Annotation of rpl/lapack/lapack/dgelsd.f, revision 1.5
1.1 bertrand 1: SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
2: $ WORK, LWORK, IWORK, INFO )
3: *
1.5 ! bertrand 4: * -- LAPACK driver routine (version 3.2.2) --
1.1 bertrand 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.5 ! bertrand 7: * June 2010
1.1 bertrand 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))
1.5 ! bertrand 117: * LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
1.1 bertrand 118: * where MINMN = MIN( M,N ).
1.5 ! bertrand 119: * On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
1.1 bertrand 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,
1.5 ! bertrand 145: $ LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
! 146: $ MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
1.1 bertrand 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
1.5 ! bertrand 192: LIWORK = 1
1.1 bertrand 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
1.5 ! bertrand 199: LIWORK = 3*MINMN*NLVL + 11*MINMN
1.1 bertrand 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
1.5 ! bertrand 267: IWORK( 1 ) = LIWORK
! 268:
1.1 bertrand 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
1.5 ! bertrand 534: IWORK( 1 ) = LIWORK
1.1 bertrand 535: RETURN
536: *
537: * End of DGELSD
538: *
539: END
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