The R Project SVN R

Rev

Rev 17290 | Blame | Compare with Previous | Last modification | View Log | Download | RSS feed 

      SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
     $                   LDU, C, LDC, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  DBDSQR computes the singular value decomposition (SVD) of a real
*  N-by-N (upper or lower) bidiagonal matrix B:  B = Q * S * P' (P'
*  denotes the transpose of P), where S is a diagonal matrix with
*  non-negative diagonal elements (the singular values of B), and Q
*  and P are orthogonal matrices.
*
*  The routine computes S, and optionally computes U * Q, P' * VT,
*  or Q' * C, for given real input matrices U, VT, and 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.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  B is upper bidiagonal;
*          = 'L':  B is lower bidiagonal.
*
*  N       (input) INTEGER
*          The order of the matrix B.  N >= 0.
*
*  NCVT    (input) INTEGER
*          The number of columns of the matrix VT. NCVT >= 0.
*
*  NRU     (input) INTEGER
*          The number of rows of the matrix U. NRU >= 0.
*
*  NCC     (input) INTEGER
*          The number of columns of the matrix C. NCC >= 0.
*
*  D       (input/output) 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.
*
*  E       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the elements of E contain the
*          offdiagonal elements of the bidiagonal matrix whose SVD
*          is desired. On normal exit (INFO = 0), E is destroyed.
*          If the algorithm does not converge (INFO > 0), D and E
*          will contain the diagonal and superdiagonal elements of a
*          bidiagonal matrix orthogonally equivalent to the one given
*          as input. E(N) is used for workspace.
*
*  VT      (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT)
*          On entry, an N-by-NCVT matrix VT.
*          On exit, VT is overwritten by P' * VT.
*          VT is not referenced if NCVT = 0.
*
*  LDVT    (input) INTEGER
*          The leading dimension of the array VT.
*          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
*
*  U       (input/output) DOUBLE PRECISION array, dimension (LDU, N)
*          On entry, an NRU-by-N matrix U.
*          On exit, U is overwritten by U * Q.
*          U is not referenced if NRU = 0.
*
*  LDU     (input) INTEGER
*          The leading dimension of the array U.  LDU >= max(1,NRU).
*
*  C       (input/output) DOUBLE PRECISION array, dimension (LDC, NCC)
*          On entry, an N-by-NCC matrix C.
*          On exit, C is overwritten by Q' * C.
*          C is not referenced if NCC = 0.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C.
*          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension
*            4*N  if only singular values wanted (NCVT = NRU = NCC = 0)
*            max( 1, 4*N-4 ) otherwise
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  If INFO = -i, the i-th argument had an illegal value
*          > 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.
*
*  Internal Parameters
*  ===================
*
*  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.
*
*  =====================================================================
*
*     .. 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, J, LL, LLL, M, MAXIT, 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, SMINL, SMINLO, 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 )
         RETURN
      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
      SMINL = 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.)
*
      MAXIT = MAXITR*N*N
      ITER = 0
      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.GT.MAXIT )
     $   GO TO 200
*
*     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 ) )
      SMIN = SMAX
      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
         SMIN = MIN( SMIN, ABSS )
         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 ) )
            SMINL = MU
            DO 100 LLL = LL, M - 1
               IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
                  E( LLL ) = ZERO
                  GO TO 60
               END IF
               SMINLO = SMINL
               MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
               SMINL = MIN( SMINL, 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 ) )
            SMINL = MU
            DO 110 LLL = M - 1, LL, -1
               IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
                  E( LLL ) = ZERO
                  GO TO 60
               END IF
               SMINLO = SMINL
               MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
               SMINL = MIN( SMINL, 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*( SMINL / 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 ).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
      SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
     $                   INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      CHARACTER          JOB, SIDE
      INTEGER            IHI, ILO, INFO, LDV, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   SCALE( * ), V( LDV, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  JOB     (input) 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.
*
*  SIDE    (input) CHARACTER*1
*          = 'R':  V contains right eigenvectors;
*          = 'L':  V contains left eigenvectors.
*
*  N       (input) INTEGER
*          The number of rows of the matrix V.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) INTEGER
*          The integers ILO and IHI determined by DGEBAL.
*          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
*  SCALE   (input) DOUBLE PRECISION array, dimension (N)
*          Details of the permutation and scaling factors, as returned
*          by DGEBAL.
*
*  M       (input) INTEGER
*          The number of columns of the matrix V.  M >= 0.
*
*  V       (input/output) 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.
*
*  LDV     (input) INTEGER
*          The leading dimension of the array V. LDV >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  =====================================================================
*
*     .. 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 = 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 = 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
      SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          JOB
      INTEGER            IHI, ILO, INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), SCALE( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  JOB     (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  ILO     (output) INTEGER
*  IHI     (output) 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.
*
*  SCALE   (output) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  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
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   SCLFAC
      PARAMETER          ( SCLFAC = 0.8D+1 )
      DOUBLE PRECISION   FACTOR
      PARAMETER          ( FACTOR = 0.95D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOCONV
      INTEGER            I, ICA, IEXC, IRA, J, K, L, M
      DOUBLE PRECISION   C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
     $                   SFMIN2
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, IDAMAX, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. 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
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEBAL', -INFO )
         RETURN
      END IF
*
      K = 1
      L = N
*
      IF( N.EQ.0 )
     $   GO TO 210
*
      IF( LSAME( JOB, 'N' ) ) THEN
         DO 10 I = 1, N
            SCALE( I ) = ONE
   10    CONTINUE
         GO TO 210
      END IF
*
      IF( LSAME( JOB, 'S' ) )
     $   GO TO 120
*
*     Permutation to isolate eigenvalues if possible
*
      GO TO 50
*
*     Row and column exchange.
*
   20 CONTINUE
      SCALE( M ) = J
      IF( J.EQ.M )
     $   GO TO 30
*
      CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
      CALL DSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA )
*
   30 CONTINUE
      GO TO ( 40, 80 )IEXC
*
*     Search for rows isolating an eigenvalue and push them down.
*
   40 CONTINUE
      IF( L.EQ.1 )
     $   GO TO 210
      L = L - 1
*
   50 CONTINUE
      DO 70 J = L, 1, -1
*
         DO 60 I = 1, L
            IF( I.EQ.J )
     $         GO TO 60
            IF( A( J, I ).NE.ZERO )
     $         GO TO 70
   60    CONTINUE
*
         M = L
         IEXC = 1
         GO TO 20
   70 CONTINUE
*
      GO TO 90
*
*     Search for columns isolating an eigenvalue and push them left.
*
   80 CONTINUE
      K = K + 1
*
   90 CONTINUE
      DO 110 J = K, L
*
         DO 100 I = K, L
            IF( I.EQ.J )
     $         GO TO 100
            IF( A( I, J ).NE.ZERO )
     $         GO TO 110
  100    CONTINUE
*
         M = K
         IEXC = 2
         GO TO 20
  110 CONTINUE
*
  120 CONTINUE
      DO 130 I = K, L
         SCALE( I ) = ONE
  130 CONTINUE
*
      IF( LSAME( JOB, 'P' ) )
     $   GO TO 210
*
*     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
  140 CONTINUE
      NOCONV = .FALSE.
*
      DO 200 I = K, L
         C = ZERO
         R = ZERO
*
         DO 150 J = K, L
            IF( J.EQ.I )
     $         GO TO 150
            C = C + ABS( A( J, I ) )
            R = R + ABS( A( I, J ) )
  150    CONTINUE
         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 )
     $      GO TO 200
         G = R / SCLFAC
         F = ONE
         S = C + R
  160    CONTINUE
         IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR.
     $       MIN( R, G, RA ).LE.SFMIN2 )GO TO 170
         F = F*SCLFAC
         C = C*SCLFAC
         CA = CA*SCLFAC
         R = R / SCLFAC
         G = G / SCLFAC
         RA = RA / SCLFAC
         GO TO 160
*
  170    CONTINUE
         G = C / SCLFAC
  180    CONTINUE
         IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR.
     $       MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190
         F = F / SCLFAC
         C = C / SCLFAC
         G = G / SCLFAC
         CA = CA / SCLFAC
         R = R*SCLFAC
         RA = RA*SCLFAC
         GO TO 180
*
*        Now balance.
*
  190    CONTINUE
         IF( ( C+R ).GE.FACTOR*S )
     $      GO TO 200
         IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
            IF( F*SCALE( I ).LE.SFMIN1 )
     $         GO TO 200
         END IF
         IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
            IF( SCALE( I ).GE.SFMAX1 / F )
     $         GO TO 200
         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 )
*
  200 CONTINUE
*
      IF( NOCONV )
     $   GO TO 140
*
  210 CONTINUE
      ILO = K
      IHI = L
*
      RETURN
*
*     End of DGEBAL
*
      END
      SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   TAUQ( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEBD2 reduces a real general m by n matrix A to upper or lower
*  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.
*
*  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows in the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns in the matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The diagonal elements of the bidiagonal matrix B:
*          D(i) = A(i,i).
*
*  E       (output) 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.
*
*  TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix Q. See Further Details.
*
*  TAUP    (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix P. See Further Details.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (max(M,N))
*
*  INFO    (output) INTEGER
*          = 0: successful exit.
*          < 0: if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  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'  and G(i) = I - taup * u * u'
*
*  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'  and G(i) = I - taup * u * u'
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, 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 )
            A( I, I ) = ONE
*
*           Apply H(i) to A(i:m,i+1:n) from the left
*
            CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ),
     $                  A( I, I+1 ), LDA, WORK )
            A( I, I ) = D( I )
*
            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 )
               A( I, I+1 ) = ONE
*
*              Apply G(i) to A(i+1:m,i+1:n) from the right
*
               CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
     $                     TAUP( I ), A( I+1, I+1 ), LDA, WORK )
               A( I, I+1 ) = E( I )
            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 )
            A( I, I ) = ONE
*
*           Apply G(i) to A(i+1:m,i:n) from the right
*
            CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, TAUP( I ),
     $                  A( MIN( I+1, M ), I ), LDA, WORK )
            A( I, I ) = D( I )
*
            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 )
               A( I+1, I ) = ONE
*
*              Apply H(i) to A(i+1:m,i+1:n) from the left
*
               CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ),
     $                     A( I+1, I+1 ), LDA, WORK )
               A( I+1, I ) = E( I )
            ELSE
               TAUQ( I ) = ZERO
            END IF
   20    CONTINUE
      END IF
      RETURN
*
*     End of DGEBD2
*
      END
      SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
     $                   INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   TAUQ( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows in the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns in the matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The diagonal elements of the bidiagonal matrix B:
*          D(i) = A(i,i).
*
*  E       (output) 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.
*
*  TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix Q. See Further Details.
*
*  TAUP    (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix P. See Further Details.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The length of the array WORK.  LWORK >= max(1,M,N).
*          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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  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'  and G(i) = I - taup * u * u'
*
*  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'  and G(i) = I - taup * u * u'
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
     $                   NBMIN, NX
      DOUBLE PRECISION   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
      NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
      LWKOPT = ( M+N )*NB
      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.MAX( 1, M, N ) .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
*
      MINMN = MIN( M, N )
      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 = ( M+N )*NB
            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' - X*U'
*
         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
      SUBROUTINE DGECON( NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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)) ).
*
*  Arguments
*  =========
*
*  NORM    (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
*          The factors L and U from the factorization A = P*L*U
*          as computed by DGETRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  ANORM   (input) 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.
*
*  RCOND   (output) DOUBLE PRECISION
*          The reciprocal of the condition number of the matrix A,
*          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)
*
*  IWORK   (workspace) INTEGER array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, IDAMAX, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACON, DLATRS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. 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( 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
      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 DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE )
      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').
*
            CALL DLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N, A,
     $                   LDA, WORK, SU, WORK( 3*N+1 ), INFO )
*
*           Multiply by inv(L').
*
            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 )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
   20 CONTINUE
      RETURN
*
*     End of DGECON
*
      END
      SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
     $                  VS, LDVS, WORK, LWORK, BWORK, INFO )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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
*     ..
*
*  Purpose
*  =======
*
*  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).
*
*  Arguments
*  =========
*
*  JOBVS   (input) CHARACTER*1
*          = 'N': Schur vectors are not computed;
*          = 'V': Schur vectors are computed.
*
*  SORT    (input) 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).
*
*  SELECT  (input) 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).
*
*  N       (input) INTEGER
*          The order of the matrix A. N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  SDIM    (output) 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.)
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) 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.
*
*  VS      (output) 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.
*
*  LDVS    (input) INTEGER
*          The leading dimension of the array VS.  LDVS >= 1; if
*          JOBVS = 'V', LDVS >= N.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  BWORK   (workspace) LOGICAL array, dimension (N)
*          Not referenced if SORT = 'N'.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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, K, MAXB,
     $                   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, MIN, 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.)
*
      MINWRK = 1
      IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN
         MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
         MINWRK = MAX( 1, 3*N )
         IF( .NOT.WANTVS ) THEN
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 )
            K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SN', N, 1,
     $          N, -1 ) ) )
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, N+HSWORK, 1 )
         ELSE
            MAXWRK = MAX( MAXWRK, 2*N+( N-1 )*
     $               ILAENV( 1, 'DORGHR', ' ', N, 1, N, -1 ) )
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'EN', N, 1, N, -1 ), 2 )
            K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N, 1,
     $          N, -1 ) ) )
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, N+HSWORK, 1 )
         END IF
         WORK( 1 ) = MAXWRK
      END IF
      IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
         INFO = -13
      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
      CALL DLABAD( SMLNUM, BIGNUM )
      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 )
                     CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
                     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
      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 (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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
*     ..
*
*  Purpose
*  =======
*
*  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).
*
*  Arguments
*  =========
*
*  JOBVS   (input) CHARACTER*1
*          = 'N': Schur vectors are not computed;
*          = 'V': Schur vectors are computed.
*
*  SORT    (input) 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).
*
*  SELECT  (input) 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).
*
*  SENSE   (input) 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'.
*
*  N       (input) INTEGER
*          The order of the matrix A. N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  SDIM    (output) 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.)
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) 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.
*
*  VS      (output) 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.
*
*  LDVS    (input) INTEGER
*          The leading dimension of the array VS.  LDVS >= 1, and if
*          JOBVS = 'V', LDVS >= N.
*
*  RCONDE  (output) 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'.
*
*  RCONDV  (output) 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'.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*          For good performance, LWORK must generally be larger.
*
*  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
*          Not referenced if SENSE = 'N' or 'E'.
*          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
*  LIWORK  (input) INTEGER
*          The dimension of the array IWORK.
*          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
*
*  BWORK   (workspace) LOGICAL array, dimension (N)
*          Not referenced if SORT = 'N'.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, LASTSL, LST2SL, SCALEA, WANTSB, WANTSE,
     $                   WANTSN, WANTST, WANTSV, WANTVS
      INTEGER            HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
     $                   IHI, ILO, INXT, IP, ITAU, IWRK, K, MAXB,
     $                   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, MIN, 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' )
      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.)
*
      MINWRK = 1
      IF( INFO.EQ.0 .AND. LWORK.GE.1 ) THEN
         MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
         MINWRK = MAX( 1, 3*N )
         IF( .NOT.WANTVS ) THEN
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 )
            K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SN', N, 1,
     $          N, -1 ) ) )
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, N+HSWORK, 1 )
         ELSE
            MAXWRK = MAX( MAXWRK, 2*N+( N-1 )*
     $               ILAENV( 1, 'DORGHR', ' ', N, 1, N, -1 ) )
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SV', N, 1, N, -1 ), 2 )
            K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SV', N, 1,
     $          N, -1 ) ) )
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, N+HSWORK, 1 )
         END IF
         WORK( 1 ) = MAXWRK
      END IF
      IF( LWORK.LT.MINWRK ) THEN
         INFO = -16
      END IF
      IF( LIWORK.LT.1 ) THEN
         INFO = -18
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEESX', -INFO )
         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
      CALL DLABAD( SMLNUM, BIGNUM )
      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 )
                     CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
                     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 ) = SDIM*( N-SDIM )
      ELSE
         IWORK( 1 ) = 1
      END IF
*
      RETURN
*
*     End of DGEESX
*
      END
      SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
     $                  LDVR, WORK, LWORK, INFO )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  JOBVL   (input) CHARACTER*1
*          = 'N': left eigenvectors of A are not computed;
*          = 'V': left eigenvectors of A are computed.
*
*  JOBVR   (input) CHARACTER*1
*          = 'N': right eigenvectors of A are not computed;
*          = 'V': right eigenvectors of A are computed.
*
*  N       (input) INTEGER
*          The order of the matrix A. N >= 0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the N-by-N matrix A.
*          On exit, A has been overwritten.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) 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.
*
*  VL      (output) 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).
*
*  LDVL    (input) INTEGER
*          The leading dimension of the array VL.  LDVL >= 1; if
*          JOBVL = 'V', LDVL >= N.
*
*  VR      (output) 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).
*
*  LDVR    (input) INTEGER
*          The leading dimension of the array VR.  LDVR >= 1; if
*          JOBVR = 'V', LDVR >= N.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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,
     $                   MAXB, 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, DTREVC, 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, MIN, 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.)
*
      MINWRK = 1
      IF( INFO.EQ.0 ) THEN
c      IF( INFO.EQ.0 .AND. LWORK.GE.1 ) THEN
         MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
         IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
            MINWRK = MAX( 1, 3*N )
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'EN', N, 1, N, -1 ), 2 )
            K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N, 1,
     $          N, -1 ) ) )
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, N+1, N+HSWORK )
         ELSE
            MINWRK = MAX( 1, 4*N )
            MAXWRK = MAX( MAXWRK, 2*N+( N-1 )*
     $               ILAENV( 1, 'DORGHR', ' ', N, 1, N, -1 ) )
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SV', N, 1, N, -1 ), 2 )
            K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SV', N, 1,
     $          N, -1 ) ) )
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, N+1, N+HSWORK )
            MAXWRK = MAX( MAXWRK, 4*N )
         END IF
         WORK( 1 ) = MAXWRK
      END IF
      IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
         INFO = -13
      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
      CALL DLABAD( SMLNUM, BIGNUM )
      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 > 0 from DHSEQR, then quit
*
      IF( INFO.GT.0 )
     $   GO TO 50
*
      IF( WANTVL .OR. WANTVR ) THEN
*
*        Compute left and/or right eigenvectors
*        (Workspace: need 4*N)
*
         CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
     $                N, NOUT, WORK( IWRK ), 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
      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 )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  BALANC  (input) 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.
*
*  JOBVL   (input) 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'.
*
*  JOBVR   (input) 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'.
*
*  SENSE   (input) 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').
*
*  N       (input) INTEGER
*          The order of the matrix A. N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) 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.
*
*  VL      (output) 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).
*
*  LDVL    (input) INTEGER
*          The leading dimension of the array VL.  LDVL >= 1; if
*          JOBVL = 'V', LDVL >= N.
*
*  VR      (output) 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).
*
*  LDVR    (input) INTEGER
*          The leading dimension of the array VR.  LDVR >= 1, and if
*          JOBVR = 'V', LDVR >= N.
*
*  ILO,IHI (output) 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.
*
*  SCALE   (output) 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.
*
*  ABNRM   (output) DOUBLE PRECISION
*          The one-norm of the balanced matrix (the maximum
*          of the sum of absolute values of elements of any column).
*
*  RCONDE  (output) DOUBLE PRECISION array, dimension (N)
*          RCONDE(j) is the reciprocal condition number of the j-th
*          eigenvalue.
*
*  RCONDV  (output) DOUBLE PRECISION array, dimension (N)
*          RCONDV(j) is the reciprocal condition number of the j-th
*          right eigenvector.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  IWORK   (workspace) INTEGER array, dimension (2*N-2)
*          If SENSE = 'N' or 'E', not referenced.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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, MAXB,
     $                   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, DTREVC, 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, MIN, 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.)
*
      MINWRK = 1
      IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN
         MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
         IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
            MINWRK = MAX( 1, 2*N )
            IF( .NOT.WNTSNN )
     $         MINWRK = MAX( MINWRK, N*N+6*N )
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 )
            IF( WNTSNN ) THEN
               K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N,
     $             1, N, -1 ) ) )
            ELSE
               K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SN', N,
     $             1, N, -1 ) ) )
            END IF
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, 1, HSWORK )
            IF( .NOT.WNTSNN )
     $         MAXWRK = MAX( MAXWRK, N*N+6*N )
         ELSE
            MINWRK = MAX( 1, 3*N )
            IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
     $         MINWRK = MAX( MINWRK, N*N+6*N )
            MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 )
            K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N, 1,
     $          N, -1 ) ) )
            HSWORK = MAX( K*( K+2 ), 2*N )
            MAXWRK = MAX( MAXWRK, 1, 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, 1 )
         END IF
         WORK( 1 ) = MAXWRK
      END IF
      IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
         INFO = -21
      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
      CALL DLABAD( SMLNUM, BIGNUM )
      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 > 0 from DHSEQR, then quit
*
      IF( INFO.GT.0 )
     $   GO TO 50
*
      IF( WANTVL .OR. WANTVR ) THEN
*
*        Compute left and/or right eigenvectors
*        (Workspace: need 3*N)
*
         CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
     $                N, NOUT, WORK( IWRK ), 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
      SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
*  an orthogonal similarity transformation:  Q' * A * Q = H .
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) 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).
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   AII
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, 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 ) )
         AII = A( I+1, I )
         A( I+1, I ) = ONE
*
*        Apply H(i) to A(1:ihi,i+1:ihi) from the right
*
         CALL DLARF( '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 DLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
     $               A( I+1, I+1 ), LDA, WORK )
*
         A( I+1, I ) = AII
   10 CONTINUE
*
      RETURN
*
*     End of DGEHD2
*
      END
      SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEHRD reduces a real general matrix A to upper Hessenberg form H by
*  an orthogonal similarity transformation:  Q' * A * Q = H .
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) 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.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  TAU     (output) 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.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The length 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, LDWORK, LWKOPT, NB, NBMIN,
     $                   NH, NX
      DOUBLE PRECISION   EI
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   T( LDT, NBMAX )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEHD2, DGEMM, DLAHRD, DLARFB, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
      LWKOPT = N*NB
      WORK( 1 ) = LWKOPT
      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
      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
*
      NH = IHI - ILO + 1
      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
      IWS = 1
      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.
*
            IWS = N*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.
*
               NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI,
     $                 -1 ) )
               IF( LWORK.GE.N*NBMIN ) THEN
                  NB = LWORK / 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
*
         DO 30 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'
*           which performs the reduction, and also the matrix Y = A*V*T
*
            CALL DLAHRD( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
     $                   WORK, LDWORK )
*
*           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
*           right, computing  A := A - Y * V'. 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(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, T, LDT,
     $                   A( I+1, I+IB ), LDA, WORK, LDWORK )
   30    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 ) = IWS
*
      RETURN
*
*     End of DGEHRD
*
      END
      SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGELQ2 computes an LQ factorization of a real m by n matrix A:
*  A = L * Q.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  A       (input/output) 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).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (M)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*
*  Further Details
*  ===============
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
      DOUBLE PRECISION   AII
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, 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
*
            AII = A( I, I )
            A( I, I ) = ONE
            CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, TAU( I ),
     $                  A( I+1, I ), LDA, WORK )
            A( I, I ) = AII
         END IF
   10 CONTINUE
      RETURN
*
*     End of DGELQ2
*
      END
      SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGELQF computes an LQ factorization of a real M-by-N matrix A:
*  A = L * Q.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  A       (input/output) 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).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  Further Details
*  ===============
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. 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
      NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
      LWKOPT = M*NB
      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.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELQF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      K = MIN( M, N )
      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
      SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEQR2 computes a QR factorization of a real m by n matrix A:
*  A = Q * R.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  A       (input/output) 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).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*
*  Further Details
*  ===============
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
      DOUBLE PRECISION   AII
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, 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
*
            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 )
            A( I, I ) = AII
         END IF
   10 CONTINUE
      RETURN
*
*     End of DGEQR2
*
      END
      SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEQRF computes a QR factorization of a real M-by-N matrix A:
*  A = Q * R.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  A       (input/output) 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).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  Further Details
*  ===============
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. 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
*
      INFO = 0
      NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
      LWKOPT = N*NB
      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.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQRF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      K = MIN( M, N )
      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' 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
      SUBROUTINE DGESC2( N, A, LDA, RHS, IPIV, JPIV, SCALE )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            LDA, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), JPIV( * )
      DOUBLE PRECISION   A( LDA, * ), RHS( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.
*
*  A       (input) 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
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1, N).
*
*  RHS     (input/output) DOUBLE PRECISION array, dimension (N).
*          On entry, the right hand side vector b.
*          On exit, the solution vector X.
*
*  IPIV    (iput) INTEGER array, dimension (N).
*          The pivot indices; for 1 <= i <= N, row i of the
*          matrix has been interchanged with row IPIV(i).
*
*  JPIV    (iput) INTEGER array, dimension (N).
*          The pivot indices; for 1 <= j <= N, column j of the
*          matrix has been interchanged with column JPIV(j).
*
*  SCALE    (output) DOUBLE PRECISION
*           On exit, SCALE contains the scale factor. SCALE is chosen
*           0 <= SCALE <= 1 to prevent owerflow in the solution.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*     Umea University, S-901 87 Umea, Sweden.
*
*  =====================================================================
*
*     .. 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 owerflow
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      BIGNUM = ONE / SMLNUM
      CALL DLABAD( SMLNUM, BIGNUM )
*
*     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
      SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  JOBU    (input) 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.
*
*  JOBVT   (input) 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'.
*
*  M       (input) INTEGER
*          The number of rows of the input matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the input matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The singular values of A, sorted so that S(i) >= S(i+1).
*
*  U       (output) 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.
*
*  LDU     (input) INTEGER
*          The leading dimension of the array U.  LDU >= 1; if
*          JOBU = 'S' or 'A', LDU >= M.
*
*  VT      (output) 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.
*
*  LDVT    (input) INTEGER
*          The leading dimension of the array VT.  LDVT >= 1; if
*          JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (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.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK. LWORK >= 1.
*          LWORK >= MAX(3*MIN(M,N)+MAX(M,N),5*MIN(M,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.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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
      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 )
      MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
      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' )
      MINWRK = 1
      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 .AND. ( LWORK.GE.1 .OR. LQUERY ) .AND. M.GT.0 .AND.
     $    N.GT.0 ) THEN
         IF( M.GE.N ) THEN
*
*           Compute space needed for DBDSQR
*
            BDSPAC = 5*N
            IF( M.GE.MNTHR ) THEN
               IF( WNTUN ) THEN
*
*                 Path 1 (M much larger than N, JOBU='N')
*
                  MAXWRK = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1,
     $                     -1 )
                  MAXWRK = MAX( MAXWRK, 3*N+2*N*
     $                     ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  IF( WNTVO .OR. WNTVAS )
     $               MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
     $                        ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
                  MAXWRK = MAX( MAXWRK, BDSPAC )
                  MINWRK = MAX( 4*N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUO .AND. WNTVN ) THEN
*
*                 Path 2 (M much larger than N, JOBU='O', JOBVT='N')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N )
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
*                 Path 3 (M much larger than N, JOBU='O', JOBVT='S' or
*                 'A')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+( N-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N )
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUS .AND. WNTVN ) THEN
*
*                 Path 4 (M much larger than N, JOBU='S', JOBVT='N')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUS .AND. WNTVO ) THEN
*
*                 Path 5 (M much larger than N, JOBU='S', JOBVT='O')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+( N-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*N*N + WRKBL
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUS .AND. WNTVAS ) THEN
*
*                 Path 6 (M much larger than N, JOBU='S', JOBVT='S' or
*                 'A')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+( N-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUA .AND. WNTVN ) THEN
*
*                 Path 7 (M much larger than N, JOBU='A', JOBVT='N')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
     $                    M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUA .AND. WNTVO ) THEN
*
*                 Path 8 (M much larger than N, JOBU='A', JOBVT='O')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
     $                    M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+( N-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*N*N + WRKBL
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTUA .AND. WNTVAS ) THEN
*
*                 Path 9 (M much larger than N, JOBU='A', JOBVT='S' or
*                 'A')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
     $                    M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+( N-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N+M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               END IF
            ELSE
*
*              Path 10 (M at least N, but not much larger)
*
               MAXWRK = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N,
     $                  -1, -1 )
               IF( WNTUS .OR. WNTUO )
     $            MAXWRK = MAX( MAXWRK, 3*N+N*
     $                     ILAENV( 1, 'DORGBR', 'Q', M, N, N, -1 ) )
               IF( WNTUA )
     $            MAXWRK = MAX( MAXWRK, 3*N+M*
     $                     ILAENV( 1, 'DORGBR', 'Q', M, M, N, -1 ) )
               IF( .NOT.WNTVN )
     $            MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
     $                     ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
               MAXWRK = MAX( MAXWRK, BDSPAC )
               MINWRK = MAX( 3*N+M, BDSPAC )
               MAXWRK = MAX( MAXWRK, MINWRK )
            END IF
         ELSE
*
*           Compute space needed for DBDSQR
*
            BDSPAC = 5*M
            IF( N.GE.MNTHR ) THEN
               IF( WNTVN ) THEN
*
*                 Path 1t(N much larger than M, JOBVT='N')
*
                  MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
     $                     -1 )
                  MAXWRK = MAX( MAXWRK, 3*M+2*M*
     $                     ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  IF( WNTUO .OR. WNTUAS )
     $               MAXWRK = MAX( MAXWRK, 3*M+M*
     $                        ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, BDSPAC )
                  MINWRK = MAX( 4*M, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVO .AND. WNTUN ) THEN
*
*                 Path 2t(N much larger than M, JOBU='N', JOBVT='O')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M )
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
*                 Path 3t(N much larger than M, JOBU='S' or 'A',
*                 JOBVT='O')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M )
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVS .AND. WNTUN ) THEN
*
*                 Path 4t(N much larger than M, JOBU='N', JOBVT='S')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVS .AND. WNTUO ) THEN
*
*                 Path 5t(N much larger than M, JOBU='O', JOBVT='S')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*M*M + WRKBL
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVS .AND. WNTUAS ) THEN
*
*                 Path 6t(N much larger than M, JOBU='S' or 'A',
*                 JOBVT='S')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVA .AND. WNTUN ) THEN
*
*                 Path 7t(N much larger than M, JOBU='N', JOBVT='A')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVA .AND. WNTUO ) THEN
*
*                 Path 8t(N much larger than M, JOBU='O', JOBVT='A')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*M*M + WRKBL
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               ELSE IF( WNTVA .AND. WNTUAS ) THEN
*
*                 Path 9t(N much larger than M, JOBU='S' or 'A',
*                 JOBVT='A')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+( M-1 )*
     $                    ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M+N, BDSPAC )
                  MAXWRK = MAX( MAXWRK, MINWRK )
               END IF
            ELSE
*
*              Path 10t(N greater than M, but not much larger)
*
               MAXWRK = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N,
     $                  -1, -1 )
               IF( WNTVS .OR. WNTVO )
     $            MAXWRK = MAX( MAXWRK, 3*M+M*
     $                     ILAENV( 1, 'DORGBR', 'P', M, N, M, -1 ) )
               IF( WNTVA )
     $            MAXWRK = MAX( MAXWRK, 3*M+N*
     $                     ILAENV( 1, 'DORGBR', 'P', N, N, M, -1 ) )
               IF( .NOT.WNTUN )
     $            MAXWRK = MAX( MAXWRK, 3*M+( M-1 )*
     $                     ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
               MAXWRK = MAX( MAXWRK, BDSPAC )
               MINWRK = MAX( 3*M+N, BDSPAC )
               MAXWRK = MAX( MAXWRK, MINWRK )
            END IF
         END IF
         WORK( 1 ) = MAXWRK
      END IF
*
      IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
         INFO = -13
      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
         IF( LWORK.GE.1 )
     $      WORK( 1 ) = ONE
         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
*
               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
               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 )
                  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 )
                  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
*
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                            LDA )
*
*                    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
*
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                            LDA )
*
*                    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 )
                     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
*
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                            LDA )
*
*                    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
*
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                            LDA )
*
*                    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 )
                     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
      SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
     $                   LDZ, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ, JOB
      INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
     $                   Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  DHSEQR computes the eigenvalues of a real upper 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.
*
*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          = 'E':  compute eigenvalues only;
*          = 'S':  compute eigenvalues and the Schur form T.
*
*  COMPZ   (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix H.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) 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 SGEHRD
*          when the matrix output by DGEBAL is reduced to Hessenberg
*          form. Otherwise ILO and IHI should be set to 1 and N
*          respectively.
*          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
*  H       (input/output) DOUBLE PRECISION array, dimension (LDH,N)
*          On entry, the upper Hessenberg matrix H.
*          On exit, if JOB = 'S', 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 JOB = 'E',
*          the contents of H are unspecified on exit.
*
*  LDH     (input) INTEGER
*          The leading dimension of the array H. LDH >= max(1,N).
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) 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).
*
*  Z       (input/output) 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, 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 Z contains Q*Z.
*          Normally Q is the orthogonal matrix generated by DORGHR after
*          the call to DGEHRD which formed the Hessenberg matrix H.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.
*          LDZ >= max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 otherwise.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) 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 in a total of 30*(IHI-ILO+1) iterations;
*                elements 1:ilo-1 and i+1:n of WR and WI contain those
*                eigenvalues which have been successfully computed.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
      DOUBLE PRECISION   CONST
      PARAMETER          ( CONST = 1.5D+0 )
      INTEGER            NSMAX, LDS
      PARAMETER          ( NSMAX = 15, LDS = NSMAX )
*     ..
*     .. Local Scalars ..
      LOGICAL            INITZ, LQUERY, WANTT, WANTZ
      INTEGER            I, I1, I2, IERR, II, ITEMP, ITN, ITS, J, K, L,
     $                   MAXB, NH, NR, NS, NV
      DOUBLE PRECISION   ABSW, OVFL, SMLNUM, TAU, TEMP, TST1, ULP, UNFL
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   S( LDS, NSMAX ), V( NSMAX+1 ), VV( NSMAX+1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX, ILAENV
      DOUBLE PRECISION   DLAMCH, DLANHS, DLAPY2
      EXTERNAL           LSAME, IDAMAX, ILAENV, DLAMCH, DLANHS, DLAPY2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMV, DLACPY, DLAHQR, DLARFG, DLARFX,
     $                   DLASET, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      WANTT = LSAME( JOB, 'S' )
      INITZ = LSAME( COMPZ, 'I' )
      WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
*
      INFO = 0
      WORK( 1 ) = MAX( 1, N )
      LQUERY = ( LWORK.EQ.-1 )
      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
         CALL XERBLA( 'DHSEQR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Initialize Z, if necessary
*
      IF( INITZ )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
*     Store the 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
*
*     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
*
*     Set rows and columns ILO to IHI to zero below the first
*     subdiagonal.
*
      DO 40 J = ILO, IHI - 2
         DO 30 I = J + 2, N
            H( I, J ) = ZERO
   30    CONTINUE
   40 CONTINUE
      NH = IHI - ILO + 1
*
*     Determine the order of the multi-shift QR algorithm to be used.
*
      NS = ILAENV( 4, 'DHSEQR', JOB // COMPZ, N, ILO, IHI, -1 )
      MAXB = ILAENV( 8, 'DHSEQR', JOB // COMPZ, N, ILO, IHI, -1 )
      IF( NS.LE.2 .OR. NS.GT.NH .OR. MAXB.GE.NH ) THEN
*
*        Use the standard double-shift algorithm
*
         CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
     $                IHI, Z, LDZ, INFO )
         RETURN
      END IF
      MAXB = MAX( 3, MAXB )
      NS = MIN( NS, MAXB, NSMAX )
*
*     Now 2 < NS <= MAXB < NH.
*
*     Set machine-dependent constants for the stopping criterion.
*     If norm(H) <= sqrt(OVFL), overflow should not occur.
*
      UNFL = DLAMCH( 'Safe minimum' )
      OVFL = ONE / UNFL
      CALL DLABAD( UNFL, OVFL )
      ULP = DLAMCH( 'Precision' )
      SMLNUM = UNFL*( 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
*
*     ITN is the total number of multiple-shift QR iterations allowed.
*
      ITN = 30*NH
*
*     The main loop begins here. I is the loop index and decreases from
*     IHI to ILO in steps of at most MAXB. 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
   50 CONTINUE
      L = ILO
      IF( I.LT.ILO )
     $   GO TO 170
*
*     Perform multiple-shift QR iterations on rows and columns ILO to I
*     until a submatrix of order at most MAXB splits off at the bottom
*     because a subdiagonal element has become negligible.
*
      DO 150 ITS = 0, ITN
*
*        Look for a single small subdiagonal element.
*
         DO 60 K = I, L + 1, -1
            TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
            IF( TST1.EQ.ZERO )
     $         TST1 = DLANHS( '1', I-L+1, H( L, L ), LDH, WORK )
            IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) )
     $         GO TO 70
   60    CONTINUE
   70    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 <= MAXB has split off.
*
         IF( L.GE.I-MAXB+1 )
     $      GO TO 160
*
*        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( ITS.EQ.20 .OR. ITS.EQ.30 ) THEN
*
*           Exceptional shifts.
*
            DO 80 II = I - NS + 1, I
               WR( II ) = CONST*( ABS( H( II, II-1 ) )+
     $                    ABS( H( II, II ) ) )
               WI( II ) = ZERO
   80       CONTINUE
         ELSE
*
*           Use eigenvalues of trailing submatrix of order NS as shifts.
*
            CALL DLACPY( 'Full', NS, NS, H( I-NS+1, I-NS+1 ), LDH, S,
     $                   LDS )
            CALL DLAHQR( .FALSE., .FALSE., NS, 1, NS, S, LDS,
     $                   WR( I-NS+1 ), WI( I-NS+1 ), 1, NS, Z, LDZ,
     $                   IERR )
            IF( IERR.GT.0 ) THEN
*
*              If DLAHQR failed to compute all NS eigenvalues, use the
*              unconverged diagonal elements as the remaining shifts.
*
               DO 90 II = 1, IERR
                  WR( I-NS+II ) = S( II, II )
                  WI( I-NS+II ) = ZERO
   90          CONTINUE
            END IF
         END IF
*
*        Form the first column of (G-w(1)) (G-w(2)) . . . (G-w(ns))
*        where G is the Hessenberg submatrix H(L:I,L:I) and w is
*        the vector of shifts (stored in WR and WI). The result is
*        stored in the local array V.
*
         V( 1 ) = ONE
         DO 100 II = 2, NS + 1
            V( II ) = ZERO
  100    CONTINUE
         NV = 1
         DO 120 J = I - NS + 1, I
            IF( WI( J ).GE.ZERO ) THEN
               IF( WI( J ).EQ.ZERO ) THEN
*
*                 real shift
*
                  CALL DCOPY( NV+1, V, 1, VV, 1 )
                  CALL DGEMV( 'No transpose', NV+1, NV, ONE, H( L, L ),
     $                        LDH, VV, 1, -WR( J ), V, 1 )
                  NV = NV + 1
               ELSE IF( WI( J ).GT.ZERO ) THEN
*
*                 complex conjugate pair of shifts
*
                  CALL DCOPY( NV+1, V, 1, VV, 1 )
                  CALL DGEMV( 'No transpose', NV+1, NV, ONE, H( L, L ),
     $                        LDH, V, 1, -TWO*WR( J ), VV, 1 )
                  ITEMP = IDAMAX( NV+1, VV, 1 )
                  TEMP = ONE / MAX( ABS( VV( ITEMP ) ), SMLNUM )
                  CALL DSCAL( NV+1, TEMP, VV, 1 )
                  ABSW = DLAPY2( WR( J ), WI( J ) )
                  TEMP = ( TEMP*ABSW )*ABSW
                  CALL DGEMV( 'No transpose', NV+2, NV+1, ONE,
     $                        H( L, L ), LDH, VV, 1, TEMP, V, 1 )
                  NV = NV + 2
               END IF
*
*              Scale V(1:NV) so that max(abs(V(i))) = 1. If V is zero,
*              reset it to the unit vector.
*
               ITEMP = IDAMAX( NV, V, 1 )
               TEMP = ABS( V( ITEMP ) )
               IF( TEMP.EQ.ZERO ) THEN
                  V( 1 ) = ONE
                  DO 110 II = 2, NV
                     V( II ) = ZERO
  110             CONTINUE
               ELSE
                  TEMP = MAX( TEMP, SMLNUM )
                  CALL DSCAL( NV, ONE / TEMP, V, 1 )
               END IF
            END IF
  120    CONTINUE
*
*        Multiple-shift QR step
*
         DO 140 K = L, 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( NS+1, I-K+1 )
            IF( K.GT.L )
     $         CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 )
            CALL DLARFG( NR, V( 1 ), V( 2 ), 1, TAU )
            IF( K.GT.L ) THEN
               H( K, K-1 ) = V( 1 )
               DO 130 II = K + 1, I
                  H( II, K-1 ) = ZERO
  130          CONTINUE
            END IF
            V( 1 ) = ONE
*
*           Apply G from the left to transform the rows of the matrix in
*           columns K to I2.
*
            CALL DLARFX( 'Left', NR, I2-K+1, V, TAU, H( K, K ), LDH,
     $                   WORK )
*
*           Apply G from the right to transform the columns of the
*           matrix in rows I1 to min(K+NR,I).
*
            CALL DLARFX( 'Right', MIN( K+NR, I )-I1+1, NR, V, TAU,
     $                   H( I1, K ), LDH, WORK )
*
            IF( WANTZ ) THEN
*
*              Accumulate transformations in the matrix Z
*
               CALL DLARFX( 'Right', NH, NR, V, TAU, Z( ILO, K ), LDZ,
     $                      WORK )
            END IF
  140    CONTINUE
*
  150 CONTINUE
*
*     Failure to converge in remaining number of iterations
*
      INFO = I
      RETURN
*
  160 CONTINUE
*
*     A submatrix of order <= MAXB in rows and columns L to I has split
*     off. Use the double-shift QR algorithm to handle it.
*
      CALL DLAHQR( WANTT, WANTZ, N, L, I, H, LDH, WR, WI, ILO, IHI, Z,
     $             LDZ, INFO )
      IF( INFO.GT.0 )
     $   RETURN
*
*     Decrement number of remaining iterations, and return to start of
*     the main loop with a new value of I.
*
      ITN = ITN - ITS
      I = L - 1
      GO TO 50
*
  170 CONTINUE
      WORK( 1 ) = MAX( 1, N )
      RETURN
*
*     End of DHSEQR
*
      END
      SUBROUTINE DLABAD( SMALL, LARGE )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   LARGE, SMALL
*     ..
*
*  Purpose
*  =======
*
*  DLABAD takes as input the values computed by DLAMCH for underflow and
*  overflow, and returns the square root of each of these values if the
*  log of LARGE is sufficiently large.  This subroutine is intended to
*  identify machines with a large exponent range, such as the Crays, and
*  redefine the underflow and overflow limits to be the square roots of
*  the values computed by DLAMCH.  This subroutine is needed because
*  DLAMCH does not compensate for poor arithmetic in the upper half of
*  the exponent range, as is found on a Cray.
*
*  Arguments
*  =========
*
*  SMALL   (input/output) DOUBLE PRECISION
*          On entry, the underflow threshold as computed by DLAMCH.
*          On exit, if LOG10(LARGE) is sufficiently large, the square
*          root of SMALL, otherwise unchanged.
*
*  LARGE   (input/output) DOUBLE PRECISION
*          On entry, the overflow threshold as computed by DLAMCH.
*          On exit, if LOG10(LARGE) is sufficiently large, the square
*          root of LARGE, otherwise unchanged.
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
     $                   LDY )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LDX, LDY, M, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
*     ..
*
*  Purpose
*  =======
*
*  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' * 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
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows in the matrix A.
*
*  N       (input) INTEGER
*          The number of columns in the matrix A.
*
*  NB      (input) INTEGER
*          The number of leading rows and columns of A to be reduced.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  D       (output) DOUBLE PRECISION array, dimension (NB)
*          The diagonal elements of the first NB rows and columns of
*          the reduced matrix.  D(i) = A(i,i).
*
*  E       (output) DOUBLE PRECISION array, dimension (NB)
*          The off-diagonal elements of the first NB rows and columns of
*          the reduced matrix.
*
*  TAUQ    (output) DOUBLE PRECISION array dimension (NB)
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix Q. See Further Details.
*
*  TAUP    (output) DOUBLE PRECISION array, dimension (NB)
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix P. See Further Details.
*
*  X       (output) DOUBLE PRECISION array, dimension (LDX,NB)
*          The m-by-nb matrix X required to update the unreduced part
*          of A.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X. LDX >= M.
*
*  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
*          The n-by-nb matrix Y required to update the unreduced part
*          of A.
*
*  LDY     (output) INTEGER
*          The leading dimension of the array Y. LDY >= N.
*
*  Further Details
*  ===============
*
*  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'  and G(i) = I - taup * u * u'
*
*  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' 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' - X*U'.
*
*  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).
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLACON( N, V, X, ISGN, EST, KASE )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      INTEGER            KASE, N
      DOUBLE PRECISION   EST
*     ..
*     .. Array Arguments ..
      INTEGER            ISGN( * )
      DOUBLE PRECISION   V( * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  DLACON estimates the 1-norm of a square, real matrix A.
*  Reverse communication is used for evaluating matrix-vector products.
*
*  Arguments
*  =========
*
*  N      (input) INTEGER
*         The order of the matrix.  N >= 1.
*
*  V      (workspace) DOUBLE PRECISION array, dimension (N)
*         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
*         (W is not returned).
*
*  X      (input/output) DOUBLE PRECISION array, dimension (N)
*         On an intermediate return, X should be overwritten by
*               A * X,   if KASE=1,
*               A' * X,  if KASE=2,
*         and DLACON must be re-called with all the other parameters
*         unchanged.
*
*  ISGN   (workspace) INTEGER array, dimension (N)
*
*  EST    (output) DOUBLE PRECISION
*         An estimate (a lower bound) for norm(A).
*
*  KASE   (input/output) 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' * X.
*         On the final return from DLACON, KASE will again be 0.
*
*  Further Details
*  ======= =======
*
*  Contributed by Nick Higham, University of Manchester.
*  Originally named SONEST, dated March 16, 1988.
*
*  Reference: 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.
*
*  =====================================================================
*
*     .. 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 TRANDPOSE(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 TRANDPOSE(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
      SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, LDB, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  Purpose
*  =======
*
*  DLACPY copies all or part of a two-dimensional matrix A to another
*  matrix B.
*
*  Arguments
*  =========
*
*  UPLO    (input) 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
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  A       (input) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  B       (output) DOUBLE PRECISION array, dimension (LDB,N)
*          On exit, B = A in the locations specified by UPLO.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,M).
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLADIV( A, B, C, D, P, Q )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, D, P, Q
*     ..
*
*  Purpose
*  =======
*
*  DLADIV performs complex division in  real arithmetic
*
*                        a + i*b
*             p + i*q = ---------
*                        c + i*d
*
*  The algorithm is due to Robert L. Smith and can be found
*  in D. Knuth, The art of Computer Programming, Vol.2, p.195
*
*  Arguments
*  =========
*
*  A       (input) DOUBLE PRECISION
*  B       (input) DOUBLE PRECISION
*  C       (input) DOUBLE PRECISION
*  D       (input) DOUBLE PRECISION
*          The scalars a, b, c, and d in the above expression.
*
*  P       (output) DOUBLE PRECISION
*  Q       (output) DOUBLE PRECISION
*          The scalars p and q in the above expression.
*
*  =====================================================================
*
*     .. Local Scalars ..
      DOUBLE PRECISION   E, F
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
      IF( ABS( D ).LT.ABS( C ) ) THEN
         E = D / C
         F = C + D*E
         P = ( A+B*E ) / F
         Q = ( B-A*E ) / F
      ELSE
         E = C / D
         F = D + C*E
         P = ( B+A*E ) / F
         Q = ( -A+B*E ) / F
      END IF
*
      RETURN
*
*     End of DLADIV
*
      END
      SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, RT1, RT2
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  A       (input) DOUBLE PRECISION
*          The (1,1) element of the 2-by-2 matrix.
*
*  B       (input) DOUBLE PRECISION
*          The (1,2) and (2,1) elements of the 2-by-2 matrix.
*
*  C       (input) DOUBLE PRECISION
*          The (2,2) element of the 2-by-2 matrix.
*
*  RT1     (output) DOUBLE PRECISION
*          The eigenvalue of larger absolute value.
*
*  RT2     (output) DOUBLE PRECISION
*          The eigenvalue of smaller absolute value.
*
*  Further Details
*  ===============
*
*  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.
*
* =====================================================================
*
*     .. 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
      SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
*     ..
*
*  Purpose
*  =======
*
*  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 ].
*
*  Arguments
*  =========
*
*  A       (input) DOUBLE PRECISION
*          The (1,1) element of the 2-by-2 matrix.
*
*  B       (input) DOUBLE PRECISION
*          The (1,2) element and the conjugate of the (2,1) element of
*          the 2-by-2 matrix.
*
*  C       (input) DOUBLE PRECISION
*          The (2,2) element of the 2-by-2 matrix.
*
*  RT1     (output) DOUBLE PRECISION
*          The eigenvalue of larger absolute value.
*
*  RT2     (output) DOUBLE PRECISION
*          The eigenvalue of smaller absolute value.
*
*  CS1     (output) DOUBLE PRECISION
*  SN1     (output) DOUBLE PRECISION
*          The vector (CS1, SN1) is a unit right eigenvector for RT1.
*
*  Further Details
*  ===============
*
*  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.
*
* =====================================================================
*
*     .. 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
      SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
     $                   INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ
      INTEGER            INFO, J1, LDQ, LDT, N, N1, N2
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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 elemnts equal and its off-diagonal elements of
*  opposite sign.
*
*  Arguments
*  =========
*
*  WANTQ   (input) LOGICAL
*          = .TRUE. : accumulate the transformation in the matrix Q;
*          = .FALSE.: do not accumulate the transformation.
*
*  N       (input) INTEGER
*          The order of the matrix T. N >= 0.
*
*  T       (input/output) 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.
*
*  LDT     (input)  INTEGER
*          The leading dimension of the array T. LDT >= max(1,N).
*
*  Q       (input/output) 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.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.
*          LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.
*
*  J1      (input) INTEGER
*          The index of the first row of the first block T11.
*
*  N1      (input) INTEGER
*          The order of the first block T11. N1 = 0, 1 or 2.
*
*  N2      (input) INTEGER
*          The order of the second block T22. N2 = 0, 1 or 2.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILOZ, IHIZ, Z, LDZ, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      LOGICAL            WANTT, WANTZ
      INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  WANTT   (input) LOGICAL
*          = .TRUE. : the full Schur form T is required;
*          = .FALSE.: only eigenvalues are required.
*
*  WANTZ   (input) LOGICAL
*          = .TRUE. : the matrix of Schur vectors Z is required;
*          = .FALSE.: Schur vectors are not required.
*
*  N       (input) INTEGER
*          The order of the matrix H.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) 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.
*
*  H       (input/output) DOUBLE PRECISION array, dimension (LDH,N)
*          On entry, the upper Hessenberg matrix H.
*          On exit, 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 WANTT is .FALSE., the contents of H are
*          unspecified on exit.
*
*  LDH     (input) INTEGER
*          The leading dimension of the array H. LDH >= max(1,N).
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) 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).
*
*  ILOZ    (input) INTEGER
*  IHIZ    (input) INTEGER
*          Specify the rows of Z to which transformations must be
*          applied if WANTZ is .TRUE..
*          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*
*  Z       (input/output) 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.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z. LDZ >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          > 0: DLAHQR failed to compute all the eigenvalues ILO to IHI
*               in a total of 30*(IHI-ILO+1) iterations; if INFO = i,
*               elements i+1:ihi of WR and WI contain those eigenvalues
*               which have been successfully computed.
*
*  Further Details
*  ===============
*
*  2-96 Based on modifications by
*     David Day, Sandia National Laboratory, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, HALF
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D0 )
      DOUBLE PRECISION   DAT1, DAT2
      PARAMETER          ( DAT1 = 0.75D+0, DAT2 = -0.4375D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, I1, I2, ITN, ITS, J, K, L, M, NH, NR, NZ
      DOUBLE PRECISION   AVE, CS, DISC, H00, H10, H11, H12, H21, H22,
     $                   H33, H33S, H43H34, H44, H44S, OVFL, S, SMLNUM,
     $                   SN, SUM, T1, T2, T3, TST1, ULP, UNFL, V1, V2,
     $                   V3
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   V( 3 ), WORK( 1 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLANHS
      EXTERNAL           DLAMCH, DLANHS
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLANV2, DLARFG, DROT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SIGN, 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
*
      NH = IHI - ILO + 1
      NZ = IHIZ - ILOZ + 1
*
*     Set machine-dependent constants for the stopping criterion.
*     If norm(H) <= sqrt(OVFL), overflow should not occur.
*
      UNFL = DLAMCH( 'Safe minimum' )
      OVFL = ONE / UNFL
      CALL DLABAD( UNFL, OVFL )
      ULP = DLAMCH( 'Precision' )
      SMLNUM = UNFL*( 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
*
*     ITN is the total number of QR iterations allowed.
*
      ITN = 30*NH
*
*     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
   10 CONTINUE
      L = ILO
      IF( I.LT.ILO )
     $   GO TO 150
*
*     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 130 ITS = 0, ITN
*
*        Look for a single small subdiagonal element.
*
         DO 20 K = I, L + 1, -1
            TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
            IF( TST1.EQ.ZERO )
     $         TST1 = DLANHS( '1', I-L+1, H( L, L ), LDH, WORK )
            IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) )
     $         GO TO 30
   20    CONTINUE
   30    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 140
*
*        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( ITS.EQ.10 .OR. ITS.EQ.20 ) THEN
*
*           Exceptional shift.
*
            S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
            H44 = DAT1*S + H( I, I )
            H33 = H44
            H43H34 = DAT2*S*S
         ELSE
*
*           Prepare to use Francis' double shift
*           (i.e. 2nd degree generalized Rayleigh quotient)
*
            H44 = H( I, I )
            H33 = H( I-1, I-1 )
            H43H34 = H( I, I-1 )*H( I-1, I )
            S = H( I-1, I-2 )*H( I-1, I-2 )
            DISC = ( H33-H44 )*HALF
            DISC = DISC*DISC + H43H34
            IF( DISC.GT.ZERO ) THEN
*
*              Real roots: use Wilkinson's shift twice
*
               DISC = SQRT( DISC )
               AVE = HALF*( H33+H44 )
               IF( ABS( H33 )-ABS( H44 ).GT.ZERO ) THEN
                  H33 = H33*H44 - H43H34
                  H44 = H33 / ( SIGN( DISC, AVE )+AVE )
               ELSE
                  H44 = SIGN( DISC, AVE ) + AVE
               END IF
               H33 = H44
               H43H34 = ZERO
            END IF
         END IF
*
*        Look for two consecutive small subdiagonal elements.
*
         DO 40 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.
*
            H11 = H( M, M )
            H22 = H( M+1, M+1 )
            H21 = H( M+1, M )
            H12 = H( M, M+1 )
            H44S = H44 - H11
            H33S = H33 - H11
            V1 = ( H33S*H44S-H43H34 ) / H21 + H12
            V2 = H22 - H11 - H33S - H44S
            V3 = H( M+2, M+1 )
            S = ABS( V1 ) + ABS( V2 ) + ABS( V3 )
            V1 = V1 / S
            V2 = V2 / S
            V3 = V3 / S
            V( 1 ) = V1
            V( 2 ) = V2
            V( 3 ) = V3
            IF( M.EQ.L )
     $         GO TO 50
            H00 = H( M-1, M-1 )
            H10 = H( M, M-1 )
            TST1 = ABS( V1 )*( ABS( H00 )+ABS( H11 )+ABS( H22 ) )
            IF( ABS( H10 )*( ABS( V2 )+ABS( V3 ) ).LE.ULP*TST1 )
     $         GO TO 50
   40    CONTINUE
   50    CONTINUE
*
*        Double-shift QR step
*
         DO 120 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
               H( K, K-1 ) = -H( K, K-1 )
            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 60 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
   60          CONTINUE
*
*              Apply G from the right to transform the columns of the
*              matrix in rows I1 to min(K+3,I).
*
               DO 70 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
   70          CONTINUE
*
               IF( WANTZ ) THEN
*
*                 Accumulate transformations in the matrix Z
*
                  DO 80 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
   80             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 90 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
   90          CONTINUE
*
*              Apply G from the right to transform the columns of the
*              matrix in rows I1 to min(K+3,I).
*
               DO 100 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
  100          CONTINUE
*
               IF( WANTZ ) THEN
*
*                 Accumulate transformations in the matrix Z
*
                  DO 110 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
  110             CONTINUE
               END IF
            END IF
  120    CONTINUE
*
  130 CONTINUE
*
*     Failure to converge in remaining number of iterations
*
      INFO = I
      RETURN
*
  140 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
*
*     Decrement number of remaining iterations, and return to start of
*     the main loop with new value of I.
*
      ITN = ITN - ITS
      I = L - 1
      GO TO 10
*
  150 CONTINUE
      RETURN
*
*     End of DLAHQR
*
      END
      SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            K, LDA, LDT, LDY, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
     $                   Y( LDY, NB )
*     ..
*
*  Purpose
*  =======
*
*  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' * A * Q. The routine returns the matrices V and T which determine
*  Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.
*
*  This is an auxiliary routine called by DGEHRD.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.
*
*  K       (input) INTEGER
*          The offset for the reduction. Elements below the k-th
*          subdiagonal in the first NB columns are reduced to zero.
*
*  NB      (input) INTEGER
*          The number of columns to be reduced.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (NB)
*          The scalar factors of the elementary reflectors. See Further
*          Details.
*
*  T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
*          The upper triangular matrix T.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T.  LDT >= NB.
*
*  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
*          The n-by-nb matrix Y.
*
*  LDY     (input) INTEGER
*          The leading dimension of the array Y. LDY >= N.
*
*  Further Details
*  ===============
*
*  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'
*
*  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') * (A - Y*V').
*
*  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).
*
*  =====================================================================
*
*     .. 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'
*
            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' * V' 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' * 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'*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'*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
      SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
     $                   LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
*  DLALN2 solves a system of the form  (ca A - w D ) X = s B
*  or (ca A' - w D) X = s B   with possible scaling ("s") and
*  perturbation of A.  (A' 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 (.LE. 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.)
*
*  Arguments
*  ==========
*
*  LTRANS  (input) LOGICAL
*          =.TRUE.:  A-transpose will be used.
*          =.FALSE.: A will be used (not transposed.)
*
*  NA      (input) INTEGER
*          The size of the matrix A.  It may (only) be 1 or 2.
*
*  NW      (input) INTEGER
*          1 if "w" is real, 2 if "w" is complex.  It may only be 1
*          or 2.
*
*  SMIN    (input) 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.)
*
*  CA      (input) DOUBLE PRECISION
*          The coefficient c, which A is multiplied by.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,NA)
*          The NA x NA matrix A.
*
*  LDA     (input) INTEGER
*          The leading dimension of A.  It must be at least NA.
*
*  D1      (input) DOUBLE PRECISION
*          The 1,1 element in the diagonal matrix D.
*
*  D2      (input) DOUBLE PRECISION
*          The 2,2 element in the diagonal matrix D.  Not used if NW=1.
*
*  B       (input) 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.
*
*  LDB     (input) INTEGER
*          The leading dimension of B.  It must be at least NA.
*
*  WR      (input) DOUBLE PRECISION
*          The real part of the scalar "w".
*
*  WI      (input) DOUBLE PRECISION
*          The imaginary part of the scalar "w".  Not used if NW=1.
*
*  X       (output) 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.
*
*  LDX     (input) INTEGER
*          The leading dimension of X.  It must be at least NA.
*
*  SCALE   (output) 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.
*
*  XNORM   (output) DOUBLE PRECISION
*          The infinity-norm of X, when X is regarded as an NA x NW
*          real matrix.
*
*  INFO    (output) 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.
*
* =====================================================================
*
*     .. 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' - 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
      DOUBLE PRECISION FUNCTION DLAMCH( CMACH )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          CMACH
*     ..
*
*  Purpose
*  =======
*
*  DLAMCH determines double precision machine parameters.
*
*  Arguments
*  =========
*
*  CMACH   (input) CHARACTER*1
*          Specifies the value to be returned by DLAMCH:
*          = 'E' or 'e',   DLAMCH := eps
*          = 'S' or 's ,   DLAMCH := sfmin
*          = 'B' or 'b',   DLAMCH := base
*          = 'P' or 'p',   DLAMCH := eps*base
*          = 'N' or 'n',   DLAMCH := t
*          = 'R' or 'r',   DLAMCH := rnd
*          = 'M' or 'm',   DLAMCH := emin
*          = 'U' or 'u',   DLAMCH := rmin
*          = 'L' or 'l',   DLAMCH := emax
*          = 'O' or 'o',   DLAMCH := rmax
*
*          where
*
*          eps   = relative machine precision
*          sfmin = safe minimum, such that 1/sfmin does not overflow
*          base  = base of the machine
*          prec  = eps*base
*          t     = number of (base) digits in the mantissa
*          rnd   = 1.0 when rounding occurs in addition, 0.0 otherwise
*          emin  = minimum exponent before (gradual) underflow
*          rmin  = underflow threshold - base**(emin-1)
*          emax  = largest exponent before overflow
*          rmax  = overflow threshold  - (base**emax)*(1-eps)
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            FIRST, LRND
      INTEGER            BETA, IMAX, IMIN, IT
      DOUBLE PRECISION   BASE, EMAX, EMIN, EPS, PREC, RMACH, RMAX, RMIN,
     $                   RND, SFMIN, SMALL, T
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAMC2
*     ..
*     .. Save statement ..
      SAVE               FIRST, EPS, SFMIN, BASE, T, RND, EMIN, RMIN,
     $                   EMAX, RMAX, PREC
*     ..
*     .. Data statements ..
      DATA               FIRST / .TRUE. /
*     ..
*     .. Executable Statements ..
*
      IF( FIRST ) THEN
         FIRST = .FALSE.
         CALL DLAMC2( BETA, IT, LRND, EPS, IMIN, RMIN, IMAX, RMAX )
         BASE = BETA
         T = IT
         IF( LRND ) THEN
            RND = ONE
            EPS = ( BASE**( 1-IT ) ) / 2
         ELSE
            RND = ZERO
            EPS = BASE**( 1-IT )
         END IF
         PREC = EPS*BASE
         EMIN = IMIN
         EMAX = IMAX
         SFMIN = RMIN
         SMALL = ONE / RMAX
         IF( SMALL.GE.SFMIN ) THEN
*
*           Use SMALL plus a bit, to avoid the possibility of rounding
*           causing overflow when computing  1/sfmin.
*
            SFMIN = SMALL*( ONE+EPS )
         END IF
      END IF
*
      IF( LSAME( CMACH, 'E' ) ) THEN
         RMACH = EPS
      ELSE IF( LSAME( CMACH, 'S' ) ) THEN
         RMACH = SFMIN
      ELSE IF( LSAME( CMACH, 'B' ) ) THEN
         RMACH = BASE
      ELSE IF( LSAME( CMACH, 'P' ) ) THEN
         RMACH = PREC
      ELSE IF( LSAME( CMACH, 'N' ) ) THEN
         RMACH = T
      ELSE IF( LSAME( CMACH, 'R' ) ) THEN
         RMACH = RND
      ELSE IF( LSAME( CMACH, 'M' ) ) THEN
         RMACH = EMIN
      ELSE IF( LSAME( CMACH, 'U' ) ) THEN
         RMACH = RMIN
      ELSE IF( LSAME( CMACH, 'L' ) ) THEN
         RMACH = EMAX
      ELSE IF( LSAME( CMACH, 'O' ) ) THEN
         RMACH = RMAX
      END IF
*
      DLAMCH = RMACH
      RETURN
*
*     End of DLAMCH
*
      END
*
************************************************************************
*
      SUBROUTINE DLAMC1( BETA, T, RND, IEEE1 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      LOGICAL            IEEE1, RND
      INTEGER            BETA, T
*     ..
*
*  Purpose
*  =======
*
*  DLAMC1 determines the machine parameters given by BETA, T, RND, and
*  IEEE1.
*
*  Arguments
*  =========
*
*  BETA    (output) INTEGER
*          The base of the machine.
*
*  T       (output) INTEGER
*          The number of ( BETA ) digits in the mantissa.
*
*  RND     (output) LOGICAL
*          Specifies whether proper rounding  ( RND = .TRUE. )  or
*          chopping  ( RND = .FALSE. )  occurs in addition. This may not
*          be a reliable guide to the way in which the machine performs
*          its arithmetic.
*
*  IEEE1   (output) LOGICAL
*          Specifies whether rounding appears to be done in the IEEE
*          'round to nearest' style.
*
*  Further Details
*  ===============
*
*  The routine is based on the routine  ENVRON  by Malcolm and
*  incorporates suggestions by Gentleman and Marovich. See
*
*     Malcolm M. A. (1972) Algorithms to reveal properties of
*        floating-point arithmetic. Comms. of the ACM, 15, 949-951.
*
*     Gentleman W. M. and Marovich S. B. (1974) More on algorithms
*        that reveal properties of floating point arithmetic units.
*        Comms. of the ACM, 17, 276-277.
*
* =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            FIRST, LIEEE1, LRND
      INTEGER            LBETA, LT
      DOUBLE PRECISION   A, B, C, F, ONE, QTR, SAVEC, T1, T2
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMC3
      EXTERNAL           DLAMC3
*     ..
*     .. Save statement ..
      SAVE               FIRST, LIEEE1, LBETA, LRND, LT
*     ..
*     .. Data statements ..
      DATA               FIRST / .TRUE. /
*     ..
*     .. Executable Statements ..
*
      IF( FIRST ) THEN
         FIRST = .FALSE.
         ONE = 1
*
*        LBETA,  LIEEE1,  LT and  LRND  are the  local values  of  BETA,
*        IEEE1, T and RND.
*
*        Throughout this routine  we use the function  DLAMC3  to ensure
*        that relevant values are  stored and not held in registers,  or
*        are not affected by optimizers.
*
*        Compute  a = 2.0**m  with the  smallest positive integer m such
*        that
*
*           fl( a + 1.0 ) = a.
*
         A = 1
         C = 1
*
*+       WHILE( C.EQ.ONE )LOOP
   10    CONTINUE
         IF( C.EQ.ONE ) THEN
            A = 2*A
            C = DLAMC3( A, ONE )
            C = DLAMC3( C, -A )
            GO TO 10
         END IF
*+       END WHILE
*
*        Now compute  b = 2.0**m  with the smallest positive integer m
*        such that
*
*           fl( a + b ) .gt. a.
*
         B = 1
         C = DLAMC3( A, B )
*
*+       WHILE( C.EQ.A )LOOP
   20    CONTINUE
         IF( C.EQ.A ) THEN
            B = 2*B
            C = DLAMC3( A, B )
            GO TO 20
         END IF
*+       END WHILE
*
*        Now compute the base.  a and c  are neighbouring floating point
*        numbers  in the  interval  ( beta**t, beta**( t + 1 ) )  and so
*        their difference is beta. Adding 0.25 to c is to ensure that it
*        is truncated to beta and not ( beta - 1 ).
*
         QTR = ONE / 4
         SAVEC = C
         C = DLAMC3( C, -A )
         LBETA = C + QTR
*
*        Now determine whether rounding or chopping occurs,  by adding a
*        bit  less  than  beta/2  and a  bit  more  than  beta/2  to  a.
*
         B = LBETA
         F = DLAMC3( B / 2, -B / 100 )
         C = DLAMC3( F, A )
         IF( C.EQ.A ) THEN
            LRND = .TRUE.
         ELSE
            LRND = .FALSE.
         END IF
         F = DLAMC3( B / 2, B / 100 )
         C = DLAMC3( F, A )
         IF( ( LRND ) .AND. ( C.EQ.A ) )
     $      LRND = .FALSE.
*
*        Try and decide whether rounding is done in the  IEEE  'round to
*        nearest' style. B/2 is half a unit in the last place of the two
*        numbers A and SAVEC. Furthermore, A is even, i.e. has last  bit
*        zero, and SAVEC is odd. Thus adding B/2 to A should not  change
*        A, but adding B/2 to SAVEC should change SAVEC.
*
         T1 = DLAMC3( B / 2, A )
         T2 = DLAMC3( B / 2, SAVEC )
         LIEEE1 = ( T1.EQ.A ) .AND. ( T2.GT.SAVEC ) .AND. LRND
*
*        Now find  the  mantissa, t.  It should  be the  integer part of
*        log to the base beta of a,  however it is safer to determine  t
*        by powering.  So we find t as the smallest positive integer for
*        which
*
*           fl( beta**t + 1.0 ) = 1.0.
*
         LT = 0
         A = 1
         C = 1
*
*+       WHILE( C.EQ.ONE )LOOP
   30    CONTINUE
         IF( C.EQ.ONE ) THEN
            LT = LT + 1
            A = A*LBETA
            C = DLAMC3( A, ONE )
            C = DLAMC3( C, -A )
            GO TO 30
         END IF
*+       END WHILE
*
      END IF
*
      BETA = LBETA
      T = LT
      RND = LRND
      IEEE1 = LIEEE1
      RETURN
*
*     End of DLAMC1
*
      END
*
************************************************************************
*
      SUBROUTINE DLAMC2( BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      LOGICAL            RND
      INTEGER            BETA, EMAX, EMIN, T
      DOUBLE PRECISION   EPS, RMAX, RMIN
*     ..
*
*  Purpose
*  =======
*
*  DLAMC2 determines the machine parameters specified in its argument
*  list.
*
*  Arguments
*  =========
*
*  BETA    (output) INTEGER
*          The base of the machine.
*
*  T       (output) INTEGER
*          The number of ( BETA ) digits in the mantissa.
*
*  RND     (output) LOGICAL
*          Specifies whether proper rounding  ( RND = .TRUE. )  or
*          chopping  ( RND = .FALSE. )  occurs in addition. This may not
*          be a reliable guide to the way in which the machine performs
*          its arithmetic.
*
*  EPS     (output) DOUBLE PRECISION
*          The smallest positive number such that
*
*             fl( 1.0 - EPS ) .LT. 1.0,
*
*          where fl denotes the computed value.
*
*  EMIN    (output) INTEGER
*          The minimum exponent before (gradual) underflow occurs.
*
*  RMIN    (output) DOUBLE PRECISION
*          The smallest normalized number for the machine, given by
*          BASE**( EMIN - 1 ), where  BASE  is the floating point value
*          of BETA.
*
*  EMAX    (output) INTEGER
*          The maximum exponent before overflow occurs.
*
*  RMAX    (output) DOUBLE PRECISION
*          The largest positive number for the machine, given by
*          BASE**EMAX * ( 1 - EPS ), where  BASE  is the floating point
*          value of BETA.
*
*  Further Details
*  ===============
*
*  The computation of  EPS  is based on a routine PARANOIA by
*  W. Kahan of the University of California at Berkeley.
*
* =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            FIRST, IEEE, IWARN, LIEEE1, LRND
      INTEGER            GNMIN, GPMIN, I, LBETA, LEMAX, LEMIN, LT,
     $                   NGNMIN, NGPMIN
      DOUBLE PRECISION   A, B, C, HALF, LEPS, LRMAX, LRMIN, ONE, RBASE,
     $                   SIXTH, SMALL, THIRD, TWO, ZERO
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMC3
      EXTERNAL           DLAMC3
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAMC1, DLAMC4, DLAMC5
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Save statement ..
      SAVE               FIRST, IWARN, LBETA, LEMAX, LEMIN, LEPS, LRMAX,
     $                   LRMIN, LT
*     ..
*     .. Data statements ..
      DATA               FIRST / .TRUE. / , IWARN / .FALSE. /
*     ..
*     .. Executable Statements ..
*
      IF( FIRST ) THEN
         FIRST = .FALSE.
         ZERO = 0
         ONE = 1
         TWO = 2
*
*        LBETA, LT, LRND, LEPS, LEMIN and LRMIN  are the local values of
*        BETA, T, RND, EPS, EMIN and RMIN.
*
*        Throughout this routine  we use the function  DLAMC3  to ensure
*        that relevant values are stored  and not held in registers,  or
*        are not affected by optimizers.
*
*        DLAMC1 returns the parameters  LBETA, LT, LRND and LIEEE1.
*
         CALL DLAMC1( LBETA, LT, LRND, LIEEE1 )
*
*        Start to find EPS.
*
         B = LBETA
         A = B**( -LT )
         LEPS = A
*
*        Try some tricks to see whether or not this is the correct  EPS.
*
         B = TWO / 3
         HALF = ONE / 2
         SIXTH = DLAMC3( B, -HALF )
         THIRD = DLAMC3( SIXTH, SIXTH )
         B = DLAMC3( THIRD, -HALF )
         B = DLAMC3( B, SIXTH )
         B = ABS( B )
         IF( B.LT.LEPS )
     $      B = LEPS
*
         LEPS = 1
*
*+       WHILE( ( LEPS.GT.B ).AND.( B.GT.ZERO ) )LOOP
   10    CONTINUE
         IF( ( LEPS.GT.B ) .AND. ( B.GT.ZERO ) ) THEN
            LEPS = B
            C = DLAMC3( HALF*LEPS, ( TWO**5 )*( LEPS**2 ) )
            C = DLAMC3( HALF, -C )
            B = DLAMC3( HALF, C )
            C = DLAMC3( HALF, -B )
            B = DLAMC3( HALF, C )
            GO TO 10
         END IF
*+       END WHILE
*
         IF( A.LT.LEPS )
     $      LEPS = A
*
*        Computation of EPS complete.
*
*        Now find  EMIN.  Let A = + or - 1, and + or - (1 + BASE**(-3)).
*        Keep dividing  A by BETA until (gradual) underflow occurs. This
*        is detected when we cannot recover the previous A.
*
         RBASE = ONE / LBETA
         SMALL = ONE
         DO 20 I = 1, 3
            SMALL = DLAMC3( SMALL*RBASE, ZERO )
   20    CONTINUE
         A = DLAMC3( ONE, SMALL )
         CALL DLAMC4( NGPMIN, ONE, LBETA )
         CALL DLAMC4( NGNMIN, -ONE, LBETA )
         CALL DLAMC4( GPMIN, A, LBETA )
         CALL DLAMC4( GNMIN, -A, LBETA )
         IEEE = .FALSE.
*
         IF( ( NGPMIN.EQ.NGNMIN ) .AND. ( GPMIN.EQ.GNMIN ) ) THEN
            IF( NGPMIN.EQ.GPMIN ) THEN
               LEMIN = NGPMIN
*            ( Non twos-complement machines, no gradual underflow;
*              e.g.,  VAX )
            ELSE IF( ( GPMIN-NGPMIN ).EQ.3 ) THEN
               LEMIN = NGPMIN - 1 + LT
               IEEE = .TRUE.
*            ( Non twos-complement machines, with gradual underflow;
*              e.g., IEEE standard followers )
            ELSE
               LEMIN = MIN( NGPMIN, GPMIN )
*            ( A guess; no known machine )
               IWARN = .TRUE.
            END IF
*
         ELSE IF( ( NGPMIN.EQ.GPMIN ) .AND. ( NGNMIN.EQ.GNMIN ) ) THEN
            IF( ABS( NGPMIN-NGNMIN ).EQ.1 ) THEN
               LEMIN = MAX( NGPMIN, NGNMIN )
*            ( Twos-complement machines, no gradual underflow;
*              e.g., CYBER 205 )
            ELSE
               LEMIN = MIN( NGPMIN, NGNMIN )
*            ( A guess; no known machine )
               IWARN = .TRUE.
            END IF
*
         ELSE IF( ( ABS( NGPMIN-NGNMIN ).EQ.1 ) .AND.
     $            ( GPMIN.EQ.GNMIN ) ) THEN
            IF( ( GPMIN-MIN( NGPMIN, NGNMIN ) ).EQ.3 ) THEN
               LEMIN = MAX( NGPMIN, NGNMIN ) - 1 + LT
*            ( Twos-complement machines with gradual underflow;
*              no known machine )
            ELSE
               LEMIN = MIN( NGPMIN, NGNMIN )
*            ( A guess; no known machine )
               IWARN = .TRUE.
            END IF
*
         ELSE
            LEMIN = MIN( NGPMIN, NGNMIN, GPMIN, GNMIN )
*         ( A guess; no known machine )
            IWARN = .TRUE.
         END IF
***
* Comment out this if block if EMIN is ok
         IF( IWARN ) THEN
            FIRST = .TRUE.
            WRITE( 6, FMT = 9999 )LEMIN
         END IF
***
*
*        Assume IEEE arithmetic if we found denormalised  numbers above,
*        or if arithmetic seems to round in the  IEEE style,  determined
*        in routine DLAMC1. A true IEEE machine should have both  things
*        true; however, faulty machines may have one or the other.
*
         IEEE = IEEE .OR. LIEEE1
*
*        Compute  RMIN by successive division by  BETA. We could compute
*        RMIN as BASE**( EMIN - 1 ),  but some machines underflow during
*        this computation.
*
         LRMIN = 1
         DO 30 I = 1, 1 - LEMIN
            LRMIN = DLAMC3( LRMIN*RBASE, ZERO )
   30    CONTINUE
*
*        Finally, call DLAMC5 to compute EMAX and RMAX.
*
         CALL DLAMC5( LBETA, LT, LEMIN, IEEE, LEMAX, LRMAX )
      END IF
*
      BETA = LBETA
      T = LT
      RND = LRND
      EPS = LEPS
      EMIN = LEMIN
      RMIN = LRMIN
      EMAX = LEMAX
      RMAX = LRMAX
*
      RETURN
*
 9999 FORMAT( / / ' WARNING. The value EMIN may be incorrect:-',
     $      '  EMIN = ', I8, /
     $      ' If, after inspection, the value EMIN looks',
     $      ' acceptable please comment out ',
     $      / ' the IF block as marked within the code of routine',
     $      ' DLAMC2,', / ' otherwise supply EMIN explicitly.', / )
*
*     End of DLAMC2
*
      END
*
************************************************************************
*
      DOUBLE PRECISION FUNCTION DLAMC3( A, B )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B
*     ..
*
*  Purpose
*  =======
*
*  DLAMC3  is intended to force  A  and  B  to be stored prior to doing
*  the addition of  A  and  B ,  for use in situations where optimizers
*  might hold one of these in a register.
*
*  Arguments
*  =========
*
*  A, B    (input) DOUBLE PRECISION
*          The values A and B.
*
* =====================================================================
*
*     .. Executable Statements ..
*
      DLAMC3 = A + B
*
      RETURN
*
*     End of DLAMC3
*
      END
*
************************************************************************
*
      SUBROUTINE DLAMC4( EMIN, START, BASE )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      INTEGER            BASE, EMIN
      DOUBLE PRECISION   START
*     ..
*
*  Purpose
*  =======
*
*  DLAMC4 is a service routine for DLAMC2.
*
*  Arguments
*  =========
*
*  EMIN    (output) EMIN
*          The minimum exponent before (gradual) underflow, computed by
*          setting A = START and dividing by BASE until the previous A
*          can not be recovered.
*
*  START   (input) DOUBLE PRECISION
*          The starting point for determining EMIN.
*
*  BASE    (input) INTEGER
*          The base of the machine.
*
* =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   A, B1, B2, C1, C2, D1, D2, ONE, RBASE, ZERO
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMC3
      EXTERNAL           DLAMC3
*     ..
*     .. Executable Statements ..
*
      A = START
      ONE = 1
      RBASE = ONE / BASE
      ZERO = 0
      EMIN = 1
      B1 = DLAMC3( A*RBASE, ZERO )
      C1 = A
      C2 = A
      D1 = A
      D2 = A
*+    WHILE( ( C1.EQ.A ).AND.( C2.EQ.A ).AND.
*    $       ( D1.EQ.A ).AND.( D2.EQ.A )      )LOOP
   10 CONTINUE
      IF( ( C1.EQ.A ) .AND. ( C2.EQ.A ) .AND. ( D1.EQ.A ) .AND.
     $    ( D2.EQ.A ) ) THEN
         EMIN = EMIN - 1
         A = B1
         B1 = DLAMC3( A / BASE, ZERO )
         C1 = DLAMC3( B1*BASE, ZERO )
         D1 = ZERO
         DO 20 I = 1, BASE
            D1 = D1 + B1
   20    CONTINUE
         B2 = DLAMC3( A*RBASE, ZERO )
         C2 = DLAMC3( B2 / RBASE, ZERO )
         D2 = ZERO
         DO 30 I = 1, BASE
            D2 = D2 + B2
   30    CONTINUE
         GO TO 10
      END IF
*+    END WHILE
*
      RETURN
*
*     End of DLAMC4
*
      END
*
************************************************************************
*
      SUBROUTINE DLAMC5( BETA, P, EMIN, IEEE, EMAX, RMAX )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      LOGICAL            IEEE
      INTEGER            BETA, EMAX, EMIN, P
      DOUBLE PRECISION   RMAX
*     ..
*
*  Purpose
*  =======
*
*  DLAMC5 attempts to compute RMAX, the largest machine floating-point
*  number, without overflow.  It assumes that EMAX + abs(EMIN) sum
*  approximately to a power of 2.  It will fail on machines where this
*  assumption does not hold, for example, the Cyber 205 (EMIN = -28625,
*  EMAX = 28718).  It will also fail if the value supplied for EMIN is
*  too large (i.e. too close to zero), probably with overflow.
*
*  Arguments
*  =========
*
*  BETA    (input) INTEGER
*          The base of floating-point arithmetic.
*
*  P       (input) INTEGER
*          The number of base BETA digits in the mantissa of a
*          floating-point value.
*
*  EMIN    (input) INTEGER
*          The minimum exponent before (gradual) underflow.
*
*  IEEE    (input) LOGICAL
*          A logical flag specifying whether or not the arithmetic
*          system is thought to comply with the IEEE standard.
*
*  EMAX    (output) INTEGER
*          The largest exponent before overflow
*
*  RMAX    (output) DOUBLE PRECISION
*          The largest machine floating-point number.
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            EXBITS, EXPSUM, I, LEXP, NBITS, TRY, UEXP
      DOUBLE PRECISION   OLDY, RECBAS, Y, Z
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMC3
      EXTERNAL           DLAMC3
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MOD
*     ..
*     .. Executable Statements ..
*
*     First compute LEXP and UEXP, two powers of 2 that bound
*     abs(EMIN). We then assume that EMAX + abs(EMIN) will sum
*     approximately to the bound that is closest to abs(EMIN).
*     (EMAX is the exponent of the required number RMAX).
*
      LEXP = 1
      EXBITS = 1
   10 CONTINUE
      TRY = LEXP*2
      IF( TRY.LE.( -EMIN ) ) THEN
         LEXP = TRY
         EXBITS = EXBITS + 1
         GO TO 10
      END IF
      IF( LEXP.EQ.-EMIN ) THEN
         UEXP = LEXP
      ELSE
         UEXP = TRY
         EXBITS = EXBITS + 1
      END IF
*
*     Now -LEXP is less than or equal to EMIN, and -UEXP is greater
*     than or equal to EMIN. EXBITS is the number of bits needed to
*     store the exponent.
*
      IF( ( UEXP+EMIN ).GT.( -LEXP-EMIN ) ) THEN
         EXPSUM = 2*LEXP
      ELSE
         EXPSUM = 2*UEXP
      END IF
*
*     EXPSUM is the exponent range, approximately equal to
*     EMAX - EMIN + 1 .
*
      EMAX = EXPSUM + EMIN - 1
      NBITS = 1 + EXBITS + P
*
*     NBITS is the total number of bits needed to store a
*     floating-point number.
*
      IF( ( MOD( NBITS, 2 ).EQ.1 ) .AND. ( BETA.EQ.2 ) ) THEN
*
*        Either there are an odd number of bits used to store a
*        floating-point number, which is unlikely, or some bits are
*        not used in the representation of numbers, which is possible,
*        (e.g. Cray machines) or the mantissa has an implicit bit,
*        (e.g. IEEE machines, Dec Vax machines), which is perhaps the
*        most likely. We have to assume the last alternative.
*        If this is true, then we need to reduce EMAX by one because
*        there must be some way of representing zero in an implicit-bit
*        system. On machines like Cray, we are reducing EMAX by one
*        unnecessarily.
*
         EMAX = EMAX - 1
      END IF
*
      IF( IEEE ) THEN
*
*        Assume we are on an IEEE machine which reserves one exponent
*        for infinity and NaN.
*
         EMAX = EMAX - 1
      END IF
*
*     Now create RMAX, the largest machine number, which should
*     be equal to (1.0 - BETA**(-P)) * BETA**EMAX .
*
*     First compute 1.0 - BETA**(-P), being careful that the
*     result is less than 1.0 .
*
      RECBAS = ONE / BETA
      Z = BETA - ONE
      Y = ZERO
      DO 20 I = 1, P
         Z = Z*RECBAS
         IF( Y.LT.ONE )
     $      OLDY = Y
         Y = DLAMC3( Y, Z )
   20 CONTINUE
      IF( Y.GE.ONE )
     $   Y = OLDY
*
*     Now multiply by BETA**EMAX to get RMAX.
*
      DO 30 I = 1, EMAX
         Y = DLAMC3( Y*BETA, ZERO )
   30 CONTINUE
*
      RMAX = Y
      RETURN
*
*     End of DLAMC5
*
      END
      DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Description
*  ===========
*
*  DLANGE returns the value
*
*     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  matrix norm.
*
*  Arguments
*  =========
*
*  NORM    (input) CHARACTER*1
*          Specifies the value to be returned in DLANGE as described
*          above.
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.  When M = 0,
*          DLANGE is set to zero.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.  When N = 0,
*          DLANGE is set to zero.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
*          The m by n matrix A.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(M,1).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK),
*          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
*          referenced.
*
* =====================================================================
*
*     .. 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
      EXTERNAL           LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, 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
               VALUE = MAX( VALUE, ABS( A( I, J ) ) )
   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
            VALUE = MAX( 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
            VALUE = MAX( VALUE, WORK( I ) )
   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
      DOUBLE PRECISION FUNCTION DLANHS( NORM, N, A, LDA, WORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Description
*  ===========
*
*  DLANHS returns the value
*
*     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  matrix norm.
*
*  Arguments
*  =========
*
*  NORM    (input) CHARACTER*1
*          Specifies the value to be returned in DLANHS as described
*          above.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.  When N = 0, DLANHS is
*          set to zero.
*
*  A       (input) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(N,1).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK),
*          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*          referenced.
*
* =====================================================================
*
*     .. 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
      EXTERNAL           LSAME
*     ..
*     .. 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 = 1, MIN( N, J+1 )
               VALUE = MAX( VALUE, ABS( A( I, J ) ) )
   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
            VALUE = MAX( 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
            VALUE = MAX( VALUE, WORK( I ) )
   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
      DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Description
*  ===========
*
*  DLANST returns the value
*
*     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  matrix norm.
*
*  Arguments
*  =========
*
*  NORM    (input) CHARACTER*1
*          Specifies the value to be returned in DLANST as described
*          above.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.  When N = 0, DLANST is
*          set to zero.
*
*  D       (input) DOUBLE PRECISION array, dimension (N)
*          The diagonal elements of A.
*
*  E       (input) DOUBLE PRECISION array, dimension (N-1)
*          The (n-1) sub-diagonal or super-diagonal elements of A.
*
*  =====================================================================
*
*     .. 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
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, 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
            ANORM = MAX( ANORM, ABS( D( I ) ) )
            ANORM = MAX( ANORM, ABS( E( I ) ) )
   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 = MAX( ABS( D( 1 ) )+ABS( E( 1 ) ),
     $              ABS( E( N-1 ) )+ABS( D( N ) ) )
            DO 20 I = 2, N - 1
               ANORM = MAX( ANORM, ABS( D( I ) )+ABS( E( I ) )+
     $                 ABS( E( I-1 ) ) )
   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
      DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          NORM, UPLO
      INTEGER            LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Description
*  ===========
*
*  DLANSY returns the value
*
*     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  matrix norm.
*
*  Arguments
*  =========
*
*  NORM    (input) CHARACTER*1
*          Specifies the value to be returned in DLANSY as described
*          above.
*
*  UPLO    (input) 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
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.  When N = 0, DLANSY is
*          set to zero.
*
*  A       (input) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(N,1).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK),
*          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*          WORK is not referenced.
*
* =====================================================================
*
*     .. 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
      EXTERNAL           LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, 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
                  VALUE = MAX( VALUE, ABS( A( I, J ) ) )
   10          CONTINUE
   20       CONTINUE
         ELSE
            DO 40 J = 1, N
               DO 30 I = J, N
                  VALUE = MAX( VALUE, ABS( A( I, J ) ) )
   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
               VALUE = MAX( VALUE, WORK( I ) )
   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
               VALUE = MAX( 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
      SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  A       (input/output) DOUBLE PRECISION
*  B       (input/output) DOUBLE PRECISION
*  C       (input/output) DOUBLE PRECISION
*  D       (input/output) DOUBLE PRECISION
*          On entry, the elements of the input matrix.
*          On exit, they are overwritten by the elements of the
*          standardised Schur form.
*
*  RT1R    (output) DOUBLE PRECISION
*  RT1I    (output) DOUBLE PRECISION
*  RT2R    (output) DOUBLE PRECISION
*  RT2I    (output) DOUBLE PRECISION
*          The real and imaginary parts of the eigenvalues. If the
*          eigenvalues are a complex conjugate pair, RT1I > 0.
*
*  CS      (output) DOUBLE PRECISION
*  SN      (output) DOUBLE PRECISION
*          Parameters of the rotation matrix.
*
*  Further Details
*  ===============
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
      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
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           DLAMCH, DLAPY2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
      EPS = DLAMCH( 'P' )
      IF( C.EQ.ZERO ) THEN
         CS = ONE
         SN = ZERO
         GO TO 10
*
      ELSE IF( B.EQ.ZERO ) THEN
*
*        Swap rows and columns
*
         CS = ZERO
         SN = ONE
         TEMP = D
         D = A
         A = TEMP
         B = -C
         C = ZERO
         GO TO 10
      ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
     $          THEN
         CS = ONE
         SN = ZERO
         GO TO 10
      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.
*
            SIGMA = B + C
            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 ]
*
            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
*
   10 CONTINUE
*
*     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
      DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   X, Y
*     ..
*
*  Purpose
*  =======
*
*  DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary
*  overflow.
*
*  Arguments
*  =========
*
*  X       (input) DOUBLE PRECISION
*  Y       (input) DOUBLE PRECISION
*          X and Y specify the values x and y.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   W, XABS, YABS, Z
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      XABS = ABS( X )
      YABS = ABS( Y )
      W = MAX( XABS, YABS )
      Z = MIN( XABS, YABS )
      IF( Z.EQ.ZERO ) THEN
         DLAPY2 = W
      ELSE
         DLAPY2 = W*SQRT( ONE+( Z / W )**2 )
      END IF
      RETURN
*
*     End of DLAPY2
*
      END
      SUBROUTINE DLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK,
     $                   INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      LOGICAL            LREAL, LTRAN
      INTEGER            INFO, LDT, N
      DOUBLE PRECISION   SCALE, W
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( * ), T( LDT, * ), WORK( * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  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', A' denotes the conjugate 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.
*
*  Arguments
*  =========
*
*  LTRAN   (input) 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)'.
*
*  LREAL   (input) LOGICAL
*          On entry, LREAL specifies the input matrix structure:
*             = .FALSE.,    the input is complex
*             = .TRUE.,     the input is real
*
*  N       (input) INTEGER
*          On entry, N specifies the order of T+i*B. N >= 0.
*
*  T       (input) 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.
*
*  LDT     (input) INTEGER
*          The leading dimension of the matrix T. LDT >= max(1,N).
*
*  B       (input) 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.
*
*  W       (input) DOUBLE PRECISION
*          On entry, W is the diagonal element of the matrix B.
*          If LREAL = .TRUE., W is not referenced.
*
*  SCALE   (output) DOUBLE PRECISION
*          On exit, SCALE is the scale factor.
*
*  X       (input/output) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) 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.
*
* =====================================================================
*
*     .. 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'*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)'*(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
      SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            INCV, LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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'
*
*  where tau is a real scalar and v is a real vector.
*
*  If tau = 0, then H is taken to be the unit matrix.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': form  H * C
*          = 'R': form  C * H
*
*  M       (input) INTEGER
*          The number of rows of the matrix C.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C.
*
*  V       (input) 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.
*
*  INCV    (input) INTEGER
*          The increment between elements of v. INCV <> 0.
*
*  TAU     (input) DOUBLE PRECISION
*          The value tau in the representation of H.
*
*  C       (input/output) 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'.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension
*                         (N) if SIDE = 'L'
*                      or (M) if SIDE = 'R'
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DGER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
      IF( LSAME( SIDE, 'L' ) ) THEN
*
*        Form  H * C
*
         IF( TAU.NE.ZERO ) THEN
*
*           w := C' * v
*
            CALL DGEMV( 'Transpose', M, N, ONE, C, LDC, V, INCV, ZERO,
     $                  WORK, 1 )
*
*           C := C - v * w'
*
            CALL DGER( M, N, -TAU, V, INCV, WORK, 1, C, LDC )
         END IF
      ELSE
*
*        Form  C * H
*
         IF( TAU.NE.ZERO ) THEN
*
*           w := C * v
*
            CALL DGEMV( 'No transpose', M, N, ONE, C, LDC, V, INCV,
     $                  ZERO, WORK, 1 )
*
*           C := C - w * v'
*
            CALL DGER( M, N, -TAU, WORK, 1, V, INCV, C, LDC )
         END IF
      END IF
      RETURN
*
*     End of DLARF
*
      END
      SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
     $                   T, LDT, C, LDC, WORK, LDWORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
*  DLARFB applies a real block reflector H or its transpose H' to a
*  real m by n matrix C, from either the left or the right.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply H or H' from the Left
*          = 'R': apply H or H' from the Right
*
*  TRANS   (input) CHARACTER*1
*          = 'N': apply H (No transpose)
*          = 'T': apply H' (Transpose)
*
*  DIRECT  (input) 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)
*
*  STOREV  (input) CHARACTER*1
*          Indicates how the vectors which define the elementary
*          reflectors are stored:
*          = 'C': Columnwise
*          = 'R': Rowwise
*
*  M       (input) INTEGER
*          The number of rows of the matrix C.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C.
*
*  K       (input) INTEGER
*          The order of the matrix T (= the number of elementary
*          reflectors whose product defines the block reflector).
*
*  V       (input) 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.
*
*  LDV     (input) 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.
*
*  T       (input) DOUBLE PRECISION array, dimension (LDT,K)
*          The triangular k by k matrix T in the representation of the
*          block reflector.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T. LDT >= K.
*
*  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
*          On entry, the m by n matrix C.
*          On exit, C is overwritten by H*C or H'*C or C*H or C*H'.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDA >= max(1,M).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (LDWORK,K)
*
*  LDWORK  (input) INTEGER
*          The leading dimension of the array WORK.
*          If SIDE = 'L', LDWORK >= max(1,N);
*          if SIDE = 'R', LDWORK >= max(1,M).
*
*  =====================================================================
*
*     .. 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' * C  where  C = ( C1 )
*                                                  ( C2 )
*
*              W := C' * V  =  (C1'*V1 + C2'*V2)  (stored in WORK)
*
*              W := C1'
*
               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'*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'  or  W * T
*
               CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V * W'
*
               IF( M.GT.K ) THEN
*
*                 C2 := C2 - V2 * W'
*
                  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'
*
               CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K,
     $                     ONE, V, LDV, WORK, LDWORK )
*
*              C1 := C1 - W'
*
               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'  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'
*
               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', 'Transpose', M, N-K, K,
     $                        -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE,
     $                        C( 1, K+1 ), LDC )
               END IF
*
*              W := W * V1'
*
               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' * C  where  C = ( C1 )
*                                                  ( C2 )
*
*              W := C' * V  =  (C1'*V1 + C2'*V2)  (stored in WORK)
*
*              W := C2'
*
               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'*V1
*
                  CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K,
     $                        ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
               END IF
*
*              W := W * T'  or  W * T
*
               CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V * W'
*
               IF( M.GT.K ) THEN
*
*                 C1 := C1 - V1 * W'
*
                  CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K,
     $                        -ONE, V, LDV, WORK, LDWORK, ONE, C, LDC )
               END IF
*
*              W := W * V2'
*
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K,
     $                     ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
*
*              C2 := C2 - W'
*
               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'  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'
*
               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', 'Transpose', M, N-K, K,
     $                        -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
               END IF
*
*              W := W * V2'
*
               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' * C  where  C = ( C1 )
*                                                  ( C2 )
*
*              W := C' * V'  =  (C1'*V1' + C2'*V2') (stored in WORK)
*
*              W := C1'
*
               DO 130 J = 1, K
                  CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
  130          CONTINUE
*
*              W := W * V1'
*
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K,
     $                     ONE, V, LDV, WORK, LDWORK )
               IF( M.GT.K ) THEN
*
*                 W := W + C2'*V2'
*
                  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'  or  W * T
*
               CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V' * W'
*
               IF( M.GT.K ) THEN
*
*                 C2 := C2 - V2' * W'
*
                  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'
*
               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'  where  C = ( C1  C2 )
*
*              W := C * V'  =  (C1*V1' + C2*V2')  (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'
*
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K,
     $                     ONE, V, LDV, WORK, LDWORK )
               IF( N.GT.K ) THEN
*
*                 W := W + C2 * V2'
*
                  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'
*
               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' * C  where  C = ( C1 )
*                                                  ( C2 )
*
*              W := C' * V'  =  (C1'*V1' + C2'*V2') (stored in WORK)
*
*              W := C2'
*
               DO 190 J = 1, K
                  CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 )
  190          CONTINUE
*
*              W := W * V2'
*
               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'*V1'
*
                  CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE,
     $                        C, LDC, V, LDV, ONE, WORK, LDWORK )
               END IF
*
*              W := W * T'  or  W * T
*
               CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V' * W'
*
               IF( M.GT.K ) THEN
*
*                 C1 := C1 - V1' * W'
*
                  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'
*
               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'  =  (C1*V1' + C2*V2')  (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'
*
               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'
*
                  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'
*
               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
      SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   ALPHA, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * )
*     ..
*
*  Purpose
*  =======
*
*  DLARFG generates a real elementary reflector H of order n, such
*  that
*
*        H * ( alpha ) = ( beta ),   H' * 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' ) ,
*                      ( 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.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the elementary reflector.
*
*  ALPHA   (input/output) DOUBLE PRECISION
*          On entry, the value alpha.
*          On exit, it is overwritten with the value beta.
*
*  X       (input/output) DOUBLE PRECISION array, dimension
*                         (1+(N-2)*abs(INCX))
*          On entry, the vector x.
*          On exit, it is overwritten with the vector v.
*
*  INCX    (input) INTEGER
*          The increment between elements of X. INCX > 0.
*
*  TAU     (output) DOUBLE PRECISION
*          The value tau.
*
*  =====================================================================
*
*     .. 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' )
         IF( ABS( BETA ).LT.SAFMIN ) THEN
*
*           XNORM, BETA may be inaccurate; scale X and recompute them
*
            RSAFMN = ONE / SAFMIN
            KNT = 0
   10       CONTINUE
            KNT = KNT + 1
            CALL DSCAL( N-1, RSAFMN, X, INCX )
            BETA = BETA*RSAFMN
            ALPHA = ALPHA*RSAFMN
            IF( ABS( BETA ).LT.SAFMIN )
     $         GO TO 10
*
*           New BETA is at most 1, at least SAFMIN
*
            XNORM = DNRM2( N-1, X, INCX )
            BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
            TAU = ( BETA-ALPHA ) / BETA
            CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
*
*           If ALPHA is subnormal, it may lose relative accuracy
*
            ALPHA = BETA
            DO 20 J = 1, KNT
               ALPHA = ALPHA*SAFMIN
   20       CONTINUE
         ELSE
            TAU = ( BETA-ALPHA ) / BETA
            CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
            ALPHA = BETA
         END IF
      END IF
*
      RETURN
*
*     End of DLARFG
*
      END
      SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          DIRECT, STOREV
      INTEGER            K, LDT, LDV, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )
*     ..
*
*  Purpose
*  =======
*
*  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'
*
*  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 * V
*
*  Arguments
*  =========
*
*  DIRECT  (input) 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)
*
*  STOREV  (input) CHARACTER*1
*          Specifies how the vectors which define the elementary
*          reflectors are stored (see also Further Details):
*          = 'C': columnwise
*          = 'R': rowwise
*
*  N       (input) INTEGER
*          The order of the block reflector H. N >= 0.
*
*  K       (input) INTEGER
*          The order of the triangular factor T (= the number of
*          elementary reflectors). K >= 1.
*
*  V       (input/output) DOUBLE PRECISION array, dimension
*                               (LDV,K) if STOREV = 'C'
*                               (LDV,N) if STOREV = 'R'
*          The matrix V. See further details.
*
*  LDV     (input) INTEGER
*          The leading dimension of the array V.
*          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i).
*
*  T       (output) 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.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T. LDT >= K.
*
*  Further Details
*  ===============
*
*  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 = (  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 )
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   VII
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DTRMV
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( LSAME( DIRECT, 'F' ) ) THEN
         DO 20 I = 1, K
            IF( TAU( I ).EQ.ZERO ) THEN
*
*              H(i)  =  I
*
               DO 10 J = 1, I
                  T( J, I ) = ZERO
   10          CONTINUE
            ELSE
*
*              general case
*
               VII = V( I, I )
               V( I, I ) = ONE
               IF( LSAME( STOREV, 'C' ) ) THEN
*
*                 T(1:i-1,i) := - tau(i) * V(i:n,1:i-1)' * V(i:n,i)
*
                  CALL DGEMV( 'Transpose', N-I+1, I-1, -TAU( I ),
     $                        V( I, 1 ), LDV, V( I, I ), 1, ZERO,
     $                        T( 1, I ), 1 )
               ELSE
*
*                 T(1:i-1,i) := - tau(i) * V(1:i-1,i:n) * V(i,i:n)'
*
                  CALL DGEMV( 'No transpose', I-1, N-I+1, -TAU( I ),
     $                        V( 1, I ), LDV, V( I, I ), LDV, ZERO,
     $                        T( 1, I ), 1 )
               END IF
               V( I, I ) = VII
*
*              T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
               CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
     $                     LDT, T( 1, I ), 1 )
               T( I, I ) = TAU( I )
            END IF
   20    CONTINUE
      ELSE
         DO 40 I = K, 1, -1
            IF( TAU( I ).EQ.ZERO ) THEN
*
*              H(i)  =  I
*
               DO 30 J = I, K
                  T( J, I ) = ZERO
   30          CONTINUE
            ELSE
*
*              general case
*
               IF( I.LT.K ) THEN
                  IF( LSAME( STOREV, 'C' ) ) THEN
                     VII = V( N-K+I, I )
                     V( N-K+I, I ) = ONE
*
*                    T(i+1:k,i) :=
*                            - tau(i) * V(1:n-k+i,i+1:k)' * V(1:n-k+i,i)
*
                     CALL DGEMV( 'Transpose', N-K+I, K-I, -TAU( I ),
     $                           V( 1, I+1 ), LDV, V( 1, I ), 1, ZERO,
     $                           T( I+1, I ), 1 )
                     V( N-K+I, I ) = VII
                  ELSE
                     VII = V( I, N-K+I )
                     V( I, N-K+I ) = ONE
*
*                    T(i+1:k,i) :=
*                            - tau(i) * V(i+1:k,1:n-k+i) * V(i,1:n-k+i)'
*
                     CALL DGEMV( 'No transpose', K-I, N-K+I, -TAU( I ),
     $                           V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO,
     $                           T( I+1, I ), 1 )
                     V( I, N-K+I ) = VII
                  END IF
*
*                 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
   40    CONTINUE
      END IF
      RETURN
*
*     End of DLARFT
*
      END
      SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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'
*
*  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.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': form  H * C
*          = 'R': form  C * H
*
*  M       (input) INTEGER
*          The number of rows of the matrix C.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C.
*
*  V       (input) DOUBLE PRECISION array, dimension (M) if SIDE = 'L'
*                                     or (N) if SIDE = 'R'
*          The vector v in the representation of H.
*
*  TAU     (input) DOUBLE PRECISION
*          The value tau in the representation of H.
*
*  C       (input/output) 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'.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDA >= (1,M).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension
*                      (N) if SIDE = 'L'
*                      or (M) if SIDE = 'R'
*          WORK is not referenced if H has order < 11.
*
*  =====================================================================
*
*     .. 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           DGEMV, DGER
*     ..
*     .. 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
*
*        w := C'*v
*
         CALL DGEMV( 'Transpose', M, N, ONE, C, LDC, V, 1, ZERO, WORK,
     $               1 )
*
*        C := C - tau * v * w'
*
         CALL DGER( M, N, -TAU, V, 1, WORK, 1, C, LDC )
         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
*
*        w := C * v
*
         CALL DGEMV( 'No transpose', M, N, ONE, C, LDC, V, 1, ZERO,
     $               WORK, 1 )
*
*        C := C - tau * w * v'
*
         CALL DGER( M, N, -TAU, WORK, 1, V, 1, C, LDC )
         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
      SUBROUTINE DLARTG( F, G, CS, SN, R )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   CS, F, G, R, SN
*     ..
*
*  Purpose
*  =======
*
*  DLARTG generate 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 BLAS1 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 without doing any
*        floating point operations (saves work in DBDSQR when
*        there are zeros on the diagonal).
*
*  If F exceeds G in magnitude, CS will be positive.
*
*  Arguments
*  =========
*
*  F       (input) DOUBLE PRECISION
*          The first component of vector to be rotated.
*
*  G       (input) DOUBLE PRECISION
*          The second component of vector to be rotated.
*
*  CS      (output) DOUBLE PRECISION
*          The cosine of the rotation.
*
*  SN      (output) DOUBLE PRECISION
*          The sine of the rotation.
*
*  R       (output) DOUBLE PRECISION
*          The nonzero component of the rotated vector.
*
*  =====================================================================
*
*     .. 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, SQRT
*     ..
*     .. Save statement ..
      SAVE               FIRST, SAFMX2, SAFMIN, SAFMN2
*     ..
*     .. Data statements ..
      DATA               FIRST / .TRUE. /
*     ..
*     .. Executable Statements ..
*
      IF( FIRST ) THEN
         FIRST = .FALSE.
         SAFMIN = DLAMCH( 'S' )
         EPS = DLAMCH( 'E' )
         SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
     $            LOG( DLAMCH( 'B' ) ) / TWO )
         SAFMX2 = ONE / SAFMN2
      END IF
      IF( G.EQ.ZERO ) THEN
         CS = ONE
         SN = ZERO
         R = F
      ELSE IF( F.EQ.ZERO ) THEN
         CS = ZERO
         SN = ONE
         R = 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 )
     $         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( ABS( F ).GT.ABS( G ) .AND. CS.LT.ZERO ) THEN
            CS = -CS
            SN = -SN
            R = -R
         END IF
      END IF
      RETURN
*
*     End of DLARTG
*
      END
      SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   F, G, H, SSMAX, SSMIN
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  F       (input) DOUBLE PRECISION
*          The (1,1) element of the 2-by-2 matrix.
*
*  G       (input) DOUBLE PRECISION
*          The (1,2) element of the 2-by-2 matrix.
*
*  H       (input) DOUBLE PRECISION
*          The (2,2) element of the 2-by-2 matrix.
*
*  SSMIN   (output) DOUBLE PRECISION
*          The smaller singular value.
*
*  SSMAX   (output) DOUBLE PRECISION
*          The larger singular value.
*
*  Further Details
*  ===============
*
*  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. (On machines with
*  partial overflow, like the Cray, overflow may occur if the largest
*  singular value is within a factor of 2 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.
*
*  ====================================================================
*
*     .. 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
      SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          TYPE
      INTEGER            INFO, KL, KU, LDA, M, N
      DOUBLE PRECISION   CFROM, CTO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  TYPE    (input) 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.
*
*  KL      (input) INTEGER
*          The lower bandwidth of A.  Referenced only if TYPE = 'B',
*          'Q' or 'Z'.
*
*  KU      (input) INTEGER
*          The upper bandwidth of A.  Referenced only if TYPE = 'B',
*          'Q' or 'Z'.
*
*  CFROM   (input) DOUBLE PRECISION
*  CTO     (input) 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.
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
*          The matrix to be multiplied by CTO/CFROM.  See TYPE for the
*          storage type.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  INFO    (output) INTEGER
*          0  - successful exit
*          <0 - if INFO = -i, the i-th argument had an illegal value.
*
*  =====================================================================
*
*     .. 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
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. 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 ) THEN
         INFO = -4
      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
      CTO1 = CTOC / BIGNUM
      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.
      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
      SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, M, N
      DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  DLASET initializes an m-by-n matrix A to BETA on the diagonal and
*  ALPHA on the offdiagonals.
*
*  Arguments
*  =========
*
*  UPLO    (input) 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.
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  ALPHA   (input) DOUBLE PRECISION
*          The constant to which the offdiagonal elements are to be set.
*
*  BETA    (input) DOUBLE PRECISION
*          The constant to which the diagonal elements are to be set.
*
*  A       (input/output) 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).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
* =====================================================================
*
*     .. 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
      SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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
*  dqds", LAPACK technical report.
*
*  Note : DLASQ1 works only on machines which follow ieee-754
*  floating-point standard in their handling of infinities and NaNs.
*  Normal execution of DLASQ1 may create NaNs and infinities and hence
*  may abort due to a floating point exception in environments which
*  do not conform to the ieee standard.
*
*  Arguments
*  =========
*
*  N     (input) INTEGER
*        The number of rows and columns in the matrix. N >= 0.
*
*  D     (input/output) 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.
*
*  E     (input/output) 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.
*
*  WORK  (workspace) DOUBLE PRECISION array, dimension (4*N)
*
*  INFO  (output) 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 30*N
*                   iterations (in inner while loop)
*              = 3, termination criterion of outer while loop not met
*                   (program created more than N unreduced blocks)
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IINFO
      DOUBLE PRECISION   EPS, SCALE, SFMIN, 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 = -2
         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 )
         GO TO 50
      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' )
      SFMIN = DLAMCH( 'Safe minimum' )
      SCALE = SQRT( EPS / SFMIN )
      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 )
      END IF
*
   50 CONTINUE
      RETURN
*
*     End of DLASQ1
*
      END
      SUBROUTINE DLASQ2( N, Z, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  Purpose
*  =======
*
*  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 works only on machines which follow ieee-754
*  floating-point standard in their handling of infinities and NaNs.
*  Normal execution of DLASQ2 may create NaNs and infinities and hence
*  may abort due to a floating point exception in environments which
*  do not conform to the ieee standard.
*
*  Arguments
*  =========
*
*  N     (input) INTEGER
*        The number of rows and columns in the matrix. N >= 0.
*
*  Z     (workspace) 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, Z( 2*N+2 ) holds the sum of the eigenvalues, 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.
*
*  INFO  (output) 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 30*N
*                   iterations (in inner while loop)
*              = 3, termination criterion of outer while loop not met
*                   (program created more than N unreduced blocks)
*
*  Further Details
*  ===============
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CBIAS
      PARAMETER          ( CBIAS = 1.50D0 )
      DOUBLE PRECISION   ZERO, HALF, ONE, TWO, FOUR, TEN, HNDRD
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
     $                   TWO = 2.0D0, FOUR = 4.0D0, TEN = 10.0D0,
     $                   HNDRD = 100.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I0, I4, IINFO, IPN4, ITER, IWHILA, IWHILB, K,
     $                   N0, NBIG, NDIV, NFAIL, PP, SPLT
      DOUBLE PRECISION   D, DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, E,
     $                   EMAX, EMIN, EPS, EPS2, OLDEMN, QMAX, QMIN, S,
     $                   SIGMA, T, TAU, TEMP, TRACE, ZMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASQ3, DLASQ5, DLASRT, XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. 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' )*TEN
      EPS2 = EPS**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( 2 ).LT.ZERO .OR. Z( 3 ).LT.ZERO ) THEN
            INFO = -2
            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 )*EPS2 ) 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 )
         Z( 7 ) = ZERO
         Z( 8 ) = ZERO
         Z( 9 ) = ZERO
         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
      D = ZERO
      E = ZERO
*
      DO 10 K = 1, N
         IF( Z( K ).LT.ZERO ) THEN
            INFO = -( 200+K )
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         ELSE IF( Z( N+K ).LT.ZERO ) THEN
            INFO = -( 200+N+K )
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         END IF
         D = D + Z( K )
         E = E + Z( N+K )
         QMAX = MAX( QMAX, Z( K ) )
   10 CONTINUE
      ZMAX = QMAX
      DO 20 K = 1, N - 1
         EMIN = MIN( EMIN, Z( N+K ) )
         ZMAX = MAX( ZMAX, Z( N+K ) )
   20 CONTINUE
*
*     Check for diagonality.
*
      IF( E.EQ.ZERO ) THEN
         CALL DLASRT( 'D', N, Z, IINFO )
         Z( 2*N-1 ) = D
         RETURN
      END IF
*
      TRACE = D + E
      I0 = 1
      N0 = N
*
*     Check for zero data.
*
      IF( TRACE.EQ.ZERO ) THEN
         Z( 2*N-1 ) = ZERO
         RETURN
      END IF
*
*     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
*
*     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
*
         IF( EMIN.LE.EPS2*QMAX ) THEN
*
*           Li's reverse test.
*
            D = Z( 4*N0+PP-3 )
            DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4
               IF( Z( I4-1 ).LE.EPS2*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
               IF( Z( I4-1 ).LE.EPS2*D ) THEN
                  Z( I4-1 ) = -ZERO
                  Z( I4-2*PP-2 ) = D
                  Z( I4-2*PP ) = ZERO
                  D = Z( I4+1 )
                  EMIN = ZERO
               ELSE
                  Z( I4-2*PP-2 ) = D + Z( I4-1 )
                  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 ) )
                  EMIN = MIN( EMIN, Z( I4-2*PP ) )
               END IF
   60       CONTINUE
            Z( 4*N0-PP-2 ) = D
         ELSE
            TAU = ZERO
            CALL DLASQ5( I0, N0, Z, PP, TAU, DMIN, DMIN1, DMIN2, DN,
     $                   DN1, DN2 )
*
            EMIN = Z( 4*N0 )
         END IF
*
*        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
*
      ITER = 2
      NFAIL = 0
      NDIV = 2*( N0-I0 )
*
      DO 140 IWHILA = 1, N + 1
         IF( N0.LT.1 )
     $      GO TO 150
*
*        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
*
*        Store EMIN for passing to DLASQ3.
*
         Z( 4*N0-1 ) = EMIN
*
*        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 = 0
*
         NBIG = 30*( N0-I0+1 )
         DO 120 IWHILB = 1, NBIG
            IF( I0.GT.N0 )
     $         GO TO 130
*
*           While submatrix unfinished take a good dqds step.
*
            CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
     $                   ITER, NDIV )
*
            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.EPS2*QMAX .OR. Z( 4*N0-1 ).LE.EPS2*
     $             SIGMA ) THEN
                  SPLT = I0 - 1
                  QMAX = Z( 4*I0-3 )
                  EMIN = Z( 4*I0-1 )
                  OLDEMN = Z( 4*I0 )
                  DO 110 I4 = 4*I0, 4*( N0-3 ), 4
                     IF( Z( I4 ).LE.EPS2*Z( I4-3 ) .OR. Z( I4-1 ).LE.
     $                   EPS2*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
  110             CONTINUE
                  Z( 4*N0-1 ) = EMIN
                  Z( 4*N0 ) = OLDEMN
                  I0 = SPLT + 1
               END IF
            END IF
*
  120    CONTINUE
*
         INFO = 2
         RETURN
*
*        end IWHILB
*
  130    CONTINUE
*
  140 CONTINUE
*
      INFO = 3
      RETURN
*
*     end IWHILA
*
  150 CONTINUE
*
*     Move q's to the front.
*
      DO 160 K = 2, N
         Z( K ) = Z( 4*K-3 )
  160 CONTINUE
*
*     Sort and compute sum of eigenvalues.
*
      CALL DLASRT( 'D', N, Z, IINFO )
*
      E = ZERO
      DO 170 K = N, 1, -1
         E = E + Z( K )
  170 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 ) = HNDRD*NFAIL / DBLE( ITER )
      RETURN
*
*     End of DLASQ2
*
      END
      SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
     $                   ITER, NDIV )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            I0, ITER, N0, NDIV, NFAIL, PP
      DOUBLE PRECISION   DESIG, DMIN, QMAX, SIGMA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  Purpose
*  =======
*  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.
*
*  Arguments
*  =========
*
*  I0     (input) INTEGER
*         First index.
*
*  N0     (input) INTEGER
*         Last index.
*
*  Z      (input) DOUBLE PRECISION array, dimension ( 4*N )
*         Z holds the qd array.
*
*  PP     (input) INTEGER
*         PP=0 for ping, PP=1 for pong.
*
*  DMIN   (output) DOUBLE PRECISION
*         Minimum value of d.
*
*  SIGMA  (output) DOUBLE PRECISION
*         Sum of shifts used in current segment.
*
*  DESIG  (input/output) DOUBLE PRECISION
*         Lower order part of SIGMA
*
*  QMAX   (input) DOUBLE PRECISION
*         Maximum value of q.
*
*  NFAIL  (output) INTEGER
*         Number of times shift was too big.
*
*  ITER   (output) INTEGER
*         Number of iterations.
*
*  NDIV   (output) INTEGER
*         Number of divisions.
*
*  TTYPE  (output) INTEGER
*         Shift type.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CBIAS
      PARAMETER          ( CBIAS = 1.50D0 )
      DOUBLE PRECISION   ZERO, QURTR, HALF, ONE, TWO, TEN
      PARAMETER          ( ZERO = 0.0D0, QURTR = 0.250D0, HALF = 0.5D0,
     $                   ONE = 1.0D0, TWO = 2.0D0, TEN = 10.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            IPN4, J4, N0IN, NN, TTYPE
      DOUBLE PRECISION   DMIN1, DMIN2, DN, DN1, DN2, EPS, EPS2, S,
     $                   SFMIN, T, TAU, TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASQ4, DLASQ5, DLASQ6
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Save statement ..
      SAVE               TTYPE, DMIN1, DMIN2, DN, DN1, DN2, TAU
*     ..
*     .. Data statements ..
      DATA               TTYPE / 0 /
      DATA               DMIN1 / ZERO / , DMIN2 / ZERO / , DN / ZERO / ,
     $                   DN1 / ZERO / , DN2 / ZERO / , TAU / ZERO /
*     ..
*     .. Executable Statements ..
*
      N0IN = N0
      EPS = DLAMCH( 'Precision' )*TEN
      SFMIN = DLAMCH( 'Safe minimum' )
      EPS2 = EPS**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-by-1 case.
*
      IF( Z( NN-5 ).GT.EPS2*( SIGMA+Z( NN-3 ) ) .AND. Z( NN-2*PP-4 ).GT.
     $    EPS2*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-by-2 case.
*
   30 CONTINUE
*
      IF( Z( NN-9 ).GT.EPS2*SIGMA .AND. Z( NN-2*PP-8 ).GT.EPS2*
     $    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
      IF( Z( NN-5 ).GT.Z( NN-3 )*EPS2 ) THEN
         T = HALF*( ( Z( NN-7 )-Z( NN-3 ) )+Z( NN-5 ) )
         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
*
*     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*I0-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
*
   70 CONTINUE
*
      IF( DMIN.LT.ZERO .OR. SFMIN*QMAX.LE.
     $    MIN( Z( 4*N0+PP-1 ), Z( 4*N0+PP-9 ), DMIN2+Z( 4*N0-PP ) ) )
     $     THEN
*
*        Choose a shift.
*
         CALL DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1,
     $                DN2, TAU, TTYPE )
*
*        Call dqds until DMIN > 0.
*
   80    CONTINUE
*
         CALL DLASQ5( I0, N0, Z, PP, TAU, DMIN, DMIN1, DMIN2, DN, DN1,
     $                DN2 )
*
         ITER = ITER + 1
         NDIV = NDIV + ( N0-I0+2 )
*
*        Check for NaN: "DMIN.NE.DMIN"
*
         IF( DMIN.NE.DMIN ) THEN
            Z( 4*N0+PP-1 ) = ZERO
            TAU = ZERO
            GO TO 70
         END IF
*
*        Check for convergence hidden by negative DN.
*
         IF( DMIN.LT.ZERO .AND. DMIN1.GT.ZERO .AND.
     $       Z( 4*( N0-1 )-PP ).LT.EPS*( SIGMA+DN1 ) .AND. ABS( DN ).LT.
     $       EPS*SIGMA ) THEN
            Z( 4*( N0-1 )-PP+2 ) = ZERO
            DMIN = ABS( DMIN )
         END IF
*
         IF( DMIN.LT.ZERO ) THEN
*
*           Failure. Select new TAU and try again.
*
            NFAIL = NFAIL + 1
*
*           Failed twice. Play it safe.
*
            IF( TTYPE.LT.-22 ) THEN
               TAU = ZERO
               GO TO 80
            END IF
*
            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 80
         END IF
      ELSE
         CALL DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DN1, DN2 )
         ITER = ITER + 1
         NDIV = NDIV + ( N0-I0 )
         TAU = ZERO
      END IF
*
      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
      SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
     $                   DN1, DN2, TAU, TTYPE )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            I0, N0, N0IN, PP, TTYPE
      DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DN1, DN2, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  Purpose
*  =======
*  DLASQ4 computes an approximation TAU to the smallest eigenvalue
*  using values of d from the previous transform.
*
*  I0    (input) INTEGER
*        First index.
*
*  N0    (input) INTEGER
*        Last index.
*
*  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
*        Z holds the qd array.
*
*  PP    (input) INTEGER
*        PP=0 for ping, PP=1 for pong.
*
*  NOIN  (input) INTEGER
*        The value of N0 at start of EIGTEST.
*
*  DMIN  (input) DOUBLE PRECISION
*        Minimum value of d.
*
*  DMIN1 (input) DOUBLE PRECISION
*        Minimum value of d, excluding D( N0 ).
*
*  DMIN2 (input) DOUBLE PRECISION
*        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*
*  DN    (input) DOUBLE PRECISION
*        d(N)
*
*  DN1   (input) DOUBLE PRECISION
*        d(N-1)
*
*  DN2   (input) DOUBLE PRECISION
*        d(N-2)
*
*  TAU   (output) DOUBLE PRECISION
*        This is the shift.
*
*  TTYPE (output) INTEGER
*        Shift type.
*
*  Further Details
*  ===============
*  CNST1 = 9/16
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CNST1, CNST2, CNST3
      PARAMETER          ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
     $                   CNST3 = 1.050D0 )
      DOUBLE PRECISION   QURTR, THIRD, HALF, ZERO, ONE, TWO, HNDRD
      PARAMETER          ( QURTR = 0.250D0, THIRD = 0.3330D0,
     $                   HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
     $                   TWO = 2.0D0, HNDRD = 100.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I4, NN, NP
      DOUBLE PRECISION   A2, B1, B2, G, GAM, GAP1, GAP2, S
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Save statement ..
      SAVE               G
*     ..
*     .. Data statements ..
      DATA               G / ZERO /
*     ..
*     .. 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.
*
               IF( DMIN.EQ.DN ) THEN
                  GAM = DN
                  A2 = ZERO
                  B2 = Z( NN-5 ) / Z( NN-7 )
                  NP = NN - 9
               ELSE
                  NP = NN - 2*PP
                  B2 = Z( NP-2 )
                  GAM = DN1
                  A2 = Z( NP-4 ) / Z( NP-2 )
                  B2 = Z( NN-9 ) / Z( NN-11 )
                  NP = NN - 13
               END IF
*
*              Approximate contribution to norm squared from I < NN-1.
*
               IF( B2.EQ.ZERO )
     $            GO TO 20
               A2 = A2 + B2
               DO 10 I4 = NP, 4*I0 - 1 + PP, -4
                  B1 = B2
                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
                  A2 = A2 + B2
                  IF( HNDRD*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 ) THEN
                  S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
               ELSE
                  S = QURTR*GAM
               END IF
               TTYPE = -4
            END IF
         ELSE IF( DMIN.EQ.DN2 ) THEN
*
*           Case 5.
*
*           Compute contribution to norm squared from I > NN-2.
*
            NP = NN - 2*PP
            B1 = Z( NP-2 )
            B2 = Z( NP-6 )
            GAM = DN2
            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 )
               IF( B2.EQ.ZERO )
     $            GO TO 40
               A2 = A2 + B2
               DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
                  B1 = B2
                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
                  A2 = A2 + B2
                  IF( HNDRD*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 ) THEN
               S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
            ELSE
               S = QURTR*GAM / ( ONE+A2 )
            END IF
            TTYPE = -5
         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.
*
            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
               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
               B2 = B2 + B1
               IF( HNDRD*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( A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ),
     $             THIRD*DMIN1 )
               TTYPE = -7
            ELSE
               S = MAX( A2*( ONE-CNST2*B2 ), THIRD*DMIN1 )
               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
            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
               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
               B2 = B2 + B1
               IF( HNDRD*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( A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ),
     $             THIRD*DMIN2 )
            ELSE
               S = MAX( A2*( ONE-CNST2*B2 ), THIRD*DMIN2 )
            END IF
            TTYPE = -10
         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
      SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, DMIN, DMIN1, DMIN2, DN,
     $                   DNM1, DNM2 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            I0, N0, PP
      DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  Purpose
*  =======
*  DLASQ5 computes one dqds transform in ping-pong form.
*
*  Arguments
*  =========
*
*  I0    (input) INTEGER
*        First index.
*
*  N0    (input) INTEGER
*        Last index.
*
*  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
*        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*        an extra argument.
*
*  PP    (input) INTEGER
*        PP=0 for ping, PP=1 for pong.
*
*  TAU   (input) DOUBLE PRECISION
*        This is the shift.
*
*  DMIN  (output) DOUBLE PRECISION
*        Minimum value of d.
*
*  DMIN1 (output) DOUBLE PRECISION
*        Minimum value of d, excluding D( N0 ).
*
*  DMIN2 (output) DOUBLE PRECISION
*        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*
*  DN    (output) DOUBLE PRECISION
*        d(N0), the last value of d.
*
*  DNM1  (output) DOUBLE PRECISION
*        d(N0-1).
*
*  DNM2  (output) DOUBLE PRECISION
*        d(N0-2).
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            J4, J4P2
      DOUBLE PRECISION   D, EMIN, TEMP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
      IF( ( N0-I0-1 ).LE.0 )
     $   RETURN
*
      J4 = 4*I0 + PP - 3
      EMIN = Z( J4+4 )
      D = Z( J4 ) - TAU
      DMIN = D
*
      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 )
*
      Z( J4+2 ) = DN
      Z( 4*N0-PP ) = EMIN
      RETURN
*
*     End of DLASQ5
*
      END
      SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DNM1,
     $                   DNM2 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            I0, N0, PP
      DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  Purpose
*  =======
*  DLASQ6 computes one dqds transform in ping-pong form.
*
*  Arguments
*  =========
*
*  I0    (input) INTEGER
*        First index.
*
*  N0    (input) INTEGER
*        Last index.
*
*  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
*        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*        an extra argument.
*
*  PP    (input) INTEGER
*        PP=0 for ping, PP=1 for pong.
*
*  DMIN  (output) DOUBLE PRECISION
*        Minimum value of d.
*
*  DMIN1 (output) DOUBLE PRECISION
*        Minimum value of d, excluding D( N0 ).
*
*  DMIN2 (output) DOUBLE PRECISION
*        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*
*  DN    (output) DOUBLE PRECISION
*        d(N0), the last value of d.
*
*  DNM1  (output) DOUBLE PRECISION
*        d(N0-1).
*
*  DNM2  (output) DOUBLE PRECISION
*        d(N0-2).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J4, J4P2
      DOUBLE PRECISION   D, EMIN, SFMIN, TEMP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
      IF( ( N0-I0-1 ).LE.0 )
     $   RETURN
*
      SFMIN = DLAMCH( 'Safe minimum' )
      J4 = 4*I0 + PP - 3
      EMIN = Z( J4+4 )
      D = Z( J4 )
      DMIN = D
*
      DO 10 J4 = 4*I0 - PP, 4*( N0-3 ) - PP, 4
         J4P2 = J4 + 2*PP - 1
         Z( J4-2 ) = D + Z( J4P2 )
         IF( Z( J4-2 ).EQ.ZERO ) THEN
            Z( J4 ) = ZERO
            D = Z( J4P2+2 )
            DMIN = D
            EMIN = ZERO
         ELSE IF( SFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) ) THEN
            TEMP = Z( J4P2+2 ) / Z( J4-2 )
            Z( J4 ) = Z( J4P2 )*TEMP
            D = D*TEMP
         ELSE
            Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
            D = Z( J4P2+2 )*( D / Z( J4-2 ) )
         END IF
         DMIN = MIN( DMIN, D )
         EMIN = MIN( EMIN, Z( J4 ) )
   10 CONTINUE
*
*     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( SFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) ) 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( SFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) ) 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
      SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          DIRECT, PIVOT, SIDE
      INTEGER            LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * ), S( * )
*     ..
*
*  Purpose
*  =======
*
*  DLASR   performs the transformation
*
*     A := P*A,   when SIDE = 'L' or 'l'  (  Left-hand side )
*
*     A := A*P',  when SIDE = 'R' or 'r'  ( Right-hand side )
*
*  where A is an m by n real matrix and P is an orthogonal matrix,
*  consisting of a sequence of plane rotations determined by the
*  parameters PIVOT and DIRECT as follows ( z = m when SIDE = 'L' or 'l'
*  and z = n when SIDE = 'R' or 'r' ):
*
*  When  DIRECT = 'F' or 'f'  ( Forward sequence ) then
*
*     P = P( z - 1 )*...*P( 2 )*P( 1 ),
*
*  and when DIRECT = 'B' or 'b'  ( Backward sequence ) then
*
*     P = P( 1 )*P( 2 )*...*P( z - 1 ),
*
*  where  P( k ) is a plane rotation matrix for the following planes:
*
*     when  PIVOT = 'V' or 'v'  ( Variable pivot ),
*        the plane ( k, k + 1 )
*
*     when  PIVOT = 'T' or 't'  ( Top pivot ),
*        the plane ( 1, k + 1 )
*
*     when  PIVOT = 'B' or 'b'  ( Bottom pivot ),
*        the plane ( k, z )
*
*  c( k ) and s( k )  must contain the  cosine and sine that define the
*  matrix  P( k ).  The two by two plane rotation part of the matrix
*  P( k ), R( k ), is assumed to be of the form
*
*     R( k ) = (  c( k )  s( k ) ).
*              ( -s( k )  c( k ) )
*
*  This version vectorises across rows of the array A when SIDE = 'L'.
*
*  Arguments
*  =========
*
*  SIDE    (input) 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'
*
*  DIRECT  (input) 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 )
*
*  PIVOT   (input) 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)
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  If m <= 1, an immediate
*          return is effected.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  If n <= 1, an
*          immediate return is effected.
*
*  C, S    (input) DOUBLE PRECISION arrays, dimension
*                  (M-1) if SIDE = 'L'
*                  (N-1) if SIDE = 'R'
*          c(k) and s(k) contain the cosine and sine that define the
*          matrix P(k).  The two by two plane rotation part of the
*          matrix P(k), R(k), is assumed to be of the form
*          R( k ) = (  c( k )  s( k ) ).
*                   ( -s( k )  c( k ) )
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          The m by n matrix A.  On exit, A is overwritten by P*A if
*          SIDE = 'R' or by A*P' if SIDE = 'L'.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  =====================================================================
*
*     .. 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'
*
         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
      SUBROUTINE DLASRT( ID, N, D, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      CHARACTER          ID
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  ID      (input) CHARACTER*1
*          = 'I': sort D in increasing order;
*          = 'D': sort D in decreasing order.
*
*  N       (input) INTEGER
*          The length of the array D.
*
*  D       (input/output) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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 paramters.
*
      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
      SUBROUTINE DLASSQ( N, X, INCX, SCALE, SUMSQ )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   SCALE, SUMSQ
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * )
*     ..
*
*  Purpose
*  =======
*
*  DLASSQ  returns the values  scl  and  smsq  such that
*
*     ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,
*
*  where  x( i ) = X( 1 + ( i - 1 )*INCX ). The value of  sumsq  is
*  assumed to be non-negative and  scl  returns the value
*
*     scl = max( scale, abs( x( i ) ) ).
*
*  scale and sumsq must be supplied in SCALE and SUMSQ and
*  scl and smsq are overwritten on SCALE and SUMSQ respectively.
*
*  The routine makes only one pass through the vector x.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The number of elements to be used from the vector X.
*
*  X       (input) DOUBLE PRECISION array, dimension (N)
*          The vector for which a scaled sum of squares is computed.
*             x( i )  = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.
*
*  INCX    (input) INTEGER
*          The increment between successive values of the vector X.
*          INCX > 0.
*
*  SCALE   (input/output) DOUBLE PRECISION
*          On entry, the value  scale  in the equation above.
*          On exit, SCALE is overwritten with  scl , the scaling factor
*          for the sum of squares.
*
*  SUMSQ   (input/output) DOUBLE PRECISION
*          On entry, the value  sumsq  in the equation above.
*          On exit, SUMSQ is overwritten with  smsq , the basic sum of
*          squares from which  scl  has been factored out.
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            IX
      DOUBLE PRECISION   ABSXI
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
      IF( N.GT.0 ) THEN
         DO 10 IX = 1, 1 + ( N-1 )*INCX, INCX
            IF( X( IX ).NE.ZERO ) THEN
               ABSXI = ABS( X( IX ) )
               IF( SCALE.LT.ABSXI ) THEN
                  SUMSQ = 1 + SUMSQ*( SCALE / ABSXI )**2
                  SCALE = ABSXI
               ELSE
                  SUMSQ = SUMSQ + ( ABSXI / SCALE )**2
               END IF
            END IF
   10    CONTINUE
      END IF
      RETURN
*
*     End of DLASSQ
*
      END
      SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
*     ..
*
*  Purpose
*  =======
*
*  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 ].
*
*  Arguments
*  =========
*
*  F       (input) DOUBLE PRECISION
*          The (1,1) element of the 2-by-2 matrix.
*
*  G       (input) DOUBLE PRECISION
*          The (1,2) element of the 2-by-2 matrix.
*
*  H       (input) DOUBLE PRECISION
*          The (2,2) element of the 2-by-2 matrix.
*
*  SSMIN   (output) DOUBLE PRECISION
*          abs(SSMIN) is the smaller singular value.
*
*  SSMAX   (output) DOUBLE PRECISION
*          abs(SSMAX) is the larger singular value.
*
*  SNL     (output) DOUBLE PRECISION
*  CSL     (output) DOUBLE PRECISION
*          The vector (CSL, SNL) is a unit left singular vector for the
*          singular value abs(SSMAX).
*
*  SNR     (output) DOUBLE PRECISION
*  CSR     (output) DOUBLE PRECISION
*          The vector (CSR, SNR) is a unit right singular vector for the
*          singular value abs(SSMAX).
*
*  Further Details
*  ===============
*
*  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. (On machines with
*  partial overflow, like the Cray, overflow may occur if the largest
*  singular value is within a factor of 2 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.
*
* =====================================================================
*
*     .. 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
      SUBROUTINE DLASWP( N, A, LDA, K1, K2, IPIV, INCX )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INCX, K1, K2, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.
*
*  K1      (input) INTEGER
*          The first element of IPIV for which a row interchange will
*          be done.
*
*  K2      (input) INTEGER
*          The last element of IPIV for which a row interchange will
*          be done.
*
*  IPIV    (input) INTEGER array, dimension (M*abs(INCX))
*          The vector of pivot indices.  Only the elements in positions
*          K1 through K2 of IPIV are accessed.
*          IPIV(K) = L implies rows K and L are to be interchanged.
*
*  INCX    (input) INTEGER
*          The increment between successive values of IPIV.  If IPIV
*          is negative, the pivots are applied in reverse order.
*
*  Further Details
*  ===============
*
*  Modified by
*   R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
* =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, I1, I2, INC, IP, IX, IX0, J, K, N32
      DOUBLE PRECISION   TEMP
*     ..
*     .. Executable Statements ..
*
*     Interchange row I with row IPIV(I) 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 = 1 + ( 1-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
      SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
     $                   LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
*  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', where T' denotes the transpose of T.
*
*  Arguments
*  =========
*
*  LTRANL  (input) LOGICAL
*          On entry, LTRANL specifies the op(TL):
*             = .FALSE., op(TL) = TL,
*             = .TRUE., op(TL) = TL'.
*
*  LTRANR  (input) LOGICAL
*          On entry, LTRANR specifies the op(TR):
*            = .FALSE., op(TR) = TR,
*            = .TRUE., op(TR) = TR'.
*
*  ISGN    (input) INTEGER
*          On entry, ISGN specifies the sign of the equation
*          as described before. ISGN may only be 1 or -1.
*
*  N1      (input) INTEGER
*          On entry, N1 specifies the order of matrix TL.
*          N1 may only be 0, 1 or 2.
*
*  N2      (input) INTEGER
*          On entry, N2 specifies the order of matrix TR.
*          N2 may only be 0, 1 or 2.
*
*  TL      (input) DOUBLE PRECISION array, dimension (LDTL,2)
*          On entry, TL contains an N1 by N1 matrix.
*
*  LDTL    (input) INTEGER
*          The leading dimension of the matrix TL. LDTL >= max(1,N1).
*
*  TR      (input) DOUBLE PRECISION array, dimension (LDTR,2)
*          On entry, TR contains an N2 by N2 matrix.
*
*  LDTR    (input) INTEGER
*          The leading dimension of the matrix TR. LDTR >= max(1,N2).
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB,2)
*          On entry, the N1 by N2 matrix B contains the right-hand
*          side of the equation.
*
*  LDB     (input) INTEGER
*          The leading dimension of the matrix B. LDB >= max(1,N1).
*
*  SCALE   (output) DOUBLE PRECISION
*          On exit, SCALE contains the scale factor. SCALE is chosen
*          less than or equal to 1 to prevent the solution overflowing.
*
*  X       (output) DOUBLE PRECISION array, dimension (LDX,2)
*          On exit, X contains the N1 by N2 solution.
*
*  LDX     (input) INTEGER
*          The leading dimension of the matrix X. LDX >= max(1,N1).
*
*  XNORM   (output) DOUBLE PRECISION
*          On exit, XNORM is the infinity-norm of the solution.
*
*  INFO    (output) 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.
*
* =====================================================================
*
*     .. 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 )
     $   T16( 4, 4 ) = SMIN
      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
      SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
     $                   JPIV )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            IJOB, LDZ, N
      DOUBLE PRECISION   RDSCAL, RDSUM
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), JPIV( * )
      DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  IJOB    (input) 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 choosen as either +1 or -1 (Default).
*
*  N       (input) INTEGER
*          The number of columns of the matrix Z.
*
*  Z       (input) 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
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDA >= max(1, N).
*
*  RHS     (input/output) DOUBLE PRECISION array, dimension N.
*          On entry, RHS contains contributions from other subsystems.
*          On exit, RHS contains the solution of the subsystem with
*          entries acoording to the value of IJOB (see above).
*
*  RDSUM   (input/output) 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.
*
*  RDSCAL  (input/output) 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.
*
*  IPIV    (input) INTEGER array, dimension (N).
*          The pivot indices; for 1 <= i <= N, row i of the
*          matrix has been interchanged with row IPIV(i).
*
*  JPIV    (input) INTEGER array, dimension (N).
*          The pivot indices; for 1 <= j <= N, column j of the
*          matrix has been interchanged with column JPIV(j).
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*     Umea University, S-901 87 Umea, Sweden.
*
*  This routine is a further developed implementation of algorithm
*  BSOLVE in [1] using complete pivoting in the LU factorization.
*
*  [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.
*
*  =====================================================================
*
*     .. 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 transfered 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
      SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, LDW, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
*     ..
*
*  Purpose
*  =======
*
*  DLATRD reduces NB rows and columns of a real symmetric matrix A to
*  symmetric tridiagonal form by an orthogonal similarity
*  transformation Q' * 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.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored:
*          = 'U': Upper triangular
*          = 'L': Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.
*
*  NB      (input) INTEGER
*          The number of rows and columns to be reduced.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= (1,N).
*
*  E       (output) 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.
*
*  TAU     (output) 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.
*
*  W       (output) DOUBLE PRECISION array, dimension (LDW,NB)
*          The n-by-nb matrix W required to update the unreduced part
*          of A.
*
*  LDW     (input) INTEGER
*          The leading dimension of the array W. LDW >= max(1,N).
*
*  Further Details
*  ===============
*
*  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'
*
*  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'
*
*  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' - W*V'.
*
*  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).
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
     $                   CNORM, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORMIN, TRANS, UPLO
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), CNORM( * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  DLATRS solves one of the triangular systems
*
*     A *x = s*b  or  A'*x = s*b
*
*  with scaling to prevent overflow.  Here A is an upper or lower
*  triangular matrix, A' 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.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the matrix A is upper or lower triangular.
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  TRANS   (input) CHARACTER*1
*          Specifies the operation applied to A.
*          = 'N':  Solve A * x = s*b  (No transpose)
*          = 'T':  Solve A'* x = s*b  (Transpose)
*          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)
*
*  DIAG    (input) CHARACTER*1
*          Specifies whether or not the matrix A is unit triangular.
*          = 'N':  Non-unit triangular
*          = 'U':  Unit triangular
*
*  NORMIN  (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max (1,N).
*
*  X       (input/output) 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.
*
*  SCALE   (output) DOUBLE PRECISION
*          The scaling factor s for the triangular system
*             A * x = s*b  or  A'* 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.
*
*  CNORM   (input or output) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -k, the k-th argument had an illegal value
*
*  Further Details
*  ======= =======
*
*  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'*x = b.  The basic
*  algorithm for A upper triangular is
*
*       for j = 1, ..., n
*            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
*       end
*
*  We simultaneously compute two bounds
*       G(j) = bound on ( b(i) - A[1:i-1,i]' * 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).
*
*  =====================================================================
*
*     .. 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
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM, DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, DASUM, DDOT, DLAMCH
*     ..
*     .. 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
*
      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.
*
            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
         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 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' * 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' * 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'*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
      SUBROUTINE DORG2L( M, N, K, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix Q. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix Q. M >= N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines the
*          matrix Q. N >= K >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The first dimension of the array A. LDA >= max(1,M).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQLF.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument has an illegal value
*
*  =====================================================================
*
*     .. 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.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 DLARF( '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
      SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix Q. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix Q. M >= N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines the
*          matrix Q. N >= K >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The first dimension of the array A. LDA >= max(1,M).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQRF.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument has an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, 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.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
            A( I, I ) = ONE
            CALL DLARF( '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
      SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          VECT
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  VECT    (input) 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.
*
*  M       (input) INTEGER
*          The number of rows of the matrix Q or P**T to be returned.
*          M >= 0.
*
*  N       (input) 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).
*
*  K       (input) 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.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,M).
*
*  TAU     (input) 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.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. 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
         IF( WANTQ ) THEN
            NB = ILAENV( 1, 'DORGQR', ' ', M, N, K, -1 )
         ELSE
            NB = ILAENV( 1, 'DORGLQ', ' ', M, N, K, -1 )
         END IF
         LWKOPT = MAX( 1, MN )*NB
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGBR', -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
*
      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', 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' 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'(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
      SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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).
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix Q. N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) 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.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,N).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (N-1)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEHRD.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix Q. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix Q. N >= M.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines the
*          matrix Q. M >= K >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The first dimension of the array A. LDA >= max(1,M).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGELQF.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (M)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument has an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, 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( '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
               A( I, I ) = ONE
               CALL DLARF( '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
      SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix Q. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix Q. N >= M.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines the
*          matrix Q. M >= K >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The first dimension of the array A. LDA >= max(1,M).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGELQF.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument has an illegal value
*
*  =====================================================================
*
*     .. 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' 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' 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
      SUBROUTINE DORGQL( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix Q. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix Q. M >= N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines the
*          matrix Q. N >= K >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The first dimension of the array A. LDA >= max(1,M).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQLF.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument has an illegal value
*
*  =====================================================================
*
*     .. 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
      NB = ILAENV( 1, 'DORGQL', ' ', 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( 'DORGQL', -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, '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
      SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix Q. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix Q. M >= N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines the
*          matrix Q. N >= K >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The first dimension of the array A. LDA >= max(1,M).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQRF.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument has an illegal value
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DORGTR( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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).
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U': Upper triangle of A contains elementary reflectors
*                 from DSYTRD;
*          = 'L': Lower triangle of A contains elementary reflectors
*                 from DSYTRD.
*
*  N       (input) INTEGER
*          The order of the matrix Q. N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,N).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (N-1)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DSYTRD.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DORM2R overwrites the general real m by n matrix C with
*
*        Q * C  if SIDE = 'L' and TRANS = 'N', or
*
*        Q'* C  if SIDE = 'L' and TRANS = 'T', or
*
*        C * Q  if SIDE = 'R' and TRANS = 'N', or
*
*        C * Q' 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'.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply Q or Q' from the Left
*          = 'R': apply Q or Q' from the Right
*
*  TRANS   (input) CHARACTER*1
*          = 'N': apply Q  (No transpose)
*          = 'T': apply Q' (Transpose)
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines
*          the matrix Q.
*          If SIDE = 'L', M >= K >= 0;
*          if SIDE = 'R', N >= K >= 0.
*
*  A       (input) 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.
*          A is modified by the routine but restored on exit.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.
*          If SIDE = 'L', LDA >= max(1,M);
*          if SIDE = 'R', LDA >= max(1,N).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQRF.
*
*  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
*          On entry, the m by n matrix C.
*          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension
*                                   (N) if SIDE = 'L',
*                                   (M) if SIDE = 'R'
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, NOTRAN
      INTEGER            I, I1, I2, I3, IC, 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' )
*
*     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)
*
         AII = A( I, I )
         A( I, I ) = ONE
         CALL DLARF( SIDE, MI, NI, A( I, I ), 1, TAU( I ), C( IC, JC ),
     $               LDC, WORK )
         A( I, I ) = AII
   10 CONTINUE
      RETURN
*
*     End of DORM2R
*
      END
      SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C,
     $                   LDC, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS, VECT
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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).
*
*  Arguments
*  =========
*
*  VECT    (input) CHARACTER*1
*          = 'Q': apply Q or Q**T;
*          = 'P': apply P or P**T.
*
*  SIDE    (input) 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.
*
*  TRANS   (input) CHARACTER*1
*          = 'N':  No transpose, apply Q  or P;
*          = 'T':  Transpose, apply Q**T or P**T.
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  K       (input) 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.
*
*  A       (input) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.
*          If VECT = 'Q', LDA >= max(1,nq);
*          if VECT = 'P', LDA >= max(1,min(nq,K)).
*
*  TAU     (input) 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.
*
*  C       (input/output) 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.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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 = N
      ELSE
         NQ = N
         NW = 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.MAX( 1, 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 = MAX( 1, 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
      SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DORML2 overwrites the general real m by n matrix C with
*
*        Q * C  if SIDE = 'L' and TRANS = 'N', or
*
*        Q'* C  if SIDE = 'L' and TRANS = 'T', or
*
*        C * Q  if SIDE = 'R' and TRANS = 'N', or
*
*        C * Q' 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'.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply Q or Q' from the Left
*          = 'R': apply Q or Q' from the Right
*
*  TRANS   (input) CHARACTER*1
*          = 'N': apply Q  (No transpose)
*          = 'T': apply Q' (Transpose)
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines
*          the matrix Q.
*          If SIDE = 'L', M >= K >= 0;
*          if SIDE = 'R', N >= K >= 0.
*
*  A       (input) 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.
*          A is modified by the routine but restored on exit.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,K).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGELQF.
*
*  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
*          On entry, the m by n matrix C.
*          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension
*                                   (N) if SIDE = 'L',
*                                   (M) if SIDE = 'R'
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, NOTRAN
      INTEGER            I, I1, I2, I3, IC, 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' )
*
*     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)
*
         AII = A( I, I )
         A( I, I ) = ONE
         CALL DLARF( SIDE, MI, NI, A( I, I ), LDA, TAU( I ),
     $               C( IC, JC ), LDC, WORK )
         A( I, I ) = AII
   10 CONTINUE
      RETURN
*
*     End of DORML2
*
      END
      SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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'.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply Q or Q**T from the Left;
*          = 'R': apply Q or Q**T from the Right.
*
*  TRANS   (input) CHARACTER*1
*          = 'N':  No transpose, apply Q;
*          = 'T':  Transpose, apply Q**T.
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines
*          the matrix Q.
*          If SIDE = 'L', M >= K >= 0;
*          if SIDE = 'R', N >= K >= 0.
*
*  A       (input) 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.
*          A is modified by the routine but restored on exit.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,K).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGELQF.
*
*  C       (input/output) 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.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      CHARACTER          TRANST
      INTEGER            I, I1, I2, I3, IB, IC, IINFO, IWS, JC, LDWORK,
     $                   LWKOPT, MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   T( LDT, NBMAX )
*     ..
*     .. 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 = N
      ELSE
         NQ = N
         NW = 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.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Determine the block size.  NB may be at most NBMAX, where NBMAX
*        is used to define the local array T.
*
         NB = MIN( NBMAX, ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N, K,
     $        -1 ) )
         LWKOPT = MAX( 1, NW )*NB
         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
         IWS = NW*NB
         IF( LWORK.LT.IWS ) THEN
            NB = LWORK / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMLQ', SIDE // TRANS, M, N, K,
     $              -1 ) )
         END IF
      ELSE
         IWS = NW
      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
*
         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 ), T, LDT )
            IF( LEFT ) THEN
*
*              H or H' is applied to C(i:m,1:n)
*
               MI = M - I + 1
               IC = I
            ELSE
*
*              H or H' is applied to C(1:m,i:n)
*
               NI = N - I + 1
               JC = I
            END IF
*
*           Apply H or H'
*
            CALL DLARFB( SIDE, TRANST, 'Forward', 'Rowwise', MI, NI, IB,
     $                   A( I, I ), LDA, T, LDT, C( IC, JC ), LDC, WORK,
     $                   LDWORK )
   10    CONTINUE
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMLQ
*
      END
      SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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'.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply Q or Q**T from the Left;
*          = 'R': apply Q or Q**T from the Right.
*
*  TRANS   (input) CHARACTER*1
*          = 'N':  No transpose, apply Q;
*          = 'T':  Transpose, apply Q**T.
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines
*          the matrix Q.
*          If SIDE = 'L', M >= K >= 0;
*          if SIDE = 'R', N >= K >= 0.
*
*  A       (input) 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.
*          A is modified by the routine but restored on exit.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.
*          If SIDE = 'L', LDA >= max(1,M);
*          if SIDE = 'R', LDA >= max(1,N).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQRF.
*
*  C       (input/output) 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.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      INTEGER            I, I1, I2, I3, IB, IC, IINFO, IWS, JC, LDWORK,
     $                   LWKOPT, MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   T( LDT, NBMAX )
*     ..
*     .. 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 = N
      ELSE
         NQ = N
         NW = 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.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Determine the block size.  NB may be at most NBMAX, where NBMAX
*        is used to define the local array T.
*
         NB = MIN( NBMAX, ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N, K,
     $        -1 ) )
         LWKOPT = MAX( 1, NW )*NB
         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
         IWS = NW*NB
         IF( LWORK.LT.IWS ) THEN
            NB = LWORK / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMQR', SIDE // TRANS, M, N, K,
     $              -1 ) )
         END IF
      ELSE
         IWS = NW
      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
*
         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 ), T, LDT )
            IF( LEFT ) THEN
*
*              H or H' is applied to C(i:m,1:n)
*
               MI = M - I + 1
               IC = I
            ELSE
*
*              H or H' is applied to C(1:m,i:n)
*
               NI = N - I + 1
               JC = I
            END IF
*
*           Apply H or H'
*
            CALL DLARFB( SIDE, TRANS, 'Forward', 'Columnwise', MI, NI,
     $                   IB, A( I, I ), LDA, T, LDT, C( IC, JC ), LDC,
     $                   WORK, LDWORK )
   10    CONTINUE
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMQR
*
      END
      SUBROUTINE DRSCL( N, SA, SX, INCX )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   SA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   SX( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The number of components of the vector x.
*
*  SA      (input) 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.
*
*  SX      (input/output) DOUBLE PRECISION array, dimension
*                         (1+(N-1)*abs(INCX))
*          The n-element vector x.
*
*  INCX    (input) 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
*
* =====================================================================
*
*     .. 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
      CALL DLABAD( SMLNUM, BIGNUM )
*
*     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
      SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ
      INTEGER            INFO, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  COMPZ   (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix.  N >= 0.
*
*  D       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the diagonal elements of the tridiagonal matrix.
*          On exit, if INFO = 0, the eigenvalues in ascending order.
*
*  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
*          On entry, the (n-1) subdiagonal elements of the tridiagonal
*          matrix.
*          On exit, E has been destroyed.
*
*  Z       (input/output) 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.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1, and if
*          eigenvectors are desired, then  LDZ >= max(1,N).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2))
*          If COMPZ = 'N', then WORK is not referenced.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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( 'I', 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
      SUBROUTINE DSTERF( N, D, E, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * )
*     ..
*
*  Purpose
*  =======
*
*  DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
*  using the Pal-Walker-Kahan variant of the QL or QR algorithm.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix.  N >= 0.
*
*  D       (input/output) 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.
*
*  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
*          On entry, the (n-1) subdiagonal elements of the tridiagonal
*          matrix.
*          On exit, E has been destroyed.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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
*     ..
*     .. 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
*
*     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( 'I', LEND-L+1, D( L ), E( L ) )
      ISCALE = 0
      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
      SUBROUTINE RSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DSYEV computes all eigenvalues and, optionally, eigenvectors of a
*  real symmetric matrix A.
*
*  Arguments
*  =========
*
*  JOBZ    (input) CHARACTER*1
*          = 'N':  Compute eigenvalues only;
*          = 'V':  Compute eigenvalues and eigenvectors.
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  W       (output) DOUBLE PRECISION array, dimension (N)
*          If INFO = 0, the eigenvalues in ascending order.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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, LOPT, 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
      ELSE IF( LWORK.LT.MAX( 1, 3*N-1 ) .AND. .NOT.LQUERY ) THEN
         INFO = -8
      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
      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
         WORK( 1 ) = 1
         RETURN
      END IF
*
      IF( N.EQ.1 ) THEN
         W( 1 ) = A( 1, 1 )
         WORK( 1 ) = 3
         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 )
      LOPT = 2*N + WORK( INDWRK )
*
*     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
      SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * )
*     ..
*
*  Purpose
*  =======
*
*  DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
*  form T by an orthogonal similarity transformation: Q' * A * Q = T.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored:
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  D       (output) DOUBLE PRECISION array, dimension (N)
*          The diagonal elements of the tridiagonal matrix T:
*          D(i) = A(i,i).
*
*  E       (output) 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'.
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  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'
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. 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'
*           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'*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' - w * v'
*
               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'
*           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'*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' - w * v'
*
               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
      SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * ),
     $                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DSYTRD reduces a real symmetric matrix A to real symmetric
*  tridiagonal form T by an orthogonal similarity transformation:
*  Q**T * A * Q = T.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  D       (output) DOUBLE PRECISION array, dimension (N)
*          The diagonal elements of the tridiagonal matrix T:
*          D(i) = A(i,i).
*
*  E       (output) 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'.
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  Further Details
*  ===============
*
*  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'
*
*  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'
*
*  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).
*
*  =====================================================================
*
*     .. 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 = 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' - W*V'
*
            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' - W*V'
*
            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
      SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
     $                   LDVR, MM, M, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  DTREVC computes some or all of the right and/or left eigenvectors of
*  a real upper quasi-triangular matrix T.
*
*  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 = w*y'
*
*  where y' denotes the conjugate transpose of the vector y.
*
*  If all eigenvectors are requested, the routine may either return the
*  matrices X and/or Y of right or left eigenvectors of T, or the
*  products Q*X and/or Q*Y, where Q is an input orthogonal
*  matrix. If T was obtained from the real-Schur factorization of an
*  original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of
*  right or left eigenvectors of A.
*
*  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.  Corresponding to each 2-by-2
*  diagonal block is a complex conjugate pair of eigenvalues and
*  eigenvectors; only one eigenvector of the pair is computed, namely
*  the one corresponding to the eigenvalue with positive imaginary part.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'R':  compute right eigenvectors only;
*          = 'L':  compute left eigenvectors only;
*          = 'B':  compute both right and left eigenvectors.
*
*  HOWMNY  (input) CHARACTER*1
*          = 'A':  compute all right and/or left eigenvectors;
*          = 'B':  compute all right and/or left eigenvectors,
*                  and backtransform them using the input matrices
*                  supplied in VR and/or VL;
*          = 'S':  compute selected right and/or left eigenvectors,
*                  specified by the logical array SELECT.
*
*  SELECT  (input/output) LOGICAL array, dimension (N)
*          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
*          computed.
*          If HOWMNY = 'A' or 'B', SELECT is not referenced.
*          To select the real eigenvector corresponding to a real
*          eigenvalue w(j), SELECT(j) must be set to .TRUE..  To select
*          the complex eigenvector corresponding to a complex conjugate
*          pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be
*          set to .TRUE.; then on exit SELECT(j) is .TRUE. and
*          SELECT(j+1) is .FALSE..
*
*  N       (input) INTEGER
*          The order of the matrix T. N >= 0.
*
*  T       (input) DOUBLE PRECISION array, dimension (LDT,N)
*          The upper quasi-triangular matrix T in Schur canonical form.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T. LDT >= max(1,N).
*
*  VL      (input/output) 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;
*                           VL has the same quasi-lower triangular form
*                           as T'. If T(i,i) is a real eigenvalue, then
*                           the i-th column VL(i) of VL  is its
*                           corresponding eigenvector. If T(i:i+1,i:i+1)
*                           is a 2-by-2 block whose eigenvalues are
*                           complex-conjugate eigenvalues of T, then
*                           VL(i)+sqrt(-1)*VL(i+1) is the complex
*                           eigenvector corresponding to the eigenvalue
*                           with positive real part.
*          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.
*          If SIDE = 'R', VL is not referenced.
*
*  LDVL    (input) INTEGER
*          The leading dimension of the array VL.  LDVL >= max(1,N) if
*          SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*
*  VR      (input/output) 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;
*                           VR has the same quasi-upper triangular form
*                           as T. If T(i,i) is a real eigenvalue, then
*                           the i-th column VR(i) of VR  is its
*                           corresponding eigenvector. If T(i:i+1,i:i+1)
*                           is a 2-by-2 block whose eigenvalues are
*                           complex-conjugate eigenvalues of T, then
*                           VR(i)+sqrt(-1)*VR(i+1) is the complex
*                           eigenvector corresponding to the eigenvalue
*                           with positive real part.
*          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.
*          If SIDE = 'L', VR is not referenced.
*
*  LDVR    (input) INTEGER
*          The leading dimension of the array VR.  LDVR >= max(1,N) if
*          SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*
*  MM      (input) INTEGER
*          The number of columns in the arrays VL and/or VR. MM >= M.
*
*  M       (output) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  Further Details
*  ===============
*
*  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|.
*
*  =====================================================================
*
*     .. 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
      CALL DLABAD( UNFL, OVFL )
      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)'*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)'*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)     ]'* 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) )' - (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)  ]'-(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.
*
  210          CONTINUE
               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
      SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK,
     $                   INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 31, 1993
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ
      INTEGER            IFST, ILST, INFO, LDQ, LDT, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  COMPQ   (input) CHARACTER*1
*          = 'V':  update the matrix Q of Schur vectors;
*          = 'N':  do not update Q.
*
*  N       (input) INTEGER
*          The order of the matrix T. N >= 0.
*
*  T       (input/output) 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.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T. LDT >= max(1,N).
*
*  Q       (input/output) 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.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.  LDQ >= max(1,N).
*
*  IFST    (input/output) INTEGER
*  ILST    (input/output) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. 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 ) THEN
         INFO = -7
      ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) 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
      SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
     $                   M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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 elemnts equal and its
*  off-diagonal elements of opposite sign.
*
*  Arguments
*  =========
*
*  JOB     (input) 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).
*
*  COMPQ   (input) CHARACTER*1
*          = 'V': update the matrix Q of Schur vectors;
*          = 'N': do not update Q.
*
*  SELECT  (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix T. N >= 0.
*
*  T       (input/output) 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.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T. LDT >= max(1,N).
*
*  Q       (input/output) 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.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.
*          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) 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.
*
*  M       (output) INTEGER
*          The dimension of the specified invariant subspace.
*          0 < = M <= N.
*
*  S       (output) 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.
*
*  SEP     (output) 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.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.
*          If JOB = 'N', LWORK >= max(1,N);
*          if JOB = 'E', LWORK >= M*(N-M);
*          if JOB = 'V' or 'B', LWORK >= 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.
*
*  IWORK   (workspace) INTEGER array, dimension (LIWORK)
*          IF JOB = 'N' or 'E', IWORK is not referenced.
*
*  LIWORK  (input) INTEGER
*          The dimension of the array IWORK.
*          If JOB = 'N' or 'E', LIWORK >= 1;
*          if JOB = 'V' or 'B', LIWORK >= 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.
*
*  INFO    (output) 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.
*
*  Further Details
*  ===============
*
*  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*Z = ( T11 T12 ) n1
*                         (  0  T22 ) n2
*                            n1  n2
*
*  where N = n1+n2 and Z' 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', then the reordered real Schur factorization of A is given
*  by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', 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
*
*  =====================================================================
*
*     .. 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
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLANGE
      EXTERNAL           LSAME, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACON, 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 DLACON( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE )
         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'*R - R*T22' = 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
      SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
     $                   LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK,
     $                   INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  JOB     (input) 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).
*
*  HOWMNY  (input) CHARACTER*1
*          = 'A': compute condition numbers for all eigenpairs;
*          = 'S': compute condition numbers for selected eigenpairs
*                 specified by the array SELECT.
*
*  SELECT  (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix T. N >= 0.
*
*  T       (input) DOUBLE PRECISION array, dimension (LDT,N)
*          The upper quasi-triangular matrix T, in Schur canonical form.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T. LDT >= max(1,N).
*
*  VL      (input) 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.
*
*  LDVL    (input) INTEGER
*          The leading dimension of the array VL.
*          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
*
*  VR      (input) 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.
*
*  LDVR    (input) INTEGER
*          The leading dimension of the array VR.
*          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
*
*  S       (output) 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.
*
*  SEP     (output) 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.
*
*  MM      (input) INTEGER
*          The number of elements in the arrays S (if JOB = 'E' or 'B')
*           and/or SEP (if JOB = 'V' or 'B'). MM >= M.
*
*  M       (output) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+1)
*          If JOB = 'E', WORK is not referenced.
*
*  LDWORK  (input) INTEGER
*          The leading dimension of the array WORK.
*          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
*
*  IWORK   (workspace) INTEGER array, dimension (N)
*          If JOB = 'E', IWORK is not referenced.
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*
*  Further Details
*  ===============
*
*  The reciprocal of the condition number of an eigenvalue lambda is
*  defined as
*
*          S(lambda) = |v'*u| / (norm(u)*norm(v))
*
*  where u and v are the right and left eigenvectors of T corresponding
*  to lambda; v' denotes the conjugate-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)
*
*  =====================================================================
*
*     .. 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 ..
      DOUBLE PRECISION   DUMMY( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT, DLAMCH, DLAPY2, DNRM2
      EXTERNAL           LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACON, 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
      CALL DLABAD( SMLNUM, BIGNUM )
*
      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' = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ]
*                                        [   mu                     ]
*                                        [         ..               ]
*                                        [             ..           ]
*                                        [                  mu      ]
*                 where C' is conjugate transpose of complex 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'))
*
               EST = ZERO
               KASE = 0
   50          CONTINUE
               CALL DLACON( NN, WORK( 1, N+2 ), WORK( 1, N+4 ), IWORK,
     $                      EST, KASE )
               IF( KASE.NE.0 ) THEN
                  IF( KASE.EQ.1 ) THEN
                     IF( N2.EQ.1 ) THEN
*
*                       Real eigenvalue: solve C'*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'*(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
      SUBROUTINE DTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
     $                   LDC, SCALE, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 31, 1993
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  TRANA   (input) 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)
*
*  TRANB   (input) 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)
*
*  ISGN    (input) 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
*
*  M       (input) INTEGER
*          The order of the matrix A, and the number of rows in the
*          matrices X and C. M >= 0.
*
*  N       (input) INTEGER
*          The order of the matrix B, and the number of columns in the
*          matrices X and C. N >= 0.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,M)
*          The upper quasi-triangular matrix A, in Schur canonical form.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,M).
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
*          The upper quasi-triangular matrix B, in Schur canonical form.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B. LDB >= max(1,N).
*
*  C       (input/output) 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.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M)
*
*  SCALE   (output) DOUBLE PRECISION
*          The scale factor, scale, set <= 1 to avoid overflow in X.
*
*  INFO    (output) 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).
*
*  =====================================================================
*
*     .. 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
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
*     Set constants to control overflow
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      CALL DLABAD( SMLNUM, BIGNUM )
      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 ) )
*
      SCALE = ONE
      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' *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)'*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
*
*        Where
*                   K-1                        L-1
*          R(K,L) = SUM [A(I,K)'*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'*X + ISGN*X*B' = scale*C.
*
*        The (K,L)th block of X is determined starting from
*        top-right 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
*                     K-1                          N
*            R(K,L) = SUM [A(I,K)'*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)'].
*                     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' = 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)' = 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)'].
*                    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
      INTEGER          FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1998
*
*     .. Scalar Arguments ..
      INTEGER            ISPEC
      REAL               ONE, ZERO
*     ..
*
*  Purpose
*  =======
*
*  IEEECK is called from the ILAENV to verify that Infinity and
*  possibly NaN arithmetic is safe (i.e. will not trap).
*
*  Arguments
*  =========
*
*  ISPEC   (input) INTEGER
*          Specifies whether to test just for inifinity arithmetic
*          or whether to test for infinity and NaN arithmetic.
*          = 0: Verify infinity arithmetic only.
*          = 1: Verify infinity and NaN arithmetic.
*
*  ZERO    (input) REAL
*          Must contain the value 0.0
*          This is passed to prevent the compiler from optimizing
*          away this code.
*
*  ONE     (input) 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
*
*     .. 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*0.0
*
      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
      INTEGER          FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3,
     $                 N4 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER*( * )    NAME, OPTS
      INTEGER            ISPEC, N1, N2, N3, N4
*     ..
*
*  Purpose
*  =======
*
*  ILAENV is called from the LAPACK routines to choose problem-dependent
*  parameters for the local environment.  See ISPEC for a description of
*  the parameters.
*
*  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.
*
*  Arguments
*  =========
*
*  ISPEC   (input) 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
*          = 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 and QZ methods
*               for nonsymmetric eigenvalue problems.
*          = 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 NaN arithmetic can be trusted not to trap
*          =11: infinity arithmetic can be trusted not to trap
*
*  NAME    (input) CHARACTER*(*)
*          The name of the calling subroutine, in either upper case or
*          lower case.
*
*  OPTS    (input) 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'.
*
*  N1      (input) INTEGER
*  N2      (input) INTEGER
*  N3      (input) INTEGER
*  N4      (input) INTEGER
*          Problem dimensions for the subroutine NAME; these may not all
*          be required.
*
* (ILAENV) (output) INTEGER
*          >= 0: the value of the parameter specified by ISPEC
*          < 0:  if ILAENV = -k, the k-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  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 )
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            CNAME, SNAME
      CHARACTER*1        C1
      CHARACTER*2        C2, C4
      CHARACTER*3        C3
      CHARACTER*6        SUBNAM
      INTEGER            I, IC, IZ, NB, NBMIN, NX
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          CHAR, ICHAR, INT, MIN, REAL
*     ..
*     .. External Functions ..
      INTEGER            IEEECK
      EXTERNAL           IEEECK
*     ..
*     .. Executable Statements ..
*
      GO TO ( 100, 100, 100, 400, 500, 600, 700, 800, 900, 1000,
     $        1100 ) ISPEC
*
*     Invalid value for ISPEC
*
      ILAENV = -1
      RETURN
*
  100 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 10 I = 2, 6
               IC = ICHAR( SUBNAM( I:I ) )
               IF( IC.GE.97 .AND. IC.LE.122 )
     $            SUBNAM( I:I ) = CHAR( IC-32 )
   10       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 20 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 )
   20       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 30 I = 2, 6
               IC = ICHAR( SUBNAM( I:I ) )
               IF( IC.GE.225 .AND. IC.LE.250 )
     $            SUBNAM( I:I ) = CHAR( IC-32 )
   30       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 )
*
      GO TO ( 110, 200, 300 ) ISPEC
*
  110 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( 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.'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
         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
               NB = 64
            ELSE
               NB = 64
            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
            NB = 64
         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
         END IF
      ELSE IF( C2.EQ.'LA' ) THEN
         IF( C3.EQ.'UUM' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
         IF( C3.EQ.'EBZ' ) THEN
            NB = 1
         END IF
      END IF
      ILAENV = NB
      RETURN
*
  200 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
         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
      END IF
      ILAENV = NBMIN
      RETURN
*
  300 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
         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
      END IF
      ILAENV = NX
      RETURN
*
  400 CONTINUE
*
*     ISPEC = 4:  number of shifts (used by xHSEQR)
*
      ILAENV = 6
      RETURN
*
  500 CONTINUE
*
*     ISPEC = 5:  minimum column dimension (not used)
*
      ILAENV = 2
      RETURN
*
  600 CONTINUE 
*
*     ISPEC = 6:  crossover point for SVD (used by xGELSS and xGESVD)
*
      ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
      RETURN
*
  700 CONTINUE
*
*     ISPEC = 7:  number of processors (not used)
*
      ILAENV = 1
      RETURN
*
  800 CONTINUE
*
*     ISPEC = 8:  crossover point for multishift (used by xHSEQR)
*
      ILAENV = 50
      RETURN
*
  900 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
*
 1000 CONTINUE
*
*     ISPEC = 10: ieee NaN arithmetic can be trusted not to trap
*
C     ILAENV = 0
      ILAENV = 1
      IF( ILAENV.EQ.1 ) THEN
         ILAENV = IEEECK( 0, 0.0, 1.0 ) 
      END IF
      RETURN
*
 1100 CONTINUE
*
*     ISPEC = 11: infinity arithmetic can be trusted not to trap
*
C     ILAENV = 0
      ILAENV = 1
      IF( ILAENV.EQ.1 ) THEN
         ILAENV = IEEECK( 1, 0.0, 1.0 ) 
      END IF
      RETURN
*
*     End of ILAENV
*
      END

      SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
     $                   LWORK, IWORK, INFO )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  The divide and conquer algorithm makes very mild assumptions about
*  floating point arithmetic. It will work on machines with a guard
*  digit in add/subtract, or on those binary machines without guard
*  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*  Cray-2. It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.
*
*  Arguments
*  =========
*
*  JOBZ    (input) 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 VT;
*          = 'N':  no columns of U or rows of V**T are computed.
*
*  M       (input) INTEGER
*          The number of rows of the input matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the input matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The singular values of A, sorted so that S(i) >= S(i+1).
*
*  U       (output) 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.
*
*  LDU     (input) INTEGER
*          The leading dimension of the array U.  LDU >= 1; if
*          JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*
*  VT      (output) 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.
*
*  LDVT    (input) 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).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK. LWORK >= 1.
*          If JOBZ = 'N',
*            LWORK >= 3*min(M,N) + max(max(M,N),6*min(M,N)).
*          If JOBZ = 'O',
*            LWORK >= 3*min(M,N)*min(M,N) + 
*                     max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
*          If JOBZ = 'S' or 'A'
*            LWORK >= 3*min(M,N)*min(M,N) +
*                     max(max(M,N),4*min(M,N)*min(M,N)+4*min(M,N)).
*          For good performance, LWORK should generally be larger.
*          If LWORK < 0 but other input arguments are legal, WORK(1)
*          returns the optimal LWORK.
*
*  IWORK   (workspace) INTEGER array, dimension (8*min(M,N))
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  DBDSDC did not converge, updating process failed.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
      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
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           DLAMCH, DLANGE, ILAENV, LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, INT, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      MINMN = MIN( M, N )
      MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
      WNTQA = LSAME( JOBZ, 'A' )
      WNTQS = LSAME( JOBZ, 'S' )
      WNTQAS = WNTQA .OR. WNTQS
      WNTQO = LSAME( JOBZ, 'O' )
      WNTQN = LSAME( JOBZ, 'N' )
      MINWRK = 1
      MAXWRK = 1
      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 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 .AND. M.GT.0 .AND. N.GT.0 ) THEN
         IF( M.GE.N ) THEN
*
*           Compute space needed for DBDSDC
*
            IF( WNTQN ) THEN
               BDSPAC = 7*N
            ELSE
               BDSPAC = 3*N*N + 4*N
            END IF
            IF( M.GE.MNTHR ) THEN
               IF( WNTQN ) THEN
*
*                 Path 1 (M much larger than N, JOBZ='N')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1,
     $                    -1 )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  MAXWRK = MAX( WRKBL, BDSPAC+N )
                  MINWRK = BDSPAC + N
               ELSE IF( WNTQO ) THEN
*
*                 Path 2 (M much larger than N, JOBZ='O')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
                  MAXWRK = WRKBL + 2*N*N
                  MINWRK = BDSPAC + 2*N*N + 3*N
               ELSE IF( WNTQS ) THEN
*
*                 Path 3 (M much larger than N, JOBZ='S')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
                  MAXWRK = WRKBL + N*N
                  MINWRK = BDSPAC + N*N + 3*N
               ELSE IF( WNTQA ) THEN
*
*                 Path 4 (M much larger than N, JOBZ='A')
*
                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
     $                    M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
                  MAXWRK = WRKBL + N*N
                  MINWRK = BDSPAC + N*N + 3*N
               END IF
            ELSE
*
*              Path 5 (M at least N, but not much larger)
*
               WRKBL = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
     $                 -1 )
               IF( WNTQN ) THEN
                  MAXWRK = MAX( WRKBL, BDSPAC+3*N )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               ELSE IF( WNTQO ) THEN
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
                  MAXWRK = WRKBL + M*N
                  MINWRK = 3*N + MAX( M, N*N+BDSPAC )
               ELSE IF( WNTQS ) THEN
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                  MAXWRK = MAX( WRKBL, BDSPAC+3*N )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               ELSE IF( WNTQA ) THEN
                  WRKBL = MAX( WRKBL, 3*N+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                  MAXWRK = MAX( MAXWRK, BDSPAC+3*N )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               END IF
            END IF
         ELSE
*
*           Compute space needed for DBDSDC
*
            IF( WNTQN ) THEN
               BDSPAC = 7*M
            ELSE
               BDSPAC = 3*M*M + 4*M
            END IF
            IF( N.GE.MNTHR ) THEN
               IF( WNTQN ) THEN
*
*                 Path 1t (N much larger than M, JOBZ='N')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
     $                    -1 )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  MAXWRK = MAX( WRKBL, BDSPAC+M )
                  MINWRK = BDSPAC + M
               ELSE IF( WNTQO ) THEN
*
*                 Path 2t (N much larger than M, JOBZ='O')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
                  MAXWRK = WRKBL + 2*M*M
                  MINWRK = BDSPAC + 2*M*M + 3*M
               ELSE IF( WNTQS ) THEN
*
*                 Path 3t (N much larger than M, JOBZ='S')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
                  MAXWRK = WRKBL + M*M
                  MINWRK = BDSPAC + M*M + 3*M
               ELSE IF( WNTQA ) THEN
*
*                 Path 4t (N much larger than M, JOBZ='A')
*
                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
                  MAXWRK = WRKBL + M*M
                  MINWRK = BDSPAC + M*M + 3*M
               END IF
            ELSE
*
*              Path 5t (N greater than M, but not much larger)
*
               WRKBL = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
     $                 -1 )
               IF( WNTQN ) THEN
                  MAXWRK = MAX( WRKBL, BDSPAC+3*M )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               ELSE IF( WNTQO ) THEN
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
                  MAXWRK = WRKBL + M*N
                  MINWRK = 3*M + MAX( N, M*M+BDSPAC )
               ELSE IF( WNTQS ) THEN
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
                  MAXWRK = MAX( WRKBL, BDSPAC+3*M )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               ELSE IF( WNTQA ) THEN
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, M, -1 ) )
                  MAXWRK = MAX( WRKBL, BDSPAC+3*M )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               END IF
            END IF
         END IF
         WORK( 1 ) = MAXWRK
      END IF
*
      IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
         INFO = -12
      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
         IF( LWORK.GE.1 )
     $      WORK( 1 ) = ONE
         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( WNTQN ) THEN
*
*              Path 1 (M much larger than N, JOBZ='N')
*              No singular vectors to be computed
*
               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, 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 4*N, prefer 3*N+2*N*NB)
*
               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+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 much larger than 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+2*N, prefer N*N+N+N*NB)
*
               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+2*N, prefer N*N+N+N*NB)
*
               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 VT, copying result to 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( 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*N+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 2*N*N+3*N, prefer 2*N*N+2*N+N*NB)
*
               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 2*N*N, prefer N*N+M*N)
*
               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 much larger than 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+2*N, prefer N*N+N+N*NB)
*
               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+2*N, prefer N*N+N+N*NB)
*
               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+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( 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+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+3*N, prefer N*N+2*N+N*NB)
*
               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)
*
               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 much larger than 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+2*N, prefer N*N+N+N*NB)
*
               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+2*N, prefer N*N+N+N*NB)
               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+4*N, prefer N*N+3*N+2*N*NB)
*
               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*N+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+3*N, prefer N*N+2*N+N*NB)
*
               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)
*
               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 at least 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+M, prefer 3*N+(M+N)*NB)
*
            CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                   IERR )
            IF( WNTQN ) THEN
*
*              Perform bidiagonal SVD, only computing singular values
*              (Workspace: need N+BDSPAC)
*
               CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
            ELSE IF( WNTQO ) THEN
               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 )
               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 N+N*N+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 N*N+2*N, prefer N*N+N+N*NB)
*
               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
*
*                 Overwrite WORK(IU) by left singular vectors of A
*                 (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
                  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
*
*                 Generate Q in A
*                 (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
                  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 2*N*N, prefer N*N+M*N)
*
                  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
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              (Workspace: need N+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, prefer 2*N+N*NB)
*
               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
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              (Workspace: need N+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
*
               CALL DLASET( 'F', M-N, M-N, ZERO, ONE, U( N+1, N+1 ),
     $                      LDU )
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              (Workspace: need N*N+2*N+M, prefer N*N+2*N+M*NB)
*
               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 much larger than M, JOBZ='N')
*              No singular vectors to be computed
*
               ITAU = 1
               NWORK = ITAU + M
*
*              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, 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 4*M, prefer 3*M+2*M*NB)
*
               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+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 much larger than M, JOBZ='O')
*              M right singular vectors to be overwritten on A and
*              M left singular vectors to be computed in U
*
               IVT = 1
*
*              IVT is M by M
*
               IL = IVT + M*M
               IF( LWORK.GE.M*N+M*M+3*M+BDSPAC ) THEN
*
*                 WORK(IL) is M by N
*
                  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+2*M, prefer M*M+M+M*NB)
*
               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+2*M, prefer M*M+M+M*NB)
*
               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+4*M, prefer M*M+3*M+2*M*NB)
*
               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*M+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 2*M*M+3*M, prefer 2*M*M+2*M+M*NB)
*
               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 2*M*M, prefer M*M+M*N)
*
               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 much larger than 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+2*M, prefer M*M+M+M*NB)
*
               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+2*M, prefer M*M+M+M*NB)
*
               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), copying result to U
*              (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*
               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+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+3*M, prefer M*M+2*M+M*NB)
*
               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)
*
               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 much larger than 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+2*M, prefer M*M+M+M*NB)
*
               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+2*M, prefer M*M+M+M*NB)
*
               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+4*M, prefer M*M+3*M+2*M*NB)
*
               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*M+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+3*M, prefer M*M+2*M+M*NB)
*
               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)
*
               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 greater than 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+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,
     $                   IERR )
            IF( WNTQN ) THEN
*
*              Perform bidiagonal SVD, only computing singular values
*              (Workspace: need M+BDSPAC)
*
               CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
            ELSE IF( WNTQO ) THEN
               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
               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 M*M+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 M*M+2*M, prefer M*M+M+M*NB)
*
               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
*
*                 Overwrite WORK(IVT) by left singular vectors of A
*                 (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
                  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
*
*                 Generate P**T in A
*                 (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
                  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 2*M*M, prefer M*M+M*N)
*
                  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
*
*              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+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, prefer 2*M+M*NB)
*
               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
*
*              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+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
*
               CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT( M+1, M+1 ),
     $                      LDVT )
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              (Workspace: need 2*M+N, prefer 2*M+N*NB)
*
               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 ) = DBLE( MAXWRK )
*
      RETURN
*
*     End of DGESDD
*
      END
      SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     December 1, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  This code makes very mild assumptions about floating point
*  arithmetic. It will work on machines with a guard digit in
*  add/subtract, or on those binary machines without guard digits
*  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*  It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.  See DLASD3 for details.
*
*  The code currently call DLASDQ if singular values only are desired.
*  However, it can be slightly modified to compute singular values
*  using the divide and conquer method.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  B is upper bidiagonal.
*          = 'L':  B is lower bidiagonal.
*
*  COMPQ   (input) 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.
*
*  N       (input) INTEGER
*          The order of the matrix B.  N >= 0.
*
*  D       (input/output) 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.
*
*  E       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the elements of E contain the offdiagonal
*          elements of the bidiagonal matrix whose SVD is desired.
*          On exit, E has been destroyed.
*
*  U       (output) 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.
*
*  LDU     (input) INTEGER
*          The leading dimension of the array U.  LDU >= 1.
*          If singular vectors are desired, then LDU >= max( 1, N ).
*
*  VT      (output) DOUBLE PRECISION array, dimension (LDVT,N)
*          If  COMPQ = 'I', then:
*             On exit, if INFO = 0, VT' contains the right singular
*             vectors of the bidiagonal matrix.
*          For other values of COMPQ, VT is not referenced.
*
*  LDVT    (input) INTEGER
*          The leading dimension of the array VT.  LDVT >= 1.
*          If singular vectors are desired, then LDVT >= max( 1, N ).
*
*  Q       (output) 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.
*
*  IQ      (output) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (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).
*
*  IWORK   (workspace) INTEGER array, dimension (8*N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  The algorithm failed to compute an singular value.
*                The update process of divide and conquer failed.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
         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
         CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
     $                LDU, WORK( WSTART ), 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 = 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 )
               IF( INFO.NE.0 ) THEN
                  RETURN
               END IF
            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
      SUBROUTINE DLASDQ( UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT,
     $                   U, LDU, C, LDC, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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' (P' 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' * VT if desired.
*  The input matrix C  is changed to Q' * 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.
*
*  Arguments
*  =========
*
*  UPLO  (input) CHARACTER*1
*        On entry, UPLO specifies whether the input bidiagonal matrix
*        is upper or lower bidiagonal, and wether it is square are
*        not.
*           UPLO = 'U' or 'u'   B is upper bidiagonal.
*           UPLO = 'L' or 'l'   B is lower bidiagonal.
*
*  SQRE  (input) 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.
*
*  N     (input) INTEGER
*        On entry, N specifies the number of rows and columns
*        in the matrix. N must be at least 0.
*
*  NCVT  (input) INTEGER
*        On entry, NCVT specifies the number of columns of
*        the matrix VT. NCVT must be at least 0.
*
*  NRU   (input) INTEGER
*        On entry, NRU specifies the number of rows of
*        the matrix U. NRU must be at least 0.
*
*  NCC   (input) INTEGER
*        On entry, NCC specifies the number of columns of
*        the matrix C. NCC must be at least 0.
*
*  D     (input/output) 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.
*
*  E     (input/output) 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.
*
*  VT    (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT)
*        On entry, contains a matrix which on exit has been
*        premultiplied by P', dimension N-by-NCVT if SQRE = 0
*        and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).
*
*  LDVT  (input) 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.
*
*  U     (input/output) 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).
*
*  LDU   (input) 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 ) .
*
*  C     (input/output) DOUBLE PRECISION array, dimension (LDC, NCC)
*        On entry, contains an N-by-NCC matrix which on exit
*        has been premultiplied by Q'  dimension N-by-NCC if SQRE = 0
*        and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).
*
*  LDC   (input) 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.
*
*  WORK  (workspace) 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.
*
*  INFO  (output) 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.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK,
     $                   WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDU, LDVT, N, SMLSIZ, SQRE
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
     $                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  N      (input) INTEGER
*         On entry, the row dimension of the upper bidiagonal matrix.
*         This is also the dimension of the main diagonal array D.
*
*  SQRE   (input) 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;
*
*  D      (input/output) 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.
*
*  E      (input) DOUBLE PRECISION array, dimension (M-1)
*         Contains the subdiagonal entries of the bidiagonal matrix.
*         On exit, E has been destroyed.
*
*  U      (output) DOUBLE PRECISION array, dimension at least (LDQ, N)
*         On exit, U contains the left singular vectors.
*
*  LDU    (input) INTEGER
*         On entry, leading dimension of U.
*
*  VT     (output) DOUBLE PRECISION array, dimension at least (LDVT, M)
*         On exit, VT' contains the right singular vectors.
*
*  LDVT   (input) INTEGER
*         On entry, leading dimension of VT.
*
*  SMLSIZ (input) INTEGER
*         On entry, maximum size of the subproblems at the
*         bottom of the computation tree.
*
*  IWORK  INTEGER work array.
*         Dimension must be at least (8 * N)
*
*  WORK   DOUBLE PRECISION work array.
*         Dimension must be at least (3 * M**2 + 2 * M)
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = 1, an singular value did not converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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 )
            IF( INFO.NE.0 ) THEN
               RETURN
            END IF
   40    CONTINUE
   50 CONTINUE
*
      RETURN
*
*     End of DLASD0
*
      END
      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 (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1999
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  ICOMPQ (input) 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.
*
*  SMLSIZ (input) INTEGER
*         The maximum size of the subproblems at the bottom of the
*         computation tree.
*
*  N      (input) INTEGER
*         The row dimension of the upper bidiagonal matrix. This is
*         also the dimension of the main diagonal array D.
*
*  SQRE   (input) 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.
*
*  D      (input/output) 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.
*
*  E      (input) DOUBLE PRECISION array, dimension ( M-1 )
*         Contains the subdiagonal entries of the bidiagonal matrix.
*         On exit, E has been destroyed.
*
*  U      (output) 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.
*
*  LDU    (input) INTEGER, LDU = > N.
*         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
*         GIVNUM, and Z.
*
*  VT     (output) DOUBLE PRECISION array,
*         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right
*         singular vector matrices of all subproblems at the bottom
*         level.
*
*  K      (output) 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.
*
*  DIFL   (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
*         where NLVL = floor(log_2 (N/SMLSIZ))).
*
*  DIFR   (output) 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.
*
*  Z      (output) 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.
*
*  POLES  (output) 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.
*
*  GIVPTR (output) 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.
*
*  GIVCOL (output) 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.
*
*  LDGCOL (input) INTEGER, LDGCOL = > N.
*         The leading dimension of arrays GIVCOL and PERM.
*
*  PERM   (output) 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.
*
*  GIVNUM (output) 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.
*
*  C      (output) 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.
*
*  S      (output) 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.
*
*  WORK   (workspace) DOUBLE PRECISION array, dimension
*         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
*
*  IWORK  (workspace) INTEGER array.
*         Dimension must be at least (7 * N).
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = 1, an singular value did not converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLASDT( N, LVL, ND, INODE, NDIML, NDIMR, MSUB )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            LVL, MSUB, N, ND
*     ..
*     .. Array Arguments ..
      INTEGER            INODE( * ), NDIML( * ), NDIMR( * )
*     ..
*
*  Purpose
*  =======
*
*  DLASDT creates a tree of subproblems for bidiagonal divide and
*  conquer.
*
*  Arguments
*  =========
*
*   N      (input) INTEGER
*          On entry, the number of diagonal elements of the
*          bidiagonal matrix.
*
*   LVL    (output) INTEGER
*          On exit, the number of levels on the computation tree.
*
*   ND     (output) INTEGER
*          On exit, the number of nodes on the tree.
*
*   INODE  (output) INTEGER array, dimension ( N )
*          On exit, centers of subproblems.
*
*   NDIML  (output) INTEGER array, dimension ( N )
*          On exit, row dimensions of left children.
*
*   NDIMR  (output) INTEGER array, dimension ( N )
*          On exit, row dimensions of right children.
*
*   MSUB   (input) INTEGER.
*          On entry, the maximum row dimension each subproblem at the
*          bottom of the tree can be of.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
     $                   IDXQ, IWORK, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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'   a   Z2'    b ) * VT(in)
*                (   0     0   D2(in) 0 )
*
*      = U(out) * ( D(out) 0) * VT(out)
*
*  where Z' = (Z1' a Z2' b) = u' VT', 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 occurence 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.
*
*  Arguments
*  =========
*
*  NL     (input) INTEGER
*         The row dimension of the upper block.  NL >= 1.
*
*  NR     (input) INTEGER
*         The row dimension of the lower block.  NR >= 1.
*
*  SQRE   (input) 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.
*
*  D      (input/output) 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.
*
*  ALPHA  (input) DOUBLE PRECISION
*         Contains the diagonal element associated with the added row.
*
*  BETA   (input) DOUBLE PRECISION
*         Contains the off-diagonal element associated with the added
*         row.
*
*  U      (input/output) 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.
*
*  LDU    (input) INTEGER
*         The leading dimension of the array U.  LDU >= max( 1, N ).
*
*  VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
*         where M = N + SQRE.
*         On entry VT(1:NL+1, 1:NL+1)' contains the right singular
*         vectors of the upper block; VT(NL+2:M, NL+2:M)' contains
*         the right singular vectors of the lower block. On exit
*         VT' contains the right singular vectors of the
*         bidiagonal matrix.
*
*  LDVT   (input) INTEGER
*         The leading dimension of the array VT.  LDVT >= max( 1, M ).
*
*  IDXQ  (output) 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.
*
*  IWORK  (workspace) INTEGER array, dimension( 4 * N )
*
*  WORK   (workspace) DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = 1, an singular value did not converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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 )
      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
      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 (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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'   a   Z2'    b ) * VT(in)
*                (   0     0   D2(in) 0 )
*
*      = U(out) * ( D(out) 0) * VT(out)
*
*  where Z' = (Z1' a Z2' b) = u' VT', 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 occurence 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.
*
*  Arguments
*  =========
*
*  ICOMPQ (input) 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.
*
*  NL     (input) INTEGER
*         The row dimension of the upper block.  NL >= 1.
*
*  NR     (input) INTEGER
*         The row dimension of the lower block.  NR >= 1.
*
*  SQRE   (input) 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.
*
*  D      (input/output) 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.
*
*  VF     (input/output) 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.
*
*  VL     (input/output) 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.
*
*  ALPHA  (input) DOUBLE PRECISION
*         Contains the diagonal element associated with the added row.
*
*  BETA   (input) DOUBLE PRECISION
*         Contains the off-diagonal element associated with the added
*         row.
*
*  IDXQ   (output) 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.
*
*  PERM   (output) INTEGER array, dimension ( N )
*         The permutations (from deflation and sorting) to be applied
*         to each block. Not referenced if ICOMPQ = 0.
*
*  GIVPTR (output) INTEGER
*         The number of Givens rotations which took place in this
*         subproblem. Not referenced if ICOMPQ = 0.
*
*  GIVCOL (output) 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.
*
*  LDGCOL (input) INTEGER
*         leading dimension of GIVCOL, must be at least N.
*
*  GIVNUM (output) 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.
*
*  LDGNUM (input) INTEGER
*         The leading dimension of GIVNUM and POLES, must be at least N.
*
*  POLES  (output) 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.
*
*  DIFL   (output) 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.
*
*  DIFR   (output) DOUBLE PRECISION array,
*                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
*                  dimension ( N ) if ICOMPQ = 0.
*         On exit, DIFR(I, 1) is the distance between I-th updated
*         (undeflated) singular value and the I+1-th (undeflated) old
*         singular value.
*
*         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.
*
*  Z      (output) DOUBLE PRECISION array, dimension ( M )
*         The first elements of this array contain the components
*         of the deflation-adjusted updating row vector.
*
*  K      (output) INTEGER
*         Contains the dimension of the non-deflated matrix,
*         This is the order of the related secular equation. 1 <= K <=N.
*
*  C      (output) 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.
*
*  S      (output) 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.
*
*  WORK   (workspace) DOUBLE PRECISION array, dimension ( 4 * M )
*
*  IWORK  (workspace) INTEGER array, dimension ( 3 * N )
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = 1, an singular value did not converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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 )
*
*     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
      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 (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     October 31, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  NL     (input) INTEGER
*         The row dimension of the upper block.  NL >= 1.
*
*  NR     (input) INTEGER
*         The row dimension of the lower block.  NR >= 1.
*
*  SQRE   (input) 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.
*
*  K      (output) INTEGER
*         Contains the dimension of the non-deflated matrix,
*         This is the order of the related secular equation. 1 <= K <=N.
*
*  D      (input/output) 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.
*
*  ALPHA  (input) DOUBLE PRECISION
*         Contains the diagonal element associated with the added row.
*
*  BETA   (input) DOUBLE PRECISION
*         Contains the off-diagonal element associated with the added
*         row.
*
*  U      (input/output) 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.
*
*  LDU    (input) INTEGER
*         The leading dimension of the array U.  LDU >= N.
*
*  Z      (output) DOUBLE PRECISION array, dimension(N)
*         On exit Z contains the updating row vector in the secular
*         equation.
*
*  DSIGMA (output) DOUBLE PRECISION array, dimension (N)
*         Contains a copy of the diagonal elements (K-1 singular values
*         and one zero) in the secular equation.
*
*  U2     (output) 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.
*
*  LDU2   (input) INTEGER
*         The leading dimension of the array U2.  LDU2 >= N.
*
*  VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
*         On entry VT' 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' 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.
*
*  LDVT   (input) INTEGER
*         The leading dimension of the array VT.  LDVT >= M.
*
*  VT2    (output) DOUBLE PRECISION array, dimension(LDVT2,N)
*         VT2' 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.
*
*  LDVT2  (input) INTEGER
*         The leading dimension of the array VT2.  LDVT2 >= M.
*
*  IDXP   (workspace) 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.
*
*  IDX    (workspace) INTEGER array, dimension(N)
*         This will contain the permutation used to sort the contents of
*         D into ascending order.
*
*  IDXC   (output) 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.
*
*  COLTYP (workspace/output) 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.
*
*  IDXQ   (input) 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.
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
      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 (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     October 31, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  This code makes very mild assumptions about floating point
*  arithmetic. It will work on machines with a guard digit in
*  add/subtract, or on those binary machines without guard digits
*  which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
*  It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.
*
*  DLASD3 is called from DLASD1.
*
*  Arguments
*  =========
*
*  NL     (input) INTEGER
*         The row dimension of the upper block.  NL >= 1.
*
*  NR     (input) INTEGER
*         The row dimension of the lower block.  NR >= 1.
*
*  SQRE   (input) 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.
*
*  K      (input) INTEGER
*         The size of the secular equation, 1 =< K = < N.
*
*  D      (output) DOUBLE PRECISION array, dimension(K)
*         On exit the square roots of the roots of the secular equation,
*         in ascending order.
*
*  Q      (workspace) DOUBLE PRECISION array,
*                     dimension at least (LDQ,K).
*
*  LDQ    (input) INTEGER
*         The leading dimension of the array Q.  LDQ >= K.
*
*  DSIGMA (input) 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.
*
*  U      (input) DOUBLE PRECISION array, dimension (LDU, N)
*         The last N - K columns of this matrix contain the deflated
*         left singular vectors.
*
*  LDU    (input) INTEGER
*         The leading dimension of the array U.  LDU >= N.
*
*  U2     (input) DOUBLE PRECISION array, dimension (LDU2, N)
*         The first K columns of this matrix contain the non-deflated
*         left singular vectors for the split problem.
*
*  LDU2   (input) INTEGER
*         The leading dimension of the array U2.  LDU2 >= N.
*
*  VT     (input) DOUBLE PRECISION array, dimension (LDVT, M)
*         The last M - K columns of VT' contain the deflated
*         right singular vectors.
*
*  LDVT   (input) INTEGER
*         The leading dimension of the array VT.  LDVT >= N.
*
*  VT2    (input) DOUBLE PRECISION array, dimension (LDVT2, N)
*         The first K columns of VT2' contain the non-deflated
*         right singular vectors for the split problem.
*
*  LDVT2  (input) INTEGER
*         The leading dimension of the array VT2.  LDVT2 >= N.
*
*  IDXC   (input) 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.
*
*  CTOT   (input) 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.
*
*  Z      (input) DOUBLE PRECISION array, dimension (K)
*         The first K elements of this array contain the components
*         of the deflation-adjusted updating row vector.
*
*  INFO   (output) INTEGER
*         = 0:  successful exit.
*         < 0:  if INFO = -i, the i-th argument had an illegal value.
*         > 0:  if INFO = 1, an singular value did not converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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   DLAMC3, DNRM2
      EXTERNAL           DLAMC3, 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
*
*     Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
*     be computed with high relative accuracy (barring over/underflow).
*     This is a problem on machines without a guard digit in
*     add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
*     The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
*     which on any of these machines zeros out the bottommost
*     bit of DSIGMA(I) if it is 1; this makes the subsequent
*     subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
*     occurs. On binary machines with a guard digit (almost all
*     machines) it does not change DSIGMA(I) at all. On hexadecimal
*     and decimal machines with a guard digit, it slightly
*     changes the bottommost bits of DSIGMA(I). It does not account
*     for hexadecimal or decimal machines without guard digits
*     (we know of none). We use a subroutine call to compute
*     2*DLAMBDA(I) to prevent optimizing compilers from eliminating
*     this code.
*
      DO 20 I = 1, K
         DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
   20 CONTINUE
*
*     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, the computation is terminated.
*
         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
      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 (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  ICOMPQ  (input) 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.
*
*  NL     (input) INTEGER
*         The row dimension of the upper block. NL >= 1.
*
*  NR     (input) INTEGER
*         The row dimension of the lower block. NR >= 1.
*
*  SQRE   (input) 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.
*
*  K      (output) INTEGER
*         Contains the dimension of the non-deflated matrix, this is
*         the order of the related secular equation. 1 <= K <=N.
*
*  D      (input/output) 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.
*
*  Z      (output) DOUBLE PRECISION array, dimension ( M )
*         On exit Z contains the updating row vector in the secular
*         equation.
*
*  ZW     (workspace) DOUBLE PRECISION array, dimension ( M )
*         Workspace for Z.
*
*  VF     (input/output) 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.
*
*  VFW    (workspace) DOUBLE PRECISION array, dimension ( M )
*         Workspace for VF.
*
*  VL     (input/output) 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.
*
*  VLW    (workspace) DOUBLE PRECISION array, dimension ( M )
*         Workspace for VL.
*
*  ALPHA  (input) DOUBLE PRECISION
*         Contains the diagonal element associated with the added row.
*
*  BETA   (input) DOUBLE PRECISION
*         Contains the off-diagonal element associated with the added
*         row.
*
*  DSIGMA (output) DOUBLE PRECISION array, dimension ( N )
*         Contains a copy of the diagonal elements (K-1 singular values
*         and one zero) in the secular equation.
*
*  IDX    (workspace) INTEGER array, dimension ( N )
*         This will contain the permutation used to sort the contents of
*         D into ascending order.
*
*  IDXP   (workspace) 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.
*
*  IDXQ   (input) 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.
*
*  PERM   (output) INTEGER array, dimension ( N )
*         The permutations (from deflation and sorting) to be applied
*         to each singular block. Not referenced if ICOMPQ = 0.
*
*  GIVPTR (output) INTEGER
*         The number of Givens rotations which took place in this
*         subproblem. Not referenced if ICOMPQ = 0.
*
*  GIVCOL (output) 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.
*
*  LDGCOL (input) INTEGER
*         The leading dimension of GIVCOL, must be at least N.
*
*  GIVNUM (output) 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.
*
*  LDGNUM (input) INTEGER
*         The leading dimension of GIVNUM, must be at least N.
*
*  C      (output) 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.
*
*  S      (output) 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.
*
*  INFO   (output) INTEGER
*         = 0:  successful exit.
*         < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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 tolerence
*
      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
      SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
     $                   DSIGMA, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            ICOMPQ, INFO, K, LDDIFR
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( LDDIFR, * ),
     $                   DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
     $                   Z( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  ICOMPQ  (input) 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.
*
*  K       (input) INTEGER
*          The number of terms in the rational function to be solved
*          by DLASD4.  K >= 1.
*
*  D       (output) DOUBLE PRECISION array, dimension ( K )
*          On output, D contains the updated singular values.
*
*  Z       (input) DOUBLE PRECISION array, dimension ( K )
*          The first K elements of this array contain the components
*          of the deflation-adjusted updating row vector.
*
*  VF      (input/output) 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.
*
*  VL      (input/output) 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.
*
*  DIFL    (output) DOUBLE PRECISION array, dimension ( K )
*          On exit, DIFL(I) = D(I) - DSIGMA(I).
*
*  DIFR    (output) 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.
*
*  LDDIFR  (input) INTEGER
*          The leading dimension of DIFR, must be at least K.
*
*  DSIGMA  (input) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension at least 3 * K
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = 1, an singular value did not converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
*
*     Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
*     be computed with high relative accuracy (barring over/underflow).
*     This is a problem on machines without a guard digit in
*     add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
*     The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
*     which on any of these machines zeros out the bottommost
*     bit of DSIGMA(I) if it is 1; this makes the subsequent
*     subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
*     occurs. On binary machines with a guard digit (almost all
*     machines) it does not change DSIGMA(I) at all. On hexadecimal
*     and decimal machines with a guard digit, it slightly
*     changes the bottommost bits of DSIGMA(I). It does not account
*     for hexadecimal or decimal machines without guard digits
*     (we know of none). We use a subroutine call to compute
*     2*DLAMBDA(I) to prevent optimizing compilers from eliminating
*     this code.
*
      DO 10 I = 1, K
         DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
   10 CONTINUE
*
*     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, the computation is terminated.
*
         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 )
         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
      SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      INTEGER            DTRD1, DTRD2, N1, N2
*     ..
*     .. Array Arguments ..
      INTEGER            INDEX( * )
      DOUBLE PRECISION   A( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  N1     (input) INTEGER
*  N2     (input) INTEGER
*         These arguements contain the respective lengths of the two
*         sorted lists to be merged.
*
*  A      (input) 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.
*
*  DTRD1  (input) INTEGER
*  DTRD2  (input) 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.
*
*  INDEX  (output) 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.
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     October 31, 1999
*
*     .. Scalar Arguments ..
      INTEGER            I, INFO, N
      DOUBLE PRECISION   RHO, SIGMA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DELTA( * ), WORK( * ), Z( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  N      (input) INTEGER
*         The length of all arrays.
*
*  I      (input) INTEGER
*         The index of the eigenvalue to be computed.  1 <= I <= N.
*
*  D      (input) DOUBLE PRECISION array, dimension ( N )
*         The original eigenvalues.  It is assumed that they are in
*         order, 0 <= D(I) < D(J)  for I < J.
*
*  Z      (input) DOUBLE PRECISION array, dimension ( N )
*         The components of the updating vector.
*
*  DELTA  (output) 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.
*
*  RHO    (input) DOUBLE PRECISION
*         The scalar in the symmetric updating formula.
*
*  SIGMA  (output) DOUBLE PRECISION
*         The computed lambda_I, the I-th updated eigenvalue.
*
*  WORK   (workspace) 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.
*
*  INFO   (output) INTEGER
*         = 0:  successful exit
*         > 0:  if INFO = 1, the updating process failed.
*
*  Internal Parameters
*  ===================
*
*  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.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ren-Cang Li, Computer Science Division, University of California
*     at Berkeley, USA
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 20 )
      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
      INTEGER            II, IIM1, IIP1, IP1, ITER, J, NITER
      DOUBLE PRECISION   A, B, C, DELSQ, DELSQ2, DPHI, DPSI, DTIIM,
     $                   DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS,
     $                   ERRETM, ETA, PHI, PREW, PSI, RHOINV, SG2LB,
     $                   SG2UB, TAU, 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
*
*     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 TAU 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
                  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)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU <= 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 TAU is to approximate
*           SIGMA_n^2 - D( N )*D( N )
*
            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)^2 < D(N)^2+TAU < SIGMA(N)^2 < D(N)^2+RHO/2
*
         END IF
*
*        The following ETA is to approximate SIGMA_n - D( N )
*
         ETA = TAU / ( D( N )+SQRT( D( N )*D( N )+TAU ) )
*
         SIGMA = D( N ) + ETA
         DO 30 J = 1, N
            DELTA( J ) = ( D( J )-D( I ) ) - ETA
            WORK( J ) = D( J ) + D( I ) + ETA
   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( TAU )*( 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
*
         TAU = TAU + ETA
         ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
         DO 50 J = 1, N
            DELTA( J ) = DELTA( J ) - ETA
            WORK( J ) = WORK( J ) + ETA
   50    CONTINUE
*
         SIGMA = SIGMA + ETA
*
*        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
*
         TEMP = Z( N ) / ( WORK( 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
               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
*
            TAU = TAU + ETA
            ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
            DO 70 J = 1, N
               DELTA( J ) = DELTA( J ) - ETA
               WORK( J ) = WORK( J ) + ETA
   70       CONTINUE
*
            SIGMA = SIGMA + ETA
*
*           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
*
            TEMP = Z( N ) / ( WORK( 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
         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
         TEMP = DELSQ2 / ( D( I )+SQRT( D( I )*D( I )+DELSQ2 ) )
         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 ) )
*
         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.
            SG2LB = ZERO
            SG2UB = DELSQ2
            A = C*DELSQ + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
            B = Z( I )*Z( I )*DELSQ
            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
*
*           TAU now is an estimation of SIGMA^2 - D( I )^2. The
*           following, however, is the corresponding estimation of
*           SIGMA - D( I ).
*
            ETA = TAU / ( D( I )+SQRT( D( I )*D( I )+TAU ) )
         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.
            SG2LB = -DELSQ2
            SG2UB = ZERO
            A = C*DELSQ - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
            B = Z( IP1 )*Z( IP1 )*DELSQ
            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
*
*           TAU now is an estimation of SIGMA^2 - D( IP1 )^2. The
*           following, however, is the corresponding estimation of
*           SIGMA - D( IP1 ).
*
            ETA = TAU / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+
     $            TAU ) ) )
         END IF
*
         IF( ORGATI ) THEN
            II = I
            SIGMA = D( I ) + ETA
            DO 130 J = 1, N
               WORK( J ) = D( J ) + D( I ) + ETA
               DELTA( J ) = ( D( J )-D( I ) ) - ETA
  130       CONTINUE
         ELSE
            II = I + 1
            SIGMA = D( IP1 ) + ETA
            DO 140 J = 1, N
               WORK( J ) = D( J ) + D( IP1 ) + ETA
               DELTA( J ) = ( D( J )-D( IP1 ) ) - ETA
  140       CONTINUE
         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 ) / ( 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( TAU )*DW
*
*        Test for convergence
*
         IF( ABS( W ).LE.EPS*ERRETM ) THEN
            GO TO 240
         END IF
*
         IF( W.LE.ZERO ) THEN
            SG2LB = MAX( SG2LB, TAU )
         ELSE
            SG2UB = MIN( SG2UB, 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 )
     $         GO TO 240
         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
         IF( ORGATI ) THEN
            TEMP1 = WORK( I )*DELTA( I )
            TEMP = ETA - TEMP1
         ELSE
            TEMP1 = WORK( IP1 )*DELTA( IP1 )
            TEMP = ETA - TEMP1
         END IF
         IF( TEMP.GT.SG2UB .OR. TEMP.LT.SG2LB ) THEN
            IF( W.LT.ZERO ) THEN
               ETA = ( SG2UB-TAU ) / TWO
            ELSE
               ETA = ( SG2LB-TAU ) / TWO
            END IF
         END IF
*
         TAU = TAU + ETA
         ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
*
         PREW = W
*
         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
*
         TEMP = Z( II ) / ( WORK( 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.LE.ZERO ) THEN
            SG2LB = MAX( SG2LB, TAU )
         ELSE
            SG2UB = MIN( SG2UB, TAU )
         END IF
*
         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
               GO TO 240
            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 )
     $            GO TO 240
            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
            IF( ORGATI ) THEN
               TEMP1 = WORK( I )*DELTA( I )
               TEMP = ETA - TEMP1
            ELSE
               TEMP1 = WORK( IP1 )*DELTA( IP1 )
               TEMP = ETA - TEMP1
            END IF
            IF( TEMP.GT.SG2UB .OR. TEMP.LT.SG2LB ) THEN
               IF( W.LT.ZERO ) THEN
                  ETA = ( SG2UB-TAU ) / TWO
               ELSE
                  ETA = ( SG2LB-TAU ) / TWO
               END IF
            END IF
*
            TAU = TAU + ETA
            ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
*
            SIGMA = SIGMA + ETA
            DO 200 J = 1, N
               WORK( J ) = WORK( J ) + ETA
               DELTA( J ) = DELTA( J ) - ETA
  200       CONTINUE
*
            PREW = W
*
*           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
*
            TEMP = Z( II ) / ( WORK( 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
*
            IF( W.LE.ZERO ) THEN
               SG2LB = MAX( SG2LB, TAU )
            ELSE
               SG2UB = MIN( SG2UB, TAU )
            END IF
*
  230    CONTINUE
*
*        Return with INFO = 1, NITER = MAXIT and not converged
*
         INFO = 1
*
      END IF
*
  240 CONTINUE
      RETURN
*
*     End of DLASD4
*
      END
      SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            I
      DOUBLE PRECISION   DSIGMA, RHO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  I      (input) INTEGER
*         The index of the eigenvalue to be computed.  I = 1 or I = 2.
*
*  D      (input) DOUBLE PRECISION array, dimension ( 2 )
*         The original eigenvalues.  We assume 0 <= D(1) < D(2).
*
*  Z      (input) DOUBLE PRECISION array, dimension ( 2 )
*         The components of the updating vector.
*
*  DELTA  (output) DOUBLE PRECISION array, dimension ( 2 )
*         Contains (D(j) - lambda_I) in its  j-th component.
*         The vector DELTA contains the information necessary
*         to construct the eigenvectors.
*
*  RHO    (input) DOUBLE PRECISION
*         The scalar in the symmetric updating formula.
*
*  DSIGMA (output) DOUBLE PRECISION
*         The computed lambda_I, the I-th updated eigenvalue.
*
*  WORK   (workspace) DOUBLE PRECISION array, dimension ( 2 )
*         WORK contains (D(j) + sigma_I) in its  j-th component.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ren-Cang Li, Computer Science Division, University of California
*     at Berkeley, USA
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      LOGICAL            ORGATI
      INTEGER            INFO, KNITER
      DOUBLE PRECISION   FINIT, RHO, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( 3 ), Z( 3 )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  KNITER       (input) INTEGER
*               Refer to DLAED4 for its significance.
*
*  ORGATI       (input) 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.
*
*  RHO          (input) DOUBLE PRECISION
*               Refer to the equation f(x) above.
*
*  D            (input) DOUBLE PRECISION array, dimension (3)
*               D satisfies d(1) < d(2) < d(3).
*
*  Z            (input) DOUBLE PRECISION array, dimension (3)
*               Each of the elements in z must be positive.
*
*  FINIT        (input) 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).
*
*  TAU          (output) DOUBLE PRECISION
*               The root of the equation f(x).
*
*  INFO         (output) INTEGER
*               = 0: successful exit
*               > 0: if INFO = 1, failure to converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ren-Cang Li, Computer Science Division, University of California
*     at Berkeley, USA
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 20 )
      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            FIRST, 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
*     ..
*     .. Save statement ..
      SAVE               FIRST, SMALL1, SMINV1, SMALL2, SMINV2, EPS
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, MIN, SQRT
*     ..
*     .. Data statements ..
      DATA               FIRST / .TRUE. /
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
      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
         TEMP = RHO + Z( 1 ) / ( D( 1 )-TAU ) +
     $          Z( 2 ) / ( D( 2 )-TAU ) + Z( 3 ) / ( D( 3 )-TAU )
         IF( ABS( FINIT ).LE.ABS( TEMP ) )
     $      TAU = ZERO
      END IF
*
*     On first call to routine, get machine parameters for
*     possible scaling to avoid overflow
*
      IF( FIRST ) THEN
         EPS = DLAMCH( 'Epsilon' )
         BASE = DLAMCH( 'Base' )
         SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) /
     $            THREE ) )
         SMINV1 = ONE / SMALL1
         SMALL2 = SMALL1*SMALL1
         SMINV2 = SMINV1*SMINV1
         FIRST = .FALSE.
      END IF
*
*     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
      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
*
*        Iteration begins
*
*     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
*
         TEMP = ETA + TAU
         IF( ORGATI ) THEN
            IF( ETA.GT.ZERO .AND. TEMP.GE.DSCALE( 3 ) )
     $         ETA = ( DSCALE( 3 )-TAU ) / TWO
            IF( ETA.LT.ZERO .AND. TEMP.LE.DSCALE( 2 ) )
     $         ETA = ( DSCALE( 2 )-TAU ) / TWO
         ELSE
            IF( ETA.GT.ZERO .AND. TEMP.GE.DSCALE( 2 ) )
     $         ETA = ( DSCALE( 2 )-TAU ) / TWO
            IF( ETA.LT.ZERO .AND. TEMP.LE.DSCALE( 1 ) )
     $         ETA = ( DSCALE( 1 )-TAU ) / TWO
         END IF
         TAU = TAU + ETA
*
         FC = ZERO
         ERRETM = ZERO
         DF = ZERO
         DDF = ZERO
         DO 40 I = 1, 3
            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
   40    CONTINUE
         F = FINIT + TAU*FC
         ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) +
     $            ABS( TAU )*DF
         IF( ABS( F ).LE.EPS*ERRETM )
     $      GO TO 60
   50 CONTINUE
      INFO = 1
   60 CONTINUE
*
*     Undo scaling
*
      IF( SCALE )
     $   TAU = TAU*SCLINV
      RETURN
*
*     End of DLAED6
*
      END
      SUBROUTINE RSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
     $                   ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
     $                   IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 20, 2000
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
*  DSYEVR computes selected eigenvalues and, optionally, eigenvectors
*  of a real symmetric 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, DSYEVR calls DSTEGR to compute the
*  eigenspectrum using Relatively Robust Representations.  DSTEGR
*  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 : DSYEVR calls DSTEGR when the full spectrum is requested
*  on machines which conform to the ieee-754 floating point standard.
*  DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and
*  when partial spectrum requests are made.
*
*  Normal execution of DSTEGR 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.
*
*  Arguments
*  =========
*
*  JOBZ    (input) CHARACTER*1
*          = 'N':  Compute eigenvalues only;
*          = 'V':  Compute eigenvalues and eigenvectors.
*
*  RANGE   (input) 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
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  VL      (input) DOUBLE PRECISION
*  VU      (input) DOUBLE PRECISION
*          If RANGE='V', the lower and upper bounds of the interval to
*          be searched for eigenvalues. VL < VU.
*          Not referenced if RANGE = 'A' or 'I'.
*
*  IL      (input) INTEGER
*  IU      (input) INTEGER
*          If RANGE='I', the indices (in ascending order) of the
*          smallest and largest eigenvalues 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'.
*
*  ABSTOL  (input) 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
*          furutre 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.
*
*  M       (output) INTEGER
*          The total number of eigenvalues found.  0 <= M <= N.
*          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
*  W       (output) DOUBLE PRECISION array, dimension (N)
*          The first M elements contain the selected eigenvalues in
*          ascending order.
*
*  Z       (output) 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.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1, and if
*          JOBZ = 'V', LDZ >= max(1,N).
*
*  ISUPPZ  (output) 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
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.  LWORK >= max(1,26*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.
*
*  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
*          On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
*
*  LIWORK  (input) 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 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  Internal error
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Inderjit Dhillon, IBM Almaden, USA
*     Osni Marques, LBNL/NERSC, USA
*     Ken Stanley, Computer Science Division, University of
*       California at Berkeley, USA
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ
      CHARACTER          ORDER
      INTEGER            I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
     $                   INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
     $                   INDWK, INDWKN, ISCALE, ITMP1, 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, DSTEGR, 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 ) )
*
      LWMIN = MAX( 1, 26*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( .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, 'ZHETRD', UPLO, N, -1, -1, -1 )
         NB = MAX( NB, ILAENV( 1, 'ZUNMTR', 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 ) = 7
         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
      VLL = VL
      VUU = VU
      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
      INDEE = INDD + N
      INDDD = INDEE + N
      INDIFL = INDDD + N
      INDWK = INDIFL + N
      LLWORK = LWORK - INDWK + 1
      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 SSTEGR 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 )
*
            CALL DSTEGR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ),
     $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
     $                   WORK( INDWK ), LWORK, IWORK, LIWORK, INFO )
*
*
*
*        Apply orthogonal matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEIN.
*
            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
            M = N
            GO TO 30
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*     Also call DSTEBZ and SSTEIN if SSTEGR fails.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF
      INDIFL = 1
      INDIBL = INDIFL + N
      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( 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.
*
   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 )
            END IF
   50    CONTINUE
      END IF
*
*     Set WORK(1) to optimal workspace size.
*
      WORK( 1 ) = LWKOPT
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DSYEVR
*
      END
      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 (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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, * )
*     ..
*
*  Purpose
*  =======
*
* DSTEGR 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. The eigenvalues
* are computed 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 : Currently DSTEGR is only set up to find ALL the n
*  eigenvalues and eigenvectors of T in O(n^2) time
*  Note 2 : Currently the routine DSTEIN is called when an appropriate
*  sigma_i cannot be chosen in step (c) above. DSTEIN invokes modified
*  Gram-Schmidt when eigenvalues are close.
*  Note 3 : DSTEGR works only on machines which follow ieee-754
*  floating-point standard in their handling of infinities and NaNs.
*  Normal execution of DSTEGR may create NaNs and infinities and hence
*  may abort due to a floating point exception in environments which
*  do not conform to the ieee standard.
*
*  Arguments
*  =========
*
*  JOBZ    (input) CHARACTER*1
*          = 'N':  Compute eigenvalues only;
*          = 'V':  Compute eigenvalues and eigenvectors.
*
*  RANGE   (input) 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.
********** Only RANGE = 'A' is currently supported *********************
*
*  N       (input) INTEGER
*          The order of the matrix.  N >= 0.
*
*  D       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the n diagonal elements of the tridiagonal matrix
*          T. On exit, D is overwritten.
*
*  E       (input/output) 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 exit, E is overwritten.
*
*  VL      (input) DOUBLE PRECISION
*  VU      (input) DOUBLE PRECISION
*          If RANGE='V', the lower and upper bounds of the interval to
*          be searched for eigenvalues. VL < VU.
*          Not referenced if RANGE = 'A' or 'I'.
*
*  IL      (input) INTEGER
*  IU      (input) INTEGER
*          If RANGE='I', the indices (in ascending order) of the
*          smallest and largest eigenvalues 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'.
*
*  ABSTOL  (input) DOUBLE PRECISION
*          The absolute error tolerance for the
*          eigenvalues/eigenvectors. IF JOBZ = 'V', the eigenvalues and
*          eigenvectors output have residual norms bounded by ABSTOL,
*          and the dot products between different eigenvectors are
*          bounded by ABSTOL. If ABSTOL is less than N*EPS*|T|, then
*          N*EPS*|T| will be used in its place, where EPS is the
*          machine precision and |T| is the 1-norm of the tridiagonal
*          matrix. The eigenvalues are computed to an accuracy of
*          EPS*|T| irrespective of ABSTOL. If high relative accuracy
*          is important, set ABSTOL to DLAMCH( 'Safe minimum' ).
*          See Barlow and 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.
*
*  M       (output) INTEGER
*          The total number of eigenvalues found.  0 <= M <= N.
*          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
*  W       (output) DOUBLE PRECISION array, dimension (N)
*          The first M elements contain the selected eigenvalues in
*          ascending order.
*
*  Z       (output) 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 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.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1, and if
*          JOBZ = 'V', LDZ >= max(1,N).
*
*  ISUPPZ  (output) 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 ).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal
*          (and minimal) LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.  LWORK >= max(1,18*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.
*
*  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
*          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
*  LIWORK  (input) 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 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = 1, internal error in DLARRE,
*                if INFO = 2, internal error in DLARRV.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Inderjit Dhillon, IBM Almaden, USA
*     Osni Marques, LBNL/NERSC, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ
      INTEGER            I, IBEGIN, IEND, IINDBL, IINDWK, IINFO, IINSPL,
     $                   INDGRS, INDWOF, INDWRK, ITMP, J, JJ, LIWMIN,
     $                   LWMIN, NSPLIT
      DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SCALE, SMLNUM,
     $                   THRESH, TMP, TNRM, TOL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARRE, DLARRV, DLASET, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, 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 ) )
      LWMIN = 18*N
      LIWMIN = 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
*
*     The following two lines need to be removed once the
*     RANGE = 'V' and RANGE = 'I' options are provided.
*
      ELSE IF( VALEIG .OR. INDEIG ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN
         INFO = -7
      ELSE IF( INDEIG .AND. IL.LT.1 ) THEN
         INFO = -8
*     The following change should be made in DSTEVX also, otherwise
*     IL can be specified as N+1 and IU as N.
*     ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN
      ELSE IF( INDEIG .AND. ( IU.LT.IL .OR. IU.GT.N ) ) THEN
         INFO = -9
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
         INFO = -17
      ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
         INFO = -19
      END IF
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEGR', -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.
*
      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
      END IF
      INDGRS = 1
      INDWOF = 2*N + 1
      INDWRK = 3*N + 1
*
      IINSPL = 1
      IINDBL = N + 1
      IINDWK = 2*N + 1
*
      CALL DLASET( 'Full', N, N, ZERO, ZERO, Z, LDZ )
*
*     Compute the desired eigenvalues of the tridiagonal after splitting
*     into smaller subblocks if the corresponding of-diagonal elements
*     are small
*
      THRESH = EPS*TNRM
      CALL DLARRE( N, D, E, THRESH, NSPLIT, IWORK( IINSPL ), M, W,
     $             WORK( INDWOF ), WORK( INDGRS ), WORK( INDWRK ),
     $             IINFO )
      IF( IINFO.NE.0 ) THEN
         INFO = 1
         RETURN
      END IF
*
      IF( WANTZ ) THEN
*
*        Compute the desired eigenvectors corresponding to the computed
*        eigenvalues
*
         TOL = MAX( ABSTOL, DBLE( N )*THRESH )
         IBEGIN = 1
         DO 20 I = 1, NSPLIT
            IEND = IWORK( IINSPL+I-1 )
            DO 10 J = IBEGIN, IEND
               IWORK( IINDBL+J-1 ) = I
   10       CONTINUE
            IBEGIN = IEND + 1
   20    CONTINUE
*
         CALL DLARRV( N, D, E, IWORK( IINSPL ), M, W, IWORK( IINDBL ),
     $                WORK( INDGRS ), TOL, Z, LDZ, ISUPPZ,
     $                WORK( INDWRK ), IWORK( IINDWK ), IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = 2
            RETURN
         END IF
*
      END IF
*
      IBEGIN = 1
      DO 40 I = 1, NSPLIT
         IEND = IWORK( IINSPL+I-1 )
         DO 30 J = IBEGIN, IEND
            W( J ) = W( J ) + WORK( INDWOF+I-1 )
   30    CONTINUE
         IBEGIN = IEND + 1
   40 CONTINUE
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( SCALE.NE.ONE ) THEN
         CALL DSCAL( M, ONE / SCALE, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( NSPLIT.GT.1 ) THEN
         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
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of DSTEGR
*
      END
      SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS, UPLO
      INTEGER            INFO, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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).
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply Q or Q**T from the Left;
*          = 'R': apply Q or Q**T from the Right.
*
*  UPLO    (input) CHARACTER*1
*          = 'U': Upper triangle of A contains elementary reflectors
*                 from DSYTRD;
*          = 'L': Lower triangle of A contains elementary reflectors
*                 from DSYTRD.
*
*  TRANS   (input) CHARACTER*1
*          = 'N':  No transpose, apply Q;
*          = 'T':  Transpose, apply Q**T.
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  A       (input) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.
*          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.
*
*  TAU     (input) 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.
*
*  C       (input/output) 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.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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 = N
      ELSE
         NQ = N
         NW = 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.MAX( 1, 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 = MAX( 1, 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
      SUBROUTINE DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E,
     $                   M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  RANGE   (input) CHARACTER
*          = '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.
*
*  ORDER   (input) CHARACTER
*          = '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.
*
*  N       (input) INTEGER
*          The order of the tridiagonal matrix T.  N >= 0.
*
*  VL      (input) DOUBLE PRECISION
*  VU      (input) DOUBLE PRECISION
*          If RANGE='V', the lower and upper bounds 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'.
*
*  IL      (input) INTEGER
*  IU      (input) INTEGER
*          If RANGE='I', the indices (in ascending order) of the
*          smallest and largest eigenvalues 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'.
*
*  ABSTOL  (input) 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.
*
*  D       (input) DOUBLE PRECISION array, dimension (N)
*          The n diagonal elements of the tridiagonal matrix T.
*
*  E       (input) DOUBLE PRECISION array, dimension (N-1)
*          The (n-1) off-diagonal elements of the tridiagonal matrix T.
*
*  M       (output) INTEGER
*          The actual number of eigenvalues found. 0 <= M <= N.
*          (See also the description of INFO=2,3.)
*
*  NSPLIT  (output) INTEGER
*          The number of diagonal blocks in the matrix T.
*          1 <= NSPLIT <= N.
*
*  W       (output) 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.)
*
*  IBLOCK  (output) 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.)
*
*  ISPLIT  (output) 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.)
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)
*
*  IWORK   (workspace) INTEGER array, dimension (3*N)
*
*  INFO    (output) 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.
*
*  Internal Parameters
*  ===================
*
*  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.
*
*  =====================================================================
*
*     .. 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 = 2.0D0, 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
*
*DIR$ NOVECTOR
      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
      SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
     $                   IWORK, IFAIL, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDZ, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
     $                   IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  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).
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix.  N >= 0.
*
*  D       (input) DOUBLE PRECISION array, dimension (N)
*          The n diagonal elements of the tridiagonal matrix T.
*
*  E       (input) DOUBLE PRECISION array, dimension (N)
*          The (n-1) subdiagonal elements of the tridiagonal matrix
*          T, in elements 1 to N-1.  E(N) need not be set.
*
*  M       (input) INTEGER
*          The number of eigenvectors to be found.  0 <= M <= N.
*
*  W       (input) 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. )
*
*  IBLOCK  (input) 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. )
*
*  ISPLIT  (input) 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. )
*
*  Z       (output) 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.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= max(1,N).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (5*N)
*
*  IWORK   (workspace) INTEGER array, dimension (N)
*
*  IFAIL   (output) 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.
*
*  INFO    (output) 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.
*
*  Internal Parameters
*  ===================
*
*  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.
*
*  =====================================================================
*
*     .. 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   DASUM, DDOT, DLAMCH, DNRM2
      EXTERNAL           IDAMAX, DASUM, 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 = B1
*
*        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.
*
            SCL = BLKSIZ*ONENRM*MAX( EPS,
     $            ABS( WORK( INDRV4+BLKSIZ ) ) ) /
     $            DASUM( BLKSIZ, WORK( INDRV1+1 ), 1 )
            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
      SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  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'.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply Q or Q**T from the Left;
*          = 'R': apply Q or Q**T from the Right.
*
*  TRANS   (input) CHARACTER*1
*          = 'N':  No transpose, apply Q;
*          = 'T':  Transpose, apply Q**T.
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines
*          the matrix Q.
*          If SIDE = 'L', M >= K >= 0;
*          if SIDE = 'R', N >= K >= 0.
*
*  A       (input) 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.
*          A is modified by the routine but restored on exit.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.
*          If SIDE = 'L', LDA >= max(1,M);
*          if SIDE = 'R', LDA >= max(1,N).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQLF.
*
*  C       (input/output) 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.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      INTEGER            I, I1, I2, I3, IB, IINFO, IWS, LDWORK, LWKOPT,
     $                   MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   T( LDT, NBMAX )
*     ..
*     .. 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 = N
      ELSE
         NQ = N
         NW = 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.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Determine the block size.  NB may be at most NBMAX, where NBMAX
*        is used to define the local array T.
*
         NB = MIN( NBMAX, ILAENV( 1, 'DORMQL', SIDE // TRANS, M, N, K,
     $        -1 ) )
         LWKOPT = MAX( 1, NW )*NB
         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 .OR. K.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = NW
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
         IWS = NW*NB
         IF( LWORK.LT.IWS ) THEN
            NB = LWORK / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMQL', SIDE // TRANS, M, N, K,
     $              -1 ) )
         END IF
      ELSE
         IWS = NW
      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
*
         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 ), T, LDT )
            IF( LEFT ) THEN
*
*              H or H' is applied to C(1:m-k+i+ib-1,1:n)
*
               MI = M - K + I + IB - 1
            ELSE
*
*              H or H' is applied to C(1:m,1:n-k+i+ib-1)
*
               NI = N - K + I + IB - 1
            END IF
*
*           Apply H or H'
*
            CALL DLARFB( SIDE, TRANS, 'Backward', 'Columnwise', MI, NI,
     $                   IB, A( 1, I ), LDA, T, LDT, C, LDC, WORK,
     $                   LDWORK )
   10    CONTINUE
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMQL
*
      END
      SUBROUTINE DLARRE( N, D, E, TOL, NSPLIT, ISPLIT, M, W, WOFF,
     $                   GERSCH, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, M, N, NSPLIT
      DOUBLE PRECISION   TOL
*     ..
*     .. Array Arguments ..
      INTEGER            ISPLIT( * )
      DOUBLE PRECISION   D( * ), E( * ), GERSCH( * ), W( * ), WOFF( * ),
     $                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  Given the tridiagonal matrix T, DLARRE sets "small" off-diagonal
*  elements to zero, and for each unreduced block T_i, it finds
*  (i) the numbers sigma_i
*  (ii) the base T_i - sigma_i I = L_i D_i L_i^T representations and
*  (iii) eigenvalues of each L_i D_i L_i^T.
*  The representations and eigenvalues found are then used by
*  DSTEGR to compute the eigenvectors of a symmetric tridiagonal
*  matrix. Currently, the base representations are limited to being
*  positive or negative definite, and the eigenvalues of the definite
*  matrices are found by the dqds algorithm (subroutine DLASQ2). As
*  an added benefit, DLARRE also outputs the n Gerschgorin
*  intervals for each L_i D_i L_i^T.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix.
*
*  D       (input/output) 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.
*
*  E       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the (n-1) subdiagonal elements of the tridiagonal
*          matrix T; E(N) need not be set.
*          On exit, the subdiagonal elements of the unit bidiagonal
*          matrices L_i.
*
*  TOL     (input) DOUBLE PRECISION
*          The threshold for splitting. If on input |E(i)| < TOL, then
*          the matrix T is split into smaller blocks.
*
*  NSPLIT  (input) INTEGER
*          The number of blocks T splits into. 1 <= NSPLIT <= N.
*
*  ISPLIT  (output) INTEGER array, dimension (2*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.
*
*  M       (output) INTEGER
*          The total number of eigenvalues (of all the L_i D_i L_i^T)
*          found.
*
*  W       (output) 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.
*
*  WOFF    (output) DOUBLE PRECISION array, dimension (N)
*          The NSPLIT base points sigma_i.
*
*  GERSCH  (output) DOUBLE PRECISION array, dimension (2*N)
*          The n Gerschgorin intervals.
*
*  WORK    (input) DOUBLE PRECISION array, dimension (4*N???)
*          Workspace.
*
*  INFO    (output) INTEGER
*          Output error code from DLASQ2
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Inderjit Dhillon, IBM Almaden, USA
*     Osni Marques, LBNL/NERSC, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, FOUR, FOURTH
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   FOUR = 4.0D0, FOURTH = ONE / FOUR )
*     ..
*     .. Local Scalars ..
      INTEGER            CNT, I, IBEGIN, IEND, IN, J, JBLK, MAXCNT
      DOUBLE PRECISION   DELTA, EPS, GL, GU, NRM, OFFD, S, SGNDEF,
     $                   SIGMA, TAU, TMP1, WIDTH
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLASQ2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      EPS = DLAMCH( 'Precision' )
*
*     Compute Splitting Points
*
      NSPLIT = 1
      DO 10 I = 1, N - 1
         IF( ABS( E( I ) ).LE.TOL ) THEN
            ISPLIT( NSPLIT ) = I
            NSPLIT = NSPLIT + 1
         END IF
   10 CONTINUE
      ISPLIT( NSPLIT ) = N
*
      IBEGIN = 1
      DO 170 JBLK = 1, NSPLIT
         IEND = ISPLIT( JBLK )
         IF( IBEGIN.EQ.IEND ) THEN
            W( IBEGIN ) = D( IBEGIN )
            WOFF( JBLK ) = ZERO
            IBEGIN = IEND + 1
            GO TO 170
         END IF
         IN = IEND - IBEGIN + 1
*
*        Form the n Gerschgorin intervals
*
         GL = D( IBEGIN ) - ABS( E( IBEGIN ) )
         GU = D( IBEGIN ) + ABS( E( IBEGIN ) )
         GERSCH( 2*IBEGIN-1 ) = GL
         GERSCH( 2*IBEGIN ) = GU
         GERSCH( 2*IEND-1 ) = D( IEND ) - ABS( E( IEND-1 ) )
         GERSCH( 2*IEND ) = D( IEND ) + ABS( E( IEND-1 ) )
         GL = MIN( GERSCH( 2*IEND-1 ), GL )
         GU = MAX( GERSCH( 2*IEND ), GU )
         DO 20 I = IBEGIN + 1, IEND - 1
            OFFD = ABS( E( I-1 ) ) + ABS( E( I ) )
            GERSCH( 2*I-1 ) = D( I ) - OFFD
            GL = MIN( GERSCH( 2*I-1 ), GL )
            GERSCH( 2*I ) = D( I ) + OFFD
            GU = MAX( GERSCH( 2*I ), GU )
   20    CONTINUE
         NRM = MAX( ABS( GL ), ABS( GU ) )
*
*        Find the number SIGMA where the base representation
*        T - sigma I = L D L^T is to be formed.
*
         WIDTH = GU - GL
         DO 30 I = IBEGIN, IEND - 1
            WORK( I ) = E( I )*E( I )
   30    CONTINUE
         DO 50 J = 1, 2
            IF( J.EQ.1 ) THEN
               TAU = GL + FOURTH*WIDTH
            ELSE
               TAU = GU - FOURTH*WIDTH
            END IF
            TMP1 = D( IBEGIN ) - TAU
            IF( TMP1.LT.ZERO ) THEN
               CNT = 1
            ELSE
               CNT = 0
            END IF
            DO 40 I = IBEGIN + 1, IEND
               TMP1 = D( I ) - TAU - WORK( I-1 ) / TMP1
               IF( TMP1.LT.ZERO )
     $            CNT = CNT + 1
   40       CONTINUE
            IF( CNT.EQ.0 ) THEN
               GL = TAU
            ELSE IF( CNT.EQ.IN ) THEN
               GU = TAU
            END IF
            IF( J.EQ.1 ) THEN
               MAXCNT = CNT
               SIGMA = GL
               SGNDEF = ONE
            ELSE
               IF( IN-CNT.GT.MAXCNT ) THEN
                  SIGMA = GU
                  SGNDEF = -ONE
               END IF
            END IF
   50    CONTINUE
*
*        Find the base L D L^T representation
*
         WORK( 3*IN ) = ONE
         DELTA = EPS
         TAU = SGNDEF*NRM
   60    CONTINUE
         SIGMA = SIGMA - DELTA*TAU
         WORK( 1 ) = D( IBEGIN ) - SIGMA
         J = IBEGIN
         DO 70 I = 1, IN - 1
            WORK( 2*IN+I ) = ONE / WORK( 2*I-1 )
            TMP1 = E( J )*WORK( 2*IN+I )
            WORK( 2*I+1 ) = ( D( J+1 )-SIGMA ) - TMP1*E( J )
            WORK( 2*I ) = TMP1
            J = J + 1
   70    CONTINUE
         DO 80 I = IN, 1, -1
            TMP1 = SGNDEF*WORK( 2*I-1 )
            IF( TMP1.LT.ZERO .OR. WORK( 2*IN+I ).EQ.ZERO .OR. .NOT.
     $          ( TMP1.GT.ZERO .OR. TMP1.LT.ONE ) ) THEN
               DELTA = TWO*DELTA
               GO TO 60
            END IF
            J = J - 1
   80    CONTINUE
*
         J = IBEGIN
         D( IBEGIN ) = WORK( 1 )
         WORK( 1 ) = ABS( WORK( 1 ) )
         DO 90 I = 1, IN - 1
            TMP1 = E( J )
            E( J ) = WORK( 2*I )
            WORK( 2*I ) = ABS( TMP1*WORK( 2*I ) )
            J = J + 1
            D( J ) = WORK( 2*I+1 )
            WORK( 2*I+1 ) = ABS( WORK( 2*I+1 ) )
   90    CONTINUE
*
         CALL DLASQ2( IN, WORK, INFO )
*
         TAU = SGNDEF*WORK( IN )
         WORK( 3*IN ) = ONE
         DELTA = TWO*EPS
  100    CONTINUE
         TAU = TAU*( ONE-DELTA )
*
         S = -TAU
         J = IBEGIN
         DO 110 I = 1, IN - 1
            WORK( I ) = D( J ) + S
            WORK( 2*IN+I ) = ONE / WORK( I )
*           WORK( N+I ) = ( E( I ) * D( I ) ) / WORK( I )
            WORK( IN+I ) = ( E( J )*D( J ) )*WORK( 2*IN+I )
            S = S*WORK( IN+I )*E( J ) - TAU
            J = J + 1
  110    CONTINUE
         WORK( IN ) = D( IEND ) + S
*
*        Checking to see if all the diagonal elements of the new
*        L D L^T representation have the same sign
*
         DO 120 I = IN, 1, -1
            TMP1 = SGNDEF*WORK( I )
            IF( TMP1.LT.ZERO .OR. WORK( 2*IN+I ).EQ.ZERO .OR. .NOT.
     $          ( TMP1.GT.ZERO .OR. TMP1.LT.ONE ) ) THEN
               DELTA = TWO*DELTA
               GO TO 100
            END IF
  120    CONTINUE
*
         SIGMA = SIGMA + TAU
         CALL DCOPY( IN, WORK, 1, D( IBEGIN ), 1 )
         CALL DCOPY( IN-1, WORK( IN+1 ), 1, E( IBEGIN ), 1 )
         WOFF( JBLK ) = SIGMA
*
*        Update the n Gerschgorin intervals
*
         DO 130 I = IBEGIN, IEND
            GERSCH( 2*I-1 ) = GERSCH( 2*I-1 ) - SIGMA
            GERSCH( 2*I ) = GERSCH( 2*I ) - SIGMA
  130    CONTINUE
*
*        Compute the eigenvalues of L D L^T.
*
         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 ) )
*
         CALL DLASQ2( IN, WORK, INFO )
*
         J = IBEGIN
         IF( SGNDEF.GT.ZERO ) THEN
            DO 150 I = 1, IN
               W( J ) = WORK( IN-I+1 )
               J = J + 1
  150       CONTINUE
         ELSE
            DO 160 I = 1, IN
               W( J ) = -WORK( I )
               J = J + 1
  160       CONTINUE
         END IF
         IBEGIN = IEND + 1
  170 CONTINUE
      M = N
*
      RETURN
*
*     End of DLARRE
*
      END
      SUBROUTINE DLARRV( N, D, L, ISPLIT, M, W, IBLOCK, GERSCH, TOL, Z,
     $                   LDZ, ISUPPZ, WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDZ, M, N
      DOUBLE PRECISION   TOL
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), ISPLIT( * ), ISUPPZ( * ),
     $                   IWORK( * )
      DOUBLE PRECISION   D( * ), GERSCH( * ), L( * ), W( * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  DLARRV computes the eigenvectors of the tridiagonal matrix
*  T = L D L^T given L, D and the eigenvalues of L D L^T.
*  The input eigenvalues should have high relative accuracy with
*  respect to the entries of L and D. The desired accuracy of the
*  output can be specified by the input parameter TOL.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix.  N >= 0.
*
*  D       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the n diagonal elements of the diagonal matrix D.
*          On exit, D may be overwritten.
*
*  L       (input/output) DOUBLE PRECISION array, dimension (N-1)
*          On entry, the (n-1) subdiagonal elements of the unit
*          bidiagonal matrix L in elements 1 to N-1 of L. L(N) need
*          not be set. On exit, L is overwritten.
*
*  ISPLIT  (input) 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.
*
*  TOL     (input) DOUBLE PRECISION
*          The absolute error tolerance for the
*          eigenvalues/eigenvectors.
*          Errors in the input eigenvalues must be bounded by TOL.
*          The eigenvectors output have residual norms
*          bounded by TOL, and the dot products between different
*          eigenvectors are bounded by TOL. TOL must be at least
*          N*EPS*|T|, where EPS is the machine precision and |T| is
*          the 1-norm of the tridiagonal matrix.
*
*  M       (input) INTEGER
*          The total number of eigenvalues found.  0 <= M <= N.
*          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
*  W       (input) 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 DLARRE is expected here ).
*          Errors in W must be bounded by TOL (see above).
*
*  IBLOCK  (input) 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.
*
*  Z       (output) 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 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.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1, and if
*          JOBZ = 'V', LDZ >= max(1,N).
*
*  ISUPPZ  (output) 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 ).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (13*N)
*
*  IWORK   (workspace) INTEGER array, dimension (6*N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = 1, internal error in DLARRB
*                if INFO = 2, internal error in DSTEIN
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Inderjit Dhillon, IBM Almaden, USA
*     Osni Marques, LBNL/NERSC, USA
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MGSSIZ
      PARAMETER          ( MGSSIZ = 20 )
      DOUBLE PRECISION   ZERO, ONE, FOUR
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, FOUR = 4.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            MGSCLS
      INTEGER            I, IBEGIN, IEND, IINDC1, IINDC2, IINDR, IINDWK,
     $                   IINFO, IM, IN, INDERR, INDGAP, INDLD, INDLLD,
     $                   INDWRK, ITER, ITMP1, ITMP2, J, JBLK, K, KTOT,
     $                   LSBDPT, MAXITR, NCLUS, NDEPTH, NDONE, NEWCLS,
     $                   NEWFRS, NEWFTT, NEWLST, NEWSIZ, NSPLIT, OLDCLS,
     $                   OLDFST, OLDIEN, OLDLST, OLDNCL, P, Q
      DOUBLE PRECISION   EPS, GAP, LAMBDA, MGSTOL, MINGMA, MINRGP,
     $                   NRMINV, RELGAP, RELTOL, RESID, RQCORR, SIGMA,
     $                   TMP1, ZTZ
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DDOT, DLAMCH, DNRM2
      EXTERNAL           DDOT, DLAMCH, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLAR1V, DLARRB, DLARRF, DLASET,
     $                   DSCAL, DSTEIN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, SQRT
*     ..
*     .. Local Arrays ..
      INTEGER            TEMP( 1 )
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INDERR = N + 1
      INDLD = 2*N
      INDLLD = 3*N
      INDGAP = 4*N
      INDWRK = 5*N + 1
*
      IINDR = N
      IINDC1 = 2*N
      IINDC2 = 3*N
      IINDWK = 4*N + 1
*
      EPS = DLAMCH( 'Precision' )
*
      DO 10 I = 1, 2*N
         IWORK( I ) = 0
   10 CONTINUE
      DO 20 I = 1, M
         WORK( INDERR+I-1 ) = EPS*ABS( W( I ) )
   20 CONTINUE
      CALL DLASET( 'Full', N, N, ZERO, ZERO, Z, LDZ )
      MGSTOL = 5.0D0*EPS
*
      NSPLIT = IBLOCK( M )
      IBEGIN = 1
      DO 170 JBLK = 1, NSPLIT
         IEND = ISPLIT( JBLK )
*
*        Find the eigenvectors of the submatrix indexed IBEGIN
*        through IEND.
*
         IF( IBEGIN.EQ.IEND ) THEN
            Z( IBEGIN, IBEGIN ) = ONE
            ISUPPZ( 2*IBEGIN-1 ) = IBEGIN
            ISUPPZ( 2*IBEGIN ) = IBEGIN
            IBEGIN = IEND + 1
            GO TO 170
         END IF
         OLDIEN = IBEGIN - 1
         IN = IEND - OLDIEN
         RELTOL = MIN( 1.0D-2, ONE / DBLE( IN ) )
         IM = IN
         CALL DCOPY( IM, W( IBEGIN ), 1, WORK, 1 )
         DO 30 I = 1, IN - 1
            WORK( INDGAP+I ) = WORK( I+1 ) - WORK( I )
   30    CONTINUE
         WORK( INDGAP+IN ) = MAX( ABS( WORK( IN ) ), EPS )
         NDONE = 0
*
         NDEPTH = 0
         LSBDPT = 1
         NCLUS = 1
         IWORK( IINDC1+1 ) = 1
         IWORK( IINDC1+2 ) = IN
*
*        While( NDONE.LT.IM ) do
*
   40    CONTINUE
         IF( NDONE.LT.IM ) THEN
            OLDNCL = NCLUS
            NCLUS = 0
            LSBDPT = 1 - LSBDPT
            DO 150 I = 1, OLDNCL
               IF( LSBDPT.EQ.0 ) THEN
                  OLDCLS = IINDC1
                  NEWCLS = IINDC2
               ELSE
                  OLDCLS = IINDC2
                  NEWCLS = IINDC1
               END IF
*
*              If NDEPTH > 1, retrieve the relatively robust
*              representation (RRR) and perform limited bisection
*              (if necessary) to get approximate eigenvalues.
*
               J = OLDCLS + 2*I
               OLDFST = IWORK( J-1 )
               OLDLST = IWORK( J )
               IF( NDEPTH.GT.0 ) THEN
                  J = OLDIEN + OLDFST
                  CALL DCOPY( IN, Z( IBEGIN, J ), 1, D( IBEGIN ), 1 )
                  CALL DCOPY( IN, Z( IBEGIN, J+1 ), 1, L( IBEGIN ), 1 )
                  SIGMA = L( IEND )
               END IF
               K = IBEGIN
               DO 50 J = 1, IN - 1
                  WORK( INDLD+J ) = D( K )*L( K )
                  WORK( INDLLD+J ) = WORK( INDLD+J )*L( K )
                  K = K + 1
   50          CONTINUE
               IF( NDEPTH.GT.0 ) THEN
                  CALL DLARRB( IN, D( IBEGIN ), L( IBEGIN ),
     $                         WORK( INDLD+1 ), WORK( INDLLD+1 ),
     $                         OLDFST, OLDLST, SIGMA, RELTOL, WORK,
     $                         WORK( INDGAP+1 ), WORK( INDERR ),
     $                         WORK( INDWRK ), IWORK( IINDWK ), IINFO )
                  IF( IINFO.NE.0 ) THEN
                     INFO = 1
                     RETURN
                  END IF
               END IF
*
*              Classify eigenvalues of the current representation (RRR)
*              as (i) isolated, (ii) loosely clustered or (iii) tightly
*              clustered
*
               NEWFRS = OLDFST
               DO 140 J = OLDFST, OLDLST
                  IF( J.EQ.OLDLST .OR. WORK( INDGAP+J ).GE.RELTOL*
     $                ABS( WORK( J ) ) ) THEN
                     NEWLST = J
                  ELSE
*
*                    continue (to the next loop)
*
                     RELGAP = WORK( INDGAP+J ) / ABS( WORK( J ) )
                     IF( J.EQ.NEWFRS ) THEN
                        MINRGP = RELGAP
                     ELSE
                        MINRGP = MIN( MINRGP, RELGAP )
                     END IF
                     GO TO 140
                  END IF
                  NEWSIZ = NEWLST - NEWFRS + 1
                  MAXITR = 10
                  NEWFTT = OLDIEN + NEWFRS
                  IF( NEWSIZ.GT.1 ) THEN
                     MGSCLS = NEWSIZ.LE.MGSSIZ .AND. MINRGP.GE.MGSTOL
                     IF( .NOT.MGSCLS ) THEN
                        CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
     $                               WORK( INDLD+1 ), WORK( INDLLD+1 ),
     $                               NEWFRS, NEWLST, WORK,
     $                               Z( IBEGIN, NEWFTT ),
     $                               Z( IBEGIN, NEWFTT+1 ),
     $                               WORK( INDWRK ), IWORK( IINDWK ),
     $                               INFO )
                        IF( INFO.EQ.0 ) THEN
                           NCLUS = NCLUS + 1
                           K = NEWCLS + 2*NCLUS
                           IWORK( K-1 ) = NEWFRS
                           IWORK( K ) = NEWLST
                        ELSE
                           INFO = 0
                           IF( MINRGP.GE.MGSTOL ) THEN
                              MGSCLS = .TRUE.
                           ELSE
*
*                             Call DSTEIN to process this tight cluster.
*                             This happens only if MINRGP <= MGSTOL
*                             and DLARRF returns INFO = 1. The latter
*                             means that a new RRR to "break" the
*                             cluster could not be found.
*
                              WORK( INDWRK ) = D( IBEGIN )
                              DO 60 K = 1, IN - 1
                                 WORK( INDWRK+K ) = D( IBEGIN+K ) +
     $                                              WORK( INDLLD+K )
   60                         CONTINUE
                              DO 70 K = 1, NEWSIZ
                                 IWORK( IINDWK+K-1 ) = 1
   70                         CONTINUE
                              DO 80 K = NEWFRS, NEWLST
                                 ISUPPZ( 2*( IBEGIN+K )-3 ) = 1
                                 ISUPPZ( 2*( IBEGIN+K )-2 ) = IN
   80                         CONTINUE
                              TEMP( 1 ) = IN
                              CALL DSTEIN( IN, WORK( INDWRK ),
     $                                     WORK( INDLD+1 ), NEWSIZ,
     $                                     WORK( NEWFRS ),
     $                                     IWORK( IINDWK ), TEMP( 1 ),
     $                                     Z( IBEGIN, NEWFTT ), LDZ,
     $                                     WORK( INDWRK+IN ),
     $                                     IWORK( IINDWK+IN ),
     $                                     IWORK( IINDWK+2*IN ), IINFO )
                              IF( IINFO.NE.0 ) THEN
                                 INFO = 2
                                 RETURN
                              END IF
                              NDONE = NDONE + NEWSIZ
                           END IF
                        END IF
                     END IF
                  ELSE
                     MGSCLS = .FALSE.
                  END IF
                  IF( NEWSIZ.EQ.1 .OR. MGSCLS ) THEN
                     KTOT = NEWFTT
                     DO 100 K = NEWFRS, NEWLST
                        ITER = 0
   90                   CONTINUE
                        LAMBDA = WORK( K )
                        CALL DLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
     $                               L( IBEGIN ), WORK( INDLD+1 ),
     $                               WORK( INDLLD+1 ),
     $                               GERSCH( 2*OLDIEN+1 ),
     $                               Z( IBEGIN, KTOT ), ZTZ, MINGMA,
     $                               IWORK( IINDR+KTOT ),
     $                               ISUPPZ( 2*KTOT-1 ),
     $                               WORK( INDWRK ) )
                        TMP1 = ONE / ZTZ
                        NRMINV = SQRT( TMP1 )
                        RESID = ABS( MINGMA )*NRMINV
                        RQCORR = MINGMA*TMP1
                        IF( K.EQ.IN ) THEN
                           GAP = WORK( INDGAP+K-1 )
                        ELSE IF( K.EQ.1 ) THEN
                           GAP = WORK( INDGAP+K )
                        ELSE
                           GAP = MIN( WORK( INDGAP+K-1 ),
     $                           WORK( INDGAP+K ) )
                        END IF
                        ITER = ITER + 1
                        IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
     $                      FOUR*EPS*ABS( LAMBDA ) ) THEN
                           WORK( K ) = LAMBDA + RQCORR
                           IF( ITER.LT.MAXITR ) THEN
                              GO TO 90
                           END IF
                        END IF
                        IWORK( KTOT ) = 1
                        IF( NEWSIZ.EQ.1 )
     $                     NDONE = NDONE + 1
                        CALL DSCAL( IN, NRMINV, Z( IBEGIN, KTOT ), 1 )
                        KTOT = KTOT + 1
  100                CONTINUE
                     IF( NEWSIZ.GT.1 ) THEN
                        ITMP1 = ISUPPZ( 2*NEWFTT-1 )
                        ITMP2 = ISUPPZ( 2*NEWFTT )
                        KTOT = OLDIEN + NEWLST
                        DO 120 P = NEWFTT + 1, KTOT
                           DO 110 Q = NEWFTT, P - 1
                              TMP1 = -DDOT( IN, Z( IBEGIN, P ), 1,
     $                               Z( IBEGIN, Q ), 1 )
                              CALL DAXPY( IN, TMP1, Z( IBEGIN, Q ), 1,
     $                                    Z( IBEGIN, P ), 1 )
  110                      CONTINUE
                           TMP1 = ONE / DNRM2( IN, Z( IBEGIN, P ), 1 )
                           CALL DSCAL( IN, TMP1, Z( IBEGIN, P ), 1 )
                           ITMP1 = MIN( ITMP1, ISUPPZ( 2*P-1 ) )
                           ITMP2 = MAX( ITMP2, ISUPPZ( 2*P ) )
  120                   CONTINUE
                        DO 130 P = NEWFTT, KTOT
                           ISUPPZ( 2*P-1 ) = ITMP1
                           ISUPPZ( 2*P ) = ITMP2
  130                   CONTINUE
                        NDONE = NDONE + NEWSIZ
                     END IF
                  END IF
                  NEWFRS = J + 1
  140          CONTINUE
  150       CONTINUE
            NDEPTH = NDEPTH + 1
            GO TO 40
         END IF
         J = 2*IBEGIN
         DO 160 I = IBEGIN, IEND
            ISUPPZ( J-1 ) = ISUPPZ( J-1 ) + OLDIEN
            ISUPPZ( J ) = ISUPPZ( J ) + OLDIEN
            J = J + 2
  160    CONTINUE
         IBEGIN = IEND + 1
  170 CONTINUE
*
      RETURN
*
*     End of DLARRV
*
      END
      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 (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  IJOB    (input) 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.
*
*  NITMAX  (input) 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.
*
*  N       (input) INTEGER
*          The dimension n of the tridiagonal matrix T.  It must be at
*          least 1.
*
*  MMAX    (input) INTEGER
*          The maximum number of intervals.  If more than MMAX intervals
*          are generated, then DLAEBZ will quit with INFO=MMAX+1.
*
*  MINP    (input) INTEGER
*          The initial number of intervals.  It may not be greater than
*          MMAX.
*
*  NBMIN   (input) INTEGER
*          The smallest number of intervals that should be processed
*          using a vector loop.  If zero, then only the scalar loop
*          will be used.
*
*  ABSTOL  (input) 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.
*
*  RELTOL  (input) 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.
*
*  PIVMIN  (input) 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.
*
*  D       (input) DOUBLE PRECISION array, dimension (N)
*          The diagonal elements of the tridiagonal matrix T.
*
*  E       (input) DOUBLE PRECISION array, dimension (N)
*          The offdiagonal elements of the tridiagonal matrix T in
*          positions 1 through N-1.  E(N) is arbitrary.
*
*  E2      (input) DOUBLE PRECISION array, dimension (N)
*          The squares of the offdiagonal elements of the tridiagonal
*          matrix T.  E2(N) is ignored.
*
*  NVAL    (input/output) 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.
*
*  AB      (input/output) 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.
*
*  C       (input/output) 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.
*
*  MOUT    (output) 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.
*
*  NAB     (input/output) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (MMAX)
*          Workspace.
*
*  IWORK   (workspace) INTEGER array, dimension (MMAX)
*          Workspace.
*
*  INFO    (output) INTEGER
*          = 0:       All intervals converged.
*          = 1--MMAX: The last INFO intervals did not converge.
*          = MMAX+1:  More than MMAX intervals were generated.
*
*  Further Details
*  ===============
*
*      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.
*
*  =====================================================================
*
*     .. 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
*DIR$ NOVECTOR
         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
*
*              A series of compiler directives to defeat vectorization
*              for the next loop
*
*$PL$ CMCHAR=' '
CDIR$          NEXTSCALAR
C$DIR          SCALAR
CDIR$          NEXT SCALAR
CVD$L          NOVECTOR
CDEC$          NOVECTOR
CVD$           NOVECTOR
*VDIR          NOVECTOR
*VOCL          LOOP,SCALAR
CIBM           PREFER SCALAR
*$PL$ CMCHAR='*'
*
               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 of Serial Version of the loop
*
         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
      SUBROUTINE DLARNV( IDIST, ISEED, N, X )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      INTEGER            IDIST, N
*     ..
*     .. Array Arguments ..
      INTEGER            ISEED( 4 )
      DOUBLE PRECISION   X( * )
*     ..
*
*  Purpose
*  =======
*
*  DLARNV returns a vector of n random real numbers from a uniform or
*  normal distribution.
*
*  Arguments
*  =========
*
*  IDIST   (input) INTEGER
*          Specifies the distribution of the random numbers:
*          = 1:  uniform (0,1)
*          = 2:  uniform (-1,1)
*          = 3:  normal (0,1)
*
*  ISEED   (input/output) 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.
*
*  N       (input) INTEGER
*          The number of random numbers to be generated.
*
*  X       (output) DOUBLE PRECISION array, dimension (N)
*          The generated random numbers.
*
*  Further Details
*  ===============
*
*  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.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, TWO
      PARAMETER          ( ONE = 1.0D+0, TWO = 2.0D+0 )
      INTEGER            LV
      PARAMETER          ( LV = 128 )
      DOUBLE PRECISION   TWOPI
      PARAMETER          ( TWOPI = 6.2831853071795864769252867663D+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
      SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
      DOUBLE PRECISION   LAMBDA, TOL
*     ..
*     .. Array Arguments ..
      INTEGER            IN( * )
      DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix T.
*
*  A       (input/output) 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.
*
*  LAMBDA  (input) DOUBLE PRECISION
*          On entry, the scalar lambda.
*
*  B       (input/output) 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.
*
*  C       (input/output) 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.
*
*  TOL     (input) 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.
*
*  D       (output) 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.
*
*  IN      (output) 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) ).le. 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,
*
*  INFO    (output) INTEGER
*          = 0   : successful exit
*          .lt. 0: if INFO = -k, the kth argument had an illegal value
*
* =====================================================================
*
*     .. 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
      SUBROUTINE DLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      INTEGER            INFO, JOB, N
      DOUBLE PRECISION   TOL
*     ..
*     .. Array Arguments ..
      INTEGER            IN( * )
      DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * ), Y( * )
*     ..
*
*  Purpose
*  =======
*
*  DLAGTS may be used to solve one of the systems of equations
*
*     (T - lambda*I)*x = y   or   (T - lambda*I)'*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.
*
*  Arguments
*  =========
*
*  JOB     (input) 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)'x = y  are to be solved,
*                but diagonal elements of U are not to be perturbed.
*          = -2: 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.
*
*  N       (input) INTEGER
*          The order of the matrix T.
*
*  A       (input) DOUBLE PRECISION array, dimension (N)
*          On entry, A must contain the diagonal elements of U as
*          returned from DLAGTF.
*
*  B       (input) DOUBLE PRECISION array, dimension (N-1)
*          On entry, B must contain the first super-diagonal elements of
*          U as returned from DLAGTF.
*
*  C       (input) DOUBLE PRECISION array, dimension (N-1)
*          On entry, C must contain the sub-diagonal elements of L as
*          returned from DLAGTF.
*
*  D       (input) DOUBLE PRECISION array, dimension (N-2)
*          On entry, D must contain the second super-diagonal elements
*          of U as returned from DLAGTF.
*
*  IN      (input) INTEGER array, dimension (N)
*          On entry, IN must contain details of the matrix P as returned
*          from DLAGTF.
*
*  Y       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the right hand side vector y.
*          On exit, Y is overwritten by the solution vector x.
*
*  TOL     (input/output) DOUBLE PRECISION
*          On entry, with  JOB .lt. 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 .gt. 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.
*
*  INFO    (output) INTEGER
*          = 0   : successful exit
*          .lt. 0: if INFO = -i, the i-th argument had an illegal value
*          .gt. 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.
*
*  =====================================================================
*
*     .. 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
      SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DORM2L overwrites the general real m by n matrix C with
*
*        Q * C  if SIDE = 'L' and TRANS = 'N', or
*
*        Q'* C  if SIDE = 'L' and TRANS = 'T', or
*
*        C * Q  if SIDE = 'R' and TRANS = 'N', or
*
*        C * Q' 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'.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'L': apply Q or Q' from the Left
*          = 'R': apply Q or Q' from the Right
*
*  TRANS   (input) CHARACTER*1
*          = 'N': apply Q  (No transpose)
*          = 'T': apply Q' (Transpose)
*
*  M       (input) INTEGER
*          The number of rows of the matrix C. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix C. N >= 0.
*
*  K       (input) INTEGER
*          The number of elementary reflectors whose product defines
*          the matrix Q.
*          If SIDE = 'L', M >= K >= 0;
*          if SIDE = 'R', N >= K >= 0.
*
*  A       (input) 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.
*          A is modified by the routine but restored on exit.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.
*          If SIDE = 'L', LDA >= max(1,M);
*          if SIDE = 'R', LDA >= max(1,N).
*
*  TAU     (input) DOUBLE PRECISION array, dimension (K)
*          TAU(i) must contain the scalar factor of the elementary
*          reflector H(i), as returned by DGEQLF.
*
*  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
*          On entry, the m by n matrix C.
*          On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C. LDC >= max(1,M).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension
*                                   (N) if SIDE = 'L',
*                                   (M) if SIDE = 'R'
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. 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, 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)
*
         AII = A( NQ-K+I, I )
         A( NQ-K+I, I ) = ONE
         CALL DLARF( SIDE, MI, NI, A( 1, I ), 1, TAU( I ), C, LDC,
     $               WORK )
         A( NQ-K+I, I ) = AII
   10 CONTINUE
      RETURN
*
*     End of DORM2L
*
      END
      SUBROUTINE DLARRB( N, D, L, LD, LLD, IFIRST, ILAST, SIGMA, RELTOL,
     $                   W, WGAP, WERR, WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            IFIRST, ILAST, INFO, N
      DOUBLE PRECISION   RELTOL, SIGMA
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), L( * ), LD( * ), LLD( * ), W( * ),
     $                   WERR( * ), WGAP( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  Given the relatively robust representation(RRR) L D L^T, DLARRB
*  does ``limited'' bisection to locate the eigenvalues of L D L^T,
*  W( IFIRST ) thru' W( ILAST ), to more accuracy. Intervals
*  [left, right] are maintained by storing their mid-points and
*  semi-widths in the arrays W and WERR respectively.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix.
*
*  D       (input) DOUBLE PRECISION array, dimension (N)
*          The n diagonal elements of the diagonal matrix D.
*
*  L       (input) DOUBLE PRECISION array, dimension (N-1)
*          The n-1 subdiagonal elements of the unit bidiagonal matrix L.
*
*  LD      (input) DOUBLE PRECISION array, dimension (N-1)
*          The n-1 elements L(i)*D(i).
*
*  LLD     (input) DOUBLE PRECISION array, dimension (N-1)
*          The n-1 elements L(i)*L(i)*D(i).
*
*  IFIRST  (input) INTEGER
*          The index of the first eigenvalue in the cluster.
*
*  ILAST   (input) INTEGER
*          The index of the last eigenvalue in the cluster.
*
*  SIGMA   (input) DOUBLE PRECISION
*          The shift used to form L D L^T (see DLARRF).
*
*  RELTOL  (input) DOUBLE PRECISION
*          The relative tolerance.
*
*  W       (input/output) DOUBLE PRECISION array, dimension (N)
*          On input, W( IFIRST ) thru' W( ILAST ) are estimates of the
*          corresponding eigenvalues of L D L^T.
*          On output, these estimates are ``refined''.
*
*  WGAP    (input/output) DOUBLE PRECISION array, dimension (N)
*          The gaps between the eigenvalues of L D L^T. Very small
*          gaps are changed on output.
*
*  WERR    (input/output) DOUBLE PRECISION array, dimension (N)
*          On input, WERR( IFIRST ) thru' WERR( ILAST ) are the errors
*          in the estimates W( IFIRST ) thru' W( ILAST ).
*          On output, these are the ``refined'' errors.
*
*****Reminder to Inder --- WORK is never used in this subroutine *****
*  WORK    (input) DOUBLE PRECISION array, dimension (???)
*          Workspace.
*
*  IWORK   (input) INTEGER array, dimension (2*N)
*          Workspace.
*
*****Reminder to Inder --- INFO is never set in this subroutine ******
*  INFO    (output) INTEGER
*          Error flag.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Inderjit Dhillon, IBM Almaden, USA
*     Osni Marques, LBNL/NERSC, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, TWO, HALF
      PARAMETER          ( ZERO = 0.0D0, TWO = 2.0D0, HALF = 0.5D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            CNT, I, I1, I2, INITI1, INITI2, J, K, NCNVRG,
     $                   NEIG, NINT, NRIGHT, OLNINT
      DOUBLE PRECISION   DELTA, EPS, GAP, LEFT, MID, PERT, RIGHT, S,
     $                   THRESH, TMP, WIDTH
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      EPS = DLAMCH( 'Precision' )
      I1 = IFIRST
      I2 = IFIRST
      NEIG = ILAST - IFIRST + 1
      NCNVRG = 0
      THRESH = RELTOL
      DO 10 I = IFIRST, ILAST
         IWORK( I ) = 0
         PERT = EPS*( ABS( SIGMA )+ABS( W( I ) ) )
         WERR( I ) = WERR( I ) + PERT
         IF( WGAP( I ).LT.PERT )
     $      WGAP( I ) = PERT
   10 CONTINUE
      DO 20 I = I1, ILAST
         IF( I.EQ.1 ) THEN
            GAP = WGAP( I )
         ELSE IF( I.EQ.N ) THEN
            GAP = WGAP( I-1 )
         ELSE
            GAP = MIN( WGAP( I-1 ), WGAP( I ) )
         END IF
         IF( WERR( I ).LT.THRESH*GAP ) THEN
            NCNVRG = NCNVRG + 1
            IWORK( I ) = 1
            IF( I1.EQ.I )
     $         I1 = I1 + 1
         ELSE
            I2 = I
         END IF
   20 CONTINUE
*
*     Initialize the unconverged intervals.
*
      I = I1
      NINT = 0
      RIGHT = ZERO
   30 CONTINUE
      IF( I.LE.I2 ) THEN
         IF( IWORK( I ).EQ.0 ) THEN
            DELTA = EPS
            LEFT = W( I ) - WERR( I )
*
*           Do while( CNT(LEFT).GT.I-1 )
*
   40       CONTINUE
            IF( I.GT.I1 .AND. LEFT.LE.RIGHT ) THEN
               LEFT = RIGHT
               CNT = I - 1
            ELSE
               S = -LEFT
               CNT = 0
               DO 50 J = 1, N - 1
                  TMP = D( J ) + S
                  S = S*( LD( J ) / TMP )*L( J ) - LEFT
                  IF( TMP.LT.ZERO )
     $               CNT = CNT + 1
   50          CONTINUE
               TMP = D( N ) + S
               IF( TMP.LT.ZERO )
     $            CNT = CNT + 1
               IF( CNT.GT.I-1 ) THEN
                  DELTA = TWO*DELTA
                  LEFT = LEFT - ( ABS( SIGMA )+ABS( LEFT ) )*DELTA
                  GO TO 40
               END IF
            END IF
            DELTA = EPS
            RIGHT = W( I ) + WERR( I )
*
*           Do while( CNT(RIGHT).LT.I )
*
   60       CONTINUE
            S = -RIGHT
            CNT = 0
            DO 70 J = 1, N - 1
               TMP = D( J ) + S
               S = S*( LD( J ) / TMP )*L( J ) - RIGHT
               IF( TMP.LT.ZERO )
     $            CNT = CNT + 1
   70       CONTINUE
            TMP = D( N ) + S
            IF( TMP.LT.ZERO )
     $         CNT = CNT + 1
            IF( CNT.LT.I ) THEN
               DELTA = TWO*DELTA
               RIGHT = RIGHT + ( ABS( SIGMA )+ABS( RIGHT ) )*DELTA
               GO TO 60
            END IF
            WERR( I ) = LEFT
            W( I ) = RIGHT
            IWORK( N+I ) = CNT
            NINT = NINT + 1
            I = CNT + 1
         ELSE
            I = I + 1
         END IF
         GO TO 30
      END IF
*
*     While( NCNVRG.LT.NEIG )
*
      INITI1 = I1
      INITI2 = I2
   80 CONTINUE
      IF( NCNVRG.LT.NEIG ) THEN
         OLNINT = NINT
         I = I1
         DO 100 K = 1, OLNINT
            NRIGHT = IWORK( N+I )
            IF( IWORK( I ).EQ.0 ) THEN
               MID = HALF*( WERR( I )+W( I ) )
               S = -MID
               CNT = 0
               DO 90 J = 1, N - 1
                  TMP = D( J ) + S
                  S = S*( LD( J ) / TMP )*L( J ) - MID
                  IF( TMP.LT.ZERO )
     $               CNT = CNT + 1
   90          CONTINUE
               TMP = D( N ) + S
               IF( TMP.LT.ZERO )
     $            CNT = CNT + 1
               CNT = MAX( I-1, MIN( NRIGHT, CNT ) )
               IF( I.EQ.NRIGHT ) THEN
                  IF( I.EQ.IFIRST ) THEN
                     GAP = WERR( I+1 ) - W( I )
                  ELSE IF( I.EQ.ILAST ) THEN
                     GAP = WERR( I ) - W( I-1 )
                  ELSE
                     GAP = MIN( WERR( I+1 )-W( I ), WERR( I )-W( I-1 ) )
                  END IF
                  WIDTH = W( I ) - MID
                  IF( WIDTH.LT.THRESH*GAP ) THEN
                     NCNVRG = NCNVRG + 1
                     IWORK( I ) = 1
                     IF( I1.EQ.I ) THEN
                        I1 = I1 + 1
                        NINT = NINT - 1
                     END IF
                  END IF
               END IF
               IF( IWORK( I ).EQ.0 )
     $            I2 = K
               IF( CNT.EQ.I-1 ) THEN
                  WERR( I ) = MID
               ELSE IF( CNT.EQ.NRIGHT ) THEN
                  W( I ) = MID
               ELSE
                  IWORK( N+I ) = CNT
                  NINT = NINT + 1
                  WERR( CNT+1 ) = MID
                  W( CNT+1 ) = W( I )
                  W( I ) = MID
                  I = CNT + 1
                  IWORK( N+I ) = NRIGHT
               END IF
            END IF
            I = NRIGHT + 1
  100    CONTINUE
         NINT = NINT - OLNINT + I2
         GO TO 80
      END IF
      DO 110 I = INITI1, INITI2
         W( I ) = HALF*( WERR( I )+W( I ) )
         WERR( I ) = W( I ) - WERR( I )
  110 CONTINUE
*
      RETURN
*
*     End of DLARRB
*
      END
      SUBROUTINE DLARRF( N, D, L, LD, LLD, IFIRST, ILAST, W, DPLUS,
     $                   LPLUS, WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            IFIRST, ILAST, INFO, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), DPLUS( * ), L( * ), LD( * ), LLD( * ),
     $                   LPLUS( * ), W( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  Given the initial representation L D L^T and its cluster of close
*  eigenvalues (in a relative measure), W( IFIRST ), W( IFIRST+1 ), ...
*  W( ILAST ), 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.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix.
*
*  D       (input) DOUBLE PRECISION array, dimension (N)
*          The n diagonal elements of the diagonal matrix D.
*
*  L       (input) DOUBLE PRECISION array, dimension (N-1)
*          The (n-1) subdiagonal elements of the unit bidiagonal
*          matrix L.
*
*  LD      (input) DOUBLE PRECISION array, dimension (N-1)
*          The n-1 elements L(i)*D(i).
*
*  LLD     (input) DOUBLE PRECISION array, dimension (N-1)
*          The n-1 elements L(i)*L(i)*D(i).
*
*  IFIRST  (input) INTEGER
*          The index of the first eigenvalue in the cluster.
*
*  ILAST   (input) INTEGER
*          The index of the last eigenvalue in the cluster.
*
*  W       (input/output) DOUBLE PRECISION array, dimension (N)
*          On input, the eigenvalues of L D L^T in ascending order.
*          W( IFIRST ) through W( ILAST ) form the cluster of relatively
*          close eigenalues.
*          On output, W( IFIRST ) thru' W( ILAST ) are estimates of the
*          corresponding eigenvalues of L(+) D(+) L(+)^T.
*
*  SIGMA   (input) DOUBLE PRECISION
*          The shift used to form L(+) D(+) L(+)^T.
*
*  DPLUS   (output) DOUBLE PRECISION array, dimension (N)
*          The n diagonal elements of the diagonal matrix D(+).
*
*  LPLUS   (output) DOUBLE PRECISION array, dimension (N)
*          The first (n-1) elements of LPLUS contain the subdiagonal
*          elements of the unit bidiagonal matrix L(+). LPLUS( N ) is
*          set to SIGMA.
*
*  WORK    (input) DOUBLE PRECISION array, dimension (???)
*          Workspace.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Inderjit Dhillon, IBM Almaden, USA
*     Osni Marques, LBNL/NERSC, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, TWO
      PARAMETER          ( ZERO = 0.0D0, TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   DELTA, EPS, S, SIGMA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      EPS = DLAMCH( 'Precision' )
      IF( IFIRST.EQ.1 ) THEN
         SIGMA = W( IFIRST )
      ELSE IF( ILAST.EQ.N ) THEN
         SIGMA = W( ILAST )
      ELSE
         INFO = 1
         RETURN
      END IF
*
*     Compute the new relatively robust representation (RRR)
*
      DELTA = TWO*EPS
   10 CONTINUE
      IF( IFIRST.EQ.1 ) THEN
         SIGMA = SIGMA - ABS( SIGMA )*DELTA
      ELSE
         SIGMA = SIGMA + ABS( SIGMA )*DELTA
      END IF
      S = -SIGMA
      DO 20 I = 1, N - 1
         DPLUS( I ) = D( I ) + S
         LPLUS( I ) = LD( I ) / DPLUS( I )
         S = S*LPLUS( I )*L( I ) - SIGMA
   20 CONTINUE
      DPLUS( N ) = D( N ) + S
      IF( IFIRST.EQ.1 ) THEN
         DO 30 I = 1, N
            IF( DPLUS( I ).LT.ZERO ) THEN
               DELTA = TWO*DELTA
               GO TO 10
            END IF
   30    CONTINUE
      ELSE
         DO 40 I = 1, N
            IF( DPLUS( I ).GT.ZERO ) THEN
               DELTA = TWO*DELTA
               GO TO 10
            END IF
   40    CONTINUE
      END IF
      DO 50 I = IFIRST, ILAST
         W( I ) = W( I ) - SIGMA
   50 CONTINUE
      LPLUS( N ) = SIGMA
*
      RETURN
*
*     End of DLARRF
*
      END
      SUBROUTINE DLAR1V( N, B1, BN, SIGMA, D, L, LD, LLD, GERSCH, Z,
     $                   ZTZ, MINGMA, R, ISUPPZ, WORK )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            B1, BN, N, R
      DOUBLE PRECISION   MINGMA, SIGMA, ZTZ
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * )
      DOUBLE PRECISION   D( * ), GERSCH( * ), L( * ), LD( * ), LLD( * ),
     $                   WORK( * ), Z( * )
*     ..
*
*  Purpose
*  =======
*
*  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. 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.
*
*  Arguments
*  =========
*
*  N        (input) INTEGER
*           The order of the matrix L D L^T.
*
*  B1       (input) INTEGER
*           First index of the submatrix of L D L^T.
*
*  BN       (input) INTEGER
*           Last index of the submatrix of L D L^T.
*
*  SIGMA    (input) DOUBLE PRECISION
*           The shift. Initially, when R = 0, SIGMA should be a good
*           approximation to an eigenvalue of L D L^T.
*
*  L        (input) DOUBLE PRECISION array, dimension (N-1)
*           The (n-1) subdiagonal elements of the unit bidiagonal matrix
*           L, in elements 1 to N-1.
*
*  D        (input) DOUBLE PRECISION array, dimension (N)
*           The n diagonal elements of the diagonal matrix D.
*
*  LD       (input) DOUBLE PRECISION array, dimension (N-1)
*           The n-1 elements L(i)*D(i).
*
*  LLD      (input) DOUBLE PRECISION array, dimension (N-1)
*           The n-1 elements L(i)*L(i)*D(i).
*
*  GERSCH   (input) DOUBLE PRECISION array, dimension (2*N)
*           The n Gerschgorin intervals. These are used to restrict
*           the initial search for R, when R is input as 0.
*
*  Z        (output) DOUBLE PRECISION array, dimension (N)
*           The (scaled) r-th column of the inverse. Z(R) is returned
*           to be 1.
*
*  ZTZ      (output) DOUBLE PRECISION
*           The square of the norm of Z.
*
*  MINGMA   (output) DOUBLE PRECISION
*           The reciprocal of the largest (in magnitude) diagonal
*           element of the inverse of L D L^T - sigma I.
*
*  R        (input/output) INTEGER
*           Initially, R should be input to be 0 and is then output as
*           the index where the diagonal element of the inverse is
*           largest in magnitude. In later iterations, this same value
*           of R should be input.
*
*  ISUPPZ   (output) 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 ).
*
*  WORK     (workspace) DOUBLE PRECISION array, dimension (4*N)
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Inderjit Dhillon, IBM Almaden, USA
*     Osni Marques, LBNL/NERSC, USA
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            BLKSIZ
      PARAMETER          ( BLKSIZ = 32 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            SAWNAN
      INTEGER            FROM, I, INDP, INDS, INDUMN, J, R1, R2, TO
      DOUBLE PRECISION   DMINUS, DPLUS, EPS, S, TMP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      EPS = DLAMCH( 'Precision' )
      IF( R.EQ.0 ) THEN
*
*        Eliminate the top and bottom indices from the possible values
*        of R where the desired eigenvector is largest in magnitude.
*
         R1 = B1
         DO 10 I = B1, BN
            IF( SIGMA.GE.GERSCH( 2*I-1 ) .OR. SIGMA.LE.GERSCH( 2*I ) )
     $           THEN
               R1 = I
               GO TO 20
            END IF
   10    CONTINUE
   20    CONTINUE
         R2 = BN
         DO 30 I = BN, B1, -1
            IF( SIGMA.GE.GERSCH( 2*I-1 ) .OR. SIGMA.LE.GERSCH( 2*I ) )
     $           THEN
               R2 = I
               GO TO 40
            END IF
   30    CONTINUE
   40    CONTINUE
      ELSE
         R1 = R
         R2 = R
      END IF
*
      INDUMN = N
      INDS = 2*N + 1
      INDP = 3*N + 1
      SAWNAN = .FALSE.
*
*     Compute the stationary transform (using the differential form)
*     untill the index R2
*
      IF( B1.EQ.1 ) THEN
         WORK( INDS ) = ZERO
      ELSE
         WORK( INDS ) = LLD( B1-1 )
      END IF
      S = WORK( INDS ) - SIGMA
      DO 50 I = B1, R2 - 1
         DPLUS = D( I ) + S
         WORK( I ) = LD( I ) / DPLUS
         WORK( INDS+I ) = S*WORK( I )*L( I )
         S = WORK( INDS+I ) - SIGMA
   50 CONTINUE
*
      IF( .NOT.( S.GT.ZERO .OR. S.LT.ONE ) ) THEN
*
*        Run a slower version of the above loop if a NaN is detected
*
         SAWNAN = .TRUE.
         J = B1 + 1
   60    CONTINUE
         IF( WORK( INDS+J ).GT.ZERO .OR. WORK( INDS+J ).LT.ONE ) THEN
            J = J + 1
            GO TO 60
         END IF
         WORK( INDS+J ) = LLD( J )
         S = WORK( INDS+J ) - SIGMA
         DO 70 I = J + 1, R2 - 1
            DPLUS = D( I ) + S
            WORK( I ) = LD( I ) / DPLUS
            IF( WORK( I ).EQ.ZERO ) THEN
               WORK( INDS+I ) = LLD( I )
            ELSE
               WORK( INDS+I ) = S*WORK( I )*L( I )
            END IF
            S = WORK( INDS+I ) - SIGMA
   70    CONTINUE
      END IF
      WORK( INDP+BN-1 ) = D( BN ) - SIGMA
      DO 80 I = BN - 1, R1, -1
         DMINUS = LLD( I ) + WORK( INDP+I )
         TMP = D( I ) / DMINUS
         WORK( INDUMN+I ) = L( I )*TMP
         WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - SIGMA
   80 CONTINUE
      TMP = WORK( INDP+R1-1 )
      IF( .NOT.( TMP.GT.ZERO .OR. TMP.LT.ONE ) ) THEN
*
*        Run a slower version of the above loop if a NaN is detected
*
         SAWNAN = .TRUE.
         J = BN - 3
   90    CONTINUE
         IF( WORK( INDP+J ).GT.ZERO .OR. WORK( INDP+J ).LT.ONE ) THEN
            J = J - 1
            GO TO 90
         END IF
         WORK( INDP+J ) = D( J+1 ) - SIGMA
         DO 100 I = J, R1, -1
            DMINUS = LLD( I ) + WORK( INDP+I )
            TMP = D( I ) / DMINUS
            WORK( INDUMN+I ) = L( I )*TMP
            IF( TMP.EQ.ZERO ) THEN
               WORK( INDP+I-1 ) = D( I ) - SIGMA
            ELSE
               WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - SIGMA
            END IF
  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.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 ).LT.ABS( MINGMA ) ) THEN
            MINGMA = TMP
            R = I + 1
         END IF
  110 CONTINUE
*
*     Compute the (scaled) r-th column of the inverse
*
      ISUPPZ( 1 ) = B1
      ISUPPZ( 2 ) = BN
      Z( R ) = ONE
      ZTZ = ONE
      IF( .NOT.SAWNAN ) THEN
         FROM = R - 1
         TO = MAX( R-BLKSIZ, B1 )
  120    CONTINUE
         IF( FROM.GE.B1 ) THEN
            DO 130 I = FROM, TO, -1
               Z( I ) = -( WORK( I )*Z( I+1 ) )
               ZTZ = ZTZ + Z( I )*Z( I )
  130       CONTINUE
            IF( ABS( Z( TO ) ).LE.EPS .AND. ABS( Z( TO+1 ) ).LE.EPS )
     $           THEN
               ISUPPZ( 1 ) = TO + 2
            ELSE
               FROM = TO - 1
               TO = MAX( TO-BLKSIZ, B1 )
               GO TO 120
            END IF
         END IF
         FROM = R + 1
         TO = MIN( R+BLKSIZ, BN )
  140    CONTINUE
         IF( FROM.LE.BN ) THEN
            DO 150 I = FROM, TO
               Z( I ) = -( WORK( INDUMN+I-1 )*Z( I-1 ) )
               ZTZ = ZTZ + Z( I )*Z( I )
  150       CONTINUE
            IF( ABS( Z( TO ) ).LE.EPS .AND. ABS( Z( TO-1 ) ).LE.EPS )
     $           THEN
               ISUPPZ( 2 ) = TO - 2
            ELSE
               FROM = TO + 1
               TO = MIN( TO+BLKSIZ, BN )
               GO TO 140
            END IF
         END IF
      ELSE
         DO 160 I = R - 1, B1, -1
            IF( Z( I+1 ).EQ.ZERO ) THEN
               Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 )
            ELSE IF( ABS( Z( I+1 ) ).LE.EPS .AND. ABS( Z( I+2 ) ).LE.
     $               EPS ) THEN
               ISUPPZ( 1 ) = I + 3
               GO TO 170
            ELSE
               Z( I ) = -( WORK( I )*Z( I+1 ) )
            END IF
            ZTZ = ZTZ + Z( I )*Z( I )
  160    CONTINUE
  170    CONTINUE
         DO 180 I = R, BN - 1
            IF( Z( I ).EQ.ZERO ) THEN
               Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 )
            ELSE IF( ABS( Z( I ) ).LE.EPS .AND. ABS( Z( I-1 ) ).LE.EPS )
     $                THEN
               ISUPPZ( 2 ) = I - 2
               GO TO 190
            ELSE
               Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
            END IF
            ZTZ = ZTZ + Z( I+1 )*Z( I+1 )
  180    CONTINUE
  190    CONTINUE
      END IF
      DO 200 I = B1, ISUPPZ( 1 ) - 3
         Z( I ) = ZERO
  200 CONTINUE
      DO 210 I = ISUPPZ( 2 ) + 3, BN
         Z( I ) = ZERO
  210 CONTINUE
*
      RETURN
*
*     End of DLAR1V
*
      END
      SUBROUTINE DLARUV( ISEED, N, X )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      INTEGER            N
*     ..
*     .. Array Arguments ..
      INTEGER            ISEED( 4 )
      DOUBLE PRECISION   X( N )
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  ISEED   (input/output) 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.
*
*  N       (input) INTEGER
*          The number of random numbers to be generated. N <= 128.
*
*  X       (output) DOUBLE PRECISION array, dimension (N)
*          The generated random numbers.
*
*  Further Details
*  ===============
*
*  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.
*
*  =====================================================================
*
*     .. 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 ..
*
      I1 = ISEED( 1 )
      I2 = ISEED( 2 )
      I3 = ISEED( 3 )
      I4 = ISEED( 4 )
*
      DO 10 I = 1, MIN( N, LV )
*
*        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 ) ) ) )
   10 CONTINUE
*
*     Return final value of seed
*
      ISEED( 1 ) = IT1
      ISEED( 2 ) = IT2
      ISEED( 3 ) = IT3
      ISEED( 4 ) = IT4
      RETURN
*
*     End of DLARUV
*
      END