3 /* Table of constant values */
5 static integer c__1 = 1;
6 static integer c_n1 = -1;
7 static integer c__0 = 0;
8 static doublereal c_b227 = 0.;
9 static doublereal c_b248 = 1.;
11 /* Subroutine */ int dgesdd_(char *jobz, integer *m, integer *n, doublereal *
12 a, integer *lda, doublereal *s, doublereal *u, integer *ldu,
13 doublereal *vt, integer *ldvt, doublereal *work, integer *lwork,
14 integer *iwork, integer *info)
16 /* System generated locals */
17 integer a_dim1, a_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1,
20 /* Builtin functions */
21 double sqrt(doublereal);
24 integer i__, ie, il, ir, iu, blk;
25 doublereal dum[1], eps;
28 integer idum[1], ierr, itau;
29 extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
30 integer *, doublereal *, doublereal *, integer *, doublereal *,
31 integer *, doublereal *, doublereal *, integer *);
32 extern logical lsame_(char *, char *);
33 integer chunk, minmn, wrkbl, itaup, itauq, mnthr;
36 logical wntqn, wntqo, wntqs;
37 extern /* Subroutine */ int dbdsdc_(char *, char *, integer *, doublereal
38 *, doublereal *, doublereal *, integer *, doublereal *, integer *,
39 doublereal *, integer *, doublereal *, integer *, integer *), dgebrd_(integer *, integer *, doublereal *,
40 integer *, doublereal *, doublereal *, doublereal *, doublereal *,
41 doublereal *, integer *, integer *);
42 extern doublereal dlamch_(char *), dlange_(char *, integer *,
43 integer *, doublereal *, integer *, doublereal *);
45 extern /* Subroutine */ int dgelqf_(integer *, integer *, doublereal *,
46 integer *, doublereal *, doublereal *, integer *, integer *),
47 dlascl_(char *, integer *, integer *, doublereal *, doublereal *,
48 integer *, integer *, doublereal *, integer *, integer *),
49 dgeqrf_(integer *, integer *, doublereal *, integer *,
50 doublereal *, doublereal *, integer *, integer *), dlacpy_(char *,
51 integer *, integer *, doublereal *, integer *, doublereal *,
52 integer *), dlaset_(char *, integer *, integer *,
53 doublereal *, doublereal *, doublereal *, integer *),
54 xerbla_(char *, integer *), dorgbr_(char *, integer *,
55 integer *, integer *, doublereal *, integer *, doublereal *,
56 doublereal *, integer *, integer *);
57 extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
58 integer *, integer *);
60 extern /* Subroutine */ int dormbr_(char *, char *, char *, integer *,
61 integer *, integer *, doublereal *, integer *, doublereal *,
62 doublereal *, integer *, doublereal *, integer *, integer *), dorglq_(integer *, integer *, integer *,
63 doublereal *, integer *, doublereal *, doublereal *, integer *,
64 integer *), dorgqr_(integer *, integer *, integer *, doublereal *,
65 integer *, doublereal *, doublereal *, integer *, integer *);
66 integer ldwrkl, ldwrkr, minwrk, ldwrku, maxwrk, ldwkvt;
68 logical wntqas, lquery;
71 /* -- LAPACK driver routine (version 3.1) -- */
72 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
75 /* .. Scalar Arguments .. */
77 /* .. Array Arguments .. */
83 /* DGESDD computes the singular value decomposition (SVD) of a real */
84 /* M-by-N matrix A, optionally computing the left and right singular */
85 /* vectors. If singular vectors are desired, it uses a */
86 /* divide-and-conquer algorithm. */
88 /* The SVD is written */
90 /* A = U * SIGMA * transpose(V) */
92 /* where SIGMA is an M-by-N matrix which is zero except for its */
93 /* min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and */
94 /* V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA */
95 /* are the singular values of A; they are real and non-negative, and */
96 /* are returned in descending order. The first min(m,n) columns of */
97 /* U and V are the left and right singular vectors of A. */
99 /* Note that the routine returns VT = V**T, not V. */
101 /* The divide and conquer algorithm makes very mild assumptions about */
102 /* floating point arithmetic. It will work on machines with a guard */
103 /* digit in add/subtract, or on those binary machines without guard */
104 /* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or */
105 /* Cray-2. It could conceivably fail on hexadecimal or decimal machines */
106 /* without guard digits, but we know of none. */
111 /* JOBZ (input) CHARACTER*1 */
112 /* Specifies options for computing all or part of the matrix U: */
113 /* = 'A': all M columns of U and all N rows of V**T are */
114 /* returned in the arrays U and VT; */
115 /* = 'S': the first min(M,N) columns of U and the first */
116 /* min(M,N) rows of V**T are returned in the arrays U */
118 /* = 'O': If M >= N, the first N columns of U are overwritten */
119 /* on the array A and all rows of V**T are returned in */
121 /* otherwise, all columns of U are returned in the */
122 /* array U and the first M rows of V**T are overwritten */
123 /* in the array A; */
124 /* = 'N': no columns of U or rows of V**T are computed. */
126 /* M (input) INTEGER */
127 /* The number of rows of the input matrix A. M >= 0. */
129 /* N (input) INTEGER */
130 /* The number of columns of the input matrix A. N >= 0. */
132 /* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) */
133 /* On entry, the M-by-N matrix A. */
135 /* if JOBZ = 'O', A is overwritten with the first N columns */
136 /* of U (the left singular vectors, stored */
137 /* columnwise) if M >= N; */
138 /* A is overwritten with the first M rows */
139 /* of V**T (the right singular vectors, stored */
140 /* rowwise) otherwise. */
141 /* if JOBZ .ne. 'O', the contents of A are destroyed. */
143 /* LDA (input) INTEGER */
144 /* The leading dimension of the array A. LDA >= max(1,M). */
146 /* S (output) DOUBLE PRECISION array, dimension (min(M,N)) */
147 /* The singular values of A, sorted so that S(i) >= S(i+1). */
149 /* U (output) DOUBLE PRECISION array, dimension (LDU,UCOL) */
150 /* UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; */
151 /* UCOL = min(M,N) if JOBZ = 'S'. */
152 /* If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M */
153 /* orthogonal matrix U; */
154 /* if JOBZ = 'S', U contains the first min(M,N) columns of U */
155 /* (the left singular vectors, stored columnwise); */
156 /* if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced. */
158 /* LDU (input) INTEGER */
159 /* The leading dimension of the array U. LDU >= 1; if */
160 /* JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M. */
162 /* VT (output) DOUBLE PRECISION array, dimension (LDVT,N) */
163 /* If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the */
164 /* N-by-N orthogonal matrix V**T; */
165 /* if JOBZ = 'S', VT contains the first min(M,N) rows of */
166 /* V**T (the right singular vectors, stored rowwise); */
167 /* if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced. */
169 /* LDVT (input) INTEGER */
170 /* The leading dimension of the array VT. LDVT >= 1; if */
171 /* JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N; */
172 /* if JOBZ = 'S', LDVT >= min(M,N). */
174 /* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
175 /* On exit, if INFO = 0, WORK(1) returns the optimal LWORK; */
177 /* LWORK (input) INTEGER */
178 /* The dimension of the array WORK. LWORK >= 1. */
180 /* LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)). */
182 /* LWORK >= 3*min(M,N)*min(M,N) + */
183 /* max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)). */
184 /* If JOBZ = 'S' or 'A' */
185 /* LWORK >= 3*min(M,N)*min(M,N) + */
186 /* max(max(M,N),4*min(M,N)*min(M,N)+4*min(M,N)). */
187 /* For good performance, LWORK should generally be larger. */
188 /* If LWORK = -1 but other input arguments are legal, WORK(1) */
189 /* returns the optimal LWORK. */
191 /* IWORK (workspace) INTEGER array, dimension (8*min(M,N)) */
193 /* INFO (output) INTEGER */
194 /* = 0: successful exit. */
195 /* < 0: if INFO = -i, the i-th argument had an illegal value. */
196 /* > 0: DBDSDC did not converge, updating process failed. */
198 /* Further Details */
199 /* =============== */
201 /* Based on contributions by */
202 /* Ming Gu and Huan Ren, Computer Science Division, University of */
203 /* California at Berkeley, USA */
205 /* ===================================================================== */
207 /* .. Parameters .. */
209 /* .. Local Scalars .. */
211 /* .. Local Arrays .. */
213 /* .. External Subroutines .. */
215 /* .. External Functions .. */
217 /* .. Intrinsic Functions .. */
219 /* .. Executable Statements .. */
221 /* Test the input arguments */
223 /* Parameter adjustments */
225 a_offset = 1 + a_dim1;
229 u_offset = 1 + u_dim1;
232 vt_offset = 1 + vt_dim1;
240 wntqa = lsame_(jobz, "A");
241 wntqs = lsame_(jobz, "S");
242 wntqas = wntqa || wntqs;
243 wntqo = lsame_(jobz, "O");
244 wntqn = lsame_(jobz, "N");
245 lquery = *lwork == -1;
247 if (! (wntqa || wntqs || wntqo || wntqn)) {
253 } else if (*lda < max(1,*m)) {
255 } else if (*ldu < 1 || wntqas && *ldu < *m || wntqo && *m < *n && *ldu < *
258 } else if (*ldvt < 1 || wntqa && *ldvt < *n || wntqs && *ldvt < minmn ||
259 wntqo && *m >= *n && *ldvt < *n) {
263 /* Compute workspace */
264 /* (Note: Comments in the code beginning "Workspace:" describe the */
265 /* minimal amount of workspace needed at that point in the code, */
266 /* as well as the preferred amount for good performance. */
267 /* NB refers to the optimal block size for the immediately */
268 /* following subroutine, as returned by ILAENV.) */
273 if (*m >= *n && minmn > 0) {
275 /* Compute space needed for DBDSDC */
277 mnthr = (integer) (minmn * 11. / 6.);
281 bdspac = *n * 3 * *n + (*n << 2);
286 /* Path 1 (M much larger than N, JOBZ='N') */
288 wrkbl = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", m, n, &
291 i__1 = wrkbl, i__2 = *n * 3 + (*n << 1) * ilaenv_(&c__1,
292 "DGEBRD", " ", n, n, &c_n1, &c_n1);
293 wrkbl = max(i__1,i__2);
295 i__1 = wrkbl, i__2 = bdspac + *n;
296 maxwrk = max(i__1,i__2);
297 minwrk = bdspac + *n;
300 /* Path 2 (M much larger than N, JOBZ='O') */
302 wrkbl = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", m, n, &
305 i__1 = wrkbl, i__2 = *n + *n * ilaenv_(&c__1, "DORGQR",
306 " ", m, n, n, &c_n1);
307 wrkbl = max(i__1,i__2);
309 i__1 = wrkbl, i__2 = *n * 3 + (*n << 1) * ilaenv_(&c__1,
310 "DGEBRD", " ", n, n, &c_n1, &c_n1);
311 wrkbl = max(i__1,i__2);
313 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
314 , "QLN", n, n, n, &c_n1);
315 wrkbl = max(i__1,i__2);
317 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
318 , "PRT", n, n, n, &c_n1);
319 wrkbl = max(i__1,i__2);
321 i__1 = wrkbl, i__2 = bdspac + *n * 3;
322 wrkbl = max(i__1,i__2);
323 maxwrk = wrkbl + (*n << 1) * *n;
324 minwrk = bdspac + (*n << 1) * *n + *n * 3;
327 /* Path 3 (M much larger than N, JOBZ='S') */
329 wrkbl = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", m, n, &
332 i__1 = wrkbl, i__2 = *n + *n * ilaenv_(&c__1, "DORGQR",
333 " ", m, n, n, &c_n1);
334 wrkbl = max(i__1,i__2);
336 i__1 = wrkbl, i__2 = *n * 3 + (*n << 1) * ilaenv_(&c__1,
337 "DGEBRD", " ", n, n, &c_n1, &c_n1);
338 wrkbl = max(i__1,i__2);
340 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
341 , "QLN", n, n, n, &c_n1);
342 wrkbl = max(i__1,i__2);
344 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
345 , "PRT", n, n, n, &c_n1);
346 wrkbl = max(i__1,i__2);
348 i__1 = wrkbl, i__2 = bdspac + *n * 3;
349 wrkbl = max(i__1,i__2);
350 maxwrk = wrkbl + *n * *n;
351 minwrk = bdspac + *n * *n + *n * 3;
354 /* Path 4 (M much larger than N, JOBZ='A') */
356 wrkbl = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", m, n, &
359 i__1 = wrkbl, i__2 = *n + *m * ilaenv_(&c__1, "DORGQR",
360 " ", m, m, n, &c_n1);
361 wrkbl = max(i__1,i__2);
363 i__1 = wrkbl, i__2 = *n * 3 + (*n << 1) * ilaenv_(&c__1,
364 "DGEBRD", " ", n, n, &c_n1, &c_n1);
365 wrkbl = max(i__1,i__2);
367 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
368 , "QLN", n, n, n, &c_n1);
369 wrkbl = max(i__1,i__2);
371 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
372 , "PRT", n, n, n, &c_n1);
373 wrkbl = max(i__1,i__2);
375 i__1 = wrkbl, i__2 = bdspac + *n * 3;
376 wrkbl = max(i__1,i__2);
377 maxwrk = wrkbl + *n * *n;
378 minwrk = bdspac + *n * *n + *n * 3;
382 /* Path 5 (M at least N, but not much larger) */
384 wrkbl = *n * 3 + (*m + *n) * ilaenv_(&c__1, "DGEBRD", " ", m,
388 i__1 = wrkbl, i__2 = bdspac + *n * 3;
389 maxwrk = max(i__1,i__2);
390 minwrk = *n * 3 + max(*m,bdspac);
393 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
394 , "QLN", m, n, n, &c_n1);
395 wrkbl = max(i__1,i__2);
397 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
398 , "PRT", n, n, n, &c_n1);
399 wrkbl = max(i__1,i__2);
401 i__1 = wrkbl, i__2 = bdspac + *n * 3;
402 wrkbl = max(i__1,i__2);
403 maxwrk = wrkbl + *m * *n;
405 i__1 = *m, i__2 = *n * *n + bdspac;
406 minwrk = *n * 3 + max(i__1,i__2);
409 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
410 , "QLN", m, n, n, &c_n1);
411 wrkbl = max(i__1,i__2);
413 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
414 , "PRT", n, n, n, &c_n1);
415 wrkbl = max(i__1,i__2);
417 i__1 = wrkbl, i__2 = bdspac + *n * 3;
418 maxwrk = max(i__1,i__2);
419 minwrk = *n * 3 + max(*m,bdspac);
422 i__1 = wrkbl, i__2 = *n * 3 + *m * ilaenv_(&c__1, "DORMBR"
423 , "QLN", m, m, n, &c_n1);
424 wrkbl = max(i__1,i__2);
426 i__1 = wrkbl, i__2 = *n * 3 + *n * ilaenv_(&c__1, "DORMBR"
427 , "PRT", n, n, n, &c_n1);
428 wrkbl = max(i__1,i__2);
430 i__1 = maxwrk, i__2 = bdspac + *n * 3;
431 maxwrk = max(i__1,i__2);
432 minwrk = *n * 3 + max(*m,bdspac);
435 } else if (minmn > 0) {
437 /* Compute space needed for DBDSDC */
439 mnthr = (integer) (minmn * 11. / 6.);
443 bdspac = *m * 3 * *m + (*m << 2);
448 /* Path 1t (N much larger than M, JOBZ='N') */
450 wrkbl = *m + *m * ilaenv_(&c__1, "DGELQF", " ", m, n, &
453 i__1 = wrkbl, i__2 = *m * 3 + (*m << 1) * ilaenv_(&c__1,
454 "DGEBRD", " ", m, m, &c_n1, &c_n1);
455 wrkbl = max(i__1,i__2);
457 i__1 = wrkbl, i__2 = bdspac + *m;
458 maxwrk = max(i__1,i__2);
459 minwrk = bdspac + *m;
462 /* Path 2t (N much larger than M, JOBZ='O') */
464 wrkbl = *m + *m * ilaenv_(&c__1, "DGELQF", " ", m, n, &
467 i__1 = wrkbl, i__2 = *m + *m * ilaenv_(&c__1, "DORGLQ",
468 " ", m, n, m, &c_n1);
469 wrkbl = max(i__1,i__2);
471 i__1 = wrkbl, i__2 = *m * 3 + (*m << 1) * ilaenv_(&c__1,
472 "DGEBRD", " ", m, m, &c_n1, &c_n1);
473 wrkbl = max(i__1,i__2);
475 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
476 , "QLN", m, m, m, &c_n1);
477 wrkbl = max(i__1,i__2);
479 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
480 , "PRT", m, m, m, &c_n1);
481 wrkbl = max(i__1,i__2);
483 i__1 = wrkbl, i__2 = bdspac + *m * 3;
484 wrkbl = max(i__1,i__2);
485 maxwrk = wrkbl + (*m << 1) * *m;
486 minwrk = bdspac + (*m << 1) * *m + *m * 3;
489 /* Path 3t (N much larger than M, JOBZ='S') */
491 wrkbl = *m + *m * ilaenv_(&c__1, "DGELQF", " ", m, n, &
494 i__1 = wrkbl, i__2 = *m + *m * ilaenv_(&c__1, "DORGLQ",
495 " ", m, n, m, &c_n1);
496 wrkbl = max(i__1,i__2);
498 i__1 = wrkbl, i__2 = *m * 3 + (*m << 1) * ilaenv_(&c__1,
499 "DGEBRD", " ", m, m, &c_n1, &c_n1);
500 wrkbl = max(i__1,i__2);
502 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
503 , "QLN", m, m, m, &c_n1);
504 wrkbl = max(i__1,i__2);
506 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
507 , "PRT", m, m, m, &c_n1);
508 wrkbl = max(i__1,i__2);
510 i__1 = wrkbl, i__2 = bdspac + *m * 3;
511 wrkbl = max(i__1,i__2);
512 maxwrk = wrkbl + *m * *m;
513 minwrk = bdspac + *m * *m + *m * 3;
516 /* Path 4t (N much larger than M, JOBZ='A') */
518 wrkbl = *m + *m * ilaenv_(&c__1, "DGELQF", " ", m, n, &
521 i__1 = wrkbl, i__2 = *m + *n * ilaenv_(&c__1, "DORGLQ",
522 " ", n, n, m, &c_n1);
523 wrkbl = max(i__1,i__2);
525 i__1 = wrkbl, i__2 = *m * 3 + (*m << 1) * ilaenv_(&c__1,
526 "DGEBRD", " ", m, m, &c_n1, &c_n1);
527 wrkbl = max(i__1,i__2);
529 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
530 , "QLN", m, m, m, &c_n1);
531 wrkbl = max(i__1,i__2);
533 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
534 , "PRT", m, m, m, &c_n1);
535 wrkbl = max(i__1,i__2);
537 i__1 = wrkbl, i__2 = bdspac + *m * 3;
538 wrkbl = max(i__1,i__2);
539 maxwrk = wrkbl + *m * *m;
540 minwrk = bdspac + *m * *m + *m * 3;
544 /* Path 5t (N greater than M, but not much larger) */
546 wrkbl = *m * 3 + (*m + *n) * ilaenv_(&c__1, "DGEBRD", " ", m,
550 i__1 = wrkbl, i__2 = bdspac + *m * 3;
551 maxwrk = max(i__1,i__2);
552 minwrk = *m * 3 + max(*n,bdspac);
555 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
556 , "QLN", m, m, n, &c_n1);
557 wrkbl = max(i__1,i__2);
559 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
560 , "PRT", m, n, m, &c_n1);
561 wrkbl = max(i__1,i__2);
563 i__1 = wrkbl, i__2 = bdspac + *m * 3;
564 wrkbl = max(i__1,i__2);
565 maxwrk = wrkbl + *m * *n;
567 i__1 = *n, i__2 = *m * *m + bdspac;
568 minwrk = *m * 3 + max(i__1,i__2);
571 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
572 , "QLN", m, m, n, &c_n1);
573 wrkbl = max(i__1,i__2);
575 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
576 , "PRT", m, n, m, &c_n1);
577 wrkbl = max(i__1,i__2);
579 i__1 = wrkbl, i__2 = bdspac + *m * 3;
580 maxwrk = max(i__1,i__2);
581 minwrk = *m * 3 + max(*n,bdspac);
584 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
585 , "QLN", m, m, n, &c_n1);
586 wrkbl = max(i__1,i__2);
588 i__1 = wrkbl, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR"
589 , "PRT", n, n, m, &c_n1);
590 wrkbl = max(i__1,i__2);
592 i__1 = wrkbl, i__2 = bdspac + *m * 3;
593 maxwrk = max(i__1,i__2);
594 minwrk = *m * 3 + max(*n,bdspac);
598 maxwrk = max(maxwrk,minwrk);
599 work[1] = (doublereal) maxwrk;
601 if (*lwork < minwrk && ! lquery) {
608 xerbla_("DGESDD", &i__1);
614 /* Quick return if possible */
616 if (*m == 0 || *n == 0) {
620 /* Get machine constants */
623 smlnum = sqrt(dlamch_("S")) / eps;
624 bignum = 1. / smlnum;
626 /* Scale A if max element outside range [SMLNUM,BIGNUM] */
628 anrm = dlange_("M", m, n, &a[a_offset], lda, dum);
630 if (anrm > 0. && anrm < smlnum) {
632 dlascl_("G", &c__0, &c__0, &anrm, &smlnum, m, n, &a[a_offset], lda, &
634 } else if (anrm > bignum) {
636 dlascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda, &
642 /* A has at least as many rows as columns. If A has sufficiently */
643 /* more rows than columns, first reduce using the QR */
644 /* decomposition (if sufficient workspace available) */
650 /* Path 1 (M much larger than N, JOBZ='N') */
651 /* No singular vectors to be computed */
657 /* (Workspace: need 2*N, prefer N+N*NB) */
659 i__1 = *lwork - nwork + 1;
660 dgeqrf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
663 /* Zero out below R */
667 dlaset_("L", &i__1, &i__2, &c_b227, &c_b227, &a[a_dim1 + 2],
674 /* Bidiagonalize R in A */
675 /* (Workspace: need 4*N, prefer 3*N+2*N*NB) */
677 i__1 = *lwork - nwork + 1;
678 dgebrd_(n, n, &a[a_offset], lda, &s[1], &work[ie], &work[
679 itauq], &work[itaup], &work[nwork], &i__1, &ierr);
682 /* Perform bidiagonal SVD, computing singular values only */
683 /* (Workspace: need N+BDSPAC) */
685 dbdsdc_("U", "N", n, &s[1], &work[ie], dum, &c__1, dum, &c__1,
686 dum, idum, &work[nwork], &iwork[1], info);
690 /* Path 2 (M much larger than N, JOBZ = 'O') */
691 /* N left singular vectors to be overwritten on A and */
692 /* N right singular vectors to be computed in VT */
696 /* WORK(IR) is LDWRKR by N */
698 if (*lwork >= *lda * *n + *n * *n + *n * 3 + bdspac) {
701 ldwrkr = (*lwork - *n * *n - *n * 3 - bdspac) / *n;
703 itau = ir + ldwrkr * *n;
707 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
709 i__1 = *lwork - nwork + 1;
710 dgeqrf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
713 /* Copy R to WORK(IR), zeroing out below it */
715 dlacpy_("U", n, n, &a[a_offset], lda, &work[ir], &ldwrkr);
718 dlaset_("L", &i__1, &i__2, &c_b227, &c_b227, &work[ir + 1], &
721 /* Generate Q in A */
722 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
724 i__1 = *lwork - nwork + 1;
725 dorgqr_(m, n, n, &a[a_offset], lda, &work[itau], &work[nwork],
732 /* Bidiagonalize R in VT, copying result to WORK(IR) */
733 /* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) */
735 i__1 = *lwork - nwork + 1;
736 dgebrd_(n, n, &work[ir], &ldwrkr, &s[1], &work[ie], &work[
737 itauq], &work[itaup], &work[nwork], &i__1, &ierr);
739 /* WORK(IU) is N by N */
742 nwork = iu + *n * *n;
744 /* Perform bidiagonal SVD, computing left singular vectors */
745 /* of bidiagonal matrix in WORK(IU) and computing right */
746 /* singular vectors of bidiagonal matrix in VT */
747 /* (Workspace: need N+N*N+BDSPAC) */
749 dbdsdc_("U", "I", n, &s[1], &work[ie], &work[iu], n, &vt[
750 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
753 /* Overwrite WORK(IU) by left singular vectors of R */
754 /* and VT by right singular vectors of R */
755 /* (Workspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB) */
757 i__1 = *lwork - nwork + 1;
758 dormbr_("Q", "L", "N", n, n, n, &work[ir], &ldwrkr, &work[
759 itauq], &work[iu], n, &work[nwork], &i__1, &ierr);
760 i__1 = *lwork - nwork + 1;
761 dormbr_("P", "R", "T", n, n, n, &work[ir], &ldwrkr, &work[
762 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__1, &
765 /* Multiply Q in A by left singular vectors of R in */
766 /* WORK(IU), storing result in WORK(IR) and copying to A */
767 /* (Workspace: need 2*N*N, prefer N*N+M*N) */
771 for (i__ = 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ +=
775 chunk = min(i__3,ldwrkr);
776 dgemm_("N", "N", &chunk, n, n, &c_b248, &a[i__ + a_dim1],
777 lda, &work[iu], n, &c_b227, &work[ir], &ldwrkr);
778 dlacpy_("F", &chunk, n, &work[ir], &ldwrkr, &a[i__ +
785 /* Path 3 (M much larger than N, JOBZ='S') */
786 /* N left singular vectors to be computed in U and */
787 /* N right singular vectors to be computed in VT */
791 /* WORK(IR) is N by N */
794 itau = ir + ldwrkr * *n;
798 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
800 i__2 = *lwork - nwork + 1;
801 dgeqrf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
804 /* Copy R to WORK(IR), zeroing out below it */
806 dlacpy_("U", n, n, &a[a_offset], lda, &work[ir], &ldwrkr);
809 dlaset_("L", &i__2, &i__1, &c_b227, &c_b227, &work[ir + 1], &
812 /* Generate Q in A */
813 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
815 i__2 = *lwork - nwork + 1;
816 dorgqr_(m, n, n, &a[a_offset], lda, &work[itau], &work[nwork],
823 /* Bidiagonalize R in WORK(IR) */
824 /* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) */
826 i__2 = *lwork - nwork + 1;
827 dgebrd_(n, n, &work[ir], &ldwrkr, &s[1], &work[ie], &work[
828 itauq], &work[itaup], &work[nwork], &i__2, &ierr);
830 /* Perform bidiagonal SVD, computing left singular vectors */
831 /* of bidiagoal matrix in U and computing right singular */
832 /* vectors of bidiagonal matrix in VT */
833 /* (Workspace: need N+BDSPAC) */
835 dbdsdc_("U", "I", n, &s[1], &work[ie], &u[u_offset], ldu, &vt[
836 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
839 /* Overwrite U by left singular vectors of R and VT */
840 /* by right singular vectors of R */
841 /* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB) */
843 i__2 = *lwork - nwork + 1;
844 dormbr_("Q", "L", "N", n, n, n, &work[ir], &ldwrkr, &work[
845 itauq], &u[u_offset], ldu, &work[nwork], &i__2, &ierr);
847 i__2 = *lwork - nwork + 1;
848 dormbr_("P", "R", "T", n, n, n, &work[ir], &ldwrkr, &work[
849 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__2, &
852 /* Multiply Q in A by left singular vectors of R in */
853 /* WORK(IR), storing result in U */
854 /* (Workspace: need N*N) */
856 dlacpy_("F", n, n, &u[u_offset], ldu, &work[ir], &ldwrkr);
857 dgemm_("N", "N", m, n, n, &c_b248, &a[a_offset], lda, &work[
858 ir], &ldwrkr, &c_b227, &u[u_offset], ldu);
862 /* Path 4 (M much larger than N, JOBZ='A') */
863 /* M left singular vectors to be computed in U and */
864 /* N right singular vectors to be computed in VT */
868 /* WORK(IU) is N by N */
871 itau = iu + ldwrku * *n;
874 /* Compute A=Q*R, copying result to U */
875 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
877 i__2 = *lwork - nwork + 1;
878 dgeqrf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
880 dlacpy_("L", m, n, &a[a_offset], lda, &u[u_offset], ldu);
882 /* Generate Q in U */
883 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
884 i__2 = *lwork - nwork + 1;
885 dorgqr_(m, m, n, &u[u_offset], ldu, &work[itau], &work[nwork],
888 /* Produce R in A, zeroing out other entries */
892 dlaset_("L", &i__2, &i__1, &c_b227, &c_b227, &a[a_dim1 + 2],
899 /* Bidiagonalize R in A */
900 /* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) */
902 i__2 = *lwork - nwork + 1;
903 dgebrd_(n, n, &a[a_offset], lda, &s[1], &work[ie], &work[
904 itauq], &work[itaup], &work[nwork], &i__2, &ierr);
906 /* Perform bidiagonal SVD, computing left singular vectors */
907 /* of bidiagonal matrix in WORK(IU) and computing right */
908 /* singular vectors of bidiagonal matrix in VT */
909 /* (Workspace: need N+N*N+BDSPAC) */
911 dbdsdc_("U", "I", n, &s[1], &work[ie], &work[iu], n, &vt[
912 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
915 /* Overwrite WORK(IU) by left singular vectors of R and VT */
916 /* by right singular vectors of R */
917 /* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB) */
919 i__2 = *lwork - nwork + 1;
920 dormbr_("Q", "L", "N", n, n, n, &a[a_offset], lda, &work[
921 itauq], &work[iu], &ldwrku, &work[nwork], &i__2, &
923 i__2 = *lwork - nwork + 1;
924 dormbr_("P", "R", "T", n, n, n, &a[a_offset], lda, &work[
925 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__2, &
928 /* Multiply Q in U by left singular vectors of R in */
929 /* WORK(IU), storing result in A */
930 /* (Workspace: need N*N) */
932 dgemm_("N", "N", m, n, n, &c_b248, &u[u_offset], ldu, &work[
933 iu], &ldwrku, &c_b227, &a[a_offset], lda);
935 /* Copy left singular vectors of A from A to U */
937 dlacpy_("F", m, n, &a[a_offset], lda, &u[u_offset], ldu);
945 /* Path 5 (M at least N, but not much larger) */
946 /* Reduce to bidiagonal form without QR decomposition */
953 /* Bidiagonalize A */
954 /* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB) */
956 i__2 = *lwork - nwork + 1;
957 dgebrd_(m, n, &a[a_offset], lda, &s[1], &work[ie], &work[itauq], &
958 work[itaup], &work[nwork], &i__2, &ierr);
961 /* Perform bidiagonal SVD, only computing singular values */
962 /* (Workspace: need N+BDSPAC) */
964 dbdsdc_("U", "N", n, &s[1], &work[ie], dum, &c__1, dum, &c__1,
965 dum, idum, &work[nwork], &iwork[1], info);
968 if (*lwork >= *m * *n + *n * 3 + bdspac) {
970 /* WORK( IU ) is M by N */
973 nwork = iu + ldwrku * *n;
974 dlaset_("F", m, n, &c_b227, &c_b227, &work[iu], &ldwrku);
977 /* WORK( IU ) is N by N */
980 nwork = iu + ldwrku * *n;
982 /* WORK(IR) is LDWRKR by N */
985 ldwrkr = (*lwork - *n * *n - *n * 3) / *n;
987 nwork = iu + ldwrku * *n;
989 /* Perform bidiagonal SVD, computing left singular vectors */
990 /* of bidiagonal matrix in WORK(IU) and computing right */
991 /* singular vectors of bidiagonal matrix in VT */
992 /* (Workspace: need N+N*N+BDSPAC) */
994 dbdsdc_("U", "I", n, &s[1], &work[ie], &work[iu], &ldwrku, &
995 vt[vt_offset], ldvt, dum, idum, &work[nwork], &iwork[
998 /* Overwrite VT by right singular vectors of A */
999 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
1001 i__2 = *lwork - nwork + 1;
1002 dormbr_("P", "R", "T", n, n, n, &a[a_offset], lda, &work[
1003 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__2, &
1006 if (*lwork >= *m * *n + *n * 3 + bdspac) {
1008 /* Overwrite WORK(IU) by left singular vectors of A */
1009 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
1011 i__2 = *lwork - nwork + 1;
1012 dormbr_("Q", "L", "N", m, n, n, &a[a_offset], lda, &work[
1013 itauq], &work[iu], &ldwrku, &work[nwork], &i__2, &
1016 /* Copy left singular vectors of A from WORK(IU) to A */
1018 dlacpy_("F", m, n, &work[iu], &ldwrku, &a[a_offset], lda);
1021 /* Generate Q in A */
1022 /* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) */
1024 i__2 = *lwork - nwork + 1;
1025 dorgbr_("Q", m, n, n, &a[a_offset], lda, &work[itauq], &
1026 work[nwork], &i__2, &ierr);
1028 /* Multiply Q in A by left singular vectors of */
1029 /* bidiagonal matrix in WORK(IU), storing result in */
1030 /* WORK(IR) and copying to A */
1031 /* (Workspace: need 2*N*N, prefer N*N+M*N) */
1035 for (i__ = 1; i__1 < 0 ? i__ >= i__2 : i__ <= i__2; i__ +=
1038 i__3 = *m - i__ + 1;
1039 chunk = min(i__3,ldwrkr);
1040 dgemm_("N", "N", &chunk, n, n, &c_b248, &a[i__ +
1041 a_dim1], lda, &work[iu], &ldwrku, &c_b227, &
1043 dlacpy_("F", &chunk, n, &work[ir], &ldwrkr, &a[i__ +
1051 /* Perform bidiagonal SVD, computing left singular vectors */
1052 /* of bidiagonal matrix in U and computing right singular */
1053 /* vectors of bidiagonal matrix in VT */
1054 /* (Workspace: need N+BDSPAC) */
1056 dlaset_("F", m, n, &c_b227, &c_b227, &u[u_offset], ldu);
1057 dbdsdc_("U", "I", n, &s[1], &work[ie], &u[u_offset], ldu, &vt[
1058 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
1061 /* Overwrite U by left singular vectors of A and VT */
1062 /* by right singular vectors of A */
1063 /* (Workspace: need 3*N, prefer 2*N+N*NB) */
1065 i__1 = *lwork - nwork + 1;
1066 dormbr_("Q", "L", "N", m, n, n, &a[a_offset], lda, &work[
1067 itauq], &u[u_offset], ldu, &work[nwork], &i__1, &ierr);
1068 i__1 = *lwork - nwork + 1;
1069 dormbr_("P", "R", "T", n, n, n, &a[a_offset], lda, &work[
1070 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__1, &
1074 /* Perform bidiagonal SVD, computing left singular vectors */
1075 /* of bidiagonal matrix in U and computing right singular */
1076 /* vectors of bidiagonal matrix in VT */
1077 /* (Workspace: need N+BDSPAC) */
1079 dlaset_("F", m, m, &c_b227, &c_b227, &u[u_offset], ldu);
1080 dbdsdc_("U", "I", n, &s[1], &work[ie], &u[u_offset], ldu, &vt[
1081 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
1084 /* Set the right corner of U to identity matrix */
1089 dlaset_("F", &i__1, &i__2, &c_b227, &c_b248, &u[*n + 1 + (
1090 *n + 1) * u_dim1], ldu);
1093 /* Overwrite U by left singular vectors of A and VT */
1094 /* by right singular vectors of A */
1095 /* (Workspace: need N*N+2*N+M, prefer N*N+2*N+M*NB) */
1097 i__1 = *lwork - nwork + 1;
1098 dormbr_("Q", "L", "N", m, m, n, &a[a_offset], lda, &work[
1099 itauq], &u[u_offset], ldu, &work[nwork], &i__1, &ierr);
1100 i__1 = *lwork - nwork + 1;
1101 dormbr_("P", "R", "T", n, n, m, &a[a_offset], lda, &work[
1102 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__1, &
1110 /* A has more columns than rows. If A has sufficiently more */
1111 /* columns than rows, first reduce using the LQ decomposition (if */
1112 /* sufficient workspace available) */
1118 /* Path 1t (N much larger than M, JOBZ='N') */
1119 /* No singular vectors to be computed */
1125 /* (Workspace: need 2*M, prefer M+M*NB) */
1127 i__1 = *lwork - nwork + 1;
1128 dgelqf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
1131 /* Zero out above L */
1135 dlaset_("U", &i__1, &i__2, &c_b227, &c_b227, &a[(a_dim1 << 1)
1142 /* Bidiagonalize L in A */
1143 /* (Workspace: need 4*M, prefer 3*M+2*M*NB) */
1145 i__1 = *lwork - nwork + 1;
1146 dgebrd_(m, m, &a[a_offset], lda, &s[1], &work[ie], &work[
1147 itauq], &work[itaup], &work[nwork], &i__1, &ierr);
1150 /* Perform bidiagonal SVD, computing singular values only */
1151 /* (Workspace: need M+BDSPAC) */
1153 dbdsdc_("U", "N", m, &s[1], &work[ie], dum, &c__1, dum, &c__1,
1154 dum, idum, &work[nwork], &iwork[1], info);
1158 /* Path 2t (N much larger than M, JOBZ='O') */
1159 /* M right singular vectors to be overwritten on A and */
1160 /* M left singular vectors to be computed in U */
1167 if (*lwork >= *m * *n + *m * *m + *m * 3 + bdspac) {
1169 /* WORK(IL) is M by N */
1175 chunk = (*lwork - *m * *m) / *m;
1177 itau = il + ldwrkl * *m;
1181 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1183 i__1 = *lwork - nwork + 1;
1184 dgelqf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
1187 /* Copy L to WORK(IL), zeroing about above it */
1189 dlacpy_("L", m, m, &a[a_offset], lda, &work[il], &ldwrkl);
1192 dlaset_("U", &i__1, &i__2, &c_b227, &c_b227, &work[il +
1195 /* Generate Q in A */
1196 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1198 i__1 = *lwork - nwork + 1;
1199 dorglq_(m, n, m, &a[a_offset], lda, &work[itau], &work[nwork],
1206 /* Bidiagonalize L in WORK(IL) */
1207 /* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) */
1209 i__1 = *lwork - nwork + 1;
1210 dgebrd_(m, m, &work[il], &ldwrkl, &s[1], &work[ie], &work[
1211 itauq], &work[itaup], &work[nwork], &i__1, &ierr);
1213 /* Perform bidiagonal SVD, computing left singular vectors */
1214 /* of bidiagonal matrix in U, and computing right singular */
1215 /* vectors of bidiagonal matrix in WORK(IVT) */
1216 /* (Workspace: need M+M*M+BDSPAC) */
1218 dbdsdc_("U", "I", m, &s[1], &work[ie], &u[u_offset], ldu, &
1219 work[ivt], m, dum, idum, &work[nwork], &iwork[1],
1222 /* Overwrite U by left singular vectors of L and WORK(IVT) */
1223 /* by right singular vectors of L */
1224 /* (Workspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB) */
1226 i__1 = *lwork - nwork + 1;
1227 dormbr_("Q", "L", "N", m, m, m, &work[il], &ldwrkl, &work[
1228 itauq], &u[u_offset], ldu, &work[nwork], &i__1, &ierr);
1229 i__1 = *lwork - nwork + 1;
1230 dormbr_("P", "R", "T", m, m, m, &work[il], &ldwrkl, &work[
1231 itaup], &work[ivt], m, &work[nwork], &i__1, &ierr);
1233 /* Multiply right singular vectors of L in WORK(IVT) by Q */
1234 /* in A, storing result in WORK(IL) and copying to A */
1235 /* (Workspace: need 2*M*M, prefer M*M+M*N) */
1239 for (i__ = 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ +=
1242 i__3 = *n - i__ + 1;
1243 blk = min(i__3,chunk);
1244 dgemm_("N", "N", m, &blk, m, &c_b248, &work[ivt], m, &a[
1245 i__ * a_dim1 + 1], lda, &c_b227, &work[il], &
1247 dlacpy_("F", m, &blk, &work[il], &ldwrkl, &a[i__ * a_dim1
1254 /* Path 3t (N much larger than M, JOBZ='S') */
1255 /* M right singular vectors to be computed in VT and */
1256 /* M left singular vectors to be computed in U */
1260 /* WORK(IL) is M by M */
1263 itau = il + ldwrkl * *m;
1267 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1269 i__2 = *lwork - nwork + 1;
1270 dgelqf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
1273 /* Copy L to WORK(IL), zeroing out above it */
1275 dlacpy_("L", m, m, &a[a_offset], lda, &work[il], &ldwrkl);
1278 dlaset_("U", &i__2, &i__1, &c_b227, &c_b227, &work[il +
1281 /* Generate Q in A */
1282 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1284 i__2 = *lwork - nwork + 1;
1285 dorglq_(m, n, m, &a[a_offset], lda, &work[itau], &work[nwork],
1292 /* Bidiagonalize L in WORK(IU), copying result to U */
1293 /* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) */
1295 i__2 = *lwork - nwork + 1;
1296 dgebrd_(m, m, &work[il], &ldwrkl, &s[1], &work[ie], &work[
1297 itauq], &work[itaup], &work[nwork], &i__2, &ierr);
1299 /* Perform bidiagonal SVD, computing left singular vectors */
1300 /* of bidiagonal matrix in U and computing right singular */
1301 /* vectors of bidiagonal matrix in VT */
1302 /* (Workspace: need M+BDSPAC) */
1304 dbdsdc_("U", "I", m, &s[1], &work[ie], &u[u_offset], ldu, &vt[
1305 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
1308 /* Overwrite U by left singular vectors of L and VT */
1309 /* by right singular vectors of L */
1310 /* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB) */
1312 i__2 = *lwork - nwork + 1;
1313 dormbr_("Q", "L", "N", m, m, m, &work[il], &ldwrkl, &work[
1314 itauq], &u[u_offset], ldu, &work[nwork], &i__2, &ierr);
1315 i__2 = *lwork - nwork + 1;
1316 dormbr_("P", "R", "T", m, m, m, &work[il], &ldwrkl, &work[
1317 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__2, &
1320 /* Multiply right singular vectors of L in WORK(IL) by */
1321 /* Q in A, storing result in VT */
1322 /* (Workspace: need M*M) */
1324 dlacpy_("F", m, m, &vt[vt_offset], ldvt, &work[il], &ldwrkl);
1325 dgemm_("N", "N", m, n, m, &c_b248, &work[il], &ldwrkl, &a[
1326 a_offset], lda, &c_b227, &vt[vt_offset], ldvt);
1330 /* Path 4t (N much larger than M, JOBZ='A') */
1331 /* N right singular vectors to be computed in VT and */
1332 /* M left singular vectors to be computed in U */
1336 /* WORK(IVT) is M by M */
1339 itau = ivt + ldwkvt * *m;
1342 /* Compute A=L*Q, copying result to VT */
1343 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1345 i__2 = *lwork - nwork + 1;
1346 dgelqf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &
1348 dlacpy_("U", m, n, &a[a_offset], lda, &vt[vt_offset], ldvt);
1350 /* Generate Q in VT */
1351 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1353 i__2 = *lwork - nwork + 1;
1354 dorglq_(n, n, m, &vt[vt_offset], ldvt, &work[itau], &work[
1355 nwork], &i__2, &ierr);
1357 /* Produce L in A, zeroing out other entries */
1361 dlaset_("U", &i__2, &i__1, &c_b227, &c_b227, &a[(a_dim1 << 1)
1368 /* Bidiagonalize L in A */
1369 /* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) */
1371 i__2 = *lwork - nwork + 1;
1372 dgebrd_(m, m, &a[a_offset], lda, &s[1], &work[ie], &work[
1373 itauq], &work[itaup], &work[nwork], &i__2, &ierr);
1375 /* Perform bidiagonal SVD, computing left singular vectors */
1376 /* of bidiagonal matrix in U and computing right singular */
1377 /* vectors of bidiagonal matrix in WORK(IVT) */
1378 /* (Workspace: need M+M*M+BDSPAC) */
1380 dbdsdc_("U", "I", m, &s[1], &work[ie], &u[u_offset], ldu, &
1381 work[ivt], &ldwkvt, dum, idum, &work[nwork], &iwork[1]
1384 /* Overwrite U by left singular vectors of L and WORK(IVT) */
1385 /* by right singular vectors of L */
1386 /* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB) */
1388 i__2 = *lwork - nwork + 1;
1389 dormbr_("Q", "L", "N", m, m, m, &a[a_offset], lda, &work[
1390 itauq], &u[u_offset], ldu, &work[nwork], &i__2, &ierr);
1391 i__2 = *lwork - nwork + 1;
1392 dormbr_("P", "R", "T", m, m, m, &a[a_offset], lda, &work[
1393 itaup], &work[ivt], &ldwkvt, &work[nwork], &i__2, &
1396 /* Multiply right singular vectors of L in WORK(IVT) by */
1397 /* Q in VT, storing result in A */
1398 /* (Workspace: need M*M) */
1400 dgemm_("N", "N", m, n, m, &c_b248, &work[ivt], &ldwkvt, &vt[
1401 vt_offset], ldvt, &c_b227, &a[a_offset], lda);
1403 /* Copy right singular vectors of A from A to VT */
1405 dlacpy_("F", m, n, &a[a_offset], lda, &vt[vt_offset], ldvt);
1413 /* Path 5t (N greater than M, but not much larger) */
1414 /* Reduce to bidiagonal form without LQ decomposition */
1421 /* Bidiagonalize A */
1422 /* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) */
1424 i__2 = *lwork - nwork + 1;
1425 dgebrd_(m, n, &a[a_offset], lda, &s[1], &work[ie], &work[itauq], &
1426 work[itaup], &work[nwork], &i__2, &ierr);
1429 /* Perform bidiagonal SVD, only computing singular values */
1430 /* (Workspace: need M+BDSPAC) */
1432 dbdsdc_("L", "N", m, &s[1], &work[ie], dum, &c__1, dum, &c__1,
1433 dum, idum, &work[nwork], &iwork[1], info);
1437 if (*lwork >= *m * *n + *m * 3 + bdspac) {
1439 /* WORK( IVT ) is M by N */
1441 dlaset_("F", m, n, &c_b227, &c_b227, &work[ivt], &ldwkvt);
1442 nwork = ivt + ldwkvt * *n;
1445 /* WORK( IVT ) is M by M */
1447 nwork = ivt + ldwkvt * *m;
1450 /* WORK(IL) is M by CHUNK */
1452 chunk = (*lwork - *m * *m - *m * 3) / *m;
1455 /* Perform bidiagonal SVD, computing left singular vectors */
1456 /* of bidiagonal matrix in U and computing right singular */
1457 /* vectors of bidiagonal matrix in WORK(IVT) */
1458 /* (Workspace: need M*M+BDSPAC) */
1460 dbdsdc_("L", "I", m, &s[1], &work[ie], &u[u_offset], ldu, &
1461 work[ivt], &ldwkvt, dum, idum, &work[nwork], &iwork[1]
1464 /* Overwrite U by left singular vectors of A */
1465 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1467 i__2 = *lwork - nwork + 1;
1468 dormbr_("Q", "L", "N", m, m, n, &a[a_offset], lda, &work[
1469 itauq], &u[u_offset], ldu, &work[nwork], &i__2, &ierr);
1471 if (*lwork >= *m * *n + *m * 3 + bdspac) {
1473 /* Overwrite WORK(IVT) by left singular vectors of A */
1474 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1476 i__2 = *lwork - nwork + 1;
1477 dormbr_("P", "R", "T", m, n, m, &a[a_offset], lda, &work[
1478 itaup], &work[ivt], &ldwkvt, &work[nwork], &i__2,
1481 /* Copy right singular vectors of A from WORK(IVT) to A */
1483 dlacpy_("F", m, n, &work[ivt], &ldwkvt, &a[a_offset], lda);
1486 /* Generate P**T in A */
1487 /* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) */
1489 i__2 = *lwork - nwork + 1;
1490 dorgbr_("P", m, n, m, &a[a_offset], lda, &work[itaup], &
1491 work[nwork], &i__2, &ierr);
1493 /* Multiply Q in A by right singular vectors of */
1494 /* bidiagonal matrix in WORK(IVT), storing result in */
1495 /* WORK(IL) and copying to A */
1496 /* (Workspace: need 2*M*M, prefer M*M+M*N) */
1500 for (i__ = 1; i__1 < 0 ? i__ >= i__2 : i__ <= i__2; i__ +=
1503 i__3 = *n - i__ + 1;
1504 blk = min(i__3,chunk);
1505 dgemm_("N", "N", m, &blk, m, &c_b248, &work[ivt], &
1506 ldwkvt, &a[i__ * a_dim1 + 1], lda, &c_b227, &
1508 dlacpy_("F", m, &blk, &work[il], m, &a[i__ * a_dim1 +
1515 /* Perform bidiagonal SVD, computing left singular vectors */
1516 /* of bidiagonal matrix in U and computing right singular */
1517 /* vectors of bidiagonal matrix in VT */
1518 /* (Workspace: need M+BDSPAC) */
1520 dlaset_("F", m, n, &c_b227, &c_b227, &vt[vt_offset], ldvt);
1521 dbdsdc_("L", "I", m, &s[1], &work[ie], &u[u_offset], ldu, &vt[
1522 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
1525 /* Overwrite U by left singular vectors of A and VT */
1526 /* by right singular vectors of A */
1527 /* (Workspace: need 3*M, prefer 2*M+M*NB) */
1529 i__1 = *lwork - nwork + 1;
1530 dormbr_("Q", "L", "N", m, m, n, &a[a_offset], lda, &work[
1531 itauq], &u[u_offset], ldu, &work[nwork], &i__1, &ierr);
1532 i__1 = *lwork - nwork + 1;
1533 dormbr_("P", "R", "T", m, n, m, &a[a_offset], lda, &work[
1534 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__1, &
1538 /* Perform bidiagonal SVD, computing left singular vectors */
1539 /* of bidiagonal matrix in U and computing right singular */
1540 /* vectors of bidiagonal matrix in VT */
1541 /* (Workspace: need M+BDSPAC) */
1543 dlaset_("F", n, n, &c_b227, &c_b227, &vt[vt_offset], ldvt);
1544 dbdsdc_("L", "I", m, &s[1], &work[ie], &u[u_offset], ldu, &vt[
1545 vt_offset], ldvt, dum, idum, &work[nwork], &iwork[1],
1548 /* Set the right corner of VT to identity matrix */
1553 dlaset_("F", &i__1, &i__2, &c_b227, &c_b248, &vt[*m + 1 +
1554 (*m + 1) * vt_dim1], ldvt);
1557 /* Overwrite U by left singular vectors of A and VT */
1558 /* by right singular vectors of A */
1559 /* (Workspace: need 2*M+N, prefer 2*M+N*NB) */
1561 i__1 = *lwork - nwork + 1;
1562 dormbr_("Q", "L", "N", m, m, n, &a[a_offset], lda, &work[
1563 itauq], &u[u_offset], ldu, &work[nwork], &i__1, &ierr);
1564 i__1 = *lwork - nwork + 1;
1565 dormbr_("P", "R", "T", n, n, m, &a[a_offset], lda, &work[
1566 itaup], &vt[vt_offset], ldvt, &work[nwork], &i__1, &
1574 /* Undo scaling if necessary */
1577 if (anrm > bignum) {
1578 dlascl_("G", &c__0, &c__0, &bignum, &anrm, &minmn, &c__1, &s[1], &
1581 if (anrm < smlnum) {
1582 dlascl_("G", &c__0, &c__0, &smlnum, &anrm, &minmn, &c__1, &s[1], &
1587 /* Return optimal workspace in WORK(1) */
1589 work[1] = (doublereal) maxwrk;