C Dqrdc2 is a *modification* of Linpack's dqrdc ('DQRDC') for R
c
c dqrdc2 uses householder transformations to compute the qr
c factorization of an n by p matrix x. a limited column
c pivoting strategy based on the 2-norms of the reduced columns
c moves columns with near-zero norm to the right-hand edge of
c the x matrix. this strategy means that sequential one
c degree-of-freedom effects can be computed in a natural way.
c
c i am very nervous about modifying linpack code in this way.
c if you are a computational linear algebra guru and you really
c understand how to solve this problem please feel free to
c suggest improvements to this code.
c
c Another change was to compute the rank.
c
c on entry
c
c x double precision(ldx,p), where ldx .ge. n.
c x contains the matrix whose decomposition is to be
c computed.
c
c ldx integer.
c ldx is the leading dimension of the array x.
c
c n integer.
c n is the number of rows of the matrix x.
c
c p integer.
c p is the number of columns of the matrix x.
c
c tol double precision
c tol is the nonnegative tolerance used to
c determine the subset of the columns of x
c included in the solution.
c
c jpvt integer(p).
c integers which are swapped in the same way as the
c the columns of x during pivoting. on entry these
c should be set equal to the column indices of the
c columns of the x matrix (typically 1 to p).
c
c work double precision(p,2).
c work is a work array.
c
c on return
c
c x x contains in its upper triangle the upper
c triangular matrix r of the qr factorization.
c below its diagonal x contains information from
c which the orthogonal part of the decomposition
c can be recovered. note that if pivoting has
c been requested, the decomposition is not that
c of the original matrix x but that of x
c with its columns permuted as described by jpvt.
c
c k integer.
c k contains the number of columns of x judged
c to be linearly independent, i.e., "the rank"
c
c qraux double precision(p).
c qraux contains further information required to recover
c the orthogonal part of the decomposition.
c
c jpvt jpvt(j) contains the index of the column of the
c original matrix that has been interchanged into
c the j-th column. Consequently, jpvt[] codes a
c permutation of 1:p; it is called 'pivot' in R
c
c original (dqrdc.f) linpack version dated 08/14/78 .
c g.w. stewart, university of maryland, argonne national lab.
c
C This version dated 22 August 1995
C Ross Ihaka
c
c bug fixes 29 September 1999 BDR (p > n case, inaccurate ranks)
c
c
c dqrdc2 uses the following functions and subprograms.
c
c blas daxpy,ddot,dscal,dnrm2
c fortran dabs,dmax1,min0,dsqrt
c
subroutine dqrdc2(x,ldx,n,p,tol,k,qraux,jpvt,work)
integer ldx,n,p
integer jpvt(p)
double precision x(ldx,p),qraux(p),work(p,2),tol
c
c internal variables
c
integer i,j,l,lp1,lup,k
double precision dnrm2,tt,ttt
double precision ddot,nrmxl,t
c
c
c compute the norms of the columns of x.
c
if (n .gt. 0) then
c avoid accessing element beyond the bound
do 70 j = 1, p
qraux(j) = dnrm2(n,x(1,j),1)
work(j,1) = qraux(j)
work(j,2) = qraux(j)
if(work(j,2) .eq. 0.0d0) work(j,2) = 1.0d0
70 continue
endif
c
c perform the householder reduction of x.
c
lup = min0(n,p)
k = p + 1
do 200 l = 1, lup
c
c previous version only cycled l to lup
c
c cycle the columns from l to p left-to-right until one
c with non-negligible norm is located. a column is considered
c to have become negligible if its norm has fallen below
c tol times its original norm. the check for l .le. k
c avoids infinite cycling.
c
80 continue
if (l .ge. k .or. qraux(l) .ge. work(l,2)*tol) go to 120
lp1 = l+1
do 100 i=1,n
t = x(i,l)
do 90 j=lp1,p
x(i,j-1) = x(i,j)
90 continue
x(i,p) = t
100 continue
i = jpvt(l)
t = qraux(l)
tt = work(l,1)
ttt = work(l,2)
do 110 j=lp1,p
jpvt(j-1) = jpvt(j)
qraux(j-1) = qraux(j)
work(j-1,1) = work(j,1)
work(j-1,2) = work(j,2)
110 continue
jpvt(p) = i
qraux(p) = t
work(p,1) = tt
work(p,2) = ttt
k = k - 1
go to 80
120 continue
if (l .eq. n) go to 190
c
c compute the householder transformation for column l.
c
nrmxl = dnrm2(n-l+1,x(l,l),1)
if (nrmxl .eq. 0.0d0) go to 180
if (x(l,l) .ne. 0.0d0) nrmxl = dsign(nrmxl,x(l,l))
call dscal(n-l+1,1.0d0/nrmxl,x(l,l),1)
x(l,l) = 1.0d0 + x(l,l)
c
c apply the transformation to the remaining columns,
c updating the norms.
c
lp1 = l + 1
if (p .lt. lp1) go to 170
do 160 j = lp1, p
t = -ddot(n-l+1,x(l,l),1,x(l,j),1)/x(l,l)
call daxpy(n-l+1,t,x(l,l),1,x(l,j),1)
if (qraux(j) .eq. 0.0d0) go to 150
tt = 1.0d0 - (dabs(x(l,j))/qraux(j))**2
tt = dmax1(tt,0.0d0)
t = tt
c
c modified 9/99 by BDR. Re-compute norms if there is large reduction
c The tolerance here is on the squared norm
c In this version we need accurate norms, so re-compute often.
c work(j,1) is only updated in one case: looks like a bug -- no longer used
c
c tt = 1.0d0 + 0.05d0*tt*(qraux(j)/work(j,1))**2
c if (tt .eq. 1.0d0) go to 130
if (dabs(t) .lt. 1d-6) go to 130
qraux(j) = qraux(j)*dsqrt(t)
go to 140
130 continue
qraux(j) = dnrm2(n-l,x(l+1,j),1)
work(j,1) = qraux(j)
140 continue
150 continue
160 continue
170 continue
c
c save the transformation.
c
qraux(l) = x(l,l)
x(l,l) = -nrmxl
180 continue
190 continue
200 continue
k = min0(k - 1, n)
return
end