Annotation of rpl/lapack/lapack/dgelss.f, revision 1.12
1.9 bertrand 1: *> \brief <b> DGELSS 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 DGELSS + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelss.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelss.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelss.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGELSS( 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: * DOUBLE PRECISION RCOND
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DGELSS 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 doubleGEsolve
170: *
171: * =====================================================================
1.1 bertrand 172: SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
173: $ WORK, LWORK, INFO )
174: *
1.9 bertrand 175: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 176: * -- LAPACK is a software package provided by Univ. of Tennessee, --
177: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 bertrand 178: * November 2011
1.1 bertrand 179: *
180: * .. Scalar Arguments ..
181: INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
182: DOUBLE PRECISION RCOND
183: * ..
184: * .. Array Arguments ..
185: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
186: * ..
187: *
188: * =====================================================================
189: *
190: * .. Parameters ..
191: DOUBLE PRECISION ZERO, ONE
192: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+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
1.9 bertrand 199: INTEGER LWORK_DGEQRF, LWORK_DORMQR, LWORK_DGEBRD,
200: $ LWORK_DORMBR, LWORK_DORGBR, LWORK_DORMLQ,
201: $ LWORK_DGELQF
1.1 bertrand 202: DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
203: * ..
204: * .. Local Arrays ..
1.9 bertrand 205: DOUBLE PRECISION DUM( 1 )
1.1 bertrand 206: * ..
207: * .. External Subroutines ..
208: EXTERNAL DBDSQR, DCOPY, DGEBRD, DGELQF, DGEMM, DGEMV,
209: $ DGEQRF, DLABAD, DLACPY, DLASCL, DLASET, DORGBR,
210: $ DORMBR, DORMLQ, DORMQR, DRSCL, XERBLA
211: * ..
212: * .. External Functions ..
213: INTEGER ILAENV
214: DOUBLE PRECISION DLAMCH, DLANGE
215: EXTERNAL ILAENV, DLAMCH, DLANGE
216: * ..
217: * .. Intrinsic Functions ..
218: INTRINSIC MAX, MIN
219: * ..
220: * .. Executable Statements ..
221: *
222: * Test the input arguments
223: *
224: INFO = 0
225: MINMN = MIN( M, N )
226: MAXMN = MAX( M, N )
227: LQUERY = ( LWORK.EQ.-1 )
228: IF( M.LT.0 ) THEN
229: INFO = -1
230: ELSE IF( N.LT.0 ) THEN
231: INFO = -2
232: ELSE IF( NRHS.LT.0 ) THEN
233: INFO = -3
234: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
235: INFO = -5
236: ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
237: INFO = -7
238: END IF
239: *
240: * Compute workspace
241: * (Note: Comments in the code beginning "Workspace:" describe the
242: * minimal amount of workspace needed at that point in the code,
243: * as well as the preferred amount for good performance.
244: * NB refers to the optimal block size for the immediately
245: * following subroutine, as returned by ILAENV.)
246: *
247: IF( INFO.EQ.0 ) THEN
248: MINWRK = 1
249: MAXWRK = 1
250: IF( MINMN.GT.0 ) THEN
251: MM = M
252: MNTHR = ILAENV( 6, 'DGELSS', ' ', M, N, NRHS, -1 )
253: IF( M.GE.N .AND. M.GE.MNTHR ) THEN
254: *
255: * Path 1a - overdetermined, with many more rows than
256: * columns
257: *
1.9 bertrand 258: * Compute space needed for DGEQRF
259: CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, INFO )
260: LWORK_DGEQRF=DUM(1)
261: * Compute space needed for DORMQR
262: CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, DUM(1), B,
263: $ LDB, DUM(1), -1, INFO )
264: LWORK_DORMQR=DUM(1)
1.1 bertrand 265: MM = N
1.9 bertrand 266: MAXWRK = MAX( MAXWRK, N + LWORK_DGEQRF )
267: MAXWRK = MAX( MAXWRK, N + LWORK_DORMQR )
1.1 bertrand 268: END IF
269: IF( M.GE.N ) THEN
270: *
271: * Path 1 - overdetermined or exactly determined
272: *
273: * Compute workspace needed for DBDSQR
274: *
275: BDSPAC = MAX( 1, 5*N )
1.9 bertrand 276: * Compute space needed for DGEBRD
277: CALL DGEBRD( MM, N, A, LDA, S, DUM(1), DUM(1),
278: $ DUM(1), DUM(1), -1, INFO )
279: LWORK_DGEBRD=DUM(1)
280: * Compute space needed for DORMBR
281: CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, DUM(1),
282: $ B, LDB, DUM(1), -1, INFO )
283: LWORK_DORMBR=DUM(1)
284: * Compute space needed for DORGBR
285: CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
286: $ DUM(1), -1, INFO )
287: LWORK_DORGBR=DUM(1)
288: * Compute total workspace needed
289: MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
290: MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORMBR )
291: MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR )
1.1 bertrand 292: MAXWRK = MAX( MAXWRK, BDSPAC )
293: MAXWRK = MAX( MAXWRK, N*NRHS )
294: MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC )
295: MAXWRK = MAX( MINWRK, MAXWRK )
296: END IF
297: IF( N.GT.M ) THEN
298: *
299: * Compute workspace needed for DBDSQR
300: *
301: BDSPAC = MAX( 1, 5*M )
302: MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC )
303: IF( N.GE.MNTHR ) THEN
304: *
305: * Path 2a - underdetermined, with many more columns
306: * than rows
307: *
1.9 bertrand 308: * Compute space needed for DGELQF
309: CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1),
310: $ -1, INFO )
311: LWORK_DGELQF=DUM(1)
312: * Compute space needed for DGEBRD
313: CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
314: $ DUM(1), DUM(1), -1, INFO )
315: LWORK_DGEBRD=DUM(1)
316: * Compute space needed for DORMBR
317: CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
318: $ DUM(1), B, LDB, DUM(1), -1, INFO )
319: LWORK_DORMBR=DUM(1)
320: * Compute space needed for DORGBR
321: CALL DORGBR( 'P', M, M, M, A, LDA, DUM(1),
322: $ DUM(1), -1, INFO )
323: LWORK_DORGBR=DUM(1)
324: * Compute space needed for DORMLQ
325: CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, DUM(1),
326: $ B, LDB, DUM(1), -1, INFO )
327: LWORK_DORMLQ=DUM(1)
328: * Compute total workspace needed
329: MAXWRK = M + LWORK_DGELQF
330: MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DGEBRD )
331: MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORMBR )
332: MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORGBR )
1.1 bertrand 333: MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC )
334: IF( NRHS.GT.1 ) THEN
335: MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
336: ELSE
337: MAXWRK = MAX( MAXWRK, M*M + 2*M )
338: END IF
1.9 bertrand 339: MAXWRK = MAX( MAXWRK, M + LWORK_DORMLQ )
1.1 bertrand 340: ELSE
341: *
342: * Path 2 - underdetermined
343: *
1.9 bertrand 344: * Compute space needed for DGEBRD
345: CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
346: $ DUM(1), DUM(1), -1, INFO )
347: LWORK_DGEBRD=DUM(1)
348: * Compute space needed for DORMBR
349: CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, A, LDA,
350: $ DUM(1), B, LDB, DUM(1), -1, INFO )
351: LWORK_DORMBR=DUM(1)
352: * Compute space needed for DORGBR
353: CALL DORGBR( 'P', M, N, M, A, LDA, DUM(1),
354: $ DUM(1), -1, INFO )
355: LWORK_DORGBR=DUM(1)
356: MAXWRK = 3*M + LWORK_DGEBRD
357: MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORMBR )
358: MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR )
1.1 bertrand 359: MAXWRK = MAX( MAXWRK, BDSPAC )
360: MAXWRK = MAX( MAXWRK, N*NRHS )
361: END IF
362: END IF
363: MAXWRK = MAX( MINWRK, MAXWRK )
364: END IF
365: WORK( 1 ) = MAXWRK
366: *
367: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
368: $ INFO = -12
369: END IF
370: *
371: IF( INFO.NE.0 ) THEN
372: CALL XERBLA( 'DGELSS', -INFO )
373: RETURN
374: ELSE IF( LQUERY ) THEN
375: RETURN
376: END IF
377: *
378: * Quick return if possible
379: *
380: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
381: RANK = 0
382: RETURN
383: END IF
384: *
385: * Get machine parameters
386: *
387: EPS = DLAMCH( 'P' )
388: SFMIN = DLAMCH( 'S' )
389: SMLNUM = SFMIN / EPS
390: BIGNUM = ONE / SMLNUM
391: CALL DLABAD( SMLNUM, BIGNUM )
392: *
393: * Scale A if max element outside range [SMLNUM,BIGNUM]
394: *
395: ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
396: IASCL = 0
397: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
398: *
399: * Scale matrix norm up to SMLNUM
400: *
401: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
402: IASCL = 1
403: ELSE IF( ANRM.GT.BIGNUM ) THEN
404: *
405: * Scale matrix norm down to BIGNUM
406: *
407: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
408: IASCL = 2
409: ELSE IF( ANRM.EQ.ZERO ) THEN
410: *
411: * Matrix all zero. Return zero solution.
412: *
413: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
1.5 bertrand 414: CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, MINMN )
1.1 bertrand 415: RANK = 0
416: GO TO 70
417: END IF
418: *
419: * Scale B if max element outside range [SMLNUM,BIGNUM]
420: *
421: BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
422: IBSCL = 0
423: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
424: *
425: * Scale matrix norm up to SMLNUM
426: *
427: CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
428: IBSCL = 1
429: ELSE IF( BNRM.GT.BIGNUM ) THEN
430: *
431: * Scale matrix norm down to BIGNUM
432: *
433: CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
434: IBSCL = 2
435: END IF
436: *
437: * Overdetermined case
438: *
439: IF( M.GE.N ) THEN
440: *
441: * Path 1 - overdetermined or exactly determined
442: *
443: MM = M
444: IF( M.GE.MNTHR ) THEN
445: *
446: * Path 1a - overdetermined, with many more rows than columns
447: *
448: MM = N
449: ITAU = 1
450: IWORK = ITAU + N
451: *
452: * Compute A=Q*R
453: * (Workspace: need 2*N, prefer N+N*NB)
454: *
455: CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
456: $ LWORK-IWORK+1, INFO )
457: *
458: * Multiply B by transpose(Q)
459: * (Workspace: need N+NRHS, prefer N+NRHS*NB)
460: *
461: CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
462: $ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
463: *
464: * Zero out below R
465: *
466: IF( N.GT.1 )
467: $ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
468: END IF
469: *
470: IE = 1
471: ITAUQ = IE + N
472: ITAUP = ITAUQ + N
473: IWORK = ITAUP + N
474: *
475: * Bidiagonalize R in A
476: * (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
477: *
478: CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
479: $ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
480: $ INFO )
481: *
482: * Multiply B by transpose of left bidiagonalizing vectors of R
483: * (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
484: *
485: CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
486: $ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
487: *
488: * Generate right bidiagonalizing vectors of R in A
489: * (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
490: *
491: CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
492: $ WORK( IWORK ), LWORK-IWORK+1, INFO )
493: IWORK = IE + N
494: *
495: * Perform bidiagonal QR iteration
496: * multiply B by transpose of left singular vectors
497: * compute right singular vectors in A
498: * (Workspace: need BDSPAC)
499: *
1.9 bertrand 500: CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
1.1 bertrand 501: $ 1, B, LDB, WORK( IWORK ), INFO )
502: IF( INFO.NE.0 )
503: $ GO TO 70
504: *
505: * Multiply B by reciprocals of singular values
506: *
507: THR = MAX( RCOND*S( 1 ), SFMIN )
508: IF( RCOND.LT.ZERO )
509: $ THR = MAX( EPS*S( 1 ), SFMIN )
510: RANK = 0
511: DO 10 I = 1, N
512: IF( S( I ).GT.THR ) THEN
513: CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
514: RANK = RANK + 1
515: ELSE
516: CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
517: END IF
518: 10 CONTINUE
519: *
520: * Multiply B by right singular vectors
521: * (Workspace: need N, prefer N*NRHS)
522: *
523: IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
524: CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB, ZERO,
525: $ WORK, LDB )
526: CALL DLACPY( 'G', N, NRHS, WORK, LDB, B, LDB )
527: ELSE IF( NRHS.GT.1 ) THEN
528: CHUNK = LWORK / N
529: DO 20 I = 1, NRHS, CHUNK
530: BL = MIN( NRHS-I+1, CHUNK )
531: CALL DGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1, I ),
532: $ LDB, ZERO, WORK, N )
533: CALL DLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB )
534: 20 CONTINUE
535: ELSE
536: CALL DGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
537: CALL DCOPY( N, WORK, 1, B, 1 )
538: END IF
539: *
540: ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
541: $ MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN
542: *
543: * Path 2a - underdetermined, with many more columns than rows
544: * and sufficient workspace for an efficient algorithm
545: *
546: LDWORK = M
547: IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
548: $ M*LDA+M+M*NRHS ) )LDWORK = LDA
549: ITAU = 1
550: IWORK = M + 1
551: *
552: * Compute A=L*Q
553: * (Workspace: need 2*M, prefer M+M*NB)
554: *
555: CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
556: $ LWORK-IWORK+1, INFO )
557: IL = IWORK
558: *
559: * Copy L to WORK(IL), zeroing out above it
560: *
561: CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
562: CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
563: $ LDWORK )
564: IE = IL + LDWORK*M
565: ITAUQ = IE + M
566: ITAUP = ITAUQ + M
567: IWORK = ITAUP + M
568: *
569: * Bidiagonalize L in WORK(IL)
570: * (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
571: *
572: CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
573: $ WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ),
574: $ LWORK-IWORK+1, INFO )
575: *
576: * Multiply B by transpose of left bidiagonalizing vectors of L
577: * (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
578: *
579: CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
580: $ WORK( ITAUQ ), B, LDB, WORK( IWORK ),
581: $ LWORK-IWORK+1, INFO )
582: *
583: * Generate right bidiagonalizing vectors of R in WORK(IL)
584: * (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
585: *
586: CALL DORGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ),
587: $ WORK( IWORK ), LWORK-IWORK+1, INFO )
588: IWORK = IE + M
589: *
590: * Perform bidiagonal QR iteration,
591: * computing right singular vectors of L in WORK(IL) and
592: * multiplying B by transpose of left singular vectors
593: * (Workspace: need M*M+M+BDSPAC)
594: *
595: CALL DBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ),
596: $ LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO )
597: IF( INFO.NE.0 )
598: $ GO TO 70
599: *
600: * Multiply B by reciprocals of singular values
601: *
602: THR = MAX( RCOND*S( 1 ), SFMIN )
603: IF( RCOND.LT.ZERO )
604: $ THR = MAX( EPS*S( 1 ), SFMIN )
605: RANK = 0
606: DO 30 I = 1, M
607: IF( S( I ).GT.THR ) THEN
608: CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
609: RANK = RANK + 1
610: ELSE
611: CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
612: END IF
613: 30 CONTINUE
614: IWORK = IE
615: *
616: * Multiply B by right singular vectors of L in WORK(IL)
617: * (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS)
618: *
619: IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN
620: CALL DGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ), LDWORK,
621: $ B, LDB, ZERO, WORK( IWORK ), LDB )
622: CALL DLACPY( 'G', M, NRHS, WORK( IWORK ), LDB, B, LDB )
623: ELSE IF( NRHS.GT.1 ) THEN
624: CHUNK = ( LWORK-IWORK+1 ) / M
625: DO 40 I = 1, NRHS, CHUNK
626: BL = MIN( NRHS-I+1, CHUNK )
627: CALL DGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ), LDWORK,
628: $ B( 1, I ), LDB, ZERO, WORK( IWORK ), M )
629: CALL DLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ),
630: $ LDB )
631: 40 CONTINUE
632: ELSE
633: CALL DGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1, 1 ),
634: $ 1, ZERO, WORK( IWORK ), 1 )
635: CALL DCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 )
636: END IF
637: *
638: * Zero out below first M rows of B
639: *
640: CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
641: IWORK = ITAU + M
642: *
643: * Multiply transpose(Q) by B
644: * (Workspace: need M+NRHS, prefer M+NRHS*NB)
645: *
646: CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
647: $ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
648: *
649: ELSE
650: *
651: * Path 2 - remaining underdetermined cases
652: *
653: IE = 1
654: ITAUQ = IE + M
655: ITAUP = ITAUQ + M
656: IWORK = ITAUP + M
657: *
658: * Bidiagonalize A
659: * (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
660: *
661: CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
662: $ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
663: $ INFO )
664: *
665: * Multiply B by transpose of left bidiagonalizing vectors
666: * (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
667: *
668: CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
669: $ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
670: *
671: * Generate right bidiagonalizing vectors in A
672: * (Workspace: need 4*M, prefer 3*M+M*NB)
673: *
674: CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
675: $ WORK( IWORK ), LWORK-IWORK+1, INFO )
676: IWORK = IE + M
677: *
678: * Perform bidiagonal QR iteration,
679: * computing right singular vectors of A in A and
680: * multiplying B by transpose of left singular vectors
681: * (Workspace: need BDSPAC)
682: *
1.9 bertrand 683: CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
1.1 bertrand 684: $ 1, B, LDB, WORK( IWORK ), INFO )
685: IF( INFO.NE.0 )
686: $ GO TO 70
687: *
688: * Multiply B by reciprocals of singular values
689: *
690: THR = MAX( RCOND*S( 1 ), SFMIN )
691: IF( RCOND.LT.ZERO )
692: $ THR = MAX( EPS*S( 1 ), SFMIN )
693: RANK = 0
694: DO 50 I = 1, M
695: IF( S( I ).GT.THR ) THEN
696: CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
697: RANK = RANK + 1
698: ELSE
699: CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
700: END IF
701: 50 CONTINUE
702: *
703: * Multiply B by right singular vectors of A
704: * (Workspace: need N, prefer N*NRHS)
705: *
706: IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
707: CALL DGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB, ZERO,
708: $ WORK, LDB )
709: CALL DLACPY( 'F', N, NRHS, WORK, LDB, B, LDB )
710: ELSE IF( NRHS.GT.1 ) THEN
711: CHUNK = LWORK / N
712: DO 60 I = 1, NRHS, CHUNK
713: BL = MIN( NRHS-I+1, CHUNK )
714: CALL DGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1, I ),
715: $ LDB, ZERO, WORK, N )
716: CALL DLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB )
717: 60 CONTINUE
718: ELSE
719: CALL DGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
720: CALL DCOPY( N, WORK, 1, B, 1 )
721: END IF
722: END IF
723: *
724: * Undo scaling
725: *
726: IF( IASCL.EQ.1 ) THEN
727: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
728: CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
729: $ INFO )
730: ELSE IF( IASCL.EQ.2 ) THEN
731: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
732: CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
733: $ INFO )
734: END IF
735: IF( IBSCL.EQ.1 ) THEN
736: CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
737: ELSE IF( IBSCL.EQ.2 ) THEN
738: CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
739: END IF
740: *
741: 70 CONTINUE
742: WORK( 1 ) = MAXWRK
743: RETURN
744: *
745: * End of DGELSS
746: *
747: END
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