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subroutine wsvdc(xr,xi,ldx,n,p,sr,si,er,ei,ur,ui,ldu,vr,vi,ldv,
* workr,worki,job,info)
integer ldx,n,p,ldu,ldv,job,info
double precision xr(ldx,*),xi(ldx,*),sr(1),si(1),er(1),ei(1),
* ur(ldu,*),ui(ldu,*),vr(ldv,*),vi(ldv,*),
* workr(1),worki(1)
c!purpose
c
c
c wsvdc is a subroutine to reduce a double-complex nxp matrix x by
c unitary transformations u and v to diagonal form. the
c diagonal elements s(i) are the singular values of x. the
c columns of u are the corresponding left singular vectors,
c and the columns of v the right singular vectors.
c
c!calling sequence
c
c subroutine wsvdc(xr,xi,ldx,n,p,sr,si,er,ei,ur,ui,ldu,vr,vi,ldv,
c on entry
c
c x double-complex(ldx,p), where ldx.ge.n.
c x contains the matrix whose singular value
c decomposition is to be computed. x is
c destroyed by wsvdc.
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 ldx integer.
c ldx is the leading dimension of the array x.
c
c ldu integer.
c ldu is the leading dimension of the array u
c (see below).
c
c ldv integer.
c ldv is the leading dimension of the array v
c (see below).
c
c work double-complex(n).
c work is a scratch array.
c
c job integer.
c job controls the computation of the singular
c vectors. it has the decimal expansion ab
c with the following meaning
c
c a.eq.0 do not compute the left singular
c vectors.
c a.eq.1 return the n left singular vectors
c in u.
c a.ge.2 returns the first min(n,p)
c left singular vectors in u.
c b.eq.0 do not compute the right singular
c vectors.
c b.eq.1 return the right singular vectors
c in v.
c
c on return
c
c s double-complex(mm), where mm=min(n+1,p).
c the first min(n,p) entries of s contain the
c singular values of x arranged in descending
c order of magnitude.
c
c e double-complex(p).
c e ordinarily contains zeros. however see the
c discussion of info for exceptions.
c
c u double-complex(ldu,k), where ldu.ge.n.
c if joba.eq.1 then k.eq.n,
c if joba.eq.2 then k.eq.min(n,p).
c u contains the matrix of right singular vectors.
c u is not referenced if joba.eq.0. if n.le.p
c or if joba.gt.2, then u may be identified with x
c in the subroutine call.
c
c v double-complex(ldv,p), where ldv.ge.p.
c v contains the matrix of right singular vectors.
c v is not referenced if jobb.eq.0. if p.le.n,
c then v may be identified whth x in the
c subroutine call.
c
c info integer.
c the singular values (and their corresponding
c singular vectors) s(info+1),s(info+2),...,s(m)
c are correct (here m=min(n,p)). thus if
c info.eq.0, all the singular values and their
c vectors are correct. in any event, the matrix
c b = ctrans(u)*x*v is the bidiagonal matrix
c with the elements of s on its diagonal and the
c elements of e on its super-diagonal (ctrans(u)
c is the conjugate-transpose of u). thus the
c singular values of x and b are the same.
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,drotg
c fortran abs,dimag,max
c fortran max,min,mod,sqrt
c
c!
c Copyright INRIA
c internal variables
c
integer i,iter,j,jobu,k,kase,kk,l,ll,lls,lm1,lp1,ls,lu,m,maxit,
* mm,mm1,mp1,nct,nctp1,ncu,nrt,nrtp1
double precision pythag,wdotcr,wdotci,tr,ti,rr,ri
double precision b,c,cs,el,emm1,f,g,wnrm2,scale,shift,sl,sm,sn,
* smm1,t1,test,ztest
logical wantu,wantv
c
double precision zdumr,zdumi
double precision cabs1
cabs1(zdumr,zdumi) = abs(zdumr) + abs(zdumi)
c
c set the maximum number of iterations.
c MODIFIED ACCORDING TO EISPACK HQR2
c
maxit = 30*min(n,p)
c
c
c determine what is to be computed.
c
wantu = .false.
wantv = .false.
jobu = mod(job,100)/10
ncu = n
if (jobu .gt. 1) ncu = min(n,p)
if (jobu .ne. 0) wantu = .true.
if (mod(job,10) .ne. 0) wantv = .true.
c
c reduce x to bidiagonal form, storing the diagonal elements
c in s and the super-diagonal elements in e.
c
info = 0
nct = min(n-1,p)
nrt = max(0,min(p-2,n))
lu = max(nct,nrt)
if (lu .lt. 1) go to 190
do 180 l = 1, lu
lp1 = l + 1
if (l .gt. nct) go to 30
c
c compute the transformation for the l-th column and
c place the l-th diagonal in s(l).
c
sr(l) = wnrm2(n-l+1,xr(l,l),xi(l,l),1)
si(l) = 0.0d+0
if (cabs1(sr(l),si(l)) .eq. 0.0d+0) go to 20
if (cabs1(xr(l,l),xi(l,l)) .eq. 0.0d+0) go to 10
call wsign(sr(l),si(l),xr(l,l),xi(l,l),sr(l),si(l))
10 continue
call wdiv(1.0d+0,0.0d+0,sr(l),si(l),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)
20 continue
sr(l) = -sr(l)
si(l) = -si(l)
30 continue
if (p .lt. lp1) go to 60
do 50 j = lp1, p
if (l .gt. nct) go to 40
if (cabs1(sr(l),si(l)) .eq. 0.0d+0) go to 40
c
c apply the transformation.
c
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)
40 continue
c
c place the l-th row of x into e for the
c subsequent calculation of the row transformation.
c
er(j) = xr(l,j)
ei(j) = -xi(l,j)
50 continue
60 continue
if (.not.wantu .or. l .gt. nct) go to 80
c
c place the transformation in u for subsequent back
c multiplication.
c
do 70 i = l, n
ur(i,l) = xr(i,l)
ui(i,l) = xi(i,l)
70 continue
80 continue
if (l .gt. nrt) go to 170
c
c compute the l-th row transformation and place the
c l-th super-diagonal in e(l).
c
er(l) = wnrm2(p-l,er(lp1),ei(lp1),1)
ei(l) = 0.0d+0
if (cabs1(er(l),ei(l)) .eq. 0.0d+0) go to 100
if (cabs1(er(lp1),ei(lp1)) .eq. 0.0d+0) go to 90
call wsign(er(l),ei(l),er(lp1),ei(lp1),er(l),ei(l))
90 continue
call wdiv(1.0d+0,0.0d+0,er(l),ei(l),tr,ti)
call wscal(p-l,tr,ti,er(lp1),ei(lp1),1)
er(lp1) = 1.0d+0 + er(lp1)
100 continue
er(l) = -er(l)
ei(l) = +ei(l)
if (lp1 .gt. n .or. cabs1(er(l),ei(l)) .eq. 0.0d+0)
* go to 140
c
c apply the transformation.
c
do 110 i = lp1, n
workr(i) = 0.0d+0
worki(i) = 0.0d+0
110 continue
do 120 j = lp1, p
call waxpy(n-l,er(j),ei(j),xr(lp1,j),xi(lp1,j),1,
* workr(lp1),worki(lp1),1)
120 continue
do 130 j = lp1, p
call wdiv(-er(j),-ei(j),er(lp1),ei(lp1),tr,ti)
call waxpy(n-l,tr,-ti,workr(lp1),worki(lp1),1,
* xr(lp1,j),xi(lp1,j),1)
130 continue
140 continue
if (.not.wantv) go to 160
c
c place the transformation in v for subsequent
c back multiplication.
c
do 150 i = lp1, p
vr(i,l) = er(i)
vi(i,l) = ei(i)
150 continue
160 continue
170 continue
180 continue
190 continue
c
c set up the final bidiagonal matrix or order m.
c
m = min(p,n+1)
nctp1 = nct + 1
nrtp1 = nrt + 1
if (nct .ge. p) go to 200
sr(nctp1) = xr(nctp1,nctp1)
si(nctp1) = xi(nctp1,nctp1)
200 continue
if (n .ge. m) go to 210
sr(m) = 0.0d+0
si(m) = 0.0d+0
210 continue
if (nrtp1 .ge. m) go to 220
er(nrtp1) = xr(nrtp1,m)
ei(nrtp1) = xi(nrtp1,m)
220 continue
er(m) = 0.0d+0
ei(m) = 0.0d+0
c
c if required, generate u.
c
if (.not.wantu) go to 350
if (ncu .lt. nctp1) go to 250
do 240 j = nctp1, ncu
do 230 i = 1, n
ur(i,j) = 0.0d+0
ui(i,j) = 0.0d+0
230 continue
ur(j,j) = 1.0d+0
ui(j,j) = 0.0d+0
240 continue
250 continue
if (nct .lt. 1) go to 340
do 330 ll = 1, nct
l = nct - ll + 1
if (cabs1(sr(l),si(l)) .eq. 0.0d+0) go to 300
lp1 = l + 1
if (ncu .lt. lp1) go to 270
do 260 j = lp1, ncu
tr = -wdotcr(n-l+1,ur(l,l),ui(l,l),1,ur(l,j),
* ui(l,j),1)
ti = -wdotci(n-l+1,ur(l,l),ui(l,l),1,ur(l,j),
* ui(l,j),1)
call wdiv(tr,ti,ur(l,l),ui(l,l),tr,ti)
call waxpy(n-l+1,tr,ti,ur(l,l),ui(l,l),1,ur(l,j),
* ui(l,j),1)
260 continue
270 continue
call wrscal(n-l+1,-1.0d+0,ur(l,l),ui(l,l),1)
ur(l,l) = 1.0d+0 + ur(l,l)
lm1 = l - 1
if (lm1 .lt. 1) go to 290
do 280 i = 1, lm1
ur(i,l) = 0.0d+0
ui(i,l) = 0.0d+0
280 continue
290 continue
go to 320
300 continue
do 310 i = 1, n
ur(i,l) = 0.0d+0
ui(i,l) = 0.0d+0
310 continue
ur(l,l) = 1.0d+0
ui(l,l) = 0.0d+0
320 continue
330 continue
340 continue
350 continue
c
c if it is required, generate v.
c
if (.not.wantv) go to 400
do 390 ll = 1, p
l = p - ll + 1
lp1 = l + 1
if (l .gt. nrt) go to 370
if (cabs1(er(l),ei(l)) .eq. 0.0d+0) go to 370
do 360 j = lp1, p
tr = -wdotcr(p-l,vr(lp1,l),vi(lp1,l),1,vr(lp1,j),
* vi(lp1,j),1)
ti = -wdotci(p-l,vr(lp1,l),vi(lp1,l),1,vr(lp1,j),
* vi(lp1,j),1)
call wdiv(tr,ti,vr(lp1,l),vi(lp1,l),tr,ti)
call waxpy(p-l,tr,ti,vr(lp1,l),vi(lp1,l),1,vr(lp1,j),
* vi(lp1,j),1)
360 continue
370 continue
do 380 i = 1, p
vr(i,l) = 0.0d+0
vi(i,l) = 0.0d+0
380 continue
vr(l,l) = 1.0d+0
vi(l,l) = 0.0d+0
390 continue
400 continue
c
c transform s and e so that they are real.
c
do 420 i = 1, m
tr = pythag(sr(i),si(i))
if (tr .eq. 0.0d+0) go to 405
rr = sr(i)/tr
ri = si(i)/tr
sr(i) = tr
si(i) = 0.0d+0
if (i .lt. m) call wdiv(er(i),ei(i),rr,ri,er(i),ei(i))
if (wantu) call wscal(n,rr,ri,ur(1,i),ui(1,i),1)
405 continue
c ...exit
if (i .eq. m) go to 430
tr = pythag(er(i),ei(i))
if (tr .eq. 0.0d+0) go to 410
call wdiv(tr,0.0d+0,er(i),ei(i),rr,ri)
er(i) = tr
ei(i) = 0.0d+0
call wmul(sr(i+1),si(i+1),rr,ri,sr(i+1),si(i+1))
if (wantv) call wscal(p,rr,ri,vr(1,i+1),vi(1,i+1),1)
410 continue
420 continue
430 continue
c
c main iteration loop for the singular values.
c
mm = m
iter = 0
440 continue
c
c quit if all the singular values have been found.
c
c ...exit
if (m .eq. 0) go to 700
c
c if too many iterations have been performed, set
c flag and return.
c
if (iter .lt. maxit) go to 450
info = m
c ......exit
go to 700
450 continue
c
c this section of the program inspects for
c negligible elements in the s and e arrays. on
c completion the variable kase is set as follows.
c
c kase = 1 if sr(m) and er(l-1) are negligible and l.lt.m
c kase = 2 if sr(l) is negligible and l.lt.m
c kase = 3 if er(l-1) is negligible, l.lt.m, and
c sr(l), ..., sr(m) are not negligible (qr step).
c kase = 4 if er(m-1) is negligible (convergence).
c
do 470 ll = 1, m
l = m - ll
c ...exit
if (l .eq. 0) go to 480
test = pythag(sr(l),si(l)) + pythag(sr(l+1),si(l+1))
ztest = test + pythag(er(l),ei(l))
if (ztest .ne. test) go to 460
er(l) = 0.0d+0
ei(l) = 0.0d+0
c ......exit
go to 480
460 continue
470 continue
480 continue
if (l .ne. m - 1) go to 490
kase = 4
go to 560
490 continue
lp1 = l + 1
mp1 = m + 1
do 510 lls = lp1, mp1
ls = m - lls + lp1
c ...exit
if (ls .eq. l) go to 520
test = 0.0d+0
if (ls .ne. m) test=test + pythag(er(ls),ei(ls))
if (ls .ne. l + 1) test=test + pythag(er(ls-1),ei(ls-1))
ztest = test + pythag(sr(ls),si(ls))
if (ztest .ne. test) go to 500
sr(ls) = 0.0d+0
si(ls) = 0.0d+0
c ......exit
go to 520
500 continue
510 continue
520 continue
if (ls .ne. l) go to 530
kase = 3
go to 550
530 continue
if (ls .ne. m) go to 540
kase = 1
go to 550
540 continue
kase = 2
l = ls
550 continue
560 continue
l = l + 1
c
c perform the task indicated by kase.
c
go to (570, 600, 620, 650), kase
c
c deflate negligible s(m).
c
570 continue
mm1 = m - 1
f = er(m-1)
er(m-1) = 0.0d+0
ei(m-1) = 0.0d+0
do 590 kk = l, mm1
k = mm1 - kk + l
t1 = sr(k)
call drotg(t1,f,cs,sn)
sr(k) = t1
si(k) = 0.0d0
if (k .eq. l) go to 580
f = -sn*er(k-1)
er(k-1) = cs*er(k-1)
ei(k-1) = cs*ei(k-1)
580 continue
if (wantv) call drot(p,vr(1,k),1,vr(1,m),1,cs,sn)
if (wantv) call drot(p,vi(1,k),1,vi(1,m),1,cs,sn)
590 continue
go to 690
c
c split at negligible s(l).
c
600 continue
f = er(l-1)
er(l-1) = 0.0d+0
ei(l-1) = 0.0d+0
do 610 k = l, m
t1 = sr(k)
call drotg(t1,f,cs,sn)
sr(k) = t1
si(k) = 0.0d0
f = -sn*er(k)
er(k) = cs*er(k)
ei(k) = cs*ei(k)
if (wantu) call drot(n,ur(1,k),1,ur(1,l-1),1,cs,sn)
if (wantu) call drot(n,ui(1,k),1,ui(1,l-1),1,cs,sn)
610 continue
go to 690
c
c perform one qr step.
c
620 continue
c
c calculate the shift.
c
scale = max(pythag(sr(m),si(m)),pythag(sr(m-1),si(m-1)),
* pythag(er(m-1),ei(m-1)),
* pythag(sr(l),si(l)),pythag(er(l),ei(l)))
sm = sr(m)/scale
smm1 = sr(m-1)/scale
emm1 = er(m-1)/scale
sl = sr(l)/scale
el = er(l)/scale
b = ((smm1 + sm)*(smm1 - sm) + emm1**2)/2.0d+0
c = (sm*emm1)**2
shift = 0.0d+0
if (b .eq. 0.0d+0 .and. c .eq. 0.0d+0) go to 630
shift = sqrt(b**2+c)
if (b .lt. 0.0d+0) shift = -shift
shift = c/(b + shift)
630 continue
c f = (sl + sm)*(sl - sm) - shift
f=(sl+sm)*(sl-sm)+shift
g = sl*el
c
c chase zeros.
c
mm1 = m - 1
do 640 k = l, mm1
call drotg(f,g,cs,sn)
if (k .ne. l) then
er(k-1) = f
ei(k-1) = 0.0d0
endif
f = cs*sr(k) + sn*er(k)
er(k) = cs*er(k) - sn*sr(k)
ei(k) = cs*ei(k) - sn*si(k)
g = sn*sr(k+1)
sr(k+1) = cs*sr(k+1)
si(k+1) = cs*si(k+1)
if (wantv) call drot(p,vr(1,k),1,vr(1,k+1),1,cs,sn)
if (wantv) call drot(p,vi(1,k),1,vi(1,k+1),1,cs,sn)
call drotg(f,g,cs,sn)
sr(k) = f
si(k) = 0.0d0
f = cs*er(k) + sn*sr(k+1)
sr(k+1) = -sn*er(k) + cs*sr(k+1)
si(k+1) = -sn*ei(k) + cs*si(k+1)
g = sn*er(k+1)
er(k+1) = cs*er(k+1)
ei(k+1) = cs*ei(k+1)
if (wantu .and. k .lt. n)
* call drot(n,ur(1,k),1,ur(1,k+1),1,cs,sn)
if (wantu .and. k .lt. n)
* call drot(n,ui(1,k),1,ui(1,k+1),1,cs,sn)
640 continue
er(m-1) = f
ei(m-1) = 0.0d0
iter = iter + 1
go to 690
c
c convergence
c
650 continue
c
c make the singular value positive
c
if (sr(l) .ge. 0.0d+0) go to 660
sr(l) = -sr(l)
si(l) = -si(l)
if (wantv) call wrscal(p,-1.0d+0,vr(1,l),vi(1,l),1)
660 continue
c
c order the singular value.
c
670 if (l .eq. mm) go to 680
c ...exit
if (sr(l) .ge. sr(l+1)) go to 680
tr = sr(l)
sr(l) = sr(l+1)
sr(l+1) = tr
tr = si(l)
si(l) = si(l+1)
si(l+1) = tr
if (wantv .and. l .lt. p)
* call wswap(p,vr(1,l),vi(1,l),1,vr(1,l+1),vi(1,l+1),1)
if (wantu .and. l .lt. n)
* call wswap(n,ur(1,l),ui(1,l),1,ur(1,l+1),ui(1,l+1),1)
l = l + 1
go to 670
680 continue
iter = 0
m = m - 1
690 continue
go to 440
700 continue
return
end
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