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sgelss.f
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1 *> \brief <b> SGELSS solves overdetermined or underdetermined systems 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
9 *> Download SGELSS + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgelss.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgelss.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgelss.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
22 * WORK, LWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
26 * REAL RCOND
27 * ..
28 * .. Array Arguments ..
29 * REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> SGELSS computes the minimum norm solution to a real linear least
39 *> squares problem:
40 *>
41 *> Minimize 2-norm(| b - A*x |).
42 *>
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 matrix
49 *> X.
50 *>
51 *> The effective rank of A is determined by treating as zero those
52 *> singular values which are less than RCOND times the largest singular
53 *> value.
54 *> \endverbatim
55 *
56 * Arguments:
57 * ==========
58 *
59 *> \param[in] M
60 *> \verbatim
61 *> M is INTEGER
62 *> The number of rows of the matrix A. M >= 0.
63 *> \endverbatim
64 *>
65 *> \param[in] N
66 *> \verbatim
67 *> N is INTEGER
68 *> The number of columns of the matrix A. N >= 0.
69 *> \endverbatim
70 *>
71 *> \param[in] NRHS
72 *> \verbatim
73 *> NRHS is INTEGER
74 *> The number of right hand sides, i.e., the number of columns
75 *> of the matrices B and X. NRHS >= 0.
76 *> \endverbatim
77 *>
78 *> \param[in,out] A
79 *> \verbatim
80 *> A is REAL array, dimension (LDA,N)
81 *> On entry, the M-by-N matrix A.
82 *> On exit, the first min(m,n) rows of A are overwritten with
83 *> its right singular vectors, stored rowwise.
84 *> \endverbatim
85 *>
86 *> \param[in] LDA
87 *> \verbatim
88 *> LDA is INTEGER
89 *> The leading dimension of the array A. LDA >= max(1,M).
90 *> \endverbatim
91 *>
92 *> \param[in,out] B
93 *> \verbatim
94 *> B is REAL array, dimension (LDB,NRHS)
95 *> On entry, the M-by-NRHS right hand side matrix B.
96 *> On exit, B is overwritten by the N-by-NRHS solution
97 *> matrix X. If m >= n and RANK = n, the residual
98 *> sum-of-squares for the solution in the i-th column is given
99 *> by the sum of squares of elements n+1:m in that column.
100 *> \endverbatim
101 *>
102 *> \param[in] LDB
103 *> \verbatim
104 *> LDB is INTEGER
105 *> The leading dimension of the array B. LDB >= max(1,max(M,N)).
106 *> \endverbatim
107 *>
108 *> \param[out] S
109 *> \verbatim
110 *> S is REAL array, dimension (min(M,N))
111 *> The singular values of A in decreasing order.
112 *> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
113 *> \endverbatim
114 *>
115 *> \param[in] RCOND
116 *> \verbatim
117 *> RCOND is REAL
118 *> RCOND is used to determine the effective rank of A.
119 *> Singular values S(i) <= RCOND*S(1) are treated as zero.
120 *> If RCOND < 0, machine precision is used instead.
121 *> \endverbatim
122 *>
123 *> \param[out] RANK
124 *> \verbatim
125 *> RANK is INTEGER
126 *> The effective rank of A, i.e., the number of singular values
127 *> which are greater than RCOND*S(1).
128 *> \endverbatim
129 *>
130 *> \param[out] WORK
131 *> \verbatim
132 *> WORK is REAL array, dimension (MAX(1,LWORK))
133 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
134 *> \endverbatim
135 *>
136 *> \param[in] LWORK
137 *> \verbatim
138 *> LWORK is INTEGER
139 *> The dimension of the array WORK. LWORK >= 1, and also:
140 *> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
141 *> For good performance, LWORK should generally be larger.
142 *>
143 *> If LWORK = -1, then a workspace query is assumed; the routine
144 *> only calculates the optimal size of the WORK array, returns
145 *> this value as the first entry of the WORK array, and no error
146 *> message related to LWORK is issued by XERBLA.
147 *> \endverbatim
148 *>
149 *> \param[out] INFO
150 *> \verbatim
151 *> INFO is INTEGER
152 *> = 0: successful exit
153 *> < 0: if INFO = -i, the i-th argument had an illegal value.
154 *> > 0: the algorithm for computing the SVD failed to converge;
155 *> if INFO = i, i off-diagonal elements of an intermediate
156 *> bidiagonal form did not converge to zero.
157 *> \endverbatim
158 *
159 * Authors:
160 * ========
161 *
162 *> \author Univ. of Tennessee
163 *> \author Univ. of California Berkeley
164 *> \author Univ. of Colorado Denver
165 *> \author NAG Ltd.
166 *
167 *> \date November 2011
168 *
169 *> \ingroup realGEsolve
170 *
171 * =====================================================================
172  SUBROUTINE sgelss( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
173  $ work, lwork, info )
174 *
175 * -- LAPACK driver routine (version 3.4.0) --
176 * -- LAPACK is a software package provided by Univ. of Tennessee, --
177 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
178 * November 2011
179 *
180 * .. Scalar Arguments ..
181  INTEGER info, lda, ldb, lwork, m, n, nrhs, rank
182  REAL rcond
183 * ..
184 * .. Array Arguments ..
185  REAL a( lda, * ), b( ldb, * ), s( * ), work( * )
186 * ..
187 *
188 * =====================================================================
189 *
190 * .. Parameters ..
191  REAL zero, one
192  parameter( zero = 0.0e+0, one = 1.0e+0 )
193 * ..
194 * .. Local Scalars ..
195  LOGICAL lquery
196  INTEGER bdspac, bl, chunk, i, iascl, ibscl, ie, il,
197  $ itau, itaup, itauq, iwork, ldwork, maxmn,
198  $ maxwrk, minmn, minwrk, mm, mnthr
199  INTEGER lwork_sgeqrf, lwork_sormqr, lwork_sgebrd,
200  $ lwork_sormbr, lwork_sorgbr, lwork_sormlq
201  REAL anrm, bignum, bnrm, eps, sfmin, smlnum, thr
202 * ..
203 * .. Local Arrays ..
204  REAL dum( 1 )
205 * ..
206 * .. External Subroutines ..
207  EXTERNAL sbdsqr, scopy, sgebrd, sgelqf, sgemm, sgemv,
210 * ..
211 * .. External Functions ..
212  INTEGER ilaenv
213  REAL slamch, slange
214  EXTERNAL ilaenv, slamch, slange
215 * ..
216 * .. Intrinsic Functions ..
217  INTRINSIC max, min
218 * ..
219 * .. Executable Statements ..
220 *
221 * Test the input arguments
222 *
223  info = 0
224  minmn = min( m, n )
225  maxmn = max( m, n )
226  lquery = ( lwork.EQ.-1 )
227  IF( m.LT.0 ) THEN
228  info = -1
229  ELSE IF( n.LT.0 ) THEN
230  info = -2
231  ELSE IF( nrhs.LT.0 ) THEN
232  info = -3
233  ELSE IF( lda.LT.max( 1, m ) ) THEN
234  info = -5
235  ELSE IF( ldb.LT.max( 1, maxmn ) ) THEN
236  info = -7
237  END IF
238 *
239 * Compute workspace
240 * (Note: Comments in the code beginning "Workspace:" describe the
241 * minimal amount of workspace needed at that point in the code,
242 * as well as the preferred amount for good performance.
243 * NB refers to the optimal block size for the immediately
244 * following subroutine, as returned by ILAENV.)
245 *
246  IF( info.EQ.0 ) THEN
247  minwrk = 1
248  maxwrk = 1
249  IF( minmn.GT.0 ) THEN
250  mm = m
251  mnthr = ilaenv( 6, 'SGELSS', ' ', m, n, nrhs, -1 )
252  IF( m.GE.n .AND. m.GE.mnthr ) THEN
253 *
254 * Path 1a - overdetermined, with many more rows than
255 * columns
256 *
257 * Compute space needed for SGEQRF
258  CALL sgeqrf( m, n, a, lda, dum(1), dum(1), -1, info )
259  lwork_sgeqrf=dum(1)
260 * Compute space needed for SORMQR
261  CALL sormqr( 'L', 'T', m, nrhs, n, a, lda, dum(1), b,
262  $ ldb, dum(1), -1, info )
263  lwork_sormqr=dum(1)
264  mm = n
265  maxwrk = max( maxwrk, n + lwork_sgeqrf )
266  maxwrk = max( maxwrk, n + lwork_sormqr )
267  END IF
268  IF( m.GE.n ) THEN
269 *
270 * Path 1 - overdetermined or exactly determined
271 *
272 * Compute workspace needed for SBDSQR
273 *
274  bdspac = max( 1, 5*n )
275 * Compute space needed for SGEBRD
276  CALL sgebrd( mm, n, a, lda, s, dum(1), dum(1),
277  $ dum(1), dum(1), -1, info )
278  lwork_sgebrd=dum(1)
279 * Compute space needed for SORMBR
280  CALL sormbr( 'Q', 'L', 'T', mm, nrhs, n, a, lda, dum(1),
281  $ b, ldb, dum(1), -1, info )
282  lwork_sormbr=dum(1)
283 * Compute space needed for SORGBR
284  CALL sorgbr( 'P', n, n, n, a, lda, dum(1),
285  $ dum(1), -1, info )
286  lwork_sorgbr=dum(1)
287 * Compute total workspace needed
288  maxwrk = max( maxwrk, 3*n + lwork_sgebrd )
289  maxwrk = max( maxwrk, 3*n + lwork_sormbr )
290  maxwrk = max( maxwrk, 3*n + lwork_sorgbr )
291  maxwrk = max( maxwrk, bdspac )
292  maxwrk = max( maxwrk, n*nrhs )
293  minwrk = max( 3*n + mm, 3*n + nrhs, bdspac )
294  maxwrk = max( minwrk, maxwrk )
295  END IF
296  IF( n.GT.m ) THEN
297 *
298 * Compute workspace needed for SBDSQR
299 *
300  bdspac = max( 1, 5*m )
301  minwrk = max( 3*m+nrhs, 3*m+n, bdspac )
302  IF( n.GE.mnthr ) THEN
303 *
304 * Path 2a - underdetermined, with many more columns
305 * than rows
306 *
307 * Compute space needed for SGEBRD
308  CALL sgebrd( m, m, a, lda, s, dum(1), dum(1),
309  $ dum(1), dum(1), -1, info )
310  lwork_sgebrd=dum(1)
311 * Compute space needed for SORMBR
312  CALL sormbr( 'Q', 'L', 'T', m, nrhs, n, a, lda,
313  $ dum(1), b, ldb, dum(1), -1, info )
314  lwork_sormbr=dum(1)
315 * Compute space needed for SORGBR
316  CALL sorgbr( 'P', m, m, m, a, lda, dum(1),
317  $ dum(1), -1, info )
318  lwork_sorgbr=dum(1)
319 * Compute space needed for SORMLQ
320  CALL sormlq( 'L', 'T', n, nrhs, m, a, lda, dum(1),
321  $ b, ldb, dum(1), -1, info )
322  lwork_sormlq=dum(1)
323 * Compute total workspace needed
324  maxwrk = m + m*ilaenv( 1, 'SGELQF', ' ', m, n, -1,
325  $ -1 )
326  maxwrk = max( maxwrk, m*m + 4*m + lwork_sgebrd )
327  maxwrk = max( maxwrk, m*m + 4*m + lwork_sormbr )
328  maxwrk = max( maxwrk, m*m + 4*m + lwork_sorgbr )
329  maxwrk = max( maxwrk, m*m + m + bdspac )
330  IF( nrhs.GT.1 ) THEN
331  maxwrk = max( maxwrk, m*m + m + m*nrhs )
332  ELSE
333  maxwrk = max( maxwrk, m*m + 2*m )
334  END IF
335  maxwrk = max( maxwrk, m + lwork_sormlq )
336  ELSE
337 *
338 * Path 2 - underdetermined
339 *
340 * Compute space needed for SGEBRD
341  CALL sgebrd( m, n, a, lda, s, dum(1), dum(1),
342  $ dum(1), dum(1), -1, info )
343  lwork_sgebrd=dum(1)
344 * Compute space needed for SORMBR
345  CALL sormbr( 'Q', 'L', 'T', m, nrhs, m, a, lda,
346  $ dum(1), b, ldb, dum(1), -1, info )
347  lwork_sormbr=dum(1)
348 * Compute space needed for SORGBR
349  CALL sorgbr( 'P', m, n, m, a, lda, dum(1),
350  $ dum(1), -1, info )
351  lwork_sorgbr=dum(1)
352  maxwrk = 3*m + lwork_sgebrd
353  maxwrk = max( maxwrk, 3*m + lwork_sormbr )
354  maxwrk = max( maxwrk, 3*m + lwork_sorgbr )
355  maxwrk = max( maxwrk, bdspac )
356  maxwrk = max( maxwrk, n*nrhs )
357  END IF
358  END IF
359  maxwrk = max( minwrk, maxwrk )
360  END IF
361  work( 1 ) = maxwrk
362 *
363  IF( lwork.LT.minwrk .AND. .NOT.lquery )
364  $ info = -12
365  END IF
366 *
367  IF( info.NE.0 ) THEN
368  CALL xerbla( 'SGELSS', -info )
369  RETURN
370  ELSE IF( lquery ) THEN
371  RETURN
372  END IF
373 *
374 * Quick return if possible
375 *
376  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
377  rank = 0
378  RETURN
379  END IF
380 *
381 * Get machine parameters
382 *
383  eps = slamch( 'P' )
384  sfmin = slamch( 'S' )
385  smlnum = sfmin / eps
386  bignum = one / smlnum
387  CALL slabad( smlnum, bignum )
388 *
389 * Scale A if max element outside range [SMLNUM,BIGNUM]
390 *
391  anrm = slange( 'M', m, n, a, lda, work )
392  iascl = 0
393  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
394 *
395 * Scale matrix norm up to SMLNUM
396 *
397  CALL slascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
398  iascl = 1
399  ELSE IF( anrm.GT.bignum ) THEN
400 *
401 * Scale matrix norm down to BIGNUM
402 *
403  CALL slascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
404  iascl = 2
405  ELSE IF( anrm.EQ.zero ) THEN
406 *
407 * Matrix all zero. Return zero solution.
408 *
409  CALL slaset( 'F', max( m, n ), nrhs, zero, zero, b, ldb )
410  CALL slaset( 'F', minmn, 1, zero, zero, s, 1 )
411  rank = 0
412  go to 70
413  END IF
414 *
415 * Scale B if max element outside range [SMLNUM,BIGNUM]
416 *
417  bnrm = slange( 'M', m, nrhs, b, ldb, work )
418  ibscl = 0
419  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
420 *
421 * Scale matrix norm up to SMLNUM
422 *
423  CALL slascl( 'G', 0, 0, bnrm, smlnum, m, nrhs, b, ldb, info )
424  ibscl = 1
425  ELSE IF( bnrm.GT.bignum ) THEN
426 *
427 * Scale matrix norm down to BIGNUM
428 *
429  CALL slascl( 'G', 0, 0, bnrm, bignum, m, nrhs, b, ldb, info )
430  ibscl = 2
431  END IF
432 *
433 * Overdetermined case
434 *
435  IF( m.GE.n ) THEN
436 *
437 * Path 1 - overdetermined or exactly determined
438 *
439  mm = m
440  IF( m.GE.mnthr ) THEN
441 *
442 * Path 1a - overdetermined, with many more rows than columns
443 *
444  mm = n
445  itau = 1
446  iwork = itau + n
447 *
448 * Compute A=Q*R
449 * (Workspace: need 2*N, prefer N+N*NB)
450 *
451  CALL sgeqrf( m, n, a, lda, work( itau ), work( iwork ),
452  $ lwork-iwork+1, info )
453 *
454 * Multiply B by transpose(Q)
455 * (Workspace: need N+NRHS, prefer N+NRHS*NB)
456 *
457  CALL sormqr( 'L', 'T', m, nrhs, n, a, lda, work( itau ), b,
458  $ ldb, work( iwork ), lwork-iwork+1, info )
459 *
460 * Zero out below R
461 *
462  IF( n.GT.1 )
463  $ CALL slaset( 'L', n-1, n-1, zero, zero, a( 2, 1 ), lda )
464  END IF
465 *
466  ie = 1
467  itauq = ie + n
468  itaup = itauq + n
469  iwork = itaup + n
470 *
471 * Bidiagonalize R in A
472 * (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
473 *
474  CALL sgebrd( mm, n, a, lda, s, work( ie ), work( itauq ),
475  $ work( itaup ), work( iwork ), lwork-iwork+1,
476  $ info )
477 *
478 * Multiply B by transpose of left bidiagonalizing vectors of R
479 * (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
480 *
481  CALL sormbr( 'Q', 'L', 'T', mm, nrhs, n, a, lda, work( itauq ),
482  $ b, ldb, work( iwork ), lwork-iwork+1, info )
483 *
484 * Generate right bidiagonalizing vectors of R in A
485 * (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
486 *
487  CALL sorgbr( 'P', n, n, n, a, lda, work( itaup ),
488  $ work( iwork ), lwork-iwork+1, info )
489  iwork = ie + n
490 *
491 * Perform bidiagonal QR iteration
492 * multiply B by transpose of left singular vectors
493 * compute right singular vectors in A
494 * (Workspace: need BDSPAC)
495 *
496  CALL sbdsqr( 'U', n, n, 0, nrhs, s, work( ie ), a, lda, dum,
497  $ 1, b, ldb, work( iwork ), info )
498  IF( info.NE.0 )
499  $ go to 70
500 *
501 * Multiply B by reciprocals of singular values
502 *
503  thr = max( rcond*s( 1 ), sfmin )
504  IF( rcond.LT.zero )
505  $ thr = max( eps*s( 1 ), sfmin )
506  rank = 0
507  DO 10 i = 1, n
508  IF( s( i ).GT.thr ) THEN
509  CALL srscl( nrhs, s( i ), b( i, 1 ), ldb )
510  rank = rank + 1
511  ELSE
512  CALL slaset( 'F', 1, nrhs, zero, zero, b( i, 1 ), ldb )
513  END IF
514  10 CONTINUE
515 *
516 * Multiply B by right singular vectors
517 * (Workspace: need N, prefer N*NRHS)
518 *
519  IF( lwork.GE.ldb*nrhs .AND. nrhs.GT.1 ) THEN
520  CALL sgemm( 'T', 'N', n, nrhs, n, one, a, lda, b, ldb, zero,
521  $ work, ldb )
522  CALL slacpy( 'G', n, nrhs, work, ldb, b, ldb )
523  ELSE IF( nrhs.GT.1 ) THEN
524  chunk = lwork / n
525  DO 20 i = 1, nrhs, chunk
526  bl = min( nrhs-i+1, chunk )
527  CALL sgemm( 'T', 'N', n, bl, n, one, a, lda, b( 1, i ),
528  $ ldb, zero, work, n )
529  CALL slacpy( 'G', n, bl, work, n, b( 1, i ), ldb )
530  20 CONTINUE
531  ELSE
532  CALL sgemv( 'T', n, n, one, a, lda, b, 1, zero, work, 1 )
533  CALL scopy( n, work, 1, b, 1 )
534  END IF
535 *
536  ELSE IF( n.GE.mnthr .AND. lwork.GE.4*m+m*m+
537  $ max( m, 2*m-4, nrhs, n-3*m ) ) THEN
538 *
539 * Path 2a - underdetermined, with many more columns than rows
540 * and sufficient workspace for an efficient algorithm
541 *
542  ldwork = m
543  IF( lwork.GE.max( 4*m+m*lda+max( m, 2*m-4, nrhs, n-3*m ),
544  $ m*lda+m+m*nrhs ) )ldwork = lda
545  itau = 1
546  iwork = m + 1
547 *
548 * Compute A=L*Q
549 * (Workspace: need 2*M, prefer M+M*NB)
550 *
551  CALL sgelqf( m, n, a, lda, work( itau ), work( iwork ),
552  $ lwork-iwork+1, info )
553  il = iwork
554 *
555 * Copy L to WORK(IL), zeroing out above it
556 *
557  CALL slacpy( 'L', m, m, a, lda, work( il ), ldwork )
558  CALL slaset( 'U', m-1, m-1, zero, zero, work( il+ldwork ),
559  $ ldwork )
560  ie = il + ldwork*m
561  itauq = ie + m
562  itaup = itauq + m
563  iwork = itaup + m
564 *
565 * Bidiagonalize L in WORK(IL)
566 * (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
567 *
568  CALL sgebrd( m, m, work( il ), ldwork, s, work( ie ),
569  $ work( itauq ), work( itaup ), work( iwork ),
570  $ lwork-iwork+1, info )
571 *
572 * Multiply B by transpose of left bidiagonalizing vectors of L
573 * (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
574 *
575  CALL sormbr( 'Q', 'L', 'T', m, nrhs, m, work( il ), ldwork,
576  $ work( itauq ), b, ldb, work( iwork ),
577  $ lwork-iwork+1, info )
578 *
579 * Generate right bidiagonalizing vectors of R in WORK(IL)
580 * (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
581 *
582  CALL sorgbr( 'P', m, m, m, work( il ), ldwork, work( itaup ),
583  $ work( iwork ), lwork-iwork+1, info )
584  iwork = ie + m
585 *
586 * Perform bidiagonal QR iteration,
587 * computing right singular vectors of L in WORK(IL) and
588 * multiplying B by transpose of left singular vectors
589 * (Workspace: need M*M+M+BDSPAC)
590 *
591  CALL sbdsqr( 'U', m, m, 0, nrhs, s, work( ie ), work( il ),
592  $ ldwork, a, lda, b, ldb, work( iwork ), info )
593  IF( info.NE.0 )
594  $ go to 70
595 *
596 * Multiply B by reciprocals of singular values
597 *
598  thr = max( rcond*s( 1 ), sfmin )
599  IF( rcond.LT.zero )
600  $ thr = max( eps*s( 1 ), sfmin )
601  rank = 0
602  DO 30 i = 1, m
603  IF( s( i ).GT.thr ) THEN
604  CALL srscl( nrhs, s( i ), b( i, 1 ), ldb )
605  rank = rank + 1
606  ELSE
607  CALL slaset( 'F', 1, nrhs, zero, zero, b( i, 1 ), ldb )
608  END IF
609  30 CONTINUE
610  iwork = ie
611 *
612 * Multiply B by right singular vectors of L in WORK(IL)
613 * (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS)
614 *
615  IF( lwork.GE.ldb*nrhs+iwork-1 .AND. nrhs.GT.1 ) THEN
616  CALL sgemm( 'T', 'N', m, nrhs, m, one, work( il ), ldwork,
617  $ b, ldb, zero, work( iwork ), ldb )
618  CALL slacpy( 'G', m, nrhs, work( iwork ), ldb, b, ldb )
619  ELSE IF( nrhs.GT.1 ) THEN
620  chunk = ( lwork-iwork+1 ) / m
621  DO 40 i = 1, nrhs, chunk
622  bl = min( nrhs-i+1, chunk )
623  CALL sgemm( 'T', 'N', m, bl, m, one, work( il ), ldwork,
624  $ b( 1, i ), ldb, zero, work( iwork ), m )
625  CALL slacpy( 'G', m, bl, work( iwork ), m, b( 1, i ),
626  $ ldb )
627  40 CONTINUE
628  ELSE
629  CALL sgemv( 'T', m, m, one, work( il ), ldwork, b( 1, 1 ),
630  $ 1, zero, work( iwork ), 1 )
631  CALL scopy( m, work( iwork ), 1, b( 1, 1 ), 1 )
632  END IF
633 *
634 * Zero out below first M rows of B
635 *
636  CALL slaset( 'F', n-m, nrhs, zero, zero, b( m+1, 1 ), ldb )
637  iwork = itau + m
638 *
639 * Multiply transpose(Q) by B
640 * (Workspace: need M+NRHS, prefer M+NRHS*NB)
641 *
642  CALL sormlq( 'L', 'T', n, nrhs, m, a, lda, work( itau ), b,
643  $ ldb, work( iwork ), lwork-iwork+1, info )
644 *
645  ELSE
646 *
647 * Path 2 - remaining underdetermined cases
648 *
649  ie = 1
650  itauq = ie + m
651  itaup = itauq + m
652  iwork = itaup + m
653 *
654 * Bidiagonalize A
655 * (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
656 *
657  CALL sgebrd( m, n, a, lda, s, work( ie ), work( itauq ),
658  $ work( itaup ), work( iwork ), lwork-iwork+1,
659  $ info )
660 *
661 * Multiply B by transpose of left bidiagonalizing vectors
662 * (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
663 *
664  CALL sormbr( 'Q', 'L', 'T', m, nrhs, n, a, lda, work( itauq ),
665  $ b, ldb, work( iwork ), lwork-iwork+1, info )
666 *
667 * Generate right bidiagonalizing vectors in A
668 * (Workspace: need 4*M, prefer 3*M+M*NB)
669 *
670  CALL sorgbr( 'P', m, n, m, a, lda, work( itaup ),
671  $ work( iwork ), lwork-iwork+1, info )
672  iwork = ie + m
673 *
674 * Perform bidiagonal QR iteration,
675 * computing right singular vectors of A in A and
676 * multiplying B by transpose of left singular vectors
677 * (Workspace: need BDSPAC)
678 *
679  CALL sbdsqr( 'L', m, n, 0, nrhs, s, work( ie ), a, lda, dum,
680  $ 1, b, ldb, work( iwork ), info )
681  IF( info.NE.0 )
682  $ go to 70
683 *
684 * Multiply B by reciprocals of singular values
685 *
686  thr = max( rcond*s( 1 ), sfmin )
687  IF( rcond.LT.zero )
688  $ thr = max( eps*s( 1 ), sfmin )
689  rank = 0
690  DO 50 i = 1, m
691  IF( s( i ).GT.thr ) THEN
692  CALL srscl( nrhs, s( i ), b( i, 1 ), ldb )
693  rank = rank + 1
694  ELSE
695  CALL slaset( 'F', 1, nrhs, zero, zero, b( i, 1 ), ldb )
696  END IF
697  50 CONTINUE
698 *
699 * Multiply B by right singular vectors of A
700 * (Workspace: need N, prefer N*NRHS)
701 *
702  IF( lwork.GE.ldb*nrhs .AND. nrhs.GT.1 ) THEN
703  CALL sgemm( 'T', 'N', n, nrhs, m, one, a, lda, b, ldb, zero,
704  $ work, ldb )
705  CALL slacpy( 'F', n, nrhs, work, ldb, b, ldb )
706  ELSE IF( nrhs.GT.1 ) THEN
707  chunk = lwork / n
708  DO 60 i = 1, nrhs, chunk
709  bl = min( nrhs-i+1, chunk )
710  CALL sgemm( 'T', 'N', n, bl, m, one, a, lda, b( 1, i ),
711  $ ldb, zero, work, n )
712  CALL slacpy( 'F', n, bl, work, n, b( 1, i ), ldb )
713  60 CONTINUE
714  ELSE
715  CALL sgemv( 'T', m, n, one, a, lda, b, 1, zero, work, 1 )
716  CALL scopy( n, work, 1, b, 1 )
717  END IF
718  END IF
719 *
720 * Undo scaling
721 *
722  IF( iascl.EQ.1 ) THEN
723  CALL slascl( 'G', 0, 0, anrm, smlnum, n, nrhs, b, ldb, info )
724  CALL slascl( 'G', 0, 0, smlnum, anrm, minmn, 1, s, minmn,
725  $ info )
726  ELSE IF( iascl.EQ.2 ) THEN
727  CALL slascl( 'G', 0, 0, anrm, bignum, n, nrhs, b, ldb, info )
728  CALL slascl( 'G', 0, 0, bignum, anrm, minmn, 1, s, minmn,
729  $ info )
730  END IF
731  IF( ibscl.EQ.1 ) THEN
732  CALL slascl( 'G', 0, 0, smlnum, bnrm, n, nrhs, b, ldb, info )
733  ELSE IF( ibscl.EQ.2 ) THEN
734  CALL slascl( 'G', 0, 0, bignum, bnrm, n, nrhs, b, ldb, info )
735  END IF
736 *
737  70 CONTINUE
738  work( 1 ) = maxwrk
739  RETURN
740 *
741 * End of SGELSS
742 *
743  END