1 SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
2 $ WORK, LWORK, IWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * June 2010
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
11 DOUBLE PRECISION RCOND
12 * ..
13 * .. Array Arguments ..
14 INTEGER IWORK( * )
15 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DGELSD computes the minimum-norm solution to a real linear least
22 * squares problem:
23 * minimize 2-norm(| b - A*x |)
24 * using the singular value decomposition (SVD) of A. A is an M-by-N
25 * matrix which may be rank-deficient.
26 *
27 * Several right hand side vectors b and solution vectors x can be
28 * handled in a single call; they are stored as the columns of the
29 * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
30 * matrix X.
31 *
32 * The problem is solved in three steps:
33 * (1) Reduce the coefficient matrix A to bidiagonal form with
34 * Householder transformations, reducing the original problem
35 * into a "bidiagonal least squares problem" (BLS)
36 * (2) Solve the BLS using a divide and conquer approach.
37 * (3) Apply back all the Householder tranformations to solve
38 * the original least squares problem.
39 *
40 * The effective rank of A is determined by treating as zero those
41 * singular values which are less than RCOND times the largest singular
42 * value.
43 *
44 * The divide and conquer algorithm makes very mild assumptions about
45 * floating point arithmetic. It will work on machines with a guard
46 * digit in add/subtract, or on those binary machines without guard
47 * digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
48 * Cray-2. It could conceivably fail on hexadecimal or decimal machines
49 * without guard digits, but we know of none.
50 *
51 * Arguments
52 * =========
53 *
54 * M (input) INTEGER
55 * The number of rows of A. M >= 0.
56 *
57 * N (input) INTEGER
58 * The number of columns of A. N >= 0.
59 *
60 * NRHS (input) INTEGER
61 * The number of right hand sides, i.e., the number of columns
62 * of the matrices B and X. NRHS >= 0.
63 *
64 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
65 * On entry, the M-by-N matrix A.
66 * On exit, A has been destroyed.
67 *
68 * LDA (input) INTEGER
69 * The leading dimension of the array A. LDA >= max(1,M).
70 *
71 * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
72 * On entry, the M-by-NRHS right hand side matrix B.
73 * On exit, B is overwritten by the N-by-NRHS solution
74 * matrix X. If m >= n and RANK = n, the residual
75 * sum-of-squares for the solution in the i-th column is given
76 * by the sum of squares of elements n+1:m in that column.
77 *
78 * LDB (input) INTEGER
79 * The leading dimension of the array B. LDB >= max(1,max(M,N)).
80 *
81 * S (output) DOUBLE PRECISION array, dimension (min(M,N))
82 * The singular values of A in decreasing order.
83 * The condition number of A in the 2-norm = S(1)/S(min(m,n)).
84 *
85 * RCOND (input) DOUBLE PRECISION
86 * RCOND is used to determine the effective rank of A.
87 * Singular values S(i) <= RCOND*S(1) are treated as zero.
88 * If RCOND < 0, machine precision is used instead.
89 *
90 * RANK (output) INTEGER
91 * The effective rank of A, i.e., the number of singular values
92 * which are greater than RCOND*S(1).
93 *
94 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
95 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
96 *
97 * LWORK (input) INTEGER
98 * The dimension of the array WORK. LWORK must be at least 1.
99 * The exact minimum amount of workspace needed depends on M,
100 * N and NRHS. As long as LWORK is at least
101 * 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
102 * if M is greater than or equal to N or
103 * 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
104 * if M is less than N, the code will execute correctly.
105 * SMLSIZ is returned by ILAENV and is equal to the maximum
106 * size of the subproblems at the bottom of the computation
107 * tree (usually about 25), and
108 * NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
109 * For good performance, LWORK should generally be larger.
110 *
111 * If LWORK = -1, then a workspace query is assumed; the routine
112 * only calculates the optimal size of the WORK array, returns
113 * this value as the first entry of the WORK array, and no error
114 * message related to LWORK is issued by XERBLA.
115 *
116 * IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
117 * LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
118 * where MINMN = MIN( M,N ).
119 * On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
120 *
121 * INFO (output) INTEGER
122 * = 0: successful exit
123 * < 0: if INFO = -i, the i-th argument had an illegal value.
124 * > 0: the algorithm for computing the SVD failed to converge;
125 * if INFO = i, i off-diagonal elements of an intermediate
126 * bidiagonal form did not converge to zero.
127 *
128 * Further Details
129 * ===============
130 *
131 * Based on contributions by
132 * Ming Gu and Ren-Cang Li, Computer Science Division, University of
133 * California at Berkeley, USA
134 * Osni Marques, LBNL/NERSC, USA
135 *
136 * =====================================================================
137 *
138 * .. Parameters ..
139 DOUBLE PRECISION ZERO, ONE, TWO
140 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
141 * ..
142 * .. Local Scalars ..
143 LOGICAL LQUERY
144 INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
145 $ LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
146 $ MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
147 DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
148 * ..
149 * .. External Subroutines ..
150 EXTERNAL DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,
151 $ DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
152 * ..
153 * .. External Functions ..
154 INTEGER ILAENV
155 DOUBLE PRECISION DLAMCH, DLANGE
156 EXTERNAL ILAENV, DLAMCH, DLANGE
157 * ..
158 * .. Intrinsic Functions ..
159 INTRINSIC DBLE, INT, LOG, MAX, MIN
160 * ..
161 * .. Executable Statements ..
162 *
163 * Test the input arguments.
164 *
165 INFO = 0
166 MINMN = MIN( M, N )
167 MAXMN = MAX( M, N )
168 MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
169 LQUERY = ( LWORK.EQ.-1 )
170 IF( M.LT.0 ) THEN
171 INFO = -1
172 ELSE IF( N.LT.0 ) THEN
173 INFO = -2
174 ELSE IF( NRHS.LT.0 ) THEN
175 INFO = -3
176 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
177 INFO = -5
178 ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
179 INFO = -7
180 END IF
181 *
182 SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
183 *
184 * Compute workspace.
185 * (Note: Comments in the code beginning "Workspace:" describe the
186 * minimal amount of workspace needed at that point in the code,
187 * as well as the preferred amount for good performance.
188 * NB refers to the optimal block size for the immediately
189 * following subroutine, as returned by ILAENV.)
190 *
191 MINWRK = 1
192 LIWORK = 1
193 MINMN = MAX( 1, MINMN )
194 NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
195 $ LOG( TWO ) ) + 1, 0 )
196 *
197 IF( INFO.EQ.0 ) THEN
198 MAXWRK = 0
199 LIWORK = 3*MINMN*NLVL + 11*MINMN
200 MM = M
201 IF( M.GE.N .AND. M.GE.MNTHR ) THEN
202 *
203 * Path 1a - overdetermined, with many more rows than columns.
204 *
205 MM = N
206 MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
207 $ -1, -1 ) )
208 MAXWRK = MAX( MAXWRK, N+NRHS*
209 $ ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
210 END IF
211 IF( M.GE.N ) THEN
212 *
213 * Path 1 - overdetermined or exactly determined.
214 *
215 MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
216 $ ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
217 MAXWRK = MAX( MAXWRK, 3*N+NRHS*
218 $ ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
219 MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
220 $ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
221 WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
222 MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
223 MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
224 END IF
225 IF( N.GT.M ) THEN
226 WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
227 IF( N.GE.MNTHR ) THEN
228 *
229 * Path 2a - underdetermined, with many more columns
230 * than rows.
231 *
232 MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
233 MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
234 $ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
235 MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
236 $ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
237 MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
238 $ ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )
239 IF( NRHS.GT.1 ) THEN
240 MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
241 ELSE
242 MAXWRK = MAX( MAXWRK, M*M+2*M )
243 END IF
244 MAXWRK = MAX( MAXWRK, M+NRHS*
245 $ ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
246 MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
247 ! XXX: Ensure the Path 2a case below is triggered. The workspace
248 ! calculation should use queries for all routines eventually.
249 MAXWRK = MAX( MAXWRK,
250 $ 4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
251 ELSE
252 *
253 * Path 2 - remaining underdetermined cases.
254 *
255 MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
256 $ -1, -1 )
257 MAXWRK = MAX( MAXWRK, 3*M+NRHS*
258 $ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )
259 MAXWRK = MAX( MAXWRK, 3*M+M*
260 $ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )
261 MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
262 END IF
263 MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
264 END IF
265 MINWRK = MIN( MINWRK, MAXWRK )
266 WORK( 1 ) = MAXWRK
267 IWORK( 1 ) = LIWORK
268
269 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
270 INFO = -12
271 END IF
272 END IF
273 *
274 IF( INFO.NE.0 ) THEN
275 CALL XERBLA( 'DGELSD', -INFO )
276 RETURN
277 ELSE IF( LQUERY ) THEN
278 GO TO 10
279 END IF
280 *
281 * Quick return if possible.
282 *
283 IF( M.EQ.0 .OR. N.EQ.0 ) THEN
284 RANK = 0
285 RETURN
286 END IF
287 *
288 * Get machine parameters.
289 *
290 EPS = DLAMCH( 'P' )
291 SFMIN = DLAMCH( 'S' )
292 SMLNUM = SFMIN / EPS
293 BIGNUM = ONE / SMLNUM
294 CALL DLABAD( SMLNUM, BIGNUM )
295 *
296 * Scale A if max entry outside range [SMLNUM,BIGNUM].
297 *
298 ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
299 IASCL = 0
300 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
301 *
302 * Scale matrix norm up to SMLNUM.
303 *
304 CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
305 IASCL = 1
306 ELSE IF( ANRM.GT.BIGNUM ) THEN
307 *
308 * Scale matrix norm down to BIGNUM.
309 *
310 CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
311 IASCL = 2
312 ELSE IF( ANRM.EQ.ZERO ) THEN
313 *
314 * Matrix all zero. Return zero solution.
315 *
316 CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
317 CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
318 RANK = 0
319 GO TO 10
320 END IF
321 *
322 * Scale B if max entry outside range [SMLNUM,BIGNUM].
323 *
324 BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
325 IBSCL = 0
326 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
327 *
328 * Scale matrix norm up to SMLNUM.
329 *
330 CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
331 IBSCL = 1
332 ELSE IF( BNRM.GT.BIGNUM ) THEN
333 *
334 * Scale matrix norm down to BIGNUM.
335 *
336 CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
337 IBSCL = 2
338 END IF
339 *
340 * If M < N make sure certain entries of B are zero.
341 *
342 IF( M.LT.N )
343 $ CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
344 *
345 * Overdetermined case.
346 *
347 IF( M.GE.N ) THEN
348 *
349 * Path 1 - overdetermined or exactly determined.
350 *
351 MM = M
352 IF( M.GE.MNTHR ) THEN
353 *
354 * Path 1a - overdetermined, with many more rows than columns.
355 *
356 MM = N
357 ITAU = 1
358 NWORK = ITAU + N
359 *
360 * Compute A=Q*R.
361 * (Workspace: need 2*N, prefer N+N*NB)
362 *
363 CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
364 $ LWORK-NWORK+1, INFO )
365 *
366 * Multiply B by transpose(Q).
367 * (Workspace: need N+NRHS, prefer N+NRHS*NB)
368 *
369 CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
370 $ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
371 *
372 * Zero out below R.
373 *
374 IF( N.GT.1 ) THEN
375 CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
376 END IF
377 END IF
378 *
379 IE = 1
380 ITAUQ = IE + N
381 ITAUP = ITAUQ + N
382 NWORK = ITAUP + N
383 *
384 * Bidiagonalize R in A.
385 * (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
386 *
387 CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
388 $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
389 $ INFO )
390 *
391 * Multiply B by transpose of left bidiagonalizing vectors of R.
392 * (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
393 *
394 CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
395 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
396 *
397 * Solve the bidiagonal least squares problem.
398 *
399 CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
400 $ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
401 IF( INFO.NE.0 ) THEN
402 GO TO 10
403 END IF
404 *
405 * Multiply B by right bidiagonalizing vectors of R.
406 *
407 CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
408 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
409 *
410 ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
411 $ MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
412 *
413 * Path 2a - underdetermined, with many more columns than rows
414 * and sufficient workspace for an efficient algorithm.
415 *
416 LDWORK = M
417 IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
418 $ M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
419 ITAU = 1
420 NWORK = M + 1
421 *
422 * Compute A=L*Q.
423 * (Workspace: need 2*M, prefer M+M*NB)
424 *
425 CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
426 $ LWORK-NWORK+1, INFO )
427 IL = NWORK
428 *
429 * Copy L to WORK(IL), zeroing out above its diagonal.
430 *
431 CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
432 CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
433 $ LDWORK )
434 IE = IL + LDWORK*M
435 ITAUQ = IE + M
436 ITAUP = ITAUQ + M
437 NWORK = ITAUP + M
438 *
439 * Bidiagonalize L in WORK(IL).
440 * (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
441 *
442 CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
443 $ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
444 $ LWORK-NWORK+1, INFO )
445 *
446 * Multiply B by transpose of left bidiagonalizing vectors of L.
447 * (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
448 *
449 CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
450 $ WORK( ITAUQ ), B, LDB, WORK( NWORK ),
451 $ LWORK-NWORK+1, INFO )
452 *
453 * Solve the bidiagonal least squares problem.
454 *
455 CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
456 $ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
457 IF( INFO.NE.0 ) THEN
458 GO TO 10
459 END IF
460 *
461 * Multiply B by right bidiagonalizing vectors of L.
462 *
463 CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
464 $ WORK( ITAUP ), B, LDB, WORK( NWORK ),
465 $ LWORK-NWORK+1, INFO )
466 *
467 * Zero out below first M rows of B.
468 *
469 CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
470 NWORK = ITAU + M
471 *
472 * Multiply transpose(Q) by B.
473 * (Workspace: need M+NRHS, prefer M+NRHS*NB)
474 *
475 CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
476 $ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
477 *
478 ELSE
479 *
480 * Path 2 - remaining underdetermined cases.
481 *
482 IE = 1
483 ITAUQ = IE + M
484 ITAUP = ITAUQ + M
485 NWORK = ITAUP + M
486 *
487 * Bidiagonalize A.
488 * (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
489 *
490 CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
491 $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
492 $ INFO )
493 *
494 * Multiply B by transpose of left bidiagonalizing vectors.
495 * (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
496 *
497 CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
498 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
499 *
500 * Solve the bidiagonal least squares problem.
501 *
502 CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
503 $ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
504 IF( INFO.NE.0 ) THEN
505 GO TO 10
506 END IF
507 *
508 * Multiply B by right bidiagonalizing vectors of A.
509 *
510 CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
511 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
512 *
513 END IF
514 *
515 * Undo scaling.
516 *
517 IF( IASCL.EQ.1 ) THEN
518 CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
519 CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
520 $ INFO )
521 ELSE IF( IASCL.EQ.2 ) THEN
522 CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
523 CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
524 $ INFO )
525 END IF
526 IF( IBSCL.EQ.1 ) THEN
527 CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
528 ELSE IF( IBSCL.EQ.2 ) THEN
529 CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
530 END IF
531 *
532 10 CONTINUE
533 WORK( 1 ) = MAXWRK
534 IWORK( 1 ) = LIWORK
535 RETURN
536 *
537 * End of DGELSD
538 *
539 END
2 $ WORK, LWORK, IWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * June 2010
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
11 DOUBLE PRECISION RCOND
12 * ..
13 * .. Array Arguments ..
14 INTEGER IWORK( * )
15 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DGELSD computes the minimum-norm solution to a real linear least
22 * squares problem:
23 * minimize 2-norm(| b - A*x |)
24 * using the singular value decomposition (SVD) of A. A is an M-by-N
25 * matrix which may be rank-deficient.
26 *
27 * Several right hand side vectors b and solution vectors x can be
28 * handled in a single call; they are stored as the columns of the
29 * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
30 * matrix X.
31 *
32 * The problem is solved in three steps:
33 * (1) Reduce the coefficient matrix A to bidiagonal form with
34 * Householder transformations, reducing the original problem
35 * into a "bidiagonal least squares problem" (BLS)
36 * (2) Solve the BLS using a divide and conquer approach.
37 * (3) Apply back all the Householder tranformations to solve
38 * the original least squares problem.
39 *
40 * The effective rank of A is determined by treating as zero those
41 * singular values which are less than RCOND times the largest singular
42 * value.
43 *
44 * The divide and conquer algorithm makes very mild assumptions about
45 * floating point arithmetic. It will work on machines with a guard
46 * digit in add/subtract, or on those binary machines without guard
47 * digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
48 * Cray-2. It could conceivably fail on hexadecimal or decimal machines
49 * without guard digits, but we know of none.
50 *
51 * Arguments
52 * =========
53 *
54 * M (input) INTEGER
55 * The number of rows of A. M >= 0.
56 *
57 * N (input) INTEGER
58 * The number of columns of A. N >= 0.
59 *
60 * NRHS (input) INTEGER
61 * The number of right hand sides, i.e., the number of columns
62 * of the matrices B and X. NRHS >= 0.
63 *
64 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
65 * On entry, the M-by-N matrix A.
66 * On exit, A has been destroyed.
67 *
68 * LDA (input) INTEGER
69 * The leading dimension of the array A. LDA >= max(1,M).
70 *
71 * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
72 * On entry, the M-by-NRHS right hand side matrix B.
73 * On exit, B is overwritten by the N-by-NRHS solution
74 * matrix X. If m >= n and RANK = n, the residual
75 * sum-of-squares for the solution in the i-th column is given
76 * by the sum of squares of elements n+1:m in that column.
77 *
78 * LDB (input) INTEGER
79 * The leading dimension of the array B. LDB >= max(1,max(M,N)).
80 *
81 * S (output) DOUBLE PRECISION array, dimension (min(M,N))
82 * The singular values of A in decreasing order.
83 * The condition number of A in the 2-norm = S(1)/S(min(m,n)).
84 *
85 * RCOND (input) DOUBLE PRECISION
86 * RCOND is used to determine the effective rank of A.
87 * Singular values S(i) <= RCOND*S(1) are treated as zero.
88 * If RCOND < 0, machine precision is used instead.
89 *
90 * RANK (output) INTEGER
91 * The effective rank of A, i.e., the number of singular values
92 * which are greater than RCOND*S(1).
93 *
94 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
95 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
96 *
97 * LWORK (input) INTEGER
98 * The dimension of the array WORK. LWORK must be at least 1.
99 * The exact minimum amount of workspace needed depends on M,
100 * N and NRHS. As long as LWORK is at least
101 * 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
102 * if M is greater than or equal to N or
103 * 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
104 * if M is less than N, the code will execute correctly.
105 * SMLSIZ is returned by ILAENV and is equal to the maximum
106 * size of the subproblems at the bottom of the computation
107 * tree (usually about 25), and
108 * NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
109 * For good performance, LWORK should generally be larger.
110 *
111 * If LWORK = -1, then a workspace query is assumed; the routine
112 * only calculates the optimal size of the WORK array, returns
113 * this value as the first entry of the WORK array, and no error
114 * message related to LWORK is issued by XERBLA.
115 *
116 * IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
117 * LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
118 * where MINMN = MIN( M,N ).
119 * On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
120 *
121 * INFO (output) INTEGER
122 * = 0: successful exit
123 * < 0: if INFO = -i, the i-th argument had an illegal value.
124 * > 0: the algorithm for computing the SVD failed to converge;
125 * if INFO = i, i off-diagonal elements of an intermediate
126 * bidiagonal form did not converge to zero.
127 *
128 * Further Details
129 * ===============
130 *
131 * Based on contributions by
132 * Ming Gu and Ren-Cang Li, Computer Science Division, University of
133 * California at Berkeley, USA
134 * Osni Marques, LBNL/NERSC, USA
135 *
136 * =====================================================================
137 *
138 * .. Parameters ..
139 DOUBLE PRECISION ZERO, ONE, TWO
140 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
141 * ..
142 * .. Local Scalars ..
143 LOGICAL LQUERY
144 INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
145 $ LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
146 $ MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
147 DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
148 * ..
149 * .. External Subroutines ..
150 EXTERNAL DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,
151 $ DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
152 * ..
153 * .. External Functions ..
154 INTEGER ILAENV
155 DOUBLE PRECISION DLAMCH, DLANGE
156 EXTERNAL ILAENV, DLAMCH, DLANGE
157 * ..
158 * .. Intrinsic Functions ..
159 INTRINSIC DBLE, INT, LOG, MAX, MIN
160 * ..
161 * .. Executable Statements ..
162 *
163 * Test the input arguments.
164 *
165 INFO = 0
166 MINMN = MIN( M, N )
167 MAXMN = MAX( M, N )
168 MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
169 LQUERY = ( LWORK.EQ.-1 )
170 IF( M.LT.0 ) THEN
171 INFO = -1
172 ELSE IF( N.LT.0 ) THEN
173 INFO = -2
174 ELSE IF( NRHS.LT.0 ) THEN
175 INFO = -3
176 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
177 INFO = -5
178 ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
179 INFO = -7
180 END IF
181 *
182 SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
183 *
184 * Compute workspace.
185 * (Note: Comments in the code beginning "Workspace:" describe the
186 * minimal amount of workspace needed at that point in the code,
187 * as well as the preferred amount for good performance.
188 * NB refers to the optimal block size for the immediately
189 * following subroutine, as returned by ILAENV.)
190 *
191 MINWRK = 1
192 LIWORK = 1
193 MINMN = MAX( 1, MINMN )
194 NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
195 $ LOG( TWO ) ) + 1, 0 )
196 *
197 IF( INFO.EQ.0 ) THEN
198 MAXWRK = 0
199 LIWORK = 3*MINMN*NLVL + 11*MINMN
200 MM = M
201 IF( M.GE.N .AND. M.GE.MNTHR ) THEN
202 *
203 * Path 1a - overdetermined, with many more rows than columns.
204 *
205 MM = N
206 MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
207 $ -1, -1 ) )
208 MAXWRK = MAX( MAXWRK, N+NRHS*
209 $ ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
210 END IF
211 IF( M.GE.N ) THEN
212 *
213 * Path 1 - overdetermined or exactly determined.
214 *
215 MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
216 $ ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
217 MAXWRK = MAX( MAXWRK, 3*N+NRHS*
218 $ ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
219 MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
220 $ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
221 WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
222 MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
223 MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
224 END IF
225 IF( N.GT.M ) THEN
226 WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
227 IF( N.GE.MNTHR ) THEN
228 *
229 * Path 2a - underdetermined, with many more columns
230 * than rows.
231 *
232 MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
233 MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
234 $ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
235 MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
236 $ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
237 MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
238 $ ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )
239 IF( NRHS.GT.1 ) THEN
240 MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
241 ELSE
242 MAXWRK = MAX( MAXWRK, M*M+2*M )
243 END IF
244 MAXWRK = MAX( MAXWRK, M+NRHS*
245 $ ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
246 MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
247 ! XXX: Ensure the Path 2a case below is triggered. The workspace
248 ! calculation should use queries for all routines eventually.
249 MAXWRK = MAX( MAXWRK,
250 $ 4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
251 ELSE
252 *
253 * Path 2 - remaining underdetermined cases.
254 *
255 MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
256 $ -1, -1 )
257 MAXWRK = MAX( MAXWRK, 3*M+NRHS*
258 $ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )
259 MAXWRK = MAX( MAXWRK, 3*M+M*
260 $ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )
261 MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
262 END IF
263 MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
264 END IF
265 MINWRK = MIN( MINWRK, MAXWRK )
266 WORK( 1 ) = MAXWRK
267 IWORK( 1 ) = LIWORK
268
269 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
270 INFO = -12
271 END IF
272 END IF
273 *
274 IF( INFO.NE.0 ) THEN
275 CALL XERBLA( 'DGELSD', -INFO )
276 RETURN
277 ELSE IF( LQUERY ) THEN
278 GO TO 10
279 END IF
280 *
281 * Quick return if possible.
282 *
283 IF( M.EQ.0 .OR. N.EQ.0 ) THEN
284 RANK = 0
285 RETURN
286 END IF
287 *
288 * Get machine parameters.
289 *
290 EPS = DLAMCH( 'P' )
291 SFMIN = DLAMCH( 'S' )
292 SMLNUM = SFMIN / EPS
293 BIGNUM = ONE / SMLNUM
294 CALL DLABAD( SMLNUM, BIGNUM )
295 *
296 * Scale A if max entry outside range [SMLNUM,BIGNUM].
297 *
298 ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
299 IASCL = 0
300 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
301 *
302 * Scale matrix norm up to SMLNUM.
303 *
304 CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
305 IASCL = 1
306 ELSE IF( ANRM.GT.BIGNUM ) THEN
307 *
308 * Scale matrix norm down to BIGNUM.
309 *
310 CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
311 IASCL = 2
312 ELSE IF( ANRM.EQ.ZERO ) THEN
313 *
314 * Matrix all zero. Return zero solution.
315 *
316 CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
317 CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
318 RANK = 0
319 GO TO 10
320 END IF
321 *
322 * Scale B if max entry outside range [SMLNUM,BIGNUM].
323 *
324 BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
325 IBSCL = 0
326 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
327 *
328 * Scale matrix norm up to SMLNUM.
329 *
330 CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
331 IBSCL = 1
332 ELSE IF( BNRM.GT.BIGNUM ) THEN
333 *
334 * Scale matrix norm down to BIGNUM.
335 *
336 CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
337 IBSCL = 2
338 END IF
339 *
340 * If M < N make sure certain entries of B are zero.
341 *
342 IF( M.LT.N )
343 $ CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
344 *
345 * Overdetermined case.
346 *
347 IF( M.GE.N ) THEN
348 *
349 * Path 1 - overdetermined or exactly determined.
350 *
351 MM = M
352 IF( M.GE.MNTHR ) THEN
353 *
354 * Path 1a - overdetermined, with many more rows than columns.
355 *
356 MM = N
357 ITAU = 1
358 NWORK = ITAU + N
359 *
360 * Compute A=Q*R.
361 * (Workspace: need 2*N, prefer N+N*NB)
362 *
363 CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
364 $ LWORK-NWORK+1, INFO )
365 *
366 * Multiply B by transpose(Q).
367 * (Workspace: need N+NRHS, prefer N+NRHS*NB)
368 *
369 CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
370 $ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
371 *
372 * Zero out below R.
373 *
374 IF( N.GT.1 ) THEN
375 CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
376 END IF
377 END IF
378 *
379 IE = 1
380 ITAUQ = IE + N
381 ITAUP = ITAUQ + N
382 NWORK = ITAUP + N
383 *
384 * Bidiagonalize R in A.
385 * (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
386 *
387 CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
388 $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
389 $ INFO )
390 *
391 * Multiply B by transpose of left bidiagonalizing vectors of R.
392 * (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
393 *
394 CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
395 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
396 *
397 * Solve the bidiagonal least squares problem.
398 *
399 CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
400 $ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
401 IF( INFO.NE.0 ) THEN
402 GO TO 10
403 END IF
404 *
405 * Multiply B by right bidiagonalizing vectors of R.
406 *
407 CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
408 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
409 *
410 ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
411 $ MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
412 *
413 * Path 2a - underdetermined, with many more columns than rows
414 * and sufficient workspace for an efficient algorithm.
415 *
416 LDWORK = M
417 IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
418 $ M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
419 ITAU = 1
420 NWORK = M + 1
421 *
422 * Compute A=L*Q.
423 * (Workspace: need 2*M, prefer M+M*NB)
424 *
425 CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
426 $ LWORK-NWORK+1, INFO )
427 IL = NWORK
428 *
429 * Copy L to WORK(IL), zeroing out above its diagonal.
430 *
431 CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
432 CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
433 $ LDWORK )
434 IE = IL + LDWORK*M
435 ITAUQ = IE + M
436 ITAUP = ITAUQ + M
437 NWORK = ITAUP + M
438 *
439 * Bidiagonalize L in WORK(IL).
440 * (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
441 *
442 CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
443 $ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
444 $ LWORK-NWORK+1, INFO )
445 *
446 * Multiply B by transpose of left bidiagonalizing vectors of L.
447 * (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
448 *
449 CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
450 $ WORK( ITAUQ ), B, LDB, WORK( NWORK ),
451 $ LWORK-NWORK+1, INFO )
452 *
453 * Solve the bidiagonal least squares problem.
454 *
455 CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
456 $ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
457 IF( INFO.NE.0 ) THEN
458 GO TO 10
459 END IF
460 *
461 * Multiply B by right bidiagonalizing vectors of L.
462 *
463 CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
464 $ WORK( ITAUP ), B, LDB, WORK( NWORK ),
465 $ LWORK-NWORK+1, INFO )
466 *
467 * Zero out below first M rows of B.
468 *
469 CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
470 NWORK = ITAU + M
471 *
472 * Multiply transpose(Q) by B.
473 * (Workspace: need M+NRHS, prefer M+NRHS*NB)
474 *
475 CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
476 $ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
477 *
478 ELSE
479 *
480 * Path 2 - remaining underdetermined cases.
481 *
482 IE = 1
483 ITAUQ = IE + M
484 ITAUP = ITAUQ + M
485 NWORK = ITAUP + M
486 *
487 * Bidiagonalize A.
488 * (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
489 *
490 CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
491 $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
492 $ INFO )
493 *
494 * Multiply B by transpose of left bidiagonalizing vectors.
495 * (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
496 *
497 CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
498 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
499 *
500 * Solve the bidiagonal least squares problem.
501 *
502 CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
503 $ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
504 IF( INFO.NE.0 ) THEN
505 GO TO 10
506 END IF
507 *
508 * Multiply B by right bidiagonalizing vectors of A.
509 *
510 CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
511 $ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
512 *
513 END IF
514 *
515 * Undo scaling.
516 *
517 IF( IASCL.EQ.1 ) THEN
518 CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
519 CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
520 $ INFO )
521 ELSE IF( IASCL.EQ.2 ) THEN
522 CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
523 CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
524 $ INFO )
525 END IF
526 IF( IBSCL.EQ.1 ) THEN
527 CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
528 ELSE IF( IBSCL.EQ.2 ) THEN
529 CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
530 END IF
531 *
532 10 CONTINUE
533 WORK( 1 ) = MAXWRK
534 IWORK( 1 ) = LIWORK
535 RETURN
536 *
537 * End of DGELSD
538 *
539 END