LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
Searching...
No Matches
zgelsd.f
Go to the documentation of this file.
1*> \brief <b> ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgelsd.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgelsd.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgelsd.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
22* WORK, LWORK, RWORK, IWORK, INFO )
23*
24* .. Scalar Arguments ..
25* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
26* DOUBLE PRECISION RCOND
27* ..
28* .. Array Arguments ..
29* INTEGER IWORK( * )
30* DOUBLE PRECISION RWORK( * ), S( * )
31* COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
32* ..
33*
34*
35*> \par Purpose:
36* =============
37*>
38*> \verbatim
39*>
40*> ZGELSD computes the minimum-norm solution to a real linear least
41*> squares problem:
42*> minimize 2-norm(| b - A*x |)
43*> using the singular value decomposition (SVD) of A. A is an M-by-N
44*> matrix which may be rank-deficient.
45*>
46*> Several right hand side vectors b and solution vectors x can be
47*> handled in a single call; they are stored as the columns of the
48*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
49*> matrix X.
50*>
51*> The problem is solved in three steps:
52*> (1) Reduce the coefficient matrix A to bidiagonal form with
53*> Householder transformations, reducing the original problem
54*> into a "bidiagonal least squares problem" (BLS)
55*> (2) Solve the BLS using a divide and conquer approach.
56*> (3) Apply back all the Householder transformations to solve
57*> the original least squares problem.
58*>
59*> The effective rank of A is determined by treating as zero those
60*> singular values which are less than RCOND times the largest singular
61*> value.
62*>
63*> The divide and conquer algorithm makes very mild assumptions about
64*> floating point arithmetic. It will work on machines with a guard
65*> digit in add/subtract, or on those binary machines without guard
66*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
67*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
68*> without guard digits, but we know of none.
69*> \endverbatim
70*
71* Arguments:
72* ==========
73*
74*> \param[in] M
75*> \verbatim
76*> M is INTEGER
77*> The number of rows of the matrix A. M >= 0.
78*> \endverbatim
79*>
80*> \param[in] N
81*> \verbatim
82*> N is INTEGER
83*> The number of columns of the matrix A. N >= 0.
84*> \endverbatim
85*>
86*> \param[in] NRHS
87*> \verbatim
88*> NRHS is INTEGER
89*> The number of right hand sides, i.e., the number of columns
90*> of the matrices B and X. NRHS >= 0.
91*> \endverbatim
92*>
93*> \param[in,out] A
94*> \verbatim
95*> A is COMPLEX*16 array, dimension (LDA,N)
96*> On entry, the M-by-N matrix A.
97*> On exit, A has been destroyed.
98*> \endverbatim
99*>
100*> \param[in] LDA
101*> \verbatim
102*> LDA is INTEGER
103*> The leading dimension of the array A. LDA >= max(1,M).
104*> \endverbatim
105*>
106*> \param[in,out] B
107*> \verbatim
108*> B is COMPLEX*16 array, dimension (LDB,NRHS)
109*> On entry, the M-by-NRHS right hand side matrix B.
110*> On exit, B is overwritten by the N-by-NRHS solution matrix X.
111*> If m >= n and RANK = n, the residual sum-of-squares for
112*> the solution in the i-th column is given by the sum of
113*> squares of the modulus of elements n+1:m in that column.
114*> \endverbatim
115*>
116*> \param[in] LDB
117*> \verbatim
118*> LDB is INTEGER
119*> The leading dimension of the array B. LDB >= max(1,M,N).
120*> \endverbatim
121*>
122*> \param[out] S
123*> \verbatim
124*> S is DOUBLE PRECISION array, dimension (min(M,N))
125*> The singular values of A in decreasing order.
126*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
127*> \endverbatim
128*>
129*> \param[in] RCOND
130*> \verbatim
131*> RCOND is DOUBLE PRECISION
132*> RCOND is used to determine the effective rank of A.
133*> Singular values S(i) <= RCOND*S(1) are treated as zero.
134*> If RCOND < 0, machine precision is used instead.
135*> \endverbatim
136*>
137*> \param[out] RANK
138*> \verbatim
139*> RANK is INTEGER
140*> The effective rank of A, i.e., the number of singular values
141*> which are greater than RCOND*S(1).
142*> \endverbatim
143*>
144*> \param[out] WORK
145*> \verbatim
146*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
147*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
148*> \endverbatim
149*>
150*> \param[in] LWORK
151*> \verbatim
152*> LWORK is INTEGER
153*> The dimension of the array WORK. LWORK must be at least 1.
154*> The exact minimum amount of workspace needed depends on M,
155*> N and NRHS. As long as LWORK is at least
156*> 2*N + N*NRHS
157*> if M is greater than or equal to N or
158*> 2*M + M*NRHS
159*> if M is less than N, the code will execute correctly.
160*> For good performance, LWORK should generally be larger.
161*>
162*> If LWORK = -1, then a workspace query is assumed; the routine
163*> only calculates the optimal size of the array WORK and the
164*> minimum sizes of the arrays RWORK and IWORK, and returns
165*> these values as the first entries of the WORK, RWORK and
166*> IWORK arrays, and no error message related to LWORK is issued
167*> by XERBLA.
168*> \endverbatim
169*>
170*> \param[out] RWORK
171*> \verbatim
172*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
173*> LRWORK >=
174*> 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
175*> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
176*> if M is greater than or equal to N or
177*> 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
178*> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
179*> if M is less than N, the code will execute correctly.
180*> SMLSIZ is returned by ILAENV and is equal to the maximum
181*> size of the subproblems at the bottom of the computation
182*> tree (usually about 25), and
183*> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
184*> On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.
185*> \endverbatim
186*>
187*> \param[out] IWORK
188*> \verbatim
189*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
190*> LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
191*> where MINMN = MIN( M,N ).
192*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
193*> \endverbatim
194*>
195*> \param[out] INFO
196*> \verbatim
197*> INFO is INTEGER
198*> = 0: successful exit
199*> < 0: if INFO = -i, the i-th argument had an illegal value.
200*> > 0: the algorithm for computing the SVD failed to converge;
201*> if INFO = i, i off-diagonal elements of an intermediate
202*> bidiagonal form did not converge to zero.
203*> \endverbatim
204*
205* Authors:
206* ========
207*
208*> \author Univ. of Tennessee
209*> \author Univ. of California Berkeley
210*> \author Univ. of Colorado Denver
211*> \author NAG Ltd.
212*
213*> \ingroup complex16GEsolve
214*
215*> \par Contributors:
216* ==================
217*>
218*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
219*> California at Berkeley, USA \n
220*> Osni Marques, LBNL/NERSC, USA \n
221*
222* =====================================================================
223 SUBROUTINE zgelsd( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
224 \$ WORK, LWORK, RWORK, IWORK, INFO )
225*
226* -- LAPACK driver routine --
227* -- LAPACK is a software package provided by Univ. of Tennessee, --
228* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
229*
230* .. Scalar Arguments ..
231 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
232 DOUBLE PRECISION RCOND
233* ..
234* .. Array Arguments ..
235 INTEGER IWORK( * )
236 DOUBLE PRECISION RWORK( * ), S( * )
237 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
238* ..
239*
240* =====================================================================
241*
242* .. Parameters ..
243 DOUBLE PRECISION ZERO, ONE, TWO
244 parameter( zero = 0.0d+0, one = 1.0d+0, two = 2.0d+0 )
245 COMPLEX*16 CZERO
246 parameter( czero = ( 0.0d+0, 0.0d+0 ) )
247* ..
248* .. Local Scalars ..
249 LOGICAL LQUERY
250 INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
251 \$ ldwork, liwork, lrwork, maxmn, maxwrk, minmn,
252 \$ minwrk, mm, mnthr, nlvl, nrwork, nwork, smlsiz
253 DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
254* ..
255* .. External Subroutines ..
256 EXTERNAL dlabad, dlascl, dlaset, xerbla, zgebrd, zgelqf,
258 \$ zunmlq, zunmqr
259* ..
260* .. External Functions ..
261 INTEGER ILAENV
262 DOUBLE PRECISION DLAMCH, ZLANGE
263 EXTERNAL ilaenv, dlamch, zlange
264* ..
265* .. Intrinsic Functions ..
266 INTRINSIC int, log, max, min, dble
267* ..
268* .. Executable Statements ..
269*
270* Test the input arguments.
271*
272 info = 0
273 minmn = min( m, n )
274 maxmn = max( m, n )
275 lquery = ( lwork.EQ.-1 )
276 IF( m.LT.0 ) THEN
277 info = -1
278 ELSE IF( n.LT.0 ) THEN
279 info = -2
280 ELSE IF( nrhs.LT.0 ) THEN
281 info = -3
282 ELSE IF( lda.LT.max( 1, m ) ) THEN
283 info = -5
284 ELSE IF( ldb.LT.max( 1, maxmn ) ) THEN
285 info = -7
286 END IF
287*
288* Compute workspace.
289* (Note: Comments in the code beginning "Workspace:" describe the
290* minimal amount of workspace needed at that point in the code,
291* as well as the preferred amount for good performance.
292* NB refers to the optimal block size for the immediately
293* following subroutine, as returned by ILAENV.)
294*
295 IF( info.EQ.0 ) THEN
296 minwrk = 1
297 maxwrk = 1
298 liwork = 1
299 lrwork = 1
300 IF( minmn.GT.0 ) THEN
301 smlsiz = ilaenv( 9, 'ZGELSD', ' ', 0, 0, 0, 0 )
302 mnthr = ilaenv( 6, 'ZGELSD', ' ', m, n, nrhs, -1 )
303 nlvl = max( int( log( dble( minmn ) / dble( smlsiz + 1 ) ) /
304 \$ log( two ) ) + 1, 0 )
305 liwork = 3*minmn*nlvl + 11*minmn
306 mm = m
307 IF( m.GE.n .AND. m.GE.mnthr ) THEN
308*
309* Path 1a - overdetermined, with many more rows than
310* columns.
311*
312 mm = n
313 maxwrk = max( maxwrk, n*ilaenv( 1, 'ZGEQRF', ' ', m, n,
314 \$ -1, -1 ) )
315 maxwrk = max( maxwrk, nrhs*ilaenv( 1, 'ZUNMQR', 'LC', m,
316 \$ nrhs, n, -1 ) )
317 END IF
318 IF( m.GE.n ) THEN
319*
320* Path 1 - overdetermined or exactly determined.
321*
322 lrwork = 10*n + 2*n*smlsiz + 8*n*nlvl + 3*smlsiz*nrhs +
323 \$ max( (smlsiz+1)**2, n*(1+nrhs) + 2*nrhs )
324 maxwrk = max( maxwrk, 2*n + ( mm + n )*ilaenv( 1,
325 \$ 'ZGEBRD', ' ', mm, n, -1, -1 ) )
326 maxwrk = max( maxwrk, 2*n + nrhs*ilaenv( 1, 'ZUNMBR',
327 \$ 'QLC', mm, nrhs, n, -1 ) )
328 maxwrk = max( maxwrk, 2*n + ( n - 1 )*ilaenv( 1,
329 \$ 'ZUNMBR', 'PLN', n, nrhs, n, -1 ) )
330 maxwrk = max( maxwrk, 2*n + n*nrhs )
331 minwrk = max( 2*n + mm, 2*n + n*nrhs )
332 END IF
333 IF( n.GT.m ) THEN
334 lrwork = 10*m + 2*m*smlsiz + 8*m*nlvl + 3*smlsiz*nrhs +
335 \$ max( (smlsiz+1)**2, n*(1+nrhs) + 2*nrhs )
336 IF( n.GE.mnthr ) THEN
337*
338* Path 2a - underdetermined, with many more columns
339* than rows.
340*
341 maxwrk = m + m*ilaenv( 1, 'ZGELQF', ' ', m, n, -1,
342 \$ -1 )
343 maxwrk = max( maxwrk, m*m + 4*m + 2*m*ilaenv( 1,
344 \$ 'ZGEBRD', ' ', m, m, -1, -1 ) )
345 maxwrk = max( maxwrk, m*m + 4*m + nrhs*ilaenv( 1,
346 \$ 'ZUNMBR', 'QLC', m, nrhs, m, -1 ) )
347 maxwrk = max( maxwrk, m*m + 4*m + ( m - 1 )*ilaenv( 1,
348 \$ 'ZUNMLQ', 'LC', n, nrhs, m, -1 ) )
349 IF( nrhs.GT.1 ) THEN
350 maxwrk = max( maxwrk, m*m + m + m*nrhs )
351 ELSE
352 maxwrk = max( maxwrk, m*m + 2*m )
353 END IF
354 maxwrk = max( maxwrk, m*m + 4*m + m*nrhs )
355! XXX: Ensure the Path 2a case below is triggered. The workspace
356! calculation should use queries for all routines eventually.
357 maxwrk = max( maxwrk,
358 \$ 4*m+m*m+max( m, 2*m-4, nrhs, n-3*m ) )
359 ELSE
360*
361* Path 2 - underdetermined.
362*
363 maxwrk = 2*m + ( n + m )*ilaenv( 1, 'ZGEBRD', ' ', m,
364 \$ n, -1, -1 )
365 maxwrk = max( maxwrk, 2*m + nrhs*ilaenv( 1, 'ZUNMBR',
366 \$ 'QLC', m, nrhs, m, -1 ) )
367 maxwrk = max( maxwrk, 2*m + m*ilaenv( 1, 'ZUNMBR',
368 \$ 'PLN', n, nrhs, m, -1 ) )
369 maxwrk = max( maxwrk, 2*m + m*nrhs )
370 END IF
371 minwrk = max( 2*m + n, 2*m + m*nrhs )
372 END IF
373 END IF
374 minwrk = min( minwrk, maxwrk )
375 work( 1 ) = maxwrk
376 iwork( 1 ) = liwork
377 rwork( 1 ) = lrwork
378*
379 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
380 info = -12
381 END IF
382 END IF
383*
384 IF( info.NE.0 ) THEN
385 CALL xerbla( 'ZGELSD', -info )
386 RETURN
387 ELSE IF( lquery ) THEN
388 RETURN
389 END IF
390*
391* Quick return if possible.
392*
393 IF( m.EQ.0 .OR. n.EQ.0 ) THEN
394 rank = 0
395 RETURN
396 END IF
397*
398* Get machine parameters.
399*
400 eps = dlamch( 'P' )
401 sfmin = dlamch( 'S' )
402 smlnum = sfmin / eps
403 bignum = one / smlnum
404 CALL dlabad( smlnum, bignum )
405*
406* Scale A if max entry outside range [SMLNUM,BIGNUM].
407*
408 anrm = zlange( 'M', m, n, a, lda, rwork )
409 iascl = 0
410 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
411*
412* Scale matrix norm up to SMLNUM
413*
414 CALL zlascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
415 iascl = 1
416 ELSE IF( anrm.GT.bignum ) THEN
417*
418* Scale matrix norm down to BIGNUM.
419*
420 CALL zlascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
421 iascl = 2
422 ELSE IF( anrm.EQ.zero ) THEN
423*
424* Matrix all zero. Return zero solution.
425*
426 CALL zlaset( 'F', max( m, n ), nrhs, czero, czero, b, ldb )
427 CALL dlaset( 'F', minmn, 1, zero, zero, s, 1 )
428 rank = 0
429 GO TO 10
430 END IF
431*
432* Scale B if max entry outside range [SMLNUM,BIGNUM].
433*
434 bnrm = zlange( 'M', m, nrhs, b, ldb, rwork )
435 ibscl = 0
436 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
437*
438* Scale matrix norm up to SMLNUM.
439*
440 CALL zlascl( 'G', 0, 0, bnrm, smlnum, m, nrhs, b, ldb, info )
441 ibscl = 1
442 ELSE IF( bnrm.GT.bignum ) THEN
443*
444* Scale matrix norm down to BIGNUM.
445*
446 CALL zlascl( 'G', 0, 0, bnrm, bignum, m, nrhs, b, ldb, info )
447 ibscl = 2
448 END IF
449*
450* If M < N make sure B(M+1:N,:) = 0
451*
452 IF( m.LT.n )
453 \$ CALL zlaset( 'F', n-m, nrhs, czero, czero, b( m+1, 1 ), ldb )
454*
455* Overdetermined case.
456*
457 IF( m.GE.n ) THEN
458*
459* Path 1 - overdetermined or exactly determined.
460*
461 mm = m
462 IF( m.GE.mnthr ) THEN
463*
464* Path 1a - overdetermined, with many more rows than columns
465*
466 mm = n
467 itau = 1
468 nwork = itau + n
469*
470* Compute A=Q*R.
471* (RWorkspace: need N)
472* (CWorkspace: need N, prefer N*NB)
473*
474 CALL zgeqrf( m, n, a, lda, work( itau ), work( nwork ),
475 \$ lwork-nwork+1, info )
476*
477* Multiply B by transpose(Q).
478* (RWorkspace: need N)
479* (CWorkspace: need NRHS, prefer NRHS*NB)
480*
481 CALL zunmqr( 'L', 'C', m, nrhs, n, a, lda, work( itau ), b,
482 \$ ldb, work( nwork ), lwork-nwork+1, info )
483*
484* Zero out below R.
485*
486 IF( n.GT.1 ) THEN
487 CALL zlaset( 'L', n-1, n-1, czero, czero, a( 2, 1 ),
488 \$ lda )
489 END IF
490 END IF
491*
492 itauq = 1
493 itaup = itauq + n
494 nwork = itaup + n
495 ie = 1
496 nrwork = ie + n
497*
498* Bidiagonalize R in A.
499* (RWorkspace: need N)
500* (CWorkspace: need 2*N+MM, prefer 2*N+(MM+N)*NB)
501*
502 CALL zgebrd( mm, n, a, lda, s, rwork( ie ), work( itauq ),
503 \$ work( itaup ), work( nwork ), lwork-nwork+1,
504 \$ info )
505*
506* Multiply B by transpose of left bidiagonalizing vectors of R.
507* (CWorkspace: need 2*N+NRHS, prefer 2*N+NRHS*NB)
508*
509 CALL zunmbr( 'Q', 'L', 'C', mm, nrhs, n, a, lda, work( itauq ),
510 \$ b, ldb, work( nwork ), lwork-nwork+1, info )
511*
512* Solve the bidiagonal least squares problem.
513*
514 CALL zlalsd( 'U', smlsiz, n, nrhs, s, rwork( ie ), b, ldb,
515 \$ rcond, rank, work( nwork ), rwork( nrwork ),
516 \$ iwork, info )
517 IF( info.NE.0 ) THEN
518 GO TO 10
519 END IF
520*
521* Multiply B by right bidiagonalizing vectors of R.
522*
523 CALL zunmbr( 'P', 'L', 'N', n, nrhs, n, a, lda, work( itaup ),
524 \$ b, ldb, work( nwork ), lwork-nwork+1, info )
525*
526 ELSE IF( n.GE.mnthr .AND. lwork.GE.4*m+m*m+
527 \$ max( m, 2*m-4, nrhs, n-3*m ) ) THEN
528*
529* Path 2a - underdetermined, with many more columns than rows
530* and sufficient workspace for an efficient algorithm.
531*
532 ldwork = m
533 IF( lwork.GE.max( 4*m+m*lda+max( m, 2*m-4, nrhs, n-3*m ),
534 \$ m*lda+m+m*nrhs ) )ldwork = lda
535 itau = 1
536 nwork = m + 1
537*
538* Compute A=L*Q.
539* (CWorkspace: need 2*M, prefer M+M*NB)
540*
541 CALL zgelqf( m, n, a, lda, work( itau ), work( nwork ),
542 \$ lwork-nwork+1, info )
543 il = nwork
544*
545* Copy L to WORK(IL), zeroing out above its diagonal.
546*
547 CALL zlacpy( 'L', m, m, a, lda, work( il ), ldwork )
548 CALL zlaset( 'U', m-1, m-1, czero, czero, work( il+ldwork ),
549 \$ ldwork )
550 itauq = il + ldwork*m
551 itaup = itauq + m
552 nwork = itaup + m
553 ie = 1
554 nrwork = ie + m
555*
556* Bidiagonalize L in WORK(IL).
557* (RWorkspace: need M)
558* (CWorkspace: need M*M+4*M, prefer M*M+4*M+2*M*NB)
559*
560 CALL zgebrd( m, m, work( il ), ldwork, s, rwork( ie ),
561 \$ work( itauq ), work( itaup ), work( nwork ),
562 \$ lwork-nwork+1, info )
563*
564* Multiply B by transpose of left bidiagonalizing vectors of L.
565* (CWorkspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
566*
567 CALL zunmbr( 'Q', 'L', 'C', m, nrhs, m, work( il ), ldwork,
568 \$ work( itauq ), b, ldb, work( nwork ),
569 \$ lwork-nwork+1, info )
570*
571* Solve the bidiagonal least squares problem.
572*
573 CALL zlalsd( 'U', smlsiz, m, nrhs, s, rwork( ie ), b, ldb,
574 \$ rcond, rank, work( nwork ), rwork( nrwork ),
575 \$ iwork, info )
576 IF( info.NE.0 ) THEN
577 GO TO 10
578 END IF
579*
580* Multiply B by right bidiagonalizing vectors of L.
581*
582 CALL zunmbr( 'P', 'L', 'N', m, nrhs, m, work( il ), ldwork,
583 \$ work( itaup ), b, ldb, work( nwork ),
584 \$ lwork-nwork+1, info )
585*
586* Zero out below first M rows of B.
587*
588 CALL zlaset( 'F', n-m, nrhs, czero, czero, b( m+1, 1 ), ldb )
589 nwork = itau + m
590*
591* Multiply transpose(Q) by B.
592* (CWorkspace: need NRHS, prefer NRHS*NB)
593*
594 CALL zunmlq( 'L', 'C', n, nrhs, m, a, lda, work( itau ), b,
595 \$ ldb, work( nwork ), lwork-nwork+1, info )
596*
597 ELSE
598*
599* Path 2 - remaining underdetermined cases.
600*
601 itauq = 1
602 itaup = itauq + m
603 nwork = itaup + m
604 ie = 1
605 nrwork = ie + m
606*
607* Bidiagonalize A.
608* (RWorkspace: need M)
609* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
610*
611 CALL zgebrd( m, n, a, lda, s, rwork( ie ), work( itauq ),
612 \$ work( itaup ), work( nwork ), lwork-nwork+1,
613 \$ info )
614*
615* Multiply B by transpose of left bidiagonalizing vectors.
616* (CWorkspace: need 2*M+NRHS, prefer 2*M+NRHS*NB)
617*
618 CALL zunmbr( 'Q', 'L', 'C', m, nrhs, n, a, lda, work( itauq ),
619 \$ b, ldb, work( nwork ), lwork-nwork+1, info )
620*
621* Solve the bidiagonal least squares problem.
622*
623 CALL zlalsd( 'L', smlsiz, m, nrhs, s, rwork( ie ), b, ldb,
624 \$ rcond, rank, work( nwork ), rwork( nrwork ),
625 \$ iwork, info )
626 IF( info.NE.0 ) THEN
627 GO TO 10
628 END IF
629*
630* Multiply B by right bidiagonalizing vectors of A.
631*
632 CALL zunmbr( 'P', 'L', 'N', n, nrhs, m, a, lda, work( itaup ),
633 \$ b, ldb, work( nwork ), lwork-nwork+1, info )
634*
635 END IF
636*
637* Undo scaling.
638*
639 IF( iascl.EQ.1 ) THEN
640 CALL zlascl( 'G', 0, 0, anrm, smlnum, n, nrhs, b, ldb, info )
641 CALL dlascl( 'G', 0, 0, smlnum, anrm, minmn, 1, s, minmn,
642 \$ info )
643 ELSE IF( iascl.EQ.2 ) THEN
644 CALL zlascl( 'G', 0, 0, anrm, bignum, n, nrhs, b, ldb, info )
645 CALL dlascl( 'G', 0, 0, bignum, anrm, minmn, 1, s, minmn,
646 \$ info )
647 END IF
648 IF( ibscl.EQ.1 ) THEN
649 CALL zlascl( 'G', 0, 0, smlnum, bnrm, n, nrhs, b, ldb, info )
650 ELSE IF( ibscl.EQ.2 ) THEN
651 CALL zlascl( 'G', 0, 0, bignum, bnrm, n, nrhs, b, ldb, info )
652 END IF
653*
654 10 CONTINUE
655 work( 1 ) = maxwrk
656 iwork( 1 ) = liwork
657 rwork( 1 ) = lrwork
658 RETURN
659*
660* End of ZGELSD
661*
662 END
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zgelqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGELQF
Definition: zgelqf.f:143
subroutine zgebrd(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
ZGEBRD
Definition: zgebrd.f:205
subroutine zgelsd(M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, IWORK, INFO)
ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
Definition: zgelsd.f:225
subroutine zlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: zlascl.f:143
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine zunmlq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMLQ
Definition: zunmlq.f:167
subroutine zlalsd(UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, RWORK, IWORK, INFO)
ZLALSD uses the singular value decomposition of A to solve the least squares problem.
Definition: zlalsd.f:187
subroutine zunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMQR
Definition: zunmqr.f:167
subroutine zunmbr(VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMBR
Definition: zunmbr.f:196
subroutine zgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm.
Definition: zgeqrf.f:152