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C/MEMBR ADD NAME=WQRDC,SSI=0
c Copyright INRIA
subroutine wqrdc(xr,xi,ldx,n,p,qrauxr,qrauxi,jpvt,workr,worki,
* job)
integer ldx,n,p,job
integer jpvt(*)
double precision xr(ldx,*),xi(ldx,*),qrauxr(*),qrauxi(*),
* workr(*),worki(*)
c!purpose
c
c wqrdc uses householder transformations to compute the qr
c factorization of an n by p matrix x. column pivoting
c based on the 2-norms of the reduced columns may be
c performed at the users option.
c
c!calling sequence
c
c subroutine wqrdc(xr,xi,ldx,n,p,qrauxr,qrauxi,jpvt,workr,worki,
c on entry
c
c x double-complex(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 jpvt integer(p).
c jpvt contains integers that control the selection
c of the pivot columns. the k-th column x(k) of x
c is placed in one of three classes according to the
c value of jpvt(k).
c
c if jpvt(k) .gt. 0, then x(k) is an initial
c column.
c
c if jpvt(k) .eq. 0, then x(k) is a free column.
c
c if jpvt(k) .lt. 0, then x(k) is a final column.
c
c before the decomposition is computed, initial columns
c are moved to the beginning of the array x and final
c columns to the end. both initial and final columns
c are frozen in place during the computation and only
c free columns are moved. at the k-th stage of the
c reduction, if x(k) is occupied by a free column
c it is interchanged with the free column of largest
c reduced norm. jpvt is not referenced if
c job .eq. 0.
c
c work double-complex(p).
c work is a work array. work is not referenced if
c job .eq. 0.
c
c job integer.
c job is an integer that initiates column pivoting.
c if job .eq. 0, no pivoting is done.
c if job .ne. 0, pivoting is done.
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 unitary 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 qraux double-complex(p).
c qraux contains further information required to recover
c the unitary part of the decomposition.
c
c jpvt jpvt(k) contains the index of the column of the
c original matrix that has been interchanged into
c the k-th column, if pivoting was requested.
c
c!originator
c linpack. this version dated 07/03/79 .
c g.w. stewart, university of maryland, argonne national lab.
c
c!auxiliary routines
c
c blas waxpy,pythag,wdotcr,wdotci,wscal,wswap,wnrm2
c fortran abs,dimag,max,min
c
c!
c internal variables
c
integer j,jp,l,lp1,lup,maxj,pl,pu
double precision maxnrm,wnrm2,tt
double precision pythag,wdotcr,wdotci,nrmxlr,nrmxli,tr,ti
logical negj,swapj
c
double precision zdumr,zdumi
double precision cabs1
cabs1(zdumr,zdumi) = abs(zdumr) + abs(zdumi)
c
pl = 1
pu = 0
if (job .eq. 0) go to 60
c
c pivoting has been requested. rearrange the columns
c according to jpvt.
c
do 20 j = 1, p
swapj = jpvt(j) .gt. 0
negj = jpvt(j) .lt. 0
jpvt(j) = j
if (negj) jpvt(j) = -j
if (.not.swapj) go to 10
if (j .ne. pl)
* call wswap(n,xr(1,pl),xi(1,pl),1,xr(1,j),xi(1,j),1)
jpvt(j) = jpvt(pl)
jpvt(pl) = j
pl = pl + 1
10 continue
20 continue
pu = p
do 50 jj = 1, p
j = p - jj + 1
if (jpvt(j) .ge. 0) go to 40
jpvt(j) = -jpvt(j)
if (j .eq. pu) go to 30
call wswap(n,xr(1,pu),xi(1,pu),1,xr(1,j),xi(1,j),1)
jp = jpvt(pu)
jpvt(pu) = jpvt(j)
jpvt(j) = jp
30 continue
pu = pu - 1
40 continue
50 continue
60 continue
c
c compute the norms of the free columns.
c
if (pu .lt. pl) go to 80
do 70 j = pl, pu
qrauxr(j) = wnrm2(n,xr(1,j),xi(1,j),1)
qrauxi(j) = 0.0d+0
workr(j) = qrauxr(j)
worki(j) = qrauxi(j)
70 continue
80 continue
c
c perform the householder reduction of x.
c
lup = min(n,p)
do 210 l = 1, lup
if (l .lt. pl .or. l .ge. pu) go to 120
c
c locate the column of largest norm and bring it
c into the pivot position.
c
maxnrm = 0.0d+0
maxj = l
do 100 j = l, pu
if (qrauxr(j) .le. maxnrm) go to 90
maxnrm = qrauxr(j)
maxj = j
90 continue
100 continue
if (maxj .eq. l) go to 110
call wswap(n,xr(1,l),xi(1,l),1,xr(1,maxj),xi(1,maxj),1)
qrauxr(maxj) = qrauxr(l)
qrauxi(maxj) = qrauxi(l)
workr(maxj) = workr(l)
worki(maxj) = worki(l)
jp = jpvt(maxj)
jpvt(maxj) = jpvt(l)
jpvt(l) = jp
110 continue
120 continue
qrauxr(l) = 0.0d+0
qrauxi(l) = 0.0d+0
if (l .eq. n) go to 200
c
c compute the householder transformation for column l.
c
nrmxlr = wnrm2(n-l+1,xr(l,l),xi(l,l),1)
nrmxli = 0.0d+0
if (cabs1(nrmxlr,nrmxli) .eq. 0.0d+0) go to 190
if (cabs1(xr(l,l),xi(l,l)) .eq. 0.0d+0) go to 130
call wsign(nrmxlr,nrmxli,xr(l,l),xi(l,l),nrmxlr,nrmxli)
130 continue
call wdiv(1.0d+0,0.0d+0,nrmxlr,nrmxli,tr,ti)
call wscal(n-l+1,tr,ti,xr(l,l),xi(l,l),1)
xr(l,l) = 1.0d+0 + xr(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 180
do 170 j = lp1, p
tr = -wdotcr(n-l+1,xr(l,l),xi(l,l),1,xr(l,j),
* xi(l,j),1)
ti = -wdotci(n-l+1,xr(l,l),xi(l,l),1,xr(l,j),
* xi(l,j),1)
call wdiv(tr,ti,xr(l,l),xi(l,l),tr,ti)
call waxpy(n-l+1,tr,ti,xr(l,l),xi(l,l),1,xr(l,j),
* xi(l,j),1)
if (j .lt. pl .or. j .gt. pu) go to 160
if (cabs1(qrauxr(j),qrauxi(j)) .eq. 0.0d+0)
* go to 160
tt = 1.0d+0-(pythag(xr(l,j),xi(l,j))/qrauxr(j))**2
tt = max(tt,0.0d+0)
tr = tt
tt = 1.0d+0+0.050d+0*tt*(qrauxr(j)/workr(j))**2
if (tt .eq. 1.0d+0) go to 140
qrauxr(j) = qrauxr(j)*sqrt(tr)
qrauxi(j) = qrauxi(j)*sqrt(tr)
go to 150
140 continue
qrauxr(j) = wnrm2(n-l,xr(l+1,j),xi(l+1,j),1)
qrauxi(j) = 0.0d+0
workr(j) = qrauxr(j)
worki(j) = qrauxi(j)
150 continue
160 continue
170 continue
180 continue
c
c save the transformation.
c
qrauxr(l) = xr(l,l)
qrauxi(l) = xi(l,l)
xr(l,l) = -nrmxlr
xi(l,l) = -nrmxli
190 continue
200 continue
210 continue
return
end
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