONTINUE
*
KASE = 0
210 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
$ FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**T).
*
CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA,
$ WORK( N+1 ),
$ 1 )
DO 220 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
220 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 230 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
230 CONTINUE
CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ),
$ 1 )
END IF
GO TO 210
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 240 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
240 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
250 CONTINUE
*
RETURN
*
* End of DTRRFS
*
END
*> \brief \b DTRSEN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRSEN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsen.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsen.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsen.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
* M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPQ, JOB
* INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
* DOUBLE PRECISION S, SEP
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
* $ WR( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRSEN reorders the real Schur factorization of a real matrix
*> A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
*> the leading diagonal blocks of the upper quasi-triangular matrix T,
*> and the leading columns of Q form an orthonormal basis of the
*> corresponding right invariant subspace.
*>
*> Optionally the routine computes the reciprocal condition numbers of
*> the cluster of eigenvalues and/or the invariant subspace.
*>
*> T must be in Schur canonical form (as returned by DHSEQR), that is,
*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*> 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies whether condition numbers are required for the
*> cluster of eigenvalues (S) or the invariant subspace (SEP):
*> = 'N': none;
*> = 'E': for eigenvalues only (S);
*> = 'V': for invariant subspace only (SEP);
*> = 'B': for both eigenvalues and invariant subspace (S and
*> SEP).
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> = 'V': update the matrix Q of Schur vectors;
*> = 'N': do not update Q.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> SELECT specifies the eigenvalues in the selected cluster. To
*> select a real eigenvalue w(j), SELECT(j) must be set to
*> .TRUE.. To select a complex conjugate pair of eigenvalues
*> w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
*> either SELECT(j) or SELECT(j+1) or both must be set to
*> .TRUE.; a complex conjugate pair of eigenvalues must be
*> either both included in the cluster or both excluded.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> On entry, the upper quasi-triangular matrix T, in Schur
*> canonical form.
*> On exit, T is overwritten by the reordered matrix T, again in
*> Schur canonical form, with the selected eigenvalues in the
*> leading diagonal blocks.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
*> On exit, if COMPQ = 'V', Q has been postmultiplied by the
*> orthogonal transformation matrix which reorders T; the
*> leading M columns of Q form an orthonormal basis for the
*> specified invariant subspace.
*> If COMPQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*>
*> The real and imaginary parts, respectively, of the reordered
*> eigenvalues of T. The eigenvalues are stored in the same
*> order as on the diagonal of T, with WR(i) = T(i,i) and, if
*> T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
*> WI(i+1) = -WI(i). Note that if a complex eigenvalue is
*> sufficiently ill-conditioned, then its value may differ
*> significantly from its value before reordering.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The dimension of the specified invariant subspace.
*> 0 < = M <= N.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION
*> If JOB = 'E' or 'B', S is a lower bound on the reciprocal
*> condition number for the selected cluster of eigenvalues.
*> S cannot underestimate the true reciprocal condition number
*> by more than a factor of sqrt(N). If M = 0 or N, S = 1.
*> If JOB = 'N' or 'V', S is not referenced.
*> \endverbatim
*>
*> \param[out] SEP
*> \verbatim
*> SEP is DOUBLE PRECISION
*> If JOB = 'V' or 'B', SEP is the estimated reciprocal
*> condition number of the specified invariant subspace. If
*> M = 0 or N, SEP = norm(T).
*> If JOB = 'N' or 'E', SEP is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If JOB = 'N', LWORK >= max(1,N);
*> if JOB = 'E', LWORK >= max(1,M*(N-M));
*> if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If JOB = 'N' or 'E', LIWORK >= 1;
*> if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> = 1: reordering of T failed because some eigenvalues are too
*> close to separate (the problem is very ill-conditioned);
*> T may have been partially reordered, and WR and WI
*> contain the eigenvalues in the same order as in T; S and
*> SEP (if requested) are set to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trsen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> DTRSEN first collects the selected eigenvalues by computing an
*> orthogonal transformation Z to move them to the top left corner of T.
*> In other words, the selected eigenvalues are the eigenvalues of T11
*> in:
*>
*> Z**T * T * Z = ( T11 T12 ) n1
*> ( 0 T22 ) n2
*> n1 n2
*>
*> where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
*> of Z span the specified invariant subspace of T.
*>
*> If T has been obtained from the real Schur factorization of a matrix
*> A = Q*T*Q**T, then the reordered real Schur factorization of A is given
*> by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
*> the corresponding invariant subspace of A.
*>
*> The reciprocal condition number of the average of the eigenvalues of
*> T11 may be returned in S. S lies between 0 (very badly conditioned)
*> and 1 (very well conditioned). It is computed as follows. First we
*> compute R so that
*>
*> P = ( I R ) n1
*> ( 0 0 ) n2
*> n1 n2
*>
*> is the projector on the invariant subspace associated with T11.
*> R is the solution of the Sylvester equation:
*>
*> T11*R - R*T22 = T12.
*>
*> Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
*> the two-norm of M. Then S is computed as the lower bound
*>
*> (1 + F-norm(R)**2)**(-1/2)
*>
*> on the reciprocal of 2-norm(P), the true reciprocal condition number.
*> S cannot underestimate 1 / 2-norm(P) by more than a factor of
*> sqrt(N).
*>
*> An approximate error bound for the computed average of the
*> eigenvalues of T11 is
*>
*> EPS * norm(T) / S
*>
*> where EPS is the machine precision.
*>
*> The reciprocal condition number of the right invariant subspace
*> spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
*> SEP is defined as the separation of T11 and T22:
*>
*> sep( T11, T22 ) = sigma-min( C )
*>
*> where sigma-min(C) is the smallest singular value of the
*> n1*n2-by-n1*n2 matrix
*>
*> C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
*>
*> I(m) is an m by m identity matrix, and kprod denotes the Kronecker
*> product. We estimate sigma-min(C) by the reciprocal of an estimate of
*> the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
*> cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
*>
*> When SEP is small, small changes in T can cause large changes in
*> the invariant subspace. An approximate bound on the maximum angular
*> error in the computed right invariant subspace is
*>
*> EPS * norm(T) / SEP
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR,
$ WI,
$ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
DOUBLE PRECISION S, SEP
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
$ WR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
$ WANTSP
INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
$ NN
DOUBLE PRECISION EST, RNORM, SCALE
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLANGE
EXTERNAL LSAME, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLACPY, DTREXC, DTRSYL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
WANTQ = LSAME( COMPQ, 'V' )
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
$ THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -8
ELSE
*
* Set M to the dimension of the specified invariant subspace,
* and test LWORK and LIWORK.
*
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( T( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
$ M = M + 2
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
*
N1 = M
N2 = N - M
NN = N1*N2
*
IF( WANTSP ) THEN
LWMIN = MAX( 1, 2*NN )
LIWMIN = MAX( 1, NN )
ELSE IF( LSAME( JOB, 'N' ) ) THEN
LWMIN = MAX( 1, N )
LIWMIN = 1
ELSE IF( LSAME( JOB, 'E' ) ) THEN
LWMIN = MAX( 1, NN )
LIWMIN = 1
END IF
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRSEN', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible.
*
IF( M.EQ.N .OR. M.EQ.0 ) THEN
IF( WANTS )
$ S = ONE
IF( WANTSP )
$ SEP = DLANGE( '1', N, N, T, LDT, WORK )
GO TO 40
END IF
*
* Collect the selected blocks at the top-left corner of T.
*
KS = 0
PAIR = .FALSE.
DO 20 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
SWAP = SELECT( K )
IF( K.LT.N ) THEN
IF( T( K+1, K ).NE.ZERO ) THEN
PAIR = .TRUE.
SWAP = SWAP .OR. SELECT( K+1 )
END IF
END IF
IF( SWAP ) THEN
KS = KS + 1
*
* Swap the K-th block to position KS.
*
IERR = 0
KK = K
IF( K.NE.KS )
$ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS,
$ WORK,
$ IERR )
IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
*
* Blocks too close to swap: exit.
*
INFO = 1
IF( WANTS )
$ S = ZERO
IF( WANTSP )
$ SEP = ZERO
GO TO 40
END IF
IF( PAIR )
$ KS = KS + 1
END IF
END IF
20 CONTINUE
*
IF( WANTS ) THEN
*
* Solve Sylvester equation for R:
*
* T11*R - R*T22 = scale*T12
*
CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
$ LDT, WORK, N1, SCALE, IERR )
*
* Estimate the reciprocal of the condition number of the cluster
* of eigenvalues.
*
RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
IF( RNORM.EQ.ZERO ) THEN
S = ONE
ELSE
S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
$ SQRT( RNORM ) )
END IF
END IF
*
IF( WANTSP ) THEN
*
* Estimate sep(T11,T22).
*
EST = ZERO
KASE = 0
30 CONTINUE
CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE,
$ ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve T11*R - R*T22 = scale*X.
*
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
$ IERR )
ELSE
*
* Solve T11**T*R - R*T22**T = scale*X.
*
CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
$ IERR )
END IF
GO TO 30
END IF
*
SEP = SCALE / EST
END IF
*
40 CONTINUE
*
* Store the output eigenvalues in WR and WI.
*
DO 50 K = 1, N
WR( K ) = T( K, K )
WI( K ) = ZERO
50 CONTINUE
DO 60 K = 1, N - 1
IF( T( K+1, K ).NE.ZERO ) THEN
WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
$ SQRT( ABS( T( K+1, K ) ) )
WI( K+1 ) = -WI( K )
END IF
60 CONTINUE
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DTRSEN
*
END
*> \brief \b DTRSNA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRSNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsna.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
* LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, JOB
* INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( LDWORK, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRSNA estimates reciprocal condition numbers for specified
*> eigenvalues and/or right eigenvectors of a real upper
*> quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
*> orthogonal).
*>
*> T must be in Schur canonical form (as returned by DHSEQR), that is,
*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*> 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies whether condition numbers are required for
*> eigenvalues (S) or eigenvectors (SEP):
*> = 'E': for eigenvalues only (S);
*> = 'V': for eigenvectors only (SEP);
*> = 'B': for both eigenvalues and eigenvectors (S and SEP).
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute condition numbers for all eigenpairs;
*> = 'S': compute condition numbers for selected eigenpairs
*> specified by the array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY = 'S', SELECT specifies the eigenpairs for which
*> condition numbers are required. To select condition numbers
*> for the eigenpair corresponding to a real eigenvalue w(j),
*> SELECT(j) must be set to .TRUE.. To select condition numbers
*> corresponding to a complex conjugate pair of eigenvalues w(j)
*> and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
*> set to .TRUE..
*> If HOWMNY = 'A', SELECT is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The upper quasi-triangular matrix T, in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,M)
*> If JOB = 'E' or 'B', VL must contain left eigenvectors of T
*> (or of any Q*T*Q**T with Q orthogonal), corresponding to the
*> eigenpairs specified by HOWMNY and SELECT. The eigenvectors
*> must be stored in consecutive columns of VL, as returned by
*> DHSEIN or DTREVC.
*> If JOB = 'V', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL.
*> LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,M)
*> If JOB = 'E' or 'B', VR must contain right eigenvectors of T
*> (or of any Q*T*Q**T with Q orthogonal), corresponding to the
*> eigenpairs specified by HOWMNY and SELECT. The eigenvectors
*> must be stored in consecutive columns of VR, as returned by
*> DHSEIN or DTREVC.
*> If JOB = 'V', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR.
*> LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (MM)
*> If JOB = 'E' or 'B', the reciprocal condition numbers of the
*> selected eigenvalues, stored in consecutive elements of the
*> array. For a complex conjugate pair of eigenvalues two
*> consecutive elements of S are set to the same value. Thus
*> S(j), SEP(j), and the j-th columns of VL and VR all
*> correspond to the same eigenpair (but not in general the
*> j-th eigenpair, unless all eigenpairs are selected).
*> If JOB = 'V', S is not referenced.
*> \endverbatim
*>
*> \param[out] SEP
*> \verbatim
*> SEP is DOUBLE PRECISION array, dimension (MM)
*> If JOB = 'V' or 'B', the estimated reciprocal condition
*> numbers of the selected eigenvectors, stored in consecutive
*> elements of the array. For a complex eigenvector two
*> consecutive elements of SEP are set to the same value. If
*> the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
*> is set to 0; this can only occur when the true value would be
*> very small anyway.
*> If JOB = 'E', SEP is not referenced.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of elements in the arrays S (if JOB = 'E' or 'B')
*> and/or SEP (if JOB = 'V' or 'B'). MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of elements of the arrays S and/or SEP actually
*> used to store the estimated condition numbers.
*> If HOWMNY = 'A', M is set to N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
*> If JOB = 'E', WORK is not referenced.
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*> LDWORK is INTEGER
*> The leading dimension of the array WORK.
*> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (2*(N-1))
*> If JOB = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trsna
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The reciprocal of the condition number of an eigenvalue lambda is
*> defined as
*>
*> S(lambda) = |v**T*u| / (norm(u)*norm(v))
*>
*> where u and v are the right and left eigenvectors of T corresponding
*> to lambda; v**T denotes the transpose of v, and norm(u)
*> denotes the Euclidean norm. These reciprocal condition numbers always
*> lie between zero (very badly conditioned) and one (very well
*> conditioned). If n = 1, S(lambda) is defined to be 1.
*>
*> An approximate error bound for a computed eigenvalue W(i) is given by
*>
*> EPS * norm(T) / S(i)
*>
*> where EPS is the machine precision.
*>
*> The reciprocal of the condition number of the right eigenvector u
*> corresponding to lambda is defined as follows. Suppose
*>
*> T = ( lambda c )
*> ( 0 T22 )
*>
*> Then the reciprocal condition number is
*>
*> SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
*>
*> where sigma-min denotes the smallest singular value. We approximate
*> the smallest singular value by the reciprocal of an estimate of the
*> one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
*> defined to be abs(T(1,1)).
*>
*> An approximate error bound for a computed right eigenvector VR(i)
*> is given by
*>
*> EPS * norm(T) / SEP(i)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
$ VR,
$ LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, JOB
INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( LDWORK, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL PAIR, SOMCON, WANTBH, WANTS, WANTSP
INTEGER I, IERR, IFST, ILST, J, K, KASE, KS, N2, NN
DOUBLE PRECISION BIGNUM, COND, CS, DELTA, DUMM, EPS, EST, LNRM,
$ MU, PROD, PROD1, PROD2, RNRM, SCALE, SMLNUM, SN
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
DOUBLE PRECISION DUMMY( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2
EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLACPY, DLAQTR, DTREXC,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
*
SOMCON = LSAME( HOWMNY, 'S' )
*
INFO = 0
IF( .NOT.WANTS .AND. .NOT.WANTSP ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDVL.LT.1 .OR. ( WANTS .AND. LDVL.LT.N ) ) THEN
INFO = -8
ELSE IF( LDVR.LT.1 .OR. ( WANTS .AND. LDVR.LT.N ) ) THEN
INFO = -10
ELSE
*
* Set M to the number of eigenpairs for which condition numbers
* are required, and test MM.
*
IF( SOMCON ) THEN
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( T( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
$ M = M + 2
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
ELSE
M = N
END IF
*
IF( MM.LT.M ) THEN
INFO = -13
ELSE IF( LDWORK.LT.1 .OR. ( WANTSP .AND. LDWORK.LT.N ) ) THEN
INFO = -16
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRSNA', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( SOMCON ) THEN
IF( .NOT.SELECT( 1 ) )
$ RETURN
END IF
IF( WANTS )
$ S( 1 ) = ONE
IF( WANTSP )
$ SEP( 1 ) = ABS( T( 1, 1 ) )
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
BIGNUM = ONE / SMLNUM
*
KS = 0
PAIR = .FALSE.
DO 60 K = 1, N
*
* Determine whether T(k,k) begins a 1-by-1 or 2-by-2 block.
*
IF( PAIR ) THEN
PAIR = .FALSE.
GO TO 60
ELSE
IF( K.LT.N )
$ PAIR = T( K+1, K ).NE.ZERO
END IF
*
* Determine whether condition numbers are required for the k-th
* eigenpair.
*
IF( SOMCON ) THEN
IF( PAIR ) THEN
IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
$ GO TO 60
ELSE
IF( .NOT.SELECT( K ) )
$ GO TO 60
END IF
END IF
*
KS = KS + 1
*
IF( WANTS ) THEN
*
* Compute the reciprocal condition number of the k-th
* eigenvalue.
*
IF( .NOT.PAIR ) THEN
*
* Real eigenvalue.
*
PROD = DDOT( N, VR( 1, KS ), 1, VL( 1, KS ), 1 )
RNRM = DNRM2( N, VR( 1, KS ), 1 )
LNRM = DNRM2( N, VL( 1, KS ), 1 )
S( KS ) = ABS( PROD ) / ( RNRM*LNRM )
ELSE
*
* Complex eigenvalue.
*
PROD1 = DDOT( N, VR( 1, KS ), 1, VL( 1, KS ), 1 )
PROD1 = PROD1 + DDOT( N, VR( 1, KS+1 ), 1, VL( 1,
$ KS+1 ),
$ 1 )
PROD2 = DDOT( N, VL( 1, KS ), 1, VR( 1, KS+1 ), 1 )
PROD2 = PROD2 - DDOT( N, VL( 1, KS+1 ), 1, VR( 1,
$ KS ),
$ 1 )
RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ),
$ DNRM2( N, VR( 1, KS+1 ), 1 ) )
LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ),
$ DNRM2( N, VL( 1, KS+1 ), 1 ) )
COND = DLAPY2( PROD1, PROD2 ) / ( RNRM*LNRM )
S( KS ) = COND
S( KS+1 ) = COND
END IF
END IF
*
IF( WANTSP ) THEN
*
* Estimate the reciprocal condition number of the k-th
* eigenvector.
*
* Copy the matrix T to the array WORK and swap the diagonal
* block beginning at T(k,k) to the (1,1) position.
*
CALL DLACPY( 'Full', N, N, T, LDT, WORK, LDWORK )
IFST = K
ILST = 1
CALL DTREXC( 'No Q', N, WORK, LDWORK, DUMMY, 1, IFST,
$ ILST,
$ WORK( 1, N+1 ), IERR )
*
IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
*
* Could not swap because blocks not well separated
*
SCALE = ONE
EST = BIGNUM
ELSE
*
* Reordering successful
*
IF( WORK( 2, 1 ).EQ.ZERO ) THEN
*
* Form C = T22 - lambda*I in WORK(2:N,2:N).
*
DO 20 I = 2, N
WORK( I, I ) = WORK( I, I ) - WORK( 1, 1 )
20 CONTINUE
N2 = 1
NN = N - 1
ELSE
*
* Triangularize the 2 by 2 block by unitary
* transformation U = [ cs i*ss ]
* [ i*ss cs ].
* such that the (1,1) position of WORK is complex
* eigenvalue lambda with positive imaginary part. (2,2)
* position of WORK is the complex eigenvalue lambda
* with negative imaginary part.
*
MU = SQRT( ABS( WORK( 1, 2 ) ) )*
$ SQRT( ABS( WORK( 2, 1 ) ) )
DELTA = DLAPY2( MU, WORK( 2, 1 ) )
CS = MU / DELTA
SN = -WORK( 2, 1 ) / DELTA
*
* Form
*
* C**T = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ]
* [ mu ]
* [ .. ]
* [ .. ]
* [ mu ]
* where C**T is transpose of matrix C,
* and RWORK is stored starting in the N+1-st column of
* WORK.
*
DO 30 J = 3, N
WORK( 2, J ) = CS*WORK( 2, J )
WORK( J, J ) = WORK( J, J ) - WORK( 1, 1 )
30 CONTINUE
WORK( 2, 2 ) = ZERO
*
WORK( 1, N+1 ) = TWO*MU
DO 40 I = 2, N - 1
WORK( I, N+1 ) = SN*WORK( 1, I+1 )
40 CONTINUE
N2 = 2
NN = 2*( N-1 )
END IF
*
* Estimate norm(inv(C**T))
*
EST = ZERO
KASE = 0
50 CONTINUE
CALL DLACN2( NN, WORK( 1, N+2 ), WORK( 1, N+4 ),
$ IWORK,
$ EST, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
IF( N2.EQ.1 ) THEN
*
* Real eigenvalue: solve C**T*x = scale*c.
*
CALL DLAQTR( .TRUE., .TRUE., N-1, WORK( 2,
$ 2 ),
$ LDWORK, DUMMY, DUMM, SCALE,
$ WORK( 1, N+4 ), WORK( 1, N+6 ),
$ IERR )
ELSE
*
* Complex eigenvalue: solve
* C**T*(p+iq) = scale*(c+id) in real arithmetic.
*
CALL DLAQTR( .TRUE., .FALSE., N-1, WORK( 2,
$ 2 ),
$ LDWORK, WORK( 1, N+1 ), MU, SCALE,
$ WORK( 1, N+4 ), WORK( 1, N+6 ),
$ IERR )
END IF
ELSE
IF( N2.EQ.1 ) THEN
*
* Real eigenvalue: solve C*x = scale*c.
*
CALL DLAQTR( .FALSE., .TRUE., N-1, WORK( 2,
$ 2 ),
$ LDWORK, DUMMY, DUMM, SCALE,
$ WORK( 1, N+4 ), WORK( 1, N+6 ),
$ IERR )
ELSE
*
* Complex eigenvalue: solve
* C*(p+iq) = scale*(c+id) in real arithmetic.
*
CALL DLAQTR( .FALSE., .FALSE., N-1,
$ WORK( 2, 2 ), LDWORK,
$ WORK( 1, N+1 ), MU, SCALE,
$ WORK( 1, N+4 ), WORK( 1, N+6 ),
$ IERR )
*
END IF
END IF
*
GO TO 50
END IF
END IF
*
SEP( KS ) = SCALE / MAX( EST, SMLNUM )
IF( PAIR )
$ SEP( KS+1 ) = SEP( KS )
END IF
*
IF( PAIR )
$ KS = KS + 1
*
60 CONTINUE
RETURN
*
* End of DTRSNA
*
END
*> \brief \b DTRSYL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRSYL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsyl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsyl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsyl.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
* LDC, SCALE, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANA, TRANB
* INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
* DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRSYL solves the real Sylvester matrix equation:
*>
*> op(A)*X + X*op(B) = scale*C or
*> op(A)*X - X*op(B) = scale*C,
*>
*> where op(A) = A or A**T, and A and B are both upper quasi-
*> triangular. A is M-by-M and B is N-by-N; the right hand side C and
*> the solution X are M-by-N; and scale is an output scale factor, set
*> <= 1 to avoid overflow in X.
*>
*> A and B must be in Schur canonical form (as returned by DHSEQR), that
*> is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
*> each 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANA
*> \verbatim
*> TRANA is CHARACTER*1
*> Specifies the option op(A):
*> = 'N': op(A) = A (No transpose)
*> = 'T': op(A) = A**T (Transpose)
*> = 'C': op(A) = A**H (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] TRANB
*> \verbatim
*> TRANB is CHARACTER*1
*> Specifies the option op(B):
*> = 'N': op(B) = B (No transpose)
*> = 'T': op(B) = B**T (Transpose)
*> = 'C': op(B) = B**H (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] ISGN
*> \verbatim
*> ISGN is INTEGER
*> Specifies the sign in the equation:
*> = +1: solve op(A)*X + X*op(B) = scale*C
*> = -1: solve op(A)*X - X*op(B) = scale*C
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The order of the matrix A, and the number of rows in the
*> matrices X and C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix B, and the number of columns in the
*> matrices X and C. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,M)
*> The upper quasi-triangular matrix A, in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The upper quasi-triangular matrix B, in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N right hand side matrix C.
*> On exit, C is overwritten by the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M)
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The scale factor, scale, set <= 1 to avoid overflow in X.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> = 1: A and B have common or very close eigenvalues; perturbed
*> values were used to solve the equation (but the matrices
*> A and B are unchanged).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trsyl
*
* =====================================================================
SUBROUTINE DTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
$ LDC, SCALE, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
DOUBLE PRECISION SCALE
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRNA, NOTRNB
INTEGER IERR, J, K, K1, K2, KNEXT, L, L1, L2, LNEXT
DOUBLE PRECISION A11, BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
$ SMLNUM, SUML, SUMR, XNORM
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUM( 1 ), VEC( 2, 2 ), X( 2, 2 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT, DLAMCH, DLANGE
EXTERNAL LSAME, DDOT, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DLALN2, DLASY2, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* Decode and Test input parameters
*
NOTRNA = LSAME( TRANA, 'N' )
NOTRNB = LSAME( TRANB, 'N' )
*
INFO = 0
IF( .NOT.NOTRNA .AND. .NOT.LSAME( TRANA, 'T' ) .AND. .NOT.
$ LSAME( TRANA, 'C' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRNB .AND. .NOT.LSAME( TRANB, 'T' ) .AND. .NOT.
$ LSAME( TRANB, 'C' ) ) THEN
INFO = -2
ELSE IF( ISGN.NE.1 .AND. ISGN.NE.-1 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRSYL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
SCALE = ONE
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SMLNUM*DBLE( M*N ) / EPS
BIGNUM = ONE / SMLNUM
*
SMIN = MAX( SMLNUM, EPS*DLANGE( 'M', M, M, A, LDA, DUM ),
$ EPS*DLANGE( 'M', N, N, B, LDB, DUM ) )
*
SGN = ISGN
*
IF( NOTRNA .AND. NOTRNB ) THEN
*
* Solve A*X + ISGN*X*B = scale*C.
*
* The (K,L)th block of X is determined starting from
* bottom-left corner column by column by
*
* A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
*
* Where
* M L-1
* R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)].
* I=K+1 J=1
*
* Start column loop (index = L)
* L1 (L2) : column index of the first (first) row of X(K,L).
*
LNEXT = 1
DO 60 L = 1, N
IF( L.LT.LNEXT )
$ GO TO 60
IF( L.EQ.N ) THEN
L1 = L
L2 = L
ELSE
IF( B( L+1, L ).NE.ZERO ) THEN
L1 = L
L2 = L + 1
LNEXT = L + 2
ELSE
L1 = L
L2 = L
LNEXT = L + 1
END IF
END IF
*
* Start row loop (index = K)
* K1 (K2): row index of the first (last) row of X(K,L).
*
KNEXT = M
DO 50 K = M, 1, -1
IF( K.GT.KNEXT )
$ GO TO 50
IF( K.EQ.1 ) THEN
K1 = K
K2 = K
ELSE
IF( A( K, K-1 ).NE.ZERO ) THEN
K1 = K - 1
K2 = K
KNEXT = K - 2
ELSE
K1 = K
K2 = K
KNEXT = K - 1
END IF
END IF
*
IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
$ C( MIN( K1+1, M ), L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
SCALOC = ONE
*
A11 = A( K1, K1 ) + SGN*B( L1, L1 )
DA11 = ABS( A11 )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( VEC( 1, 1 ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
IF( SCALOC.NE.ONE ) THEN
DO 10 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
10 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
*
ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ),
$ LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 20 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
20 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K2, L1 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
$ C( MIN( K1+1, M ), L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
$ C( MIN( K1+1, M ), L2 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ),
$ LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 30 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
30 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L2 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L2 ), 1 )
SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 )
VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
CALL DLASY2( .FALSE., .FALSE., ISGN, 2, 2,
$ A( K1, K1 ), LDA, B( L1, L1 ), LDB, VEC,
$ 2, SCALOC, X, 2, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 40 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
40 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 1, 2 )
C( K2, L1 ) = X( 2, 1 )
C( K2, L2 ) = X( 2, 2 )
END IF
*
50 CONTINUE
*
60 CONTINUE
*
ELSE IF( .NOT.NOTRNA .AND. NOTRNB ) THEN
*
* Solve A**T *X + ISGN*X*B = scale*C.
*
* The (K,L)th block of X is determined starting from
* upper-left corner column by column by
*
* A(K,K)**T*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
*
* Where
* K-1 T L-1
* R(K,L) = SUM [A(I,K)**T*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)]
* I=1 J=1
*
* Start column loop (index = L)
* L1 (L2): column index of the first (last) row of X(K,L)
*
LNEXT = 1
DO 120 L = 1, N
IF( L.LT.LNEXT )
$ GO TO 120
IF( L.EQ.N ) THEN
L1 = L
L2 = L
ELSE
IF( B( L+1, L ).NE.ZERO ) THEN
L1 = L
L2 = L + 1
LNEXT = L + 2
ELSE
L1 = L
L2 = L
LNEXT = L + 1
END IF
END IF
*
* Start row loop (index = K)
* K1 (K2): row index of the first (last) row of X(K,L)
*
KNEXT = 1
DO 110 K = 1, M
IF( K.LT.KNEXT )
$ GO TO 110
IF( K.EQ.M ) THEN
K1 = K
K2 = K
ELSE
IF( A( K+1, K ).NE.ZERO ) THEN
K1 = K
K2 = K + 1
KNEXT = K + 2
ELSE
K1 = K
K2 = K
KNEXT = K + 1
END IF
END IF
*
IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
SCALOC = ONE
*
A11 = A( K1, K1 ) + SGN*B( L1, L1 )
DA11 = ABS( A11 )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( VEC( 1, 1 ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
IF( SCALOC.NE.ONE ) THEN
DO 70 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
70 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
*
ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ),
$ LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 80 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
80 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K2, L1 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ),
$ LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 90 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
90 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 )
SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 )
VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
CALL DLASY2( .TRUE., .FALSE., ISGN, 2, 2, A( K1,
$ K1 ),
$ LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
$ 2, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 100 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
100 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 1, 2 )
C( K2, L1 ) = X( 2, 1 )
C( K2, L2 ) = X( 2, 2 )
END IF
*
110 CONTINUE
120 CONTINUE
*
ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN
*
* Solve A**T*X + ISGN*X*B**T = scale*C.
*
* The (K,L)th block of X is determined starting from
* top-right corner column by column by
*
* A(K,K)**T*X(K,L) + ISGN*X(K,L)*B(L,L)**T = C(K,L) - R(K,L)
*
* Where
* K-1 N
* R(K,L) = SUM [A(I,K)**T*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)**T].
* I=1 J=L+1
*
* Start column loop (index = L)
* L1 (L2): column index of the first (last) row of X(K,L)
*
LNEXT = N
DO 180 L = N, 1, -1
IF( L.GT.LNEXT )
$ GO TO 180
IF( L.EQ.1 ) THEN
L1 = L
L2 = L
ELSE
IF( B( L, L-1 ).NE.ZERO ) THEN
L1 = L - 1
L2 = L
LNEXT = L - 2
ELSE
L1 = L
L2 = L
LNEXT = L - 1
END IF
END IF
*
* Start row loop (index = K)
* K1 (K2): row index of the first (last) row of X(K,L)
*
KNEXT = 1
DO 170 K = 1, M
IF( K.LT.KNEXT )
$ GO TO 170
IF( K.EQ.M ) THEN
K1 = K
K2 = K
ELSE
IF( A( K+1, K ).NE.ZERO ) THEN
K1 = K
K2 = K + 1
KNEXT = K + 2
ELSE
K1 = K
K2 = K
KNEXT = K + 1
END IF
END IF
*
IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC,
$ B( L1, MIN( L1+1, N ) ), LDB )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
SCALOC = ONE
*
A11 = A( K1, K1 ) + SGN*B( L1, L1 )
DA11 = ABS( A11 )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( VEC( 1, 1 ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
IF( SCALOC.NE.ONE ) THEN
DO 130 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
130 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
*
ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ),
$ LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 140 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
140 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K2, L1 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L2, MIN( L2+1, N ) ), LDB )
VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ),
$ LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 150 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
150 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L2, MIN( L2+1, N ) ), LDB )
VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 )
SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
$ B( L2, MIN( L2+1, N ) ), LDB )
VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
CALL DLASY2( .TRUE., .TRUE., ISGN, 2, 2, A( K1,
$ K1 ),
$ LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
$ 2, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 160 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
160 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 1, 2 )
C( K2, L1 ) = X( 2, 1 )
C( K2, L2 ) = X( 2, 2 )
END IF
*
170 CONTINUE
180 CONTINUE
*
ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN
*
* Solve A*X + ISGN*X*B**T = scale*C.
*
* The (K,L)th block of X is determined starting from
* bottom-right corner column by column by
*
* A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L)**T = C(K,L) - R(K,L)
*
* Where
* M N
* R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)**T].
* I=K+1 J=L+1
*
* Start column loop (index = L)
* L1 (L2): column index of the first (last) row of X(K,L)
*
LNEXT = N
DO 240 L = N, 1, -1
IF( L.GT.LNEXT )
$ GO TO 240
IF( L.EQ.1 ) THEN
L1 = L
L2 = L
ELSE
IF( B( L, L-1 ).NE.ZERO ) THEN
L1 = L - 1
L2 = L
LNEXT = L - 2
ELSE
L1 = L
L2 = L
LNEXT = L - 1
END IF
END IF
*
* Start row loop (index = K)
* K1 (K2): row index of the first (last) row of X(K,L)
*
KNEXT = M
DO 230 K = M, 1, -1
IF( K.GT.KNEXT )
$ GO TO 230
IF( K.EQ.1 ) THEN
K1 = K
K2 = K
ELSE
IF( A( K, K-1 ).NE.ZERO ) THEN
K1 = K - 1
K2 = K
KNEXT = K - 2
ELSE
K1 = K
K2 = K
KNEXT = K - 1
END IF
END IF
*
IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
$ C( MIN( K1+1, M ), L1 ), 1 )
SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC,
$ B( L1, MIN( L1+1, N ) ), LDB )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
SCALOC = ONE
*
A11 = A( K1, K1 ) + SGN*B( L1, L1 )
DA11 = ABS( A11 )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( VEC( 1, 1 ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
$ SCALOC = ONE / DB
END IF
X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
IF( SCALOC.NE.ONE ) THEN
DO 190 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
190 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
*
ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ),
$ LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 200 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
200 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K2, L1 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
$ C( MIN( K1+1, M ), L1 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
$ C( MIN( K1+1, M ), L2 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L2, MIN( L2+1, N ) ), LDB )
VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ),
$ LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
$ ZERO, X, 2, SCALOC, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 210 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
210 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 2, 1 )
*
ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L2 ), 1 )
SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
$ B( L2, MIN( L2+1, N ) ), LDB )
VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L1 ), 1 )
SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
$ B( L1, MIN( L2+1, N ) ), LDB )
VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
$ C( MIN( K2+1, M ), L2 ), 1 )
SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
$ B( L2, MIN( L2+1, N ) ), LDB )
VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
CALL DLASY2( .FALSE., .TRUE., ISGN, 2, 2, A( K1,
$ K1 ),
$ LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
$ 2, XNORM, IERR )
IF( IERR.NE.0 )
$ INFO = 1
*
IF( SCALOC.NE.ONE ) THEN
DO 220 J = 1, N
CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
220 CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K1, L1 ) = X( 1, 1 )
C( K1, L2 ) = X( 1, 2 )
C( K2, L1 ) = X( 2, 1 )
C( K2, L2 ) = X( 2, 2 )
END IF
*
230 CONTINUE
240 CONTINUE
*
END IF
*
RETURN
*
* End of DTRSYL
*
END
*> \brief \b DTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTI2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrti2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrti2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrti2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRTI2( UPLO, DIAG, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRTI2 computes the inverse of a real upper or lower triangular
*> matrix.
*>
*> This is the Level 2 BLAS version of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the triangular matrix A. If UPLO = 'U', the
*> leading n by n upper triangular part of the array A contains
*> the upper triangular matrix, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of the array A contains
*> the lower triangular matrix, and the strictly upper
*> triangular part of A is not referenced. If DIAG = 'U', the
*> diagonal elements of A are also not referenced and are
*> assumed to be 1.
*>
*> On exit, the (triangular) inverse of the original matrix, in
*> the same storage format.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trti2
*
* =====================================================================
SUBROUTINE DTRTI2( UPLO, DIAG, N, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, UPPER
INTEGER J
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DTRMV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOUNIT = LSAME( DIAG, 'N' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRTI2', -INFO )
RETURN
END IF
*
IF( UPPER ) THEN
*
* Compute inverse of upper triangular matrix.
*
DO 10 J = 1, N
IF( NOUNIT ) THEN
A( J, J ) = ONE / A( J, J )
AJJ = -A( J, J )
ELSE
AJJ = -ONE
END IF
*
* Compute elements 1:j-1 of j-th column.
*
CALL DTRMV( 'Upper', 'No transpose', DIAG, J-1, A, LDA,
$ A( 1, J ), 1 )
CALL DSCAL( J-1, AJJ, A( 1, J ), 1 )
10 CONTINUE
ELSE
*
* Compute inverse of lower triangular matrix.
*
DO 20 J = N, 1, -1
IF( NOUNIT ) THEN
A( J, J ) = ONE / A( J, J )
AJJ = -A( J, J )
ELSE
AJJ = -ONE
END IF
IF( J.LT.N ) THEN
*
* Compute elements j+1:n of j-th column.
*
CALL DTRMV( 'Lower', 'No transpose', DIAG, N-J,
$ A( J+1, J+1 ), LDA, A( J+1, J ), 1 )
CALL DSCAL( N-J, AJJ, A( J+1, J ), 1 )
END IF
20 CONTINUE
END IF
*
RETURN
*
* End of DTRTI2
*
END
*> \brief \b DTRTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrtri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrtri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrtri.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRTRI( UPLO, DIAG, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRTRI computes the inverse of a real upper or lower triangular
*> matrix A.
*>
*> This is the Level 3 BLAS version of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the triangular matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of the array A contains
*> the upper triangular matrix, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of the array A contains
*> the lower triangular matrix, and the strictly upper
*> triangular part of A is not referenced. If DIAG = 'U', the
*> diagonal elements of A are also not referenced and are
*> assumed to be 1.
*> On exit, the (triangular) inverse of the original matrix, in
*> the same storage format.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, A(i,i) is exactly zero. The triangular
*> matrix is singular and its inverse can not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trtri
*
* =====================================================================
SUBROUTINE DTRTRI( UPLO, DIAG, N, A, LDA, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, UPPER
INTEGER J, JB, NB, NN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DTRMM, DTRSM, DTRTI2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOUNIT = LSAME( DIAG, 'N' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check for singularity if non-unit.
*
IF( NOUNIT ) THEN
DO 10 INFO = 1, N
IF( A( INFO, INFO ).EQ.ZERO )
$ RETURN
10 CONTINUE
INFO = 0
END IF
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DTRTRI', UPLO // DIAG, N, -1, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code
*
CALL DTRTI2( UPLO, DIAG, N, A, LDA, INFO )
ELSE
*
* Use blocked code
*
IF( UPPER ) THEN
*
* Compute inverse of upper triangular matrix
*
DO 20 J = 1, N, NB
JB = MIN( NB, N-J+1 )
*
* Compute rows 1:j-1 of current block column
*
CALL DTRMM( 'Left', 'Upper', 'No transpose', DIAG,
$ J-1,
$ JB, ONE, A, LDA, A( 1, J ), LDA )
CALL DTRSM( 'Right', 'Upper', 'No transpose', DIAG,
$ J-1,
$ JB, -ONE, A( J, J ), LDA, A( 1, J ), LDA )
*
* Compute inverse of current diagonal block
*
CALL DTRTI2( 'Upper', DIAG, JB, A( J, J ), LDA, INFO )
20 CONTINUE
ELSE
*
* Compute inverse of lower triangular matrix
*
NN = ( ( N-1 ) / NB )*NB + 1
DO 30 J = NN, 1, -NB
JB = MIN( NB, N-J+1 )
IF( J+JB.LE.N ) THEN
*
* Compute rows j+jb:n of current block column
*
CALL DTRMM( 'Left', 'Lower', 'No transpose', DIAG,
$ N-J-JB+1, JB, ONE, A( J+JB, J+JB ), LDA,
$ A( J+JB, J ), LDA )
CALL DTRSM( 'Right', 'Lower', 'No transpose', DIAG,
$ N-J-JB+1, JB, -ONE, A( J, J ), LDA,
$ A( J+JB, J ), LDA )
END IF
*
* Compute inverse of current diagonal block
*
CALL DTRTI2( 'Lower', DIAG, JB, A( J, J ), LDA, INFO )
30 CONTINUE
END IF
END IF
*
RETURN
*
* End of DTRTRI
*
END
*> \brief \b DTRTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRTRS solves a triangular system of the form
*>
*> A * X = B or A**T * X = B,
*>
*> where A is a triangular matrix of order N, and B is an N-by-NRHS matrix.
*>
*> This subroutine verifies that A is nonsingular, but callers should note that only exact
*> singularity is detected. It is conceivable for one or more diagonal elements of A to be
*> subnormally tiny numbers without this subroutine signalling an error.
*>
*> If a possible loss of numerical precision due to near-singular matrices is a concern, the
*> caller should verify that A is nonsingular within some tolerance before calling this subroutine.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The triangular matrix A. If UPLO = 'U', the leading N-by-N
*> upper triangular part of the array A contains the upper
*> triangular matrix, and the strictly lower triangular part of
*> A is not referenced. If UPLO = 'L', the leading N-by-N lower
*> triangular part of the array A contains the lower triangular
*> matrix, and the strictly upper triangular part of A is not
*> referenced. If DIAG = 'U', the diagonal elements of A are
*> also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, if INFO = 0, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the i-th diagonal element of A is exactly zero,
*> indicating that the matrix is singular and the solutions
*> X have not been computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trtrs
*
* =====================================================================
SUBROUTINE DTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOUNIT = LSAME( DIAG, 'N' )
IF( .NOT.LSAME( UPLO, 'U' ) .AND.
$ .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.
$ LSAME( TRANS, 'T' ) .AND.
$ .NOT.LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check for singularity.
*
IF( NOUNIT ) THEN
DO 10 INFO = 1, N
IF( A( INFO, INFO ).EQ.ZERO )
$ RETURN
10 CONTINUE
END IF
INFO = 0
*
* Solve A * x = b or A**T * x = b.
*
CALL DTRSM( 'Left', UPLO, TRANS, DIAG, N, NRHS, ONE, A, LDA, B,
$ LDB )
*
RETURN
*
* End of DTRTRS
*
END
*> \brief \b DTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrttf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrttf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrttf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRTTF( TRANSR, UPLO, N, A, LDA, ARF, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( 0: LDA-1, 0: * ), ARF( 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRTTF copies a triangular matrix A from standard full format (TR)
*> to rectangular full packed format (TF) .
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': ARF in Normal form is wanted;
*> = 'T': ARF in Transpose form is wanted.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N).
*> On entry, the triangular matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of the array A contains
*> the upper triangular matrix, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of the array A contains
*> the lower triangular matrix, and the strictly upper
*> triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the matrix A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] ARF
*> \verbatim
*> ARF is DOUBLE PRECISION array, dimension (NT).
*> NT=N*(N+1)/2. On exit, the triangular matrix A in RFP format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trttf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Rectangular Full Packed (RFP) Format when N is
*> even. We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> the transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> the transpose of the last three columns of AP lower.
*> This covers the case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 03 04 05 33 43 53
*> 13 14 15 00 44 54
*> 23 24 25 10 11 55
*> 33 34 35 20 21 22
*> 00 44 45 30 31 32
*> 01 11 55 40 41 42
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We then consider Rectangular Full Packed (RFP) Format when N is
*> odd. We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> the transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> the transpose of the last two columns of AP lower.
*> This covers the case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> 02 03 04 00 33 43
*> 12 13 14 10 11 44
*> 22 23 24 20 21 22
*> 00 33 34 30 31 32
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*> transpose of RFP A above. One therefore gets:
*>
*> RFP A RFP A
*>
*> 02 12 22 00 01 00 10 20 30 40 50
*> 03 13 23 33 11 33 11 21 31 41 51
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*
* =====================================================================
SUBROUTINE DTRTTF( TRANSR, UPLO, N, A, LDA, ARF, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANSR, UPLO
INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( 0: LDA-1, 0: * ), ARF( 0: * )
* ..
*
* =====================================================================
*
* ..
* .. Local Scalars ..
LOGICAL LOWER, NISODD, NORMALTRANSR
INTEGER I, IJ, J, K, L, N1, N2, NT, NX2, NP1X2
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRTTF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 ) THEN
IF( N.EQ.1 ) THEN
ARF( 0 ) = A( 0, 0 )
END IF
RETURN
END IF
*
* Size of array ARF(0:nt-1)
*
NT = N*( N+1 ) / 2
*
* Set N1 and N2 depending on LOWER: for N even N1=N2=K
*
IF( LOWER ) THEN
N2 = N / 2
N1 = N - N2
ELSE
N1 = N / 2
N2 = N - N1
END IF
*
* If N is odd, set NISODD = .TRUE., LDA=N+1 and A is (N+1)--by--K2.
* If N is even, set K = N/2 and NISODD = .FALSE., LDA=N and A is
* N--by--(N+1)/2.
*
IF( MOD( N, 2 ).EQ.0 ) THEN
K = N / 2
NISODD = .FALSE.
IF( .NOT.LOWER )
$ NP1X2 = N + N + 2
ELSE
NISODD = .TRUE.
IF( .NOT.LOWER )
$ NX2 = N + N
END IF
*
IF( NISODD ) THEN
*
* N is odd
*
IF( NORMALTRANSR ) THEN
*
* N is odd and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'N', and UPLO = 'L'
*
IJ = 0
DO J = 0, N2
DO I = N1, N2 + J
ARF( IJ ) = A( N2+J, I )
IJ = IJ + 1
END DO
DO I = J, N - 1
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is odd, TRANSR = 'N', and UPLO = 'U'
*
IJ = NT - N
DO J = N - 1, N1, -1
DO I = 0, J
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
DO L = J - N1, N1 - 1
ARF( IJ ) = A( J-N1, L )
IJ = IJ + 1
END DO
IJ = IJ - NX2
END DO
*
END IF
*
ELSE
*
* N is odd and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is odd, TRANSR = 'T', and UPLO = 'L'
*
IJ = 0
DO J = 0, N2 - 1
DO I = 0, J
ARF( IJ ) = A( J, I )
IJ = IJ + 1
END DO
DO I = N1 + J, N - 1
ARF( IJ ) = A( I, N1+J )
IJ = IJ + 1
END DO
END DO
DO J = N2, N - 1
DO I = 0, N1 - 1
ARF( IJ ) = A( J, I )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is odd, TRANSR = 'T', and UPLO = 'U'
*
IJ = 0
DO J = 0, N1
DO I = N1, N - 1
ARF( IJ ) = A( J, I )
IJ = IJ + 1
END DO
END DO
DO J = 0, N1 - 1
DO I = 0, J
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
DO L = N2 + J, N - 1
ARF( IJ ) = A( N2+J, L )
IJ = IJ + 1
END DO
END DO
*
END IF
*
END IF
*
ELSE
*
* N is even
*
IF( NORMALTRANSR ) THEN
*
* N is even and TRANSR = 'N'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'N', and UPLO = 'L'
*
IJ = 0
DO J = 0, K - 1
DO I = K, K + J
ARF( IJ ) = A( K+J, I )
IJ = IJ + 1
END DO
DO I = J, N - 1
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is even, TRANSR = 'N', and UPLO = 'U'
*
IJ = NT - N - 1
DO J = N - 1, K, -1
DO I = 0, J
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
DO L = J - K, K - 1
ARF( IJ ) = A( J-K, L )
IJ = IJ + 1
END DO
IJ = IJ - NP1X2
END DO
*
END IF
*
ELSE
*
* N is even and TRANSR = 'T'
*
IF( LOWER ) THEN
*
* N is even, TRANSR = 'T', and UPLO = 'L'
*
IJ = 0
J = K
DO I = K, N - 1
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
DO J = 0, K - 2
DO I = 0, J
ARF( IJ ) = A( J, I )
IJ = IJ + 1
END DO
DO I = K + 1 + J, N - 1
ARF( IJ ) = A( I, K+1+J )
IJ = IJ + 1
END DO
END DO
DO J = K - 1, N - 1
DO I = 0, K - 1
ARF( IJ ) = A( J, I )
IJ = IJ + 1
END DO
END DO
*
ELSE
*
* N is even, TRANSR = 'T', and UPLO = 'U'
*
IJ = 0
DO J = 0, K
DO I = K, N - 1
ARF( IJ ) = A( J, I )
IJ = IJ + 1
END DO
END DO
DO J = 0, K - 2
DO I = 0, J
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
DO L = K + 1 + J, N - 1
ARF( IJ ) = A( K+1+J, L )
IJ = IJ + 1
END DO
END DO
* Note that here, on exit of the loop, J = K-1
DO I = 0, J
ARF( IJ ) = A( I, J )
IJ = IJ + 1
END DO
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of DTRTTF
*
END
*> \brief \b DTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTTP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrttp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrttp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrttp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTRTTP( UPLO, N, A, LDA, AP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), AP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRTTP copies a triangular matrix A from full format (TR) to standard
*> packed format (TP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular.
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices AP and A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On exit, the triangular matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On exit, the upper or lower triangular matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trttp
*
* =====================================================================
SUBROUTINE DTRTTP( UPLO, N, A, LDA, AP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), AP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
* ..
* .. Local Scalars ..
LOGICAL LOWER
INTEGER I, J, K
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRTTP', -INFO )
RETURN
END IF
*
IF( LOWER ) THEN
K = 0
DO J = 1, N
DO I = J, N
K = K + 1
AP( K ) = A( I, J )
END DO
END DO
ELSE
K = 0
DO J = 1, N
DO I = 1, J
K = K + 1
AP( K ) = A( I, J )
END DO
END DO
END IF
*
*
RETURN
*
* End of DTRTTP
*
END
*> \brief \b DTZRQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTZRQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtzrqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtzrqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtzrqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DTZRZF.
*>
*> DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
*> to upper triangular form by means of orthogonal transformations.
*>
*> The upper trapezoidal matrix A is factored as
*>
*> A = ( R 0 ) * Z,
*>
*> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
*> triangular matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the leading M-by-N upper trapezoidal part of the
*> array A must contain the matrix to be factorized.
*> On exit, the leading M-by-M upper triangular part of A
*> contains the upper triangular matrix R, and elements M+1 to
*> N of the first M rows of A, with the array TAU, represent the
*> orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The factorization is obtained by Householder's method. The kth
*> transformation matrix, Z( k ), which is used to introduce zeros into
*> the ( m - k + 1 )th row of A, is given in the form
*>
*> Z( k ) = ( I 0 ),
*> ( 0 T( k ) )
*>
*> where
*>
*> T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
*> ( 0 )
*> ( z( k ) )
*>
*> tau is a scalar and z( k ) is an ( n - m ) element vector.
*> tau and z( k ) are chosen to annihilate the elements of the kth row
*> of X.
*>
*> The scalar tau is returned in the kth element of TAU and the vector
*> u( k ) in the kth row of A, such that the elements of z( k ) are
*> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
*> the upper triangular part of A.
*>
*> Z is given by
*>
*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K, M1
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTZRQF', -INFO )
RETURN
END IF
*
* Perform the factorization.
*
IF( M.EQ.0 )
$ RETURN
IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
ELSE
M1 = MIN( M+1, N )
DO 20 K = M, 1, -1
*
* Use a Householder reflection to zero the kth row of A.
* First set up the reflection.
*
CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA,
$ TAU( K ) )
*
IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN
*
* We now perform the operation A := A*P( k ).
*
* Use the first ( k - 1 ) elements of TAU to store a( k ),
* where a( k ) consists of the first ( k - 1 ) elements of
* the kth column of A. Also let B denote the first
* ( k - 1 ) rows of the last ( n - m ) columns of A.
*
CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 )
*
* Form w = a( k ) + B*z( k ) in TAU.
*
CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ),
$ LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
*
* Now form a( k ) := a( k ) - tau*w
* and B := B - tau*w*z( k )**T.
*
CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ),
$ LDA, A( 1, M1 ), LDA )
END IF
20 CONTINUE
END IF
*
RETURN
*
* End of DTZRQF
*
END
*> \brief \b DTZRZF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTZRZF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtzrzf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtzrzf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtzrzf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
*> to upper triangular form by means of orthogonal transformations.
*>
*> The upper trapezoidal matrix A is factored as
*>
*> A = ( R 0 ) * Z,
*>
*> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
*> triangular matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the leading M-by-N upper trapezoidal part of the
*> array A must contain the matrix to be factorized.
*> On exit, the leading M-by-M upper triangular part of A
*> contains the upper triangular matrix R, and elements M+1 to
*> N of the first M rows of A, with the array TAU, represent the
*> orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M).
*> For optimum performance LWORK >= M*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tzrzf
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The N-by-N matrix Z can be computed by
*>
*> Z = Z(1)*Z(2)* ... *Z(M)
*>
*> where each N-by-N Z(k) is given by
*>
*> Z(k) = I - tau(k)*v(k)*v(k)**T
*>
*> with v(k) is the kth row vector of the M-by-N matrix
*>
*> V = ( I A(:,M+1:N) )
*>
*> I is the M-by-M identity matrix, A(:,M+1:N)
*> is the output stored in A on exit from DTZRZF,
*> and tau(k) is the kth element of the array TAU.
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IWS, KI, KK, LDWORK, LWKMIN, LWKOPT,
$ M1, MU, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, DLARZB, DLARZT, DLATRZ
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
*
IF( INFO.EQ.0 ) THEN
IF( M.EQ.0 .OR. M.EQ.N ) THEN
LWKOPT = 1
LWKMIN = 1
ELSE
*
* Determine the block size.
*
NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
LWKOPT = M*NB
LWKMIN = MAX( 1, M )
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTZRZF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 ) THEN
RETURN
ELSE IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
RETURN
END IF
*
NBMIN = 2
NX = 1
IWS = M
IF( NB.GT.1 .AND. NB.LT.M ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
IF( NX.LT.M ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.M .AND. NX.LT.M ) THEN
*
* Use blocked code initially.
* The last kk rows are handled by the block method.
*
M1 = MIN( M+1, N )
KI = ( ( M-NX-1 ) / NB )*NB
KK = MIN( M, KI+NB )
*
DO 20 I = M - KK + KI + 1, M - KK + 1, -NB
IB = MIN( M-I+1, NB )
*
* Compute the TZ factorization of the current block
* A(i:i+ib-1,i:n)
*
CALL DLATRZ( IB, N-I+1, N-M, A( I, I ), LDA, TAU( I ),
$ WORK )
IF( I.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARZT( 'Backward', 'Rowwise', N-M, IB, A( I,
$ M1 ),
$ LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(1:i-1,i:n) from the right
*
CALL DLARZB( 'Right', 'No transpose', 'Backward',
$ 'Rowwise', I-1, N-I+1, IB, N-M, A( I, M1 ),
$ LDA, WORK, LDWORK, A( 1, I ), LDA,
$ WORK( IB+1 ), LDWORK )
END IF
20 CONTINUE
MU = I + NB - 1
ELSE
MU = M
END IF
*
* Use unblocked code to factor the last or only block
*
IF( MU.GT.0 )
$ CALL DLATRZ( MU, N, N-M, A, LDA, TAU, WORK )
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DTZRZF
*
END
*> \brief \b IEEECK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IEEECK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ieeeck.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ieeeck.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ieeeck.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
* .. Scalar Arguments ..
* INTEGER ISPEC
* REAL ONE, ZERO
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> IEEECK is called from the ILAENV to verify that Infinity and
*> possibly NaN arithmetic is safe (i.e. will not trap).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> Specifies whether to test just for infinity arithmetic
*> or whether to test for infinity and NaN arithmetic.
*> = 0: Verify infinity arithmetic only.
*> = 1: Verify infinity and NaN arithmetic.
*> \endverbatim
*>
*> \param[in] ZERO
*> \verbatim
*> ZERO is REAL
*> Must contain the value 0.0
*> This is passed to prevent the compiler from optimizing
*> away this code.
*> \endverbatim
*>
*> \param[in] ONE
*> \verbatim
*> ONE is REAL
*> Must contain the value 1.0
*> This is passed to prevent the compiler from optimizing
*> away this code.
*>
*> RETURN VALUE: INTEGER
*> = 0: Arithmetic failed to produce the correct answers
*> = 1: Arithmetic produced the correct answers
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ieeeck
*
* =====================================================================
INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER ISPEC
REAL ONE, ZERO
* ..
*
* =====================================================================
*
* .. Local Scalars ..
REAL NAN1, NAN2, NAN3, NAN4, NAN5, NAN6, NEGINF,
$ NEGZRO, NEWZRO, POSINF
* ..
* .. Executable Statements ..
IEEECK = 1
*
POSINF = ONE / ZERO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = -ONE / ZERO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGZRO = ONE / ( NEGINF+ONE )
IF( NEGZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = ONE / NEGZRO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEWZRO = NEGZRO + ZERO
IF( NEWZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = ONE / NEWZRO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = NEGINF*POSINF
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = POSINF*POSINF
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
*
*
*
* Return if we were only asked to check infinity arithmetic
*
IF( ISPEC.EQ.0 )
$ RETURN
*
NAN1 = POSINF + NEGINF
*
NAN2 = POSINF / NEGINF
*
NAN3 = POSINF / POSINF
*
NAN4 = POSINF*ZERO
*
NAN5 = NEGINF*NEGZRO
*
NAN6 = NAN5*ZERO
*
IF( NAN1.EQ.NAN1 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN2.EQ.NAN2 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN3.EQ.NAN3 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN4.EQ.NAN4 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN5.EQ.NAN5 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN6.EQ.NAN6 ) THEN
IEEECK = 0
RETURN
END IF
*
RETURN
END
*> \brief \b ILADLC scans a matrix for its last non-zero column.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iladlc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iladlc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iladlc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILADLC scans A for its last non-zero column.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilalc
*
* =====================================================================
INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER M, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. Executable Statements ..
*
* Quick test for the common case where one corner is non-zero.
IF( N.EQ.0 ) THEN
ILADLC = N
ELSE IF( A(1, N).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
ILADLC = N
ELSE
* Now scan each column from the end, returning with the first non-zero.
DO ILADLC = N, 1, -1
DO I = 1, M
IF( A(I, ILADLC).NE.ZERO ) RETURN
END DO
END DO
END IF
RETURN
END
*> \brief \b ILADLR scans a matrix for its last non-zero row.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iladlr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iladlr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iladlr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILADLR scans A for its last non-zero row.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilalr
*
* =====================================================================
INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER M, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
* ..
* .. Executable Statements ..
*
* Quick test for the common case where one corner is non-zero.
IF( M.EQ.0 ) THEN
ILADLR = M
ELSE IF( A(M, 1).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
ILADLR = M
ELSE
* Scan up each column tracking the last zero row seen.
ILADLR = 0
DO J = 1, N
I=M
DO WHILE((A(MAX(I,1),J).EQ.ZERO).AND.(I.GE.1))
I=I-1
ENDDO
ILADLR = MAX( ILADLR, I )
END DO
END IF
RETURN
END
*> \brief \b ILAENV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILAENV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ilaenv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ilaenv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ilaenv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
* .. Scalar Arguments ..
* CHARACTER*( * ) NAME, OPTS
* INTEGER ISPEC, N1, N2, N3, N4
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILAENV is called from the LAPACK routines to choose problem-dependent
*> parameters for the local environment. See ISPEC for a description of
*> the parameters.
*>
*> ILAENV returns an INTEGER
*> if ILAENV >= 0: ILAENV returns the value of the parameter specified by ISPEC
*> if ILAENV < 0: if ILAENV = -k, the k-th argument had an illegal value.
*>
*> This version provides a set of parameters which should give good,
*> but not optimal, performance on many of the currently available
*> computers. Users are encouraged to modify this subroutine to set
*> the tuning parameters for their particular machine using the option
*> and problem size information in the arguments.
*>
*> This routine will not function correctly if it is converted to all
*> lower case. Converting it to all upper case is allowed.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> Specifies the parameter to be returned as the value of
*> ILAENV.
*> = 1: the optimal blocksize; if this value is 1, an unblocked
*> algorithm will give the best performance.
*> = 2: the minimum block size for which the block routine
*> should be used; if the usable block size is less than
*> this value, an unblocked routine should be used.
*> = 3: the crossover point (in a block routine, for N less
*> than this value, an unblocked routine should be used)
*> = 4: the number of shifts, used in the nonsymmetric
*> eigenvalue routines (DEPRECATED)
*> = 5: the minimum column dimension for blocking to be used;
*> rectangular blocks must have dimension at least k by m,
*> where k is given by ILAENV(2,...) and m by ILAENV(5,...)
*> = 6: the crossover point for the SVD (when reducing an m by n
*> matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds
*> this value, a QR factorization is used first to reduce
*> the matrix to a triangular form.)
*> = 7: the number of processors
*> = 8: the crossover point for the multishift QR method
*> for nonsymmetric eigenvalue problems (DEPRECATED)
*> = 9: maximum size of the subproblems at the bottom of the
*> computation tree in the divide-and-conquer algorithm
*> (used by xGELSD and xGESDD)
*> =10: ieee infinity and NaN arithmetic can be trusted not to trap
*> =11: infinity arithmetic can be trusted not to trap
*> 12 <= ISPEC <= 17:
*> xHSEQR or related subroutines,
*> see IPARMQ for detailed explanation
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*> NAME is CHARACTER*(*)
*> The name of the calling subroutine, in either upper case or
*> lower case.
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*> OPTS is CHARACTER*(*)
*> The character options to the subroutine NAME, concatenated
*> into a single character string. For example, UPLO = 'U',
*> TRANS = 'T', and DIAG = 'N' for a triangular routine would
*> be specified as OPTS = 'UTN'.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> \endverbatim
*>
*> \param[in] N3
*> \verbatim
*> N3 is INTEGER
*> \endverbatim
*>
*> \param[in] N4
*> \verbatim
*> N4 is INTEGER
*> Problem dimensions for the subroutine NAME; these may not all
*> be required.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilaenv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The following conventions have been used when calling ILAENV from the
*> LAPACK routines:
*> 1) OPTS is a concatenation of all of the character options to
*> subroutine NAME, in the same order that they appear in the
*> argument list for NAME, even if they are not used in determining
*> the value of the parameter specified by ISPEC.
*> 2) The problem dimensions N1, N2, N3, N4 are specified in the order
*> that they appear in the argument list for NAME. N1 is used
*> first, N2 second, and so on, and unused problem dimensions are
*> passed a value of -1.
*> 3) The parameter value returned by ILAENV is checked for validity in
*> the calling subroutine. For example, ILAENV is used to retrieve
*> the optimal blocksize for STRTRI as follows:
*>
*> NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
*> IF( NB.LE.1 ) NB = MAX( 1, N )
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER*( * ) NAME, OPTS
INTEGER ISPEC, N1, N2, N3, N4
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IC, IZ, NB, NBMIN, NX
LOGICAL CNAME, SNAME, TWOSTAGE
CHARACTER C1*1, C2*2, C4*2, C3*3, SUBNAM*16
* ..
* .. Intrinsic Functions ..
INTRINSIC CHAR, ICHAR, INT, MIN, REAL
* ..
* .. External Functions ..
INTEGER IEEECK, IPARMQ, IPARAM2STAGE
EXTERNAL IEEECK, IPARMQ, IPARAM2STAGE
* ..
* .. Executable Statements ..
*
GO TO ( 10, 10, 10, 80, 90, 100, 110, 120,
$ 130, 140, 150, 160, 160, 160, 160, 160, 160)ISPEC
*
* Invalid value for ISPEC
*
ILAENV = -1
RETURN
*
10 CONTINUE
*
* Convert NAME to upper case if the first character is lower case.
*
ILAENV = 1
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 20 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
20 CONTINUE
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO 30 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
30 CONTINUE
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 40 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
40 CONTINUE
END IF
END IF
*
C1 = SUBNAM( 1: 1 )
SNAME = C1.EQ.'S' .OR. C1.EQ.'D'
CNAME = C1.EQ.'C' .OR. C1.EQ.'Z'
IF( .NOT.( CNAME .OR. SNAME ) )
$ RETURN
C2 = SUBNAM( 2: 3 )
C3 = SUBNAM( 4: 6 )
C4 = C3( 2: 3 )
TWOSTAGE = LEN( SUBNAM ).GE.11
$ .AND. SUBNAM( 11: 11 ).EQ.'2'
*
GO TO ( 50, 60, 70 )ISPEC
*
50 CONTINUE
*
* ISPEC = 1: block size
*
* In these examples, separate code is provided for setting NB for
* real and complex. We assume that NB will take the same value in
* single or double precision.
*
NB = 1
*
IF( SUBNAM(2:6).EQ.'LAORH' ) THEN
*
* This is for *LAORHR_GETRFNP routine
*
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
$ C3.EQ.'QLF' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'QR ') THEN
IF( N3 .EQ. 1) THEN
IF( SNAME ) THEN
* M*N
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
ELSE
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
END IF
ELSE
IF( SNAME ) THEN
NB = 1
ELSE
NB = 1
END IF
END IF
ELSE IF( C3.EQ.'LQ ') THEN
IF( N3 .EQ. 2) THEN
IF( SNAME ) THEN
* M*N
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
ELSE
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
END IF
ELSE
IF( SNAME ) THEN
NB = 1
ELSE
NB = 1
END IF
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
END IF
ELSE IF( C2.EQ.'PO' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( TWOSTAGE ) THEN
NB = 192
ELSE
NB = 64
END IF
ELSE
IF( TWOSTAGE ) THEN
NB = 192
ELSE
NB = 64
END IF
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( TWOSTAGE ) THEN
NB = 192
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( C2.EQ.'GB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'PB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'TR' ) THEN
IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF ( C3.EQ.'EVC' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'SYL' ) THEN
* The upper bound is to prevent overly aggressive scaling.
IF( SNAME ) THEN
NB = MIN( MAX( 48, INT( ( MIN( N1, N2 ) * 16 ) / 100) ),
$ 240 )
ELSE
NB = MIN( MAX( 24, INT( ( MIN( N1, N2 ) * 8 ) / 100) ),
$ 80 )
END IF
END IF
ELSE IF( C2.EQ.'LA' ) THEN
IF( C3.EQ.'UUM' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'TRS' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
END IF
ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
IF( C3.EQ.'EBZ' ) THEN
NB = 1
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NB = 32
IF( C3.EQ.'HD3' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
END IF
END IF
ILAENV = NB
RETURN
*
60 CONTINUE
*
* ISPEC = 2: minimum block size
*
NBMIN = 2
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NBMIN = 8
ELSE
NBMIN = 8
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NBMIN = 2
IF( C3.EQ.'HD3' ) THEN
NBMIN = 2
END IF
END IF
ILAENV = NBMIN
RETURN
*
70 CONTINUE
*
* ISPEC = 3: crossover point
*
NX = 0
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NX = 128
IF( C3.EQ.'HD3' ) THEN
NX = 128
END IF
END IF
ILAENV = NX
RETURN
*
80 CONTINUE
*
* ISPEC = 4: number of shifts (used by xHSEQR)
*
ILAENV = 6
RETURN
*
90 CONTINUE
*
* ISPEC = 5: minimum column dimension (not used)
*
ILAENV = 2
RETURN
*
100 CONTINUE
*
* ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD)
*
ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
RETURN
*
110 CONTINUE
*
* ISPEC = 7: number of processors (not used)
*
ILAENV = 1
RETURN
*
120 CONTINUE
*
* ISPEC = 8: crossover point for multishift (used by xHSEQR)
*
ILAENV = 50
RETURN
*
130 CONTINUE
*
* ISPEC = 9: maximum size of the subproblems at the bottom of the
* computation tree in the divide-and-conquer algorithm
* (used by xGELSD and xGESDD)
*
ILAENV = 25
RETURN
*
140 CONTINUE
*
* ISPEC = 10: ieee and infinity NaN arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 1, 0.0, 1.0 )
END IF
RETURN
*
150 CONTINUE
*
* ISPEC = 11: ieee infinity arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 0, 0.0, 1.0 )
END IF
RETURN
*
160 CONTINUE
*
* 12 <= ISPEC <= 17: xHSEQR or related subroutines.
*
ILAENV = IPARMQ( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
RETURN
*
* End of ILAENV
*
END
*> \brief \b ILAPREC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILAPREC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ilaprec.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ilaprec.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ilaprec.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILAPREC( PREC )
*
* .. Scalar Arguments ..
* CHARACTER PREC
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine translated from a character string specifying an
*> intermediate precision to the relevant BLAST-specified integer
*> constant.
*>
*> ILAPREC returns an INTEGER. If ILAPREC < 0, then the input is not a
*> character indicating a supported intermediate precision. Otherwise
*> ILAPREC returns the constant value corresponding to PREC.
*> \endverbatim
*
* Arguments:
* ==========
*
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilaprec
*
* =====================================================================
INTEGER FUNCTION ILAPREC( PREC )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER PREC
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER BLAS_PREC_SINGLE, BLAS_PREC_DOUBLE, BLAS_PREC_INDIGENOUS,
$ BLAS_PREC_EXTRA
PARAMETER ( BLAS_PREC_SINGLE = 211, BLAS_PREC_DOUBLE = 212,
$ BLAS_PREC_INDIGENOUS = 213, BLAS_PREC_EXTRA = 214 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
IF( LSAME( PREC, 'S' ) ) THEN
ILAPREC = BLAS_PREC_SINGLE
ELSE IF( LSAME( PREC, 'D' ) ) THEN
ILAPREC = BLAS_PREC_DOUBLE
ELSE IF( LSAME( PREC, 'I' ) ) THEN
ILAPREC = BLAS_PREC_INDIGENOUS
ELSE IF( LSAME( PREC, 'X' ) .OR. LSAME( PREC, 'E' ) ) THEN
ILAPREC = BLAS_PREC_EXTRA
ELSE
ILAPREC = -1
END IF
RETURN
*
* End of ILAPREC
*
END
*> \brief \b ILATRANS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILATRANS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ilatrans.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ilatrans.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ilatrans.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILATRANS( TRANS )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine translates from a character string specifying a
*> transposition operation to the relevant BLAST-specified integer
*> constant.
*>
*> ILATRANS returns an INTEGER. If ILATRANS < 0, then the input is not
*> a character indicating a transposition operator. Otherwise ILATRANS
*> returns the constant value corresponding to TRANS.
*> \endverbatim
*
* Arguments:
* ==========
*
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilatrans
*
* =====================================================================
INTEGER FUNCTION ILATRANS( TRANS )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER BLAS_NO_TRANS, BLAS_TRANS, BLAS_CONJ_TRANS
PARAMETER ( BLAS_NO_TRANS = 111, BLAS_TRANS = 112,
$ BLAS_CONJ_TRANS = 113 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
IF( LSAME( TRANS, 'N' ) ) THEN
ILATRANS = BLAS_NO_TRANS
ELSE IF( LSAME( TRANS, 'T' ) ) THEN
ILATRANS = BLAS_TRANS
ELSE IF( LSAME( TRANS, 'C' ) ) THEN
ILATRANS = BLAS_CONJ_TRANS
ELSE
ILATRANS = -1
END IF
RETURN
*
* End of ILATRANS
*
END
*> \brief \b ILAVER returns the LAPACK version.
**
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ILAVER( VERS_MAJOR, VERS_MINOR, VERS_PATCH )
*
* INTEGER VERS_MAJOR, VERS_MINOR, VERS_PATCH
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine returns the LAPACK version.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[out] VERS_MAJOR
*> VERS_MAJOR is INTEGER
*> return the lapack major version
*>
*> \param[out] VERS_MINOR
*> VERS_MINOR is INTEGER
*> return the lapack minor version from the major version
*>
*> \param[out] VERS_PATCH
*> VERS_PATCH is INTEGER
*> return the lapack patch version from the minor version
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilaver
*
* =====================================================================
SUBROUTINE ILAVER( VERS_MAJOR, VERS_MINOR, VERS_PATCH )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* =====================================================================
*
INTEGER VERS_MAJOR, VERS_MINOR, VERS_PATCH
* =====================================================================
VERS_MAJOR = 3
VERS_MINOR = 12
VERS_PATCH = 0
* =====================================================================
*
RETURN
END
# 1 "SRC/iparam2stage.F"
# 1 "<built-in>" 1
# 1 "<built-in>" 3
# 423 "<built-in>" 3
# 1 "<command line>" 1
# 1 "<built-in>" 2
# 1 "SRC/iparam2stage.F" 2
*> \brief \b IPARAM2STAGE
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IPARAM2STAGE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iparam2stage.F">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iparam2stage.F">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iparam2stage.F">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION IPARAM2STAGE( ISPEC, NAME, OPTS,
* NI, NBI, IBI, NXI )
* #if defined(_OPENMP)
* use omp_lib
* #endif
* IMPLICIT NONE
*
* .. Scalar Arguments ..
* CHARACTER*( * ) NAME, OPTS
* INTEGER ISPEC, NI, NBI, IBI, NXI
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This program sets problem and machine dependent parameters
*> useful for xHETRD_2STAGE, xHETRD_HE2HB, xHETRD_HB2ST,
*> xGEBRD_2STAGE, xGEBRD_GE2GB, xGEBRD_GB2BD
*> and related subroutines for eigenvalue problems.
*> It is called whenever ILAENV is called with 17 <= ISPEC <= 21.
*> It is called whenever ILAENV2STAGE is called with 1 <= ISPEC <= 5
*> with a direct conversion ISPEC + 16.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is integer scalar
*> ISPEC specifies which tunable parameter IPARAM2STAGE should
*> return.
*>
*> ISPEC=17: the optimal blocksize nb for the reduction to
*> BAND
*>
*> ISPEC=18: the optimal blocksize ib for the eigenvectors
*> singular vectors update routine
*>
*> ISPEC=19: The length of the array that store the Housholder
*> representation for the second stage
*> Band to Tridiagonal or Bidiagonal
*>
*> ISPEC=20: The workspace needed for the routine in input.
*>
*> ISPEC=21: For future release.
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*> NAME is character string
*> Name of the calling subroutine
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*> OPTS is CHARACTER*(*)
*> The character options to the subroutine NAME, concatenated
*> into a single character string. For example, UPLO = 'U',
*> TRANS = 'T', and DIAG = 'N' for a triangular routine would
*> be specified as OPTS = 'UTN'.
*> \endverbatim
*>
*> \param[in] NI
*> \verbatim
*> NI is INTEGER which is the size of the matrix
*> \endverbatim
*>
*> \param[in] NBI
*> \verbatim
*> NBI is INTEGER which is the used in the reduction,
*> (e.g., the size of the band), needed to compute workspace
*> and LHOUS2.
*> \endverbatim
*>
*> \param[in] IBI
*> \verbatim
*> IBI is INTEGER which represent the IB of the reduction,
*> needed to compute workspace and LHOUS2.
*> \endverbatim
*>
*> \param[in] NXI
*> \verbatim
*> NXI is INTEGER needed in the future release.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup iparam2stage
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Implemented by Azzam Haidar.
*>
*> All detail are available on technical report, SC11, SC13 papers.
*>
*> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
*> Parallel reduction to condensed forms for symmetric eigenvalue problems
*> using aggregated fine-grained and memory-aware kernels. In Proceedings
*> of 2011 International Conference for High Performance Computing,
*> Networking, Storage and Analysis (SC '11), New York, NY, USA,
*> Article 8 , 11 pages.
*> http://doi.acm.org/10.1145/2063384.2063394
*>
*> A. Haidar, J. Kurzak, P. Luszczek, 2013.
*> An improved parallel singular value algorithm and its implementation
*> for multicore hardware, In Proceedings of 2013 International Conference
*> for High Performance Computing, Networking, Storage and Analysis (SC '13).
*> Denver, Colorado, USA, 2013.
*> Article 90, 12 pages.
*> http://doi.acm.org/10.1145/2503210.2503292
*>
*> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
*> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
*> calculations based on fine-grained memory aware tasks.
*> International Journal of High Performance Computing Applications.
*> Volume 28 Issue 2, Pages 196-209, May 2014.
*> http://hpc.sagepub.com/content/28/2/196
*>
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION IPARAM2STAGE( ISPEC, NAME, OPTS,
$ NI, NBI, IBI, NXI )
IMPLICIT NONE
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER*( * ) NAME, OPTS
INTEGER ISPEC, NI, NBI, IBI, NXI
*
* ================================================================
* ..
* .. Local Scalars ..
INTEGER I, IC, IZ, KD, IB, LHOUS, LWORK, NTHREADS,
$ FACTOPTNB, QROPTNB, LQOPTNB
LOGICAL RPREC, CPREC
CHARACTER PREC*1, ALGO*3, STAG*5, SUBNAM*12, VECT*1
* ..
* .. Intrinsic Functions ..
INTRINSIC CHAR, ICHAR, MAX
* ..
* .. External Functions ..
INTEGER ILAENV
LOGICAL LSAME
EXTERNAL ILAENV, LSAME
* ..
* .. Executable Statements ..
*
* Invalid value for ISPEC
*
IF( (ISPEC.LT.17).OR.(ISPEC.GT.21) ) THEN
IPARAM2STAGE = -1
RETURN
ENDIF
*
* Get the number of threads
*
NTHREADS = 1
* WRITE(*,*) 'IPARAM VOICI NTHREADS ISPEC ',NTHREADS, ISPEC
*
IF( ISPEC .NE. 19 ) THEN
*
* Convert NAME to upper case if the first character is lower case.
*
IPARAM2STAGE = -1
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 100 I = 2, 12
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
100 CONTINUE
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO 110 I = 2, 12
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
110 CONTINUE
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 120 I = 2, 12
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
120 CONTINUE
END IF
END IF
*
PREC = SUBNAM( 1: 1 )
ALGO = SUBNAM( 4: 6 )
STAG = SUBNAM( 8:12 )
RPREC = PREC.EQ.'S' .OR. PREC.EQ.'D'
CPREC = PREC.EQ.'C' .OR. PREC.EQ.'Z'
*
* Invalid value for PRECISION
*
IF( .NOT.( RPREC .OR. CPREC ) ) THEN
IPARAM2STAGE = -1
RETURN
ENDIF
ENDIF
* WRITE(*,*),'RPREC,CPREC ',RPREC,CPREC,
* $ ' ALGO ',ALGO,' STAGE ',STAG
*
*
IF (( ISPEC .EQ. 17 ) .OR. ( ISPEC .EQ. 18 )) THEN
*
* ISPEC = 17, 18: block size KD, IB
* Could be also dependent from N but for now it
* depend only on sequential or parallel
*
IF( NTHREADS.GT.4 ) THEN
IF( CPREC ) THEN
KD = 128
IB = 32
ELSE
KD = 160
IB = 40
ENDIF
ELSE IF( NTHREADS.GT.1 ) THEN
IF( CPREC ) THEN
KD = 64
IB = 32
ELSE
KD = 64
IB = 32
ENDIF
ELSE
IF( CPREC ) THEN
KD = 16
IB = 16
ELSE
KD = 32
IB = 16
ENDIF
ENDIF
IF( ISPEC.EQ.17 ) IPARAM2STAGE = KD
IF( ISPEC.EQ.18 ) IPARAM2STAGE = IB
*
ELSE IF ( ISPEC .EQ. 19 ) THEN
*
* ISPEC = 19:
* LHOUS length of the Houselholder representation
* matrix (V,T) of the second stage. should be >= 1.
*
* Will add the VECT OPTION HERE next release
VECT = OPTS(1:1)
IF( LSAME( VECT, 'N' ) ) THEN
LHOUS = MAX( 1, 4*NI )
ELSE
* This is not correct, it need to call the ALGO and the stage2
LHOUS = MAX( 1, 4*NI ) + IBI
ENDIF
IF( LHOUS.GE.0 ) THEN
IPARAM2STAGE = LHOUS
ELSE
IPARAM2STAGE = -1
ENDIF
*
ELSE IF ( ISPEC .EQ. 20 ) THEN
*
* ISPEC = 20: (21 for future use)
* LWORK length of the workspace for
* either or both stages for TRD and BRD. should be >= 1.
* TRD:
* TRD_stage 1: = LT + LW + LS1 + LS2
* = LDT*KD + N*KD + N*MAX(KD,FACTOPTNB) + LDS2*KD
* where LDT=LDS2=KD
* = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
* TRD_stage 2: = (2NB+1)*N + KD*NTHREADS
* TRD_both : = max(stage1,stage2) + AB ( AB=(KD+1)*N )
* = N*KD + N*max(KD+1,FACTOPTNB)
* + max(2*KD*KD, KD*NTHREADS)
* + (KD+1)*N
LWORK = -1
SUBNAM(1:1) = PREC
SUBNAM(2:6) = 'GEQRF'
QROPTNB = ILAENV( 1, SUBNAM, ' ', NI, NBI, -1, -1 )
SUBNAM(2:6) = 'GELQF'
LQOPTNB = ILAENV( 1, SUBNAM, ' ', NBI, NI, -1, -1 )
* Could be QR or LQ for TRD and the max for BRD
FACTOPTNB = MAX(QROPTNB, LQOPTNB)
IF( ALGO.EQ.'TRD' ) THEN
IF( STAG.EQ.'2STAG' ) THEN
LWORK = NI*NBI + NI*MAX(NBI+1,FACTOPTNB)
$ + MAX(2*NBI*NBI, NBI*NTHREADS)
$ + (NBI+1)*NI
ELSE IF( (STAG.EQ.'HE2HB').OR.(STAG.EQ.'SY2SB') ) THEN
LWORK = NI*NBI + NI*MAX(NBI,FACTOPTNB) + 2*NBI*NBI
ELSE IF( (STAG.EQ.'HB2ST').OR.(STAG.EQ.'SB2ST') ) THEN
LWORK = (2*NBI+1)*NI + NBI*NTHREADS
ENDIF
ELSE IF( ALGO.EQ.'BRD' ) THEN
IF( STAG.EQ.'2STAG' ) THEN
LWORK = 2*NI*NBI + NI*MAX(NBI+1,FACTOPTNB)
$ + MAX(2*NBI*NBI, NBI*NTHREADS)
$ + (NBI+1)*NI
ELSE IF( STAG.EQ.'GE2GB' ) THEN
LWORK = NI*NBI + NI*MAX(NBI,FACTOPTNB) + 2*NBI*NBI
ELSE IF( STAG.EQ.'GB2BD' ) THEN
LWORK = (3*NBI+1)*NI + NBI*NTHREADS
ENDIF
ENDIF
LWORK = MAX ( 1, LWORK )
IF( LWORK.GT.0 ) THEN
IPARAM2STAGE = LWORK
ELSE
IPARAM2STAGE = -1
ENDIF
*
ELSE IF ( ISPEC .EQ. 21 ) THEN
*
* ISPEC = 21 for future use
IPARAM2STAGE = NXI
ENDIF
*
* ==== End of IPARAM2STAGE ====
*
END
*> \brief \b IPARMQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IPARMQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iparmq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iparmq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iparmq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, ISPEC, LWORK, N
* CHARACTER NAME*( * ), OPTS*( * )
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This program sets problem and machine dependent parameters
*> useful for xHSEQR and related subroutines for eigenvalue
*> problems. It is called whenever
*> IPARMQ is called with 12 <= ISPEC <= 16
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> ISPEC specifies which tunable parameter IPARMQ should
*> return.
*>
*> ISPEC=12: (INMIN) Matrices of order nmin or less
*> are sent directly to xLAHQR, the implicit
*> double shift QR algorithm. NMIN must be
*> at least 11.
*>
*> ISPEC=13: (INWIN) Size of the deflation window.
*> This is best set greater than or equal to
*> the number of simultaneous shifts NS.
*> Larger matrices benefit from larger deflation
*> windows.
*>
*> ISPEC=14: (INIBL) Determines when to stop nibbling and
*> invest in an (expensive) multi-shift QR sweep.
*> If the aggressive early deflation subroutine
*> finds LD converged eigenvalues from an order
*> NW deflation window and LD > (NW*NIBBLE)/100,
*> then the next QR sweep is skipped and early
*> deflation is applied immediately to the
*> remaining active diagonal block. Setting
*> IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a
*> multi-shift QR sweep whenever early deflation
*> finds a converged eigenvalue. Setting
*> IPARMQ(ISPEC=14) greater than or equal to 100
*> prevents TTQRE from skipping a multi-shift
*> QR sweep.
*>
*> ISPEC=15: (NSHFTS) The number of simultaneous shifts in
*> a multi-shift QR iteration.
*>
*> ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the
*> following meanings.
*> 0: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are not
*> accumulated when updating the
*> far-from-diagonal matrix entries.
*> 1: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are
*> accumulated, and matrix-matrix
*> multiplication is used to update the
*> far-from-diagonal matrix entries.
*> 2: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are
*> accumulated, and 2-by-2 block structure
*> is exploited during matrix-matrix
*> multiplies.
*> (If xTRMM is slower than xGEMM, then
*> IPARMQ(ISPEC=16)=1 may be more efficient than
*> IPARMQ(ISPEC=16)=2 despite the greater level of
*> arithmetic work implied by the latter choice.)
*>
*> ISPEC=17: (ICOST) An estimate of the relative cost of flops
*> within the near-the-diagonal shift chase compared
*> to flops within the BLAS calls of a QZ sweep.
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*> NAME is CHARACTER string
*> Name of the calling subroutine
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*> OPTS is CHARACTER string
*> This is a concatenation of the string arguments to
*> TTQRE.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> N is the order of the Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular
*> in rows and columns 1:ILO-1 and IHI+1:N.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The amount of workspace available.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup iparmq
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Little is known about how best to choose these parameters.
*> It is possible to use different values of the parameters
*> for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR.
*>
*> It is probably best to choose different parameters for
*> different matrices and different parameters at different
*> times during the iteration, but this has not been
*> implemented --- yet.
*>
*>
*> The best choices of most of the parameters depend
*> in an ill-understood way on the relative execution
*> rate of xLAQR3 and xLAQR5 and on the nature of each
*> particular eigenvalue problem. Experiment may be the
*> only practical way to determine which choices are most
*> effective.
*>
*> Following is a list of default values supplied by IPARMQ.
*> These defaults may be adjusted in order to attain better
*> performance in any particular computational environment.
*>
*> IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point.
*> Default: 75. (Must be at least 11.)
*>
*> IPARMQ(ISPEC=13) Recommended deflation window size.
*> This depends on ILO, IHI and NS, the
*> number of simultaneous shifts returned
*> by IPARMQ(ISPEC=15). The default for
*> (IHI-ILO+1) <= 500 is NS. The default
*> for (IHI-ILO+1) > 500 is 3*NS/2.
*>
*> IPARMQ(ISPEC=14) Nibble crossover point. Default: 14.
*>
*> IPARMQ(ISPEC=15) Number of simultaneous shifts, NS.
*> a multi-shift QR iteration.
*>
*> If IHI-ILO+1 is ...
*>
*> greater than ...but less ... the
*> or equal to ... than default is
*>
*> 0 30 NS = 2+
*> 30 60 NS = 4+
*> 60 150 NS = 10
*> 150 590 NS = **
*> 590 3000 NS = 64
*> 3000 6000 NS = 128
*> 6000 infinity NS = 256
*>
*> (+) By default matrices of this order are
*> passed to the implicit double shift routine
*> xLAHQR. See IPARMQ(ISPEC=12) above. These
*> values of NS are used only in case of a rare
*> xLAHQR failure.
*>
*> (**) The asterisks (**) indicate an ad-hoc
*> function increasing from 10 to 64.
*>
*> IPARMQ(ISPEC=16) Select structured matrix multiply.
*> (See ISPEC=16 above for details.)
*> Default: 3.
*>
*> IPARMQ(ISPEC=17) Relative cost heuristic for blocksize selection.
*> Expressed as a percentage.
*> Default: 10.
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI,
$ LWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, ISPEC, LWORK, N
CHARACTER NAME*( * ), OPTS*( * )
*
* ================================================================
* .. Parameters ..
INTEGER INMIN, INWIN, INIBL, ISHFTS, IACC22, ICOST
PARAMETER ( INMIN = 12, INWIN = 13, INIBL = 14,
$ ISHFTS = 15, IACC22 = 16, ICOST = 17 )
INTEGER NMIN, K22MIN, KACMIN, NIBBLE, KNWSWP, RCOST
PARAMETER ( NMIN = 75, K22MIN = 14, KACMIN = 14,
$ NIBBLE = 14, KNWSWP = 500, RCOST = 10 )
REAL TWO
PARAMETER ( TWO = 2.0 )
* ..
* .. Local Scalars ..
INTEGER NH, NS
INTEGER I, IC, IZ
CHARACTER SUBNAM*6
* ..
* .. Intrinsic Functions ..
INTRINSIC LOG, MAX, MOD, NINT, REAL
* ..
* .. Executable Statements ..
IF( ( ISPEC.EQ.ISHFTS ) .OR. ( ISPEC.EQ.INWIN ) .OR.
$ ( ISPEC.EQ.IACC22 ) ) THEN
*
* ==== Set the number simultaneous shifts ====
*
NH = IHI - ILO + 1
NS = 2
IF( NH.GE.30 )
$ NS = 4
IF( NH.GE.60 )
$ NS = 10
IF( NH.GE.150 )
$ NS = MAX( 10, NH / NINT( LOG( REAL( NH ) ) / LOG( TWO ) ) )
IF( NH.GE.590 )
$ NS = 64
IF( NH.GE.3000 )
$ NS = 128
IF( NH.GE.6000 )
$ NS = 256
NS = MAX( 2, NS-MOD( NS, 2 ) )
END IF
*
IF( ISPEC.EQ.INMIN ) THEN
*
*
* ===== Matrices of order smaller than NMIN get sent
* . to xLAHQR, the classic double shift algorithm.
* . This must be at least 11. ====
*
IPARMQ = NMIN
*
ELSE IF( ISPEC.EQ.INIBL ) THEN
*
* ==== INIBL: skip a multi-shift qr iteration and
* . whenever aggressive early deflation finds
* . at least (NIBBLE*(window size)/100) deflations. ====
*
IPARMQ = NIBBLE
*
ELSE IF( ISPEC.EQ.ISHFTS ) THEN
*
* ==== NSHFTS: The number of simultaneous shifts =====
*
IPARMQ = NS
*
ELSE IF( ISPEC.EQ.INWIN ) THEN
*
* ==== NW: deflation window size. ====
*
IF( NH.LE.KNWSWP ) THEN
IPARMQ = NS
ELSE
IPARMQ = 3*NS / 2
END IF
*
ELSE IF( ISPEC.EQ.IACC22 ) THEN
*
* ==== IACC22: Whether to accumulate reflections
* . before updating the far-from-diagonal elements
* . and whether to use 2-by-2 block structure while
* . doing it. A small amount of work could be saved
* . by making this choice dependent also upon the
* . NH=IHI-ILO+1.
*
*
* Convert NAME to upper case if the first character is lower case.
*
IPARMQ = 0
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
END DO
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
END DO
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
END DO
END IF
END IF
*
IF( SUBNAM( 2:6 ).EQ.'GGHRD' .OR.
$ SUBNAM( 2:6 ).EQ.'GGHD3' ) THEN
IPARMQ = 1
IF( NH.GE.K22MIN )
$ IPARMQ = 2
ELSE IF ( SUBNAM( 4:6 ).EQ.'EXC' ) THEN
IF( NH.GE.KACMIN )
$ IPARMQ = 1
IF( NH.GE.K22MIN )
$ IPARMQ = 2
ELSE IF ( SUBNAM( 2:6 ).EQ.'HSEQR' .OR.
$ SUBNAM( 2:5 ).EQ.'LAQR' ) THEN
IF( NS.GE.KACMIN )
$ IPARMQ = 1
IF( NS.GE.K22MIN )
$ IPARMQ = 2
END IF
*
ELSE IF( ISPEC.EQ.ICOST ) THEN
*
* === Relative cost of near-the-diagonal chase vs
* BLAS updates ===
*
IPARMQ = RCOST
ELSE
* ===== invalid value of ispec =====
IPARMQ = -1
*
END IF
*
* ==== End of IPARMQ ====
*
END
*> \brief \b DROUNDUP_LWORK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DROUNDUP_LWORK( LWORK )
*
* .. Scalar Arguments ..
* INTEGER LWORK
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DROUNDUP_LWORK deals with a subtle bug with returning LWORK as a Float.
*> This routine guarantees it is rounded up instead of down by
*> multiplying LWORK by 1+eps when it is necessary, where eps is the relative machine precision.
*> E.g.,
*>
*> float( 9007199254740993 ) == 9007199254740992
*> float( 9007199254740993 ) * (1.+eps) == 9007199254740994
*>
*> \return DROUNDUP_LWORK
*> \verbatim
*> DROUNDUP_LWORK >= LWORK.
*> DROUNDUP_LWORK is guaranteed to have zero decimal part.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] LWORK Workspace size.
*
* Authors:
* ========
*
*> \author Weslley Pereira, University of Colorado Denver, USA
*
*> \ingroup roundup_lwork
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> This routine was inspired in the method `magma_zmake_lwork` from MAGMA.
*> \see https://bitbucket.org/icl/magma/src/master/control/magma_zauxiliary.cpp
*> \endverbatim
*
* =====================================================================
DOUBLE PRECISION FUNCTION DROUNDUP_LWORK( LWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER LWORK
* ..
*
* =====================================================================
* ..
* .. Intrinsic Functions ..
INTRINSIC EPSILON, DBLE, INT
* ..
* .. Executable Statements ..
* ..
DROUNDUP_LWORK = DBLE( LWORK )
*
IF( INT( DROUNDUP_LWORK ) .LT. LWORK ) THEN
* Force round up of LWORK
DROUNDUP_LWORK = DROUNDUP_LWORK *
$ ( 1.0D+0 + EPSILON(0.0D+0) )
ENDIF
*
RETURN
*
* End of DROUNDUP_LWORK
*
END