LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
sgejsv.f
Go to the documentation of this file.
1 *> \brief \b SGEJSV
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgejsv.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgejsv.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgejsv.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
22 * M, N, A, LDA, SVA, U, LDU, V, LDV,
23 * WORK, LWORK, IWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * IMPLICIT NONE
27 * INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
28 * ..
29 * .. Array Arguments ..
30 * REAL A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
31 * \$ WORK( LWORK )
32 * INTEGER IWORK( * )
33 * CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> SGEJSV computes the singular value decomposition (SVD) of a real M-by-N
43 *> matrix [A], where M >= N. The SVD of [A] is written as
44 *>
45 *> [A] = [U] * [SIGMA] * [V]^t,
46 *>
47 *> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
48 *> diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
49 *> [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
50 *> the singular values of [A]. The columns of [U] and [V] are the left and
51 *> the right singular vectors of [A], respectively. The matrices [U] and [V]
52 *> are computed and stored in the arrays U and V, respectively. The diagonal
53 *> of [SIGMA] is computed and stored in the array SVA.
54 *> SGEJSV can sometimes compute tiny singular values and their singular vectors much
55 *> more accurately than other SVD routines, see below under Further Details.
56 *> \endverbatim
57 *
58 * Arguments:
59 * ==========
60 *
61 *> \param[in] JOBA
62 *> \verbatim
63 *> JOBA is CHARACTER*1
64 *> Specifies the level of accuracy:
65 *> = 'C': This option works well (high relative accuracy) if A = B * D,
66 *> with well-conditioned B and arbitrary diagonal matrix D.
67 *> The accuracy cannot be spoiled by COLUMN scaling. The
68 *> accuracy of the computed output depends on the condition of
69 *> B, and the procedure aims at the best theoretical accuracy.
70 *> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
71 *> bounded by f(M,N)*epsilon* cond(B), independent of D.
72 *> The input matrix is preprocessed with the QRF with column
73 *> pivoting. This initial preprocessing and preconditioning by
74 *> a rank revealing QR factorization is common for all values of
75 *> JOBA. Additional actions are specified as follows:
76 *> = 'E': Computation as with 'C' with an additional estimate of the
77 *> condition number of B. It provides a realistic error bound.
78 *> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
79 *> D1, D2, and well-conditioned matrix C, this option gives
80 *> higher accuracy than the 'C' option. If the structure of the
81 *> input matrix is not known, and relative accuracy is
82 *> desirable, then this option is advisable. The input matrix A
83 *> is preprocessed with QR factorization with FULL (row and
84 *> column) pivoting.
85 *> = 'G' Computation as with 'F' with an additional estimate of the
86 *> condition number of B, where A=D*B. If A has heavily weighted
87 *> rows, then using this condition number gives too pessimistic
88 *> error bound.
89 *> = 'A': Small singular values are the noise and the matrix is treated
90 *> as numerically rank defficient. The error in the computed
91 *> singular values is bounded by f(m,n)*epsilon*||A||.
92 *> The computed SVD A = U * S * V^t restores A up to
93 *> f(m,n)*epsilon*||A||.
94 *> This gives the procedure the licence to discard (set to zero)
95 *> all singular values below N*epsilon*||A||.
96 *> = 'R': Similar as in 'A'. Rank revealing property of the initial
97 *> QR factorization is used do reveal (using triangular factor)
98 *> a gap sigma_{r+1} < epsilon * sigma_r in which case the
99 *> numerical RANK is declared to be r. The SVD is computed with
100 *> absolute error bounds, but more accurately than with 'A'.
101 *> \endverbatim
102 *>
103 *> \param[in] JOBU
104 *> \verbatim
105 *> JOBU is CHARACTER*1
106 *> Specifies whether to compute the columns of U:
107 *> = 'U': N columns of U are returned in the array U.
108 *> = 'F': full set of M left sing. vectors is returned in the array U.
109 *> = 'W': U may be used as workspace of length M*N. See the description
110 *> of U.
111 *> = 'N': U is not computed.
112 *> \endverbatim
113 *>
114 *> \param[in] JOBV
115 *> \verbatim
116 *> JOBV is CHARACTER*1
117 *> Specifies whether to compute the matrix V:
118 *> = 'V': N columns of V are returned in the array V; Jacobi rotations
119 *> are not explicitly accumulated.
120 *> = 'J': N columns of V are returned in the array V, but they are
121 *> computed as the product of Jacobi rotations. This option is
122 *> allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
123 *> = 'W': V may be used as workspace of length N*N. See the description
124 *> of V.
125 *> = 'N': V is not computed.
126 *> \endverbatim
127 *>
128 *> \param[in] JOBR
129 *> \verbatim
130 *> JOBR is CHARACTER*1
131 *> Specifies the RANGE for the singular values. Issues the licence to
132 *> set to zero small positive singular values if they are outside
133 *> specified range. If A .NE. 0 is scaled so that the largest singular
134 *> value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
135 *> the licence to kill columns of A whose norm in c*A is less than
136 *> SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
137 *> where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
138 *> = 'N': Do not kill small columns of c*A. This option assumes that
139 *> BLAS and QR factorizations and triangular solvers are
140 *> implemented to work in that range. If the condition of A
141 *> is greater than BIG, use SGESVJ.
142 *> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
143 *> (roughly, as described above). This option is recommended.
144 *> ===========================
145 *> For computing the singular values in the FULL range [SFMIN,BIG]
146 *> use SGESVJ.
147 *> \endverbatim
148 *>
149 *> \param[in] JOBT
150 *> \verbatim
151 *> JOBT is CHARACTER*1
152 *> If the matrix is square then the procedure may determine to use
153 *> transposed A if A^t seems to be better with respect to convergence.
154 *> If the matrix is not square, JOBT is ignored. This is subject to
155 *> changes in the future.
156 *> The decision is based on two values of entropy over the adjoint
157 *> orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
158 *> = 'T': transpose if entropy test indicates possibly faster
159 *> convergence of Jacobi process if A^t is taken as input. If A is
160 *> replaced with A^t, then the row pivoting is included automatically.
161 *> = 'N': do not speculate.
162 *> This option can be used to compute only the singular values, or the
163 *> full SVD (U, SIGMA and V). For only one set of singular vectors
164 *> (U or V), the caller should provide both U and V, as one of the
165 *> matrices is used as workspace if the matrix A is transposed.
166 *> The implementer can easily remove this constraint and make the
167 *> code more complicated. See the descriptions of U and V.
168 *> \endverbatim
169 *>
170 *> \param[in] JOBP
171 *> \verbatim
172 *> JOBP is CHARACTER*1
173 *> Issues the licence to introduce structured perturbations to drown
174 *> denormalized numbers. This licence should be active if the
175 *> denormals are poorly implemented, causing slow computation,
176 *> especially in cases of fast convergence (!). For details see [1,2].
177 *> For the sake of simplicity, this perturbations are included only
178 *> when the full SVD or only the singular values are requested. The
179 *> implementer/user can easily add the perturbation for the cases of
180 *> computing one set of singular vectors.
181 *> = 'P': introduce perturbation
182 *> = 'N': do not perturb
183 *> \endverbatim
184 *>
185 *> \param[in] M
186 *> \verbatim
187 *> M is INTEGER
188 *> The number of rows of the input matrix A. M >= 0.
189 *> \endverbatim
190 *>
191 *> \param[in] N
192 *> \verbatim
193 *> N is INTEGER
194 *> The number of columns of the input matrix A. M >= N >= 0.
195 *> \endverbatim
196 *>
197 *> \param[in,out] A
198 *> \verbatim
199 *> A is REAL array, dimension (LDA,N)
200 *> On entry, the M-by-N matrix A.
201 *> \endverbatim
202 *>
203 *> \param[in] LDA
204 *> \verbatim
205 *> LDA is INTEGER
206 *> The leading dimension of the array A. LDA >= max(1,M).
207 *> \endverbatim
208 *>
209 *> \param[out] SVA
210 *> \verbatim
211 *> SVA is REAL array, dimension (N)
212 *> On exit,
213 *> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
214 *> computation SVA contains Euclidean column norms of the
215 *> iterated matrices in the array A.
216 *> - For WORK(1) .NE. WORK(2): The singular values of A are
217 *> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
218 *> sigma_max(A) overflows or if small singular values have been
219 *> saved from underflow by scaling the input matrix A.
220 *> - If JOBR='R' then some of the singular values may be returned
221 *> as exact zeros obtained by "set to zero" because they are
222 *> below the numerical rank threshold or are denormalized numbers.
223 *> \endverbatim
224 *>
225 *> \param[out] U
226 *> \verbatim
227 *> U is REAL array, dimension ( LDU, N )
228 *> If JOBU = 'U', then U contains on exit the M-by-N matrix of
229 *> the left singular vectors.
230 *> If JOBU = 'F', then U contains on exit the M-by-M matrix of
231 *> the left singular vectors, including an ONB
232 *> of the orthogonal complement of the Range(A).
233 *> If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
234 *> then U is used as workspace if the procedure
235 *> replaces A with A^t. In that case, [V] is computed
236 *> in U as left singular vectors of A^t and then
237 *> copied back to the V array. This 'W' option is just
238 *> a reminder to the caller that in this case U is
239 *> reserved as workspace of length N*N.
240 *> If JOBU = 'N' U is not referenced, unless JOBT='T'.
241 *> \endverbatim
242 *>
243 *> \param[in] LDU
244 *> \verbatim
245 *> LDU is INTEGER
246 *> The leading dimension of the array U, LDU >= 1.
247 *> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
248 *> \endverbatim
249 *>
250 *> \param[out] V
251 *> \verbatim
252 *> V is REAL array, dimension ( LDV, N )
253 *> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
254 *> the right singular vectors;
255 *> If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
256 *> then V is used as workspace if the pprocedure
257 *> replaces A with A^t. In that case, [U] is computed
258 *> in V as right singular vectors of A^t and then
259 *> copied back to the U array. This 'W' option is just
260 *> a reminder to the caller that in this case V is
261 *> reserved as workspace of length N*N.
262 *> If JOBV = 'N' V is not referenced, unless JOBT='T'.
263 *> \endverbatim
264 *>
265 *> \param[in] LDV
266 *> \verbatim
267 *> LDV is INTEGER
268 *> The leading dimension of the array V, LDV >= 1.
269 *> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
270 *> \endverbatim
271 *>
272 *> \param[out] WORK
273 *> \verbatim
274 *> WORK is REAL array, dimension at least LWORK.
275 *> On exit,
276 *> WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
277 *> that SCALE*SVA(1:N) are the computed singular values
278 *> of A. (See the description of SVA().)
279 *> WORK(2) = See the description of WORK(1).
280 *> WORK(3) = SCONDA is an estimate for the condition number of
281 *> column equilibrated A. (If JOBA .EQ. 'E' or 'G')
282 *> SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
283 *> It is computed using SPOCON. It holds
284 *> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
285 *> where R is the triangular factor from the QRF of A.
286 *> However, if R is truncated and the numerical rank is
287 *> determined to be strictly smaller than N, SCONDA is
288 *> returned as -1, thus indicating that the smallest
289 *> singular values might be lost.
290 *>
291 *> If full SVD is needed, the following two condition numbers are
292 *> useful for the analysis of the algorithm. They are provied for
293 *> a developer/implementer who is familiar with the details of
294 *> the method.
295 *>
296 *> WORK(4) = an estimate of the scaled condition number of the
297 *> triangular factor in the first QR factorization.
298 *> WORK(5) = an estimate of the scaled condition number of the
299 *> triangular factor in the second QR factorization.
300 *> The following two parameters are computed if JOBT .EQ. 'T'.
301 *> They are provided for a developer/implementer who is familiar
302 *> with the details of the method.
303 *>
304 *> WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
305 *> of diag(A^t*A) / Trace(A^t*A) taken as point in the
306 *> probability simplex.
307 *> WORK(7) = the entropy of A*A^t.
308 *> \endverbatim
309 *>
310 *> \param[in] LWORK
311 *> \verbatim
312 *> LWORK is INTEGER
313 *> Length of WORK to confirm proper allocation of work space.
314 *> LWORK depends on the job:
315 *>
316 *> If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
317 *> -> .. no scaled condition estimate required (JOBE.EQ.'N'):
318 *> LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
319 *> ->> For optimal performance (blocked code) the optimal value
320 *> is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
321 *> block size for DGEQP3 and DGEQRF.
322 *> In general, optimal LWORK is computed as
323 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
324 *> -> .. an estimate of the scaled condition number of A is
325 *> required (JOBA='E', 'G'). In this case, LWORK is the maximum
326 *> of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
327 *> ->> For optimal performance (blocked code) the optimal value
328 *> is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
329 *> In general, the optimal length LWORK is computed as
330 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
331 *> N+N*N+LWORK(DPOCON),7).
332 *>
333 *> If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
334 *> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
335 *> -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
336 *> where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
337 *> DORMLQ. In general, the optimal length LWORK is computed as
338 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
339 *> N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
340 *>
341 *> If SIGMA and the left singular vectors are needed
342 *> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
343 *> -> For optimal performance:
344 *> if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
345 *> if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
346 *> where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
347 *> In general, the optimal length LWORK is computed as
348 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
349 *> 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
350 *> Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or
351 *> M*NB (for JOBU.EQ.'F').
352 *>
353 *> If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
354 *> -> if JOBV.EQ.'V'
355 *> the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
356 *> -> if JOBV.EQ.'J' the minimal requirement is
357 *> LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
358 *> -> For optimal performance, LWORK should be additionally
359 *> larger than N+M*NB, where NB is the optimal block size
360 *> for DORMQR.
361 *> \endverbatim
362 *>
363 *> \param[out] IWORK
364 *> \verbatim
365 *> IWORK is INTEGER array, dimension M+3*N.
366 *> On exit,
367 *> IWORK(1) = the numerical rank determined after the initial
368 *> QR factorization with pivoting. See the descriptions
369 *> of JOBA and JOBR.
370 *> IWORK(2) = the number of the computed nonzero singular values
371 *> IWORK(3) = if nonzero, a warning message:
372 *> If IWORK(3).EQ.1 then some of the column norms of A
373 *> were denormalized floats. The requested high accuracy
374 *> is not warranted by the data.
375 *> \endverbatim
376 *>
377 *> \param[out] INFO
378 *> \verbatim
379 *> INFO is INTEGER
380 *> < 0 : if INFO = -i, then the i-th argument had an illegal value.
381 *> = 0 : successfull exit;
382 *> > 0 : SGEJSV did not converge in the maximal allowed number
383 *> of sweeps. The computed values may be inaccurate.
384 *> \endverbatim
385 *
386 * Authors:
387 * ========
388 *
389 *> \author Univ. of Tennessee
390 *> \author Univ. of California Berkeley
391 *> \author Univ. of Colorado Denver
392 *> \author NAG Ltd.
393 *
394 *> \date June 2016
395 *
396 *> \ingroup realGEsing
397 *
398 *> \par Further Details:
399 * =====================
400 *>
401 *> \verbatim
402 *>
403 *> SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,
404 *> SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an
405 *> additional row pivoting can be used as a preprocessor, which in some
406 *> cases results in much higher accuracy. An example is matrix A with the
407 *> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
408 *> diagonal matrices and C is well-conditioned matrix. In that case, complete
409 *> pivoting in the first QR factorizations provides accuracy dependent on the
410 *> condition number of C, and independent of D1, D2. Such higher accuracy is
411 *> not completely understood theoretically, but it works well in practice.
412 *> Further, if A can be written as A = B*D, with well-conditioned B and some
413 *> diagonal D, then the high accuracy is guaranteed, both theoretically and
414 *> in software, independent of D. For more details see [1], [2].
415 *> The computational range for the singular values can be the full range
416 *> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
417 *> & LAPACK routines called by SGEJSV are implemented to work in that range.
418 *> If that is not the case, then the restriction for safe computation with
419 *> the singular values in the range of normalized IEEE numbers is that the
420 *> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
421 *> overflow. This code (SGEJSV) is best used in this restricted range,
422 *> meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
423 *> returned as zeros. See JOBR for details on this.
424 *> Further, this implementation is somewhat slower than the one described
425 *> in [1,2] due to replacement of some non-LAPACK components, and because
426 *> the choice of some tuning parameters in the iterative part (SGESVJ) is
427 *> left to the implementer on a particular machine.
428 *> The rank revealing QR factorization (in this code: SGEQP3) should be
429 *> implemented as in [3]. We have a new version of SGEQP3 under development
430 *> that is more robust than the current one in LAPACK, with a cleaner cut in
431 *> rank defficient cases. It will be available in the SIGMA library [4].
432 *> If M is much larger than N, it is obvious that the inital QRF with
433 *> column pivoting can be preprocessed by the QRF without pivoting. That
434 *> well known trick is not used in SGEJSV because in some cases heavy row
435 *> weighting can be treated with complete pivoting. The overhead in cases
436 *> M much larger than N is then only due to pivoting, but the benefits in
437 *> terms of accuracy have prevailed. The implementer/user can incorporate
438 *> this extra QRF step easily. The implementer can also improve data movement
439 *> (matrix transpose, matrix copy, matrix transposed copy) - this
440 *> implementation of SGEJSV uses only the simplest, naive data movement.
441 *> \endverbatim
442 *
443 *> \par Contributors:
444 * ==================
445 *>
446 *> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
447 *
448 *> \par References:
449 * ================
450 *>
451 *> \verbatim
452 *>
453 *> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
454 *> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
455 *> LAPACK Working note 169.
456 *> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
457 *> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
458 *> LAPACK Working note 170.
459 *> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
460 *> factorization software - a case study.
461 *> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
462 *> LAPACK Working note 176.
463 *> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
464 *> QSVD, (H,K)-SVD computations.
465 *> Department of Mathematics, University of Zagreb, 2008.
466 *> \endverbatim
467 *
468 *> \par Bugs, examples and comments:
469 * =================================
470 *>
471 *> Please report all bugs and send interesting examples and/or comments to
472 *> drmac@math.hr. Thank you.
473 *>
474 * =====================================================================
475  SUBROUTINE sgejsv( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
476  \$ m, n, a, lda, sva, u, ldu, v, ldv,
477  \$ work, lwork, iwork, info )
478 *
479 * -- LAPACK computational routine (version 3.6.1) --
480 * -- LAPACK is a software package provided by Univ. of Tennessee, --
481 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
482 * June 2016
483 *
484 * .. Scalar Arguments ..
485  IMPLICIT NONE
486  INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
487 * ..
488 * .. Array Arguments ..
489  REAL A( lda, * ), SVA( n ), U( ldu, * ), V( ldv, * ),
490  \$ work( lwork )
491  INTEGER IWORK( * )
492  CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
493 * ..
494 *
495 * ===========================================================================
496 *
497 * .. Local Parameters ..
498  REAL ZERO, ONE
499  parameter( zero = 0.0e0, one = 1.0e0 )
500 * ..
501 * .. Local Scalars ..
502  REAL AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
503  \$ condr1, condr2, entra, entrat, epsln, maxprj, scalem,
504  \$ sconda, sfmin, small, temp1, uscal1, uscal2, xsc
505  INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
506  LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LSVEC,
507  \$ l2aber, l2kill, l2pert, l2rank, l2tran,
508  \$ noscal, rowpiv, rsvec, transp
509 * ..
510 * .. Intrinsic Functions ..
511  INTRINSIC abs, alog, max, min, float, nint, sign, sqrt
512 * ..
513 * .. External Functions ..
514  REAL SLAMCH, SNRM2
515  INTEGER ISAMAX
516  LOGICAL LSAME
517  EXTERNAL isamax, lsame, slamch, snrm2
518 * ..
519 * .. External Subroutines ..
520  EXTERNAL scopy, sgelqf, sgeqp3, sgeqrf, slacpy, slascl,
523 *
524  EXTERNAL sgesvj
525 * ..
526 *
527 * Test the input arguments
528 *
529  lsvec = lsame( jobu, 'U' ) .OR. lsame( jobu, 'F' )
530  jracc = lsame( jobv, 'J' )
531  rsvec = lsame( jobv, 'V' ) .OR. jracc
532  rowpiv = lsame( joba, 'F' ) .OR. lsame( joba, 'G' )
533  l2rank = lsame( joba, 'R' )
534  l2aber = lsame( joba, 'A' )
535  errest = lsame( joba, 'E' ) .OR. lsame( joba, 'G' )
536  l2tran = lsame( jobt, 'T' )
537  l2kill = lsame( jobr, 'R' )
538  defr = lsame( jobr, 'N' )
539  l2pert = lsame( jobp, 'P' )
540 *
541  IF ( .NOT.(rowpiv .OR. l2rank .OR. l2aber .OR.
542  \$ errest .OR. lsame( joba, 'C' ) )) THEN
543  info = - 1
544  ELSE IF ( .NOT.( lsvec .OR. lsame( jobu, 'N' ) .OR.
545  \$ lsame( jobu, 'W' )) ) THEN
546  info = - 2
547  ELSE IF ( .NOT.( rsvec .OR. lsame( jobv, 'N' ) .OR.
548  \$ lsame( jobv, 'W' )) .OR. ( jracc .AND. (.NOT.lsvec) ) ) THEN
549  info = - 3
550  ELSE IF ( .NOT. ( l2kill .OR. defr ) ) THEN
551  info = - 4
552  ELSE IF ( .NOT. ( l2tran .OR. lsame( jobt, 'N' ) ) ) THEN
553  info = - 5
554  ELSE IF ( .NOT. ( l2pert .OR. lsame( jobp, 'N' ) ) ) THEN
555  info = - 6
556  ELSE IF ( m .LT. 0 ) THEN
557  info = - 7
558  ELSE IF ( ( n .LT. 0 ) .OR. ( n .GT. m ) ) THEN
559  info = - 8
560  ELSE IF ( lda .LT. m ) THEN
561  info = - 10
562  ELSE IF ( lsvec .AND. ( ldu .LT. m ) ) THEN
563  info = - 13
564  ELSE IF ( rsvec .AND. ( ldv .LT. n ) ) THEN
565  info = - 14
566  ELSE IF ( (.NOT.(lsvec .OR. rsvec .OR. errest).AND.
567  \$ (lwork .LT. max(7,4*n+1,2*m+n))) .OR.
568  \$ (.NOT.(lsvec .OR. rsvec) .AND. errest .AND.
569  \$ (lwork .LT. max(7,4*n+n*n,2*m+n))) .OR.
570  \$ (lsvec .AND. (.NOT.rsvec) .AND. (lwork .LT. max(7,2*m+n,4*n+1)))
571  \$ .OR.
572  \$ (rsvec .AND. (.NOT.lsvec) .AND. (lwork .LT. max(7,2*m+n,4*n+1)))
573  \$ .OR.
574  \$ (lsvec .AND. rsvec .AND. (.NOT.jracc) .AND.
575  \$ (lwork.LT.max(2*m+n,6*n+2*n*n)))
576  \$ .OR. (lsvec .AND. rsvec .AND. jracc .AND.
577  \$ lwork.LT.max(2*m+n,4*n+n*n,2*n+n*n+6)))
578  \$ THEN
579  info = - 17
580  ELSE
581 * #:)
582  info = 0
583  END IF
584 *
585  IF ( info .NE. 0 ) THEN
586 * #:(
587  CALL xerbla( 'SGEJSV', - info )
588  RETURN
589  END IF
590 *
591 * Quick return for void matrix (Y3K safe)
592 * #:)
593  IF ( ( m .EQ. 0 ) .OR. ( n .EQ. 0 ) ) THEN
594  iwork(1:3) = 0
595  work(1:7) = 0
596  RETURN
597  ENDIF
598 *
599 * Determine whether the matrix U should be M x N or M x M
600 *
601  IF ( lsvec ) THEN
602  n1 = n
603  IF ( lsame( jobu, 'F' ) ) n1 = m
604  END IF
605 *
606 * Set numerical parameters
607 *
608 *! NOTE: Make sure SLAMCH() does not fail on the target architecture.
609 *
610  epsln = slamch('Epsilon')
611  sfmin = slamch('SafeMinimum')
612  small = sfmin / epsln
613  big = slamch('O')
614 * BIG = ONE / SFMIN
615 *
616 * Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
617 *
618 *(!) If necessary, scale SVA() to protect the largest norm from
619 * overflow. It is possible that this scaling pushes the smallest
620 * column norm left from the underflow threshold (extreme case).
621 *
622  scalem = one / sqrt(float(m)*float(n))
623  noscal = .true.
624  goscal = .true.
625  DO 1874 p = 1, n
626  aapp = zero
627  aaqq = one
628  CALL slassq( m, a(1,p), 1, aapp, aaqq )
629  IF ( aapp .GT. big ) THEN
630  info = - 9
631  CALL xerbla( 'SGEJSV', -info )
632  RETURN
633  END IF
634  aaqq = sqrt(aaqq)
635  IF ( ( aapp .LT. (big / aaqq) ) .AND. noscal ) THEN
636  sva(p) = aapp * aaqq
637  ELSE
638  noscal = .false.
639  sva(p) = aapp * ( aaqq * scalem )
640  IF ( goscal ) THEN
641  goscal = .false.
642  CALL sscal( p-1, scalem, sva, 1 )
643  END IF
644  END IF
645  1874 CONTINUE
646 *
647  IF ( noscal ) scalem = one
648 *
649  aapp = zero
650  aaqq = big
651  DO 4781 p = 1, n
652  aapp = max( aapp, sva(p) )
653  IF ( sva(p) .NE. zero ) aaqq = min( aaqq, sva(p) )
654  4781 CONTINUE
655 *
656 * Quick return for zero M x N matrix
657 * #:)
658  IF ( aapp .EQ. zero ) THEN
659  IF ( lsvec ) CALL slaset( 'G', m, n1, zero, one, u, ldu )
660  IF ( rsvec ) CALL slaset( 'G', n, n, zero, one, v, ldv )
661  work(1) = one
662  work(2) = one
663  IF ( errest ) work(3) = one
664  IF ( lsvec .AND. rsvec ) THEN
665  work(4) = one
666  work(5) = one
667  END IF
668  IF ( l2tran ) THEN
669  work(6) = zero
670  work(7) = zero
671  END IF
672  iwork(1) = 0
673  iwork(2) = 0
674  iwork(3) = 0
675  RETURN
676  END IF
677 *
678 * Issue warning if denormalized column norms detected. Override the
679 * high relative accuracy request. Issue licence to kill columns
680 * (set them to zero) whose norm is less than sigma_max / BIG (roughly).
681 * #:(
682  warning = 0
683  IF ( aaqq .LE. sfmin ) THEN
684  l2rank = .true.
685  l2kill = .true.
686  warning = 1
687  END IF
688 *
689 * Quick return for one-column matrix
690 * #:)
691  IF ( n .EQ. 1 ) THEN
692 *
693  IF ( lsvec ) THEN
694  CALL slascl( 'G',0,0,sva(1),scalem, m,1,a(1,1),lda,ierr )
695  CALL slacpy( 'A', m, 1, a, lda, u, ldu )
696 * computing all M left singular vectors of the M x 1 matrix
697  IF ( n1 .NE. n ) THEN
698  CALL sgeqrf( m, n, u,ldu, work, work(n+1),lwork-n,ierr )
699  CALL sorgqr( m,n1,1, u,ldu,work,work(n+1),lwork-n,ierr )
700  CALL scopy( m, a(1,1), 1, u(1,1), 1 )
701  END IF
702  END IF
703  IF ( rsvec ) THEN
704  v(1,1) = one
705  END IF
706  IF ( sva(1) .LT. (big*scalem) ) THEN
707  sva(1) = sva(1) / scalem
708  scalem = one
709  END IF
710  work(1) = one / scalem
711  work(2) = one
712  IF ( sva(1) .NE. zero ) THEN
713  iwork(1) = 1
714  IF ( ( sva(1) / scalem) .GE. sfmin ) THEN
715  iwork(2) = 1
716  ELSE
717  iwork(2) = 0
718  END IF
719  ELSE
720  iwork(1) = 0
721  iwork(2) = 0
722  END IF
723  iwork(3) = 0
724  IF ( errest ) work(3) = one
725  IF ( lsvec .AND. rsvec ) THEN
726  work(4) = one
727  work(5) = one
728  END IF
729  IF ( l2tran ) THEN
730  work(6) = zero
731  work(7) = zero
732  END IF
733  RETURN
734 *
735  END IF
736 *
737  transp = .false.
738  l2tran = l2tran .AND. ( m .EQ. n )
739 *
740  aatmax = -one
741  aatmin = big
742  IF ( rowpiv .OR. l2tran ) THEN
743 *
744 * Compute the row norms, needed to determine row pivoting sequence
745 * (in the case of heavily row weighted A, row pivoting is strongly
746 * advised) and to collect information needed to compare the
747 * structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).
748 *
749  IF ( l2tran ) THEN
750  DO 1950 p = 1, m
751  xsc = zero
752  temp1 = one
753  CALL slassq( n, a(p,1), lda, xsc, temp1 )
754 * SLASSQ gets both the ell_2 and the ell_infinity norm
755 * in one pass through the vector
756  work(m+n+p) = xsc * scalem
757  work(n+p) = xsc * (scalem*sqrt(temp1))
758  aatmax = max( aatmax, work(n+p) )
759  IF (work(n+p) .NE. zero) aatmin = min(aatmin,work(n+p))
760  1950 CONTINUE
761  ELSE
762  DO 1904 p = 1, m
763  work(m+n+p) = scalem*abs( a(p,isamax(n,a(p,1),lda)) )
764  aatmax = max( aatmax, work(m+n+p) )
765  aatmin = min( aatmin, work(m+n+p) )
766  1904 CONTINUE
767  END IF
768 *
769  END IF
770 *
771 * For square matrix A try to determine whether A^t would be better
772 * input for the preconditioned Jacobi SVD, with faster convergence.
773 * The decision is based on an O(N) function of the vector of column
774 * and row norms of A, based on the Shannon entropy. This should give
775 * the right choice in most cases when the difference actually matters.
776 * It may fail and pick the slower converging side.
777 *
778  entra = zero
779  entrat = zero
780  IF ( l2tran ) THEN
781 *
782  xsc = zero
783  temp1 = one
784  CALL slassq( n, sva, 1, xsc, temp1 )
785  temp1 = one / temp1
786 *
787  entra = zero
788  DO 1113 p = 1, n
789  big1 = ( ( sva(p) / xsc )**2 ) * temp1
790  IF ( big1 .NE. zero ) entra = entra + big1 * alog(big1)
791  1113 CONTINUE
792  entra = - entra / alog(float(n))
793 *
794 * Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.
795 * It is derived from the diagonal of A^t * A. Do the same with the
796 * diagonal of A * A^t, compute the entropy of the corresponding
797 * probability distribution. Note that A * A^t and A^t * A have the
798 * same trace.
799 *
800  entrat = zero
801  DO 1114 p = n+1, n+m
802  big1 = ( ( work(p) / xsc )**2 ) * temp1
803  IF ( big1 .NE. zero ) entrat = entrat + big1 * alog(big1)
804  1114 CONTINUE
805  entrat = - entrat / alog(float(m))
806 *
807 * Analyze the entropies and decide A or A^t. Smaller entropy
808 * usually means better input for the algorithm.
809 *
810  transp = ( entrat .LT. entra )
811 *
812 * If A^t is better than A, transpose A.
813 *
814  IF ( transp ) THEN
815 * In an optimal implementation, this trivial transpose
816 * should be replaced with faster transpose.
817  DO 1115 p = 1, n - 1
818  DO 1116 q = p + 1, n
819  temp1 = a(q,p)
820  a(q,p) = a(p,q)
821  a(p,q) = temp1
822  1116 CONTINUE
823  1115 CONTINUE
824  DO 1117 p = 1, n
825  work(m+n+p) = sva(p)
826  sva(p) = work(n+p)
827  1117 CONTINUE
828  temp1 = aapp
829  aapp = aatmax
830  aatmax = temp1
831  temp1 = aaqq
832  aaqq = aatmin
833  aatmin = temp1
834  kill = lsvec
835  lsvec = rsvec
836  rsvec = kill
837  IF ( lsvec ) n1 = n
838 *
839  rowpiv = .true.
840  END IF
841 *
842  END IF
843 * END IF L2TRAN
844 *
845 * Scale the matrix so that its maximal singular value remains less
846 * than SQRT(BIG) -- the matrix is scaled so that its maximal column
847 * has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
848 * SQRT(BIG) instead of BIG is the fact that SGEJSV uses LAPACK and
849 * BLAS routines that, in some implementations, are not capable of
850 * working in the full interval [SFMIN,BIG] and that they may provoke
851 * overflows in the intermediate results. If the singular values spread
852 * from SFMIN to BIG, then SGESVJ will compute them. So, in that case,
853 * one should use SGESVJ instead of SGEJSV.
854 *
855  big1 = sqrt( big )
856  temp1 = sqrt( big / float(n) )
857 *
858  CALL slascl( 'G', 0, 0, aapp, temp1, n, 1, sva, n, ierr )
859  IF ( aaqq .GT. (aapp * sfmin) ) THEN
860  aaqq = ( aaqq / aapp ) * temp1
861  ELSE
862  aaqq = ( aaqq * temp1 ) / aapp
863  END IF
864  temp1 = temp1 * scalem
865  CALL slascl( 'G', 0, 0, aapp, temp1, m, n, a, lda, ierr )
866 *
867 * To undo scaling at the end of this procedure, multiply the
868 * computed singular values with USCAL2 / USCAL1.
869 *
870  uscal1 = temp1
871  uscal2 = aapp
872 *
873  IF ( l2kill ) THEN
874 * L2KILL enforces computation of nonzero singular values in
875 * the restricted range of condition number of the initial A,
876 * sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
877  xsc = sqrt( sfmin )
878  ELSE
879  xsc = small
880 *
881 * Now, if the condition number of A is too big,
882 * sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
883 * as a precaution measure, the full SVD is computed using SGESVJ
884 * with accumulated Jacobi rotations. This provides numerically
885 * more robust computation, at the cost of slightly increased run
886 * time. Depending on the concrete implementation of BLAS and LAPACK
887 * (i.e. how they behave in presence of extreme ill-conditioning) the
888 * implementor may decide to remove this switch.
889  IF ( ( aaqq.LT.sqrt(sfmin) ) .AND. lsvec .AND. rsvec ) THEN
890  jracc = .true.
891  END IF
892 *
893  END IF
894  IF ( aaqq .LT. xsc ) THEN
895  DO 700 p = 1, n
896  IF ( sva(p) .LT. xsc ) THEN
897  CALL slaset( 'A', m, 1, zero, zero, a(1,p), lda )
898  sva(p) = zero
899  END IF
900  700 CONTINUE
901  END IF
902 *
903 * Preconditioning using QR factorization with pivoting
904 *
905  IF ( rowpiv ) THEN
906 * Optional row permutation (Bjoerck row pivoting):
907 * A result by Cox and Higham shows that the Bjoerck's
908 * row pivoting combined with standard column pivoting
909 * has similar effect as Powell-Reid complete pivoting.
910 * The ell-infinity norms of A are made nonincreasing.
911  DO 1952 p = 1, m - 1
912  q = isamax( m-p+1, work(m+n+p), 1 ) + p - 1
913  iwork(2*n+p) = q
914  IF ( p .NE. q ) THEN
915  temp1 = work(m+n+p)
916  work(m+n+p) = work(m+n+q)
917  work(m+n+q) = temp1
918  END IF
919  1952 CONTINUE
920  CALL slaswp( n, a, lda, 1, m-1, iwork(2*n+1), 1 )
921  END IF
922 *
923 * End of the preparation phase (scaling, optional sorting and
924 * transposing, optional flushing of small columns).
925 *
926 * Preconditioning
927 *
928 * If the full SVD is needed, the right singular vectors are computed
929 * from a matrix equation, and for that we need theoretical analysis
930 * of the Businger-Golub pivoting. So we use SGEQP3 as the first RR QRF.
931 * In all other cases the first RR QRF can be chosen by other criteria
932 * (eg speed by replacing global with restricted window pivoting, such
933 * as in SGEQPX from TOMS # 782). Good results will be obtained using
934 * SGEQPX with properly (!) chosen numerical parameters.
935 * Any improvement of SGEQP3 improves overal performance of SGEJSV.
936 *
937 * A * P1 = Q1 * [ R1^t 0]^t:
938  DO 1963 p = 1, n
939 * .. all columns are free columns
940  iwork(p) = 0
941  1963 CONTINUE
942  CALL sgeqp3( m,n,a,lda, iwork,work, work(n+1),lwork-n, ierr )
943 *
944 * The upper triangular matrix R1 from the first QRF is inspected for
945 * rank deficiency and possibilities for deflation, or possible
946 * ill-conditioning. Depending on the user specified flag L2RANK,
947 * the procedure explores possibilities to reduce the numerical
948 * rank by inspecting the computed upper triangular factor. If
949 * L2RANK or L2ABER are up, then SGEJSV will compute the SVD of
950 * A + dA, where ||dA|| <= f(M,N)*EPSLN.
951 *
952  nr = 1
953  IF ( l2aber ) THEN
954 * Standard absolute error bound suffices. All sigma_i with
955 * sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
956 * agressive enforcement of lower numerical rank by introducing a
957 * backward error of the order of N*EPSLN*||A||.
958  temp1 = sqrt(float(n))*epsln
959  DO 3001 p = 2, n
960  IF ( abs(a(p,p)) .GE. (temp1*abs(a(1,1))) ) THEN
961  nr = nr + 1
962  ELSE
963  GO TO 3002
964  END IF
965  3001 CONTINUE
966  3002 CONTINUE
967  ELSE IF ( l2rank ) THEN
968 * .. similarly as above, only slightly more gentle (less agressive).
969 * Sudden drop on the diagonal of R1 is used as the criterion for
970 * close-to-rank-defficient.
971  temp1 = sqrt(sfmin)
972  DO 3401 p = 2, n
973  IF ( ( abs(a(p,p)) .LT. (epsln*abs(a(p-1,p-1))) ) .OR.
974  \$ ( abs(a(p,p)) .LT. small ) .OR.
975  \$ ( l2kill .AND. (abs(a(p,p)) .LT. temp1) ) ) GO TO 3402
976  nr = nr + 1
977  3401 CONTINUE
978  3402 CONTINUE
979 *
980  ELSE
981 * The goal is high relative accuracy. However, if the matrix
982 * has high scaled condition number the relative accuracy is in
983 * general not feasible. Later on, a condition number estimator
984 * will be deployed to estimate the scaled condition number.
985 * Here we just remove the underflowed part of the triangular
986 * factor. This prevents the situation in which the code is
987 * working hard to get the accuracy not warranted by the data.
988  temp1 = sqrt(sfmin)
989  DO 3301 p = 2, n
990  IF ( ( abs(a(p,p)) .LT. small ) .OR.
991  \$ ( l2kill .AND. (abs(a(p,p)) .LT. temp1) ) ) GO TO 3302
992  nr = nr + 1
993  3301 CONTINUE
994  3302 CONTINUE
995 *
996  END IF
997 *
998  almort = .false.
999  IF ( nr .EQ. n ) THEN
1000  maxprj = one
1001  DO 3051 p = 2, n
1002  temp1 = abs(a(p,p)) / sva(iwork(p))
1003  maxprj = min( maxprj, temp1 )
1004  3051 CONTINUE
1005  IF ( maxprj**2 .GE. one - float(n)*epsln ) almort = .true.
1006  END IF
1007 *
1008 *
1009  sconda = - one
1010  condr1 = - one
1011  condr2 = - one
1012 *
1013  IF ( errest ) THEN
1014  IF ( n .EQ. nr ) THEN
1015  IF ( rsvec ) THEN
1016 * .. V is available as workspace
1017  CALL slacpy( 'U', n, n, a, lda, v, ldv )
1018  DO 3053 p = 1, n
1019  temp1 = sva(iwork(p))
1020  CALL sscal( p, one/temp1, v(1,p), 1 )
1021  3053 CONTINUE
1022  CALL spocon( 'U', n, v, ldv, one, temp1,
1023  \$ work(n+1), iwork(2*n+m+1), ierr )
1024  ELSE IF ( lsvec ) THEN
1025 * .. U is available as workspace
1026  CALL slacpy( 'U', n, n, a, lda, u, ldu )
1027  DO 3054 p = 1, n
1028  temp1 = sva(iwork(p))
1029  CALL sscal( p, one/temp1, u(1,p), 1 )
1030  3054 CONTINUE
1031  CALL spocon( 'U', n, u, ldu, one, temp1,
1032  \$ work(n+1), iwork(2*n+m+1), ierr )
1033  ELSE
1034  CALL slacpy( 'U', n, n, a, lda, work(n+1), n )
1035  DO 3052 p = 1, n
1036  temp1 = sva(iwork(p))
1037  CALL sscal( p, one/temp1, work(n+(p-1)*n+1), 1 )
1038  3052 CONTINUE
1039 * .. the columns of R are scaled to have unit Euclidean lengths.
1040  CALL spocon( 'U', n, work(n+1), n, one, temp1,
1041  \$ work(n+n*n+1), iwork(2*n+m+1), ierr )
1042  END IF
1043  sconda = one / sqrt(temp1)
1044 * SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
1045 * N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
1046  ELSE
1047  sconda = - one
1048  END IF
1049  END IF
1050 *
1051  l2pert = l2pert .AND. ( abs( a(1,1)/a(nr,nr) ) .GT. sqrt(big1) )
1052 * If there is no violent scaling, artificial perturbation is not needed.
1053 *
1054 * Phase 3:
1055 *
1056  IF ( .NOT. ( rsvec .OR. lsvec ) ) THEN
1057 *
1058 * Singular Values only
1059 *
1060 * .. transpose A(1:NR,1:N)
1061  DO 1946 p = 1, min( n-1, nr )
1062  CALL scopy( n-p, a(p,p+1), lda, a(p+1,p), 1 )
1063  1946 CONTINUE
1064 *
1065 * The following two DO-loops introduce small relative perturbation
1066 * into the strict upper triangle of the lower triangular matrix.
1067 * Small entries below the main diagonal are also changed.
1068 * This modification is useful if the computing environment does not
1069 * provide/allow FLUSH TO ZERO underflow, for it prevents many
1070 * annoying denormalized numbers in case of strongly scaled matrices.
1071 * The perturbation is structured so that it does not introduce any
1072 * new perturbation of the singular values, and it does not destroy
1073 * the job done by the preconditioner.
1074 * The licence for this perturbation is in the variable L2PERT, which
1075 * should be .FALSE. if FLUSH TO ZERO underflow is active.
1076 *
1077  IF ( .NOT. almort ) THEN
1078 *
1079  IF ( l2pert ) THEN
1080 * XSC = SQRT(SMALL)
1081  xsc = epsln / float(n)
1082  DO 4947 q = 1, nr
1083  temp1 = xsc*abs(a(q,q))
1084  DO 4949 p = 1, n
1085  IF ( ( (p.GT.q) .AND. (abs(a(p,q)).LE.temp1) )
1086  \$ .OR. ( p .LT. q ) )
1087  \$ a(p,q) = sign( temp1, a(p,q) )
1088  4949 CONTINUE
1089  4947 CONTINUE
1090  ELSE
1091  CALL slaset( 'U', nr-1,nr-1, zero,zero, a(1,2),lda )
1092  END IF
1093 *
1094 * .. second preconditioning using the QR factorization
1095 *
1096  CALL sgeqrf( n,nr, a,lda, work, work(n+1),lwork-n, ierr )
1097 *
1098 * .. and transpose upper to lower triangular
1099  DO 1948 p = 1, nr - 1
1100  CALL scopy( nr-p, a(p,p+1), lda, a(p+1,p), 1 )
1101  1948 CONTINUE
1102 *
1103  END IF
1104 *
1105 * Row-cyclic Jacobi SVD algorithm with column pivoting
1106 *
1107 * .. again some perturbation (a "background noise") is added
1108 * to drown denormals
1109  IF ( l2pert ) THEN
1110 * XSC = SQRT(SMALL)
1111  xsc = epsln / float(n)
1112  DO 1947 q = 1, nr
1113  temp1 = xsc*abs(a(q,q))
1114  DO 1949 p = 1, nr
1115  IF ( ( (p.GT.q) .AND. (abs(a(p,q)).LE.temp1) )
1116  \$ .OR. ( p .LT. q ) )
1117  \$ a(p,q) = sign( temp1, a(p,q) )
1118  1949 CONTINUE
1119  1947 CONTINUE
1120  ELSE
1121  CALL slaset( 'U', nr-1, nr-1, zero, zero, a(1,2), lda )
1122  END IF
1123 *
1124 * .. and one-sided Jacobi rotations are started on a lower
1125 * triangular matrix (plus perturbation which is ignored in
1126 * the part which destroys triangular form (confusing?!))
1127 *
1128  CALL sgesvj( 'L', 'NoU', 'NoV', nr, nr, a, lda, sva,
1129  \$ n, v, ldv, work, lwork, info )
1130 *
1131  scalem = work(1)
1132  numrank = nint(work(2))
1133 *
1134 *
1135  ELSE IF ( rsvec .AND. ( .NOT. lsvec ) ) THEN
1136 *
1137 * -> Singular Values and Right Singular Vectors <-
1138 *
1139  IF ( almort ) THEN
1140 *
1141 * .. in this case NR equals N
1142  DO 1998 p = 1, nr
1143  CALL scopy( n-p+1, a(p,p), lda, v(p,p), 1 )
1144  1998 CONTINUE
1145  CALL slaset( 'Upper', nr-1, nr-1, zero, zero, v(1,2), ldv )
1146 *
1147  CALL sgesvj( 'L','U','N', n, nr, v,ldv, sva, nr, a,lda,
1148  \$ work, lwork, info )
1149  scalem = work(1)
1150  numrank = nint(work(2))
1151
1152  ELSE
1153 *
1154 * .. two more QR factorizations ( one QRF is not enough, two require
1155 * accumulated product of Jacobi rotations, three are perfect )
1156 *
1157  CALL slaset( 'Lower', nr-1, nr-1, zero, zero, a(2,1), lda )
1158  CALL sgelqf( nr, n, a, lda, work, work(n+1), lwork-n, ierr)
1159  CALL slacpy( 'Lower', nr, nr, a, lda, v, ldv )
1160  CALL slaset( 'Upper', nr-1, nr-1, zero, zero, v(1,2), ldv )
1161  CALL sgeqrf( nr, nr, v, ldv, work(n+1), work(2*n+1),
1162  \$ lwork-2*n, ierr )
1163  DO 8998 p = 1, nr
1164  CALL scopy( nr-p+1, v(p,p), ldv, v(p,p), 1 )
1165  8998 CONTINUE
1166  CALL slaset( 'Upper', nr-1, nr-1, zero, zero, v(1,2), ldv )
1167 *
1168  CALL sgesvj( 'Lower', 'U','N', nr, nr, v,ldv, sva, nr, u,
1169  \$ ldu, work(n+1), lwork-n, info )
1170  scalem = work(n+1)
1171  numrank = nint(work(n+2))
1172  IF ( nr .LT. n ) THEN
1173  CALL slaset( 'A',n-nr, nr, zero,zero, v(nr+1,1), ldv )
1174  CALL slaset( 'A',nr, n-nr, zero,zero, v(1,nr+1), ldv )
1175  CALL slaset( 'A',n-nr,n-nr,zero,one, v(nr+1,nr+1), ldv )
1176  END IF
1177 *
1178  CALL sormlq( 'Left', 'Transpose', n, n, nr, a, lda, work,
1179  \$ v, ldv, work(n+1), lwork-n, ierr )
1180 *
1181  END IF
1182 *
1183  DO 8991 p = 1, n
1184  CALL scopy( n, v(p,1), ldv, a(iwork(p),1), lda )
1185  8991 CONTINUE
1186  CALL slacpy( 'All', n, n, a, lda, v, ldv )
1187 *
1188  IF ( transp ) THEN
1189  CALL slacpy( 'All', n, n, v, ldv, u, ldu )
1190  END IF
1191 *
1192  ELSE IF ( lsvec .AND. ( .NOT. rsvec ) ) THEN
1193 *
1194 * .. Singular Values and Left Singular Vectors ..
1195 *
1196 * .. second preconditioning step to avoid need to accumulate
1197 * Jacobi rotations in the Jacobi iterations.
1198  DO 1965 p = 1, nr
1199  CALL scopy( n-p+1, a(p,p), lda, u(p,p), 1 )
1200  1965 CONTINUE
1201  CALL slaset( 'Upper', nr-1, nr-1, zero, zero, u(1,2), ldu )
1202 *
1203  CALL sgeqrf( n, nr, u, ldu, work(n+1), work(2*n+1),
1204  \$ lwork-2*n, ierr )
1205 *
1206  DO 1967 p = 1, nr - 1
1207  CALL scopy( nr-p, u(p,p+1), ldu, u(p+1,p), 1 )
1208  1967 CONTINUE
1209  CALL slaset( 'Upper', nr-1, nr-1, zero, zero, u(1,2), ldu )
1210 *
1211  CALL sgesvj( 'Lower', 'U', 'N', nr,nr, u, ldu, sva, nr, a,
1212  \$ lda, work(n+1), lwork-n, info )
1213  scalem = work(n+1)
1214  numrank = nint(work(n+2))
1215 *
1216  IF ( nr .LT. m ) THEN
1217  CALL slaset( 'A', m-nr, nr,zero, zero, u(nr+1,1), ldu )
1218  IF ( nr .LT. n1 ) THEN
1219  CALL slaset( 'A',nr, n1-nr, zero, zero, u(1,nr+1), ldu )
1220  CALL slaset( 'A',m-nr,n1-nr,zero,one,u(nr+1,nr+1), ldu )
1221  END IF
1222  END IF
1223 *
1224  CALL sormqr( 'Left', 'No Tr', m, n1, n, a, lda, work, u,
1225  \$ ldu, work(n+1), lwork-n, ierr )
1226 *
1227  IF ( rowpiv )
1228  \$ CALL slaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1229 *
1230  DO 1974 p = 1, n1
1231  xsc = one / snrm2( m, u(1,p), 1 )
1232  CALL sscal( m, xsc, u(1,p), 1 )
1233  1974 CONTINUE
1234 *
1235  IF ( transp ) THEN
1236  CALL slacpy( 'All', n, n, u, ldu, v, ldv )
1237  END IF
1238 *
1239  ELSE
1240 *
1241 * .. Full SVD ..
1242 *
1243  IF ( .NOT. jracc ) THEN
1244 *
1245  IF ( .NOT. almort ) THEN
1246 *
1247 * Second Preconditioning Step (QRF [with pivoting])
1248 * Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
1249 * equivalent to an LQF CALL. Since in many libraries the QRF
1250 * seems to be better optimized than the LQF, we do explicit
1251 * transpose and use the QRF. This is subject to changes in an
1252 * optimized implementation of SGEJSV.
1253 *
1254  DO 1968 p = 1, nr
1255  CALL scopy( n-p+1, a(p,p), lda, v(p,p), 1 )
1256  1968 CONTINUE
1257 *
1258 * .. the following two loops perturb small entries to avoid
1259 * denormals in the second QR factorization, where they are
1260 * as good as zeros. This is done to avoid painfully slow
1261 * computation with denormals. The relative size of the perturbation
1262 * is a parameter that can be changed by the implementer.
1263 * This perturbation device will be obsolete on machines with
1264 * properly implemented arithmetic.
1265 * To switch it off, set L2PERT=.FALSE. To remove it from the
1266 * code, remove the action under L2PERT=.TRUE., leave the ELSE part.
1267 * The following two loops should be blocked and fused with the
1268 * transposed copy above.
1269 *
1270  IF ( l2pert ) THEN
1271  xsc = sqrt(small)
1272  DO 2969 q = 1, nr
1273  temp1 = xsc*abs( v(q,q) )
1274  DO 2968 p = 1, n
1275  IF ( ( p .GT. q ) .AND. ( abs(v(p,q)) .LE. temp1 )
1276  \$ .OR. ( p .LT. q ) )
1277  \$ v(p,q) = sign( temp1, v(p,q) )
1278  IF ( p .LT. q ) v(p,q) = - v(p,q)
1279  2968 CONTINUE
1280  2969 CONTINUE
1281  ELSE
1282  CALL slaset( 'U', nr-1, nr-1, zero, zero, v(1,2), ldv )
1283  END IF
1284 *
1285 * Estimate the row scaled condition number of R1
1286 * (If R1 is rectangular, N > NR, then the condition number
1287 * of the leading NR x NR submatrix is estimated.)
1288 *
1289  CALL slacpy( 'L', nr, nr, v, ldv, work(2*n+1), nr )
1290  DO 3950 p = 1, nr
1291  temp1 = snrm2(nr-p+1,work(2*n+(p-1)*nr+p),1)
1292  CALL sscal(nr-p+1,one/temp1,work(2*n+(p-1)*nr+p),1)
1293  3950 CONTINUE
1294  CALL spocon('Lower',nr,work(2*n+1),nr,one,temp1,
1295  \$ work(2*n+nr*nr+1),iwork(m+2*n+1),ierr)
1296  condr1 = one / sqrt(temp1)
1297 * .. here need a second oppinion on the condition number
1298 * .. then assume worst case scenario
1299 * R1 is OK for inverse <=> CONDR1 .LT. FLOAT(N)
1300 * more conservative <=> CONDR1 .LT. SQRT(FLOAT(N))
1301 *
1302  cond_ok = sqrt(float(nr))
1303 *[TP] COND_OK is a tuning parameter.
1304
1305  IF ( condr1 .LT. cond_ok ) THEN
1306 * .. the second QRF without pivoting. Note: in an optimized
1307 * implementation, this QRF should be implemented as the QRF
1308 * of a lower triangular matrix.
1309 * R1^t = Q2 * R2
1310  CALL sgeqrf( n, nr, v, ldv, work(n+1), work(2*n+1),
1311  \$ lwork-2*n, ierr )
1312 *
1313  IF ( l2pert ) THEN
1314  xsc = sqrt(small)/epsln
1315  DO 3959 p = 2, nr
1316  DO 3958 q = 1, p - 1
1317  temp1 = xsc * min(abs(v(p,p)),abs(v(q,q)))
1318  IF ( abs(v(q,p)) .LE. temp1 )
1319  \$ v(q,p) = sign( temp1, v(q,p) )
1320  3958 CONTINUE
1321  3959 CONTINUE
1322  END IF
1323 *
1324  IF ( nr .NE. n )
1325  \$ CALL slacpy( 'A', n, nr, v, ldv, work(2*n+1), n )
1326 * .. save ...
1327 *
1328 * .. this transposed copy should be better than naive
1329  DO 1969 p = 1, nr - 1
1330  CALL scopy( nr-p, v(p,p+1), ldv, v(p+1,p), 1 )
1331  1969 CONTINUE
1332 *
1333  condr2 = condr1
1334 *
1335  ELSE
1336 *
1337 * .. ill-conditioned case: second QRF with pivoting
1338 * Note that windowed pivoting would be equaly good
1339 * numerically, and more run-time efficient. So, in
1340 * an optimal implementation, the next call to SGEQP3
1341 * should be replaced with eg. CALL SGEQPX (ACM TOMS #782)
1342 * with properly (carefully) chosen parameters.
1343 *
1344 * R1^t * P2 = Q2 * R2
1345  DO 3003 p = 1, nr
1346  iwork(n+p) = 0
1347  3003 CONTINUE
1348  CALL sgeqp3( n, nr, v, ldv, iwork(n+1), work(n+1),
1349  \$ work(2*n+1), lwork-2*n, ierr )
1350 ** CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
1351 ** \$ LWORK-2*N, IERR )
1352  IF ( l2pert ) THEN
1353  xsc = sqrt(small)
1354  DO 3969 p = 2, nr
1355  DO 3968 q = 1, p - 1
1356  temp1 = xsc * min(abs(v(p,p)),abs(v(q,q)))
1357  IF ( abs(v(q,p)) .LE. temp1 )
1358  \$ v(q,p) = sign( temp1, v(q,p) )
1359  3968 CONTINUE
1360  3969 CONTINUE
1361  END IF
1362 *
1363  CALL slacpy( 'A', n, nr, v, ldv, work(2*n+1), n )
1364 *
1365  IF ( l2pert ) THEN
1366  xsc = sqrt(small)
1367  DO 8970 p = 2, nr
1368  DO 8971 q = 1, p - 1
1369  temp1 = xsc * min(abs(v(p,p)),abs(v(q,q)))
1370  v(p,q) = - sign( temp1, v(q,p) )
1371  8971 CONTINUE
1372  8970 CONTINUE
1373  ELSE
1374  CALL slaset( 'L',nr-1,nr-1,zero,zero,v(2,1),ldv )
1375  END IF
1376 * Now, compute R2 = L3 * Q3, the LQ factorization.
1377  CALL sgelqf( nr, nr, v, ldv, work(2*n+n*nr+1),
1378  \$ work(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, ierr )
1379 * .. and estimate the condition number
1380  CALL slacpy( 'L',nr,nr,v,ldv,work(2*n+n*nr+nr+1),nr )
1381  DO 4950 p = 1, nr
1382  temp1 = snrm2( p, work(2*n+n*nr+nr+p), nr )
1383  CALL sscal( p, one/temp1, work(2*n+n*nr+nr+p), nr )
1384  4950 CONTINUE
1385  CALL spocon( 'L',nr,work(2*n+n*nr+nr+1),nr,one,temp1,
1386  \$ work(2*n+n*nr+nr+nr*nr+1),iwork(m+2*n+1),ierr )
1387  condr2 = one / sqrt(temp1)
1388 *
1389  IF ( condr2 .GE. cond_ok ) THEN
1390 * .. save the Householder vectors used for Q3
1391 * (this overwrittes the copy of R2, as it will not be
1392 * needed in this branch, but it does not overwritte the
1393 * Huseholder vectors of Q2.).
1394  CALL slacpy( 'U', nr, nr, v, ldv, work(2*n+1), n )
1395 * .. and the rest of the information on Q3 is in
1396 * WORK(2*N+N*NR+1:2*N+N*NR+N)
1397  END IF
1398 *
1399  END IF
1400 *
1401  IF ( l2pert ) THEN
1402  xsc = sqrt(small)
1403  DO 4968 q = 2, nr
1404  temp1 = xsc * v(q,q)
1405  DO 4969 p = 1, q - 1
1406 * V(p,q) = - SIGN( TEMP1, V(q,p) )
1407  v(p,q) = - sign( temp1, v(p,q) )
1408  4969 CONTINUE
1409  4968 CONTINUE
1410  ELSE
1411  CALL slaset( 'U', nr-1,nr-1, zero,zero, v(1,2), ldv )
1412  END IF
1413 *
1414 * Second preconditioning finished; continue with Jacobi SVD
1415 * The input matrix is lower trinagular.
1416 *
1417 * Recover the right singular vectors as solution of a well
1418 * conditioned triangular matrix equation.
1419 *
1420  IF ( condr1 .LT. cond_ok ) THEN
1421 *
1422  CALL sgesvj( 'L','U','N',nr,nr,v,ldv,sva,nr,u,
1423  \$ ldu,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,info )
1424  scalem = work(2*n+n*nr+nr+1)
1425  numrank = nint(work(2*n+n*nr+nr+2))
1426  DO 3970 p = 1, nr
1427  CALL scopy( nr, v(1,p), 1, u(1,p), 1 )
1428  CALL sscal( nr, sva(p), v(1,p), 1 )
1429  3970 CONTINUE
1430
1431 * .. pick the right matrix equation and solve it
1432 *
1433  IF ( nr .EQ. n ) THEN
1434 * :)) .. best case, R1 is inverted. The solution of this matrix
1435 * equation is Q2*V2 = the product of the Jacobi rotations
1436 * used in SGESVJ, premultiplied with the orthogonal matrix
1437 * from the second QR factorization.
1438  CALL strsm( 'L','U','N','N', nr,nr,one, a,lda, v,ldv )
1439  ELSE
1440 * .. R1 is well conditioned, but non-square. Transpose(R2)
1441 * is inverted to get the product of the Jacobi rotations
1442 * used in SGESVJ. The Q-factor from the second QR
1443 * factorization is then built in explicitly.
1444  CALL strsm('L','U','T','N',nr,nr,one,work(2*n+1),
1445  \$ n,v,ldv)
1446  IF ( nr .LT. n ) THEN
1447  CALL slaset('A',n-nr,nr,zero,zero,v(nr+1,1),ldv)
1448  CALL slaset('A',nr,n-nr,zero,zero,v(1,nr+1),ldv)
1449  CALL slaset('A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv)
1450  END IF
1451  CALL sormqr('L','N',n,n,nr,work(2*n+1),n,work(n+1),
1452  \$ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr)
1453  END IF
1454 *
1455  ELSE IF ( condr2 .LT. cond_ok ) THEN
1456 *
1457 * :) .. the input matrix A is very likely a relative of
1458 * the Kahan matrix :)
1459 * The matrix R2 is inverted. The solution of the matrix equation
1460 * is Q3^T*V3 = the product of the Jacobi rotations (appplied to
1461 * the lower triangular L3 from the LQ factorization of
1462 * R2=L3*Q3), pre-multiplied with the transposed Q3.
1463  CALL sgesvj( 'L', 'U', 'N', nr, nr, v, ldv, sva, nr, u,
1464  \$ ldu, work(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, info )
1465  scalem = work(2*n+n*nr+nr+1)
1466  numrank = nint(work(2*n+n*nr+nr+2))
1467  DO 3870 p = 1, nr
1468  CALL scopy( nr, v(1,p), 1, u(1,p), 1 )
1469  CALL sscal( nr, sva(p), u(1,p), 1 )
1470  3870 CONTINUE
1471  CALL strsm('L','U','N','N',nr,nr,one,work(2*n+1),n,u,ldu)
1472 * .. apply the permutation from the second QR factorization
1473  DO 873 q = 1, nr
1474  DO 872 p = 1, nr
1475  work(2*n+n*nr+nr+iwork(n+p)) = u(p,q)
1476  872 CONTINUE
1477  DO 874 p = 1, nr
1478  u(p,q) = work(2*n+n*nr+nr+p)
1479  874 CONTINUE
1480  873 CONTINUE
1481  IF ( nr .LT. n ) THEN
1482  CALL slaset( 'A',n-nr,nr,zero,zero,v(nr+1,1),ldv )
1483  CALL slaset( 'A',nr,n-nr,zero,zero,v(1,nr+1),ldv )
1484  CALL slaset( 'A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv )
1485  END IF
1486  CALL sormqr( 'L','N',n,n,nr,work(2*n+1),n,work(n+1),
1487  \$ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
1488  ELSE
1489 * Last line of defense.
1490 * #:( This is a rather pathological case: no scaled condition
1491 * improvement after two pivoted QR factorizations. Other
1492 * possibility is that the rank revealing QR factorization
1493 * or the condition estimator has failed, or the COND_OK
1494 * is set very close to ONE (which is unnecessary). Normally,
1495 * this branch should never be executed, but in rare cases of
1496 * failure of the RRQR or condition estimator, the last line of
1497 * defense ensures that SGEJSV completes the task.
1498 * Compute the full SVD of L3 using SGESVJ with explicit
1499 * accumulation of Jacobi rotations.
1500  CALL sgesvj( 'L', 'U', 'V', nr, nr, v, ldv, sva, nr, u,
1501  \$ ldu, work(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, info )
1502  scalem = work(2*n+n*nr+nr+1)
1503  numrank = nint(work(2*n+n*nr+nr+2))
1504  IF ( nr .LT. n ) THEN
1505  CALL slaset( 'A',n-nr,nr,zero,zero,v(nr+1,1),ldv )
1506  CALL slaset( 'A',nr,n-nr,zero,zero,v(1,nr+1),ldv )
1507  CALL slaset( 'A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv )
1508  END IF
1509  CALL sormqr( 'L','N',n,n,nr,work(2*n+1),n,work(n+1),
1510  \$ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
1511 *
1512  CALL sormlq( 'L', 'T', nr, nr, nr, work(2*n+1), n,
1513  \$ work(2*n+n*nr+1), u, ldu, work(2*n+n*nr+nr+1),
1514  \$ lwork-2*n-n*nr-nr, ierr )
1515  DO 773 q = 1, nr
1516  DO 772 p = 1, nr
1517  work(2*n+n*nr+nr+iwork(n+p)) = u(p,q)
1518  772 CONTINUE
1519  DO 774 p = 1, nr
1520  u(p,q) = work(2*n+n*nr+nr+p)
1521  774 CONTINUE
1522  773 CONTINUE
1523 *
1524  END IF
1525 *
1526 * Permute the rows of V using the (column) permutation from the
1527 * first QRF. Also, scale the columns to make them unit in
1528 * Euclidean norm. This applies to all cases.
1529 *
1530  temp1 = sqrt(float(n)) * epsln
1531  DO 1972 q = 1, n
1532  DO 972 p = 1, n
1533  work(2*n+n*nr+nr+iwork(p)) = v(p,q)
1534  972 CONTINUE
1535  DO 973 p = 1, n
1536  v(p,q) = work(2*n+n*nr+nr+p)
1537  973 CONTINUE
1538  xsc = one / snrm2( n, v(1,q), 1 )
1539  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1540  \$ CALL sscal( n, xsc, v(1,q), 1 )
1541  1972 CONTINUE
1542 * At this moment, V contains the right singular vectors of A.
1543 * Next, assemble the left singular vector matrix U (M x N).
1544  IF ( nr .LT. m ) THEN
1545  CALL slaset( 'A', m-nr, nr, zero, zero, u(nr+1,1), ldu )
1546  IF ( nr .LT. n1 ) THEN
1547  CALL slaset('A',nr,n1-nr,zero,zero,u(1,nr+1),ldu)
1548  CALL slaset('A',m-nr,n1-nr,zero,one,u(nr+1,nr+1),ldu)
1549  END IF
1550  END IF
1551 *
1552 * The Q matrix from the first QRF is built into the left singular
1553 * matrix U. This applies to all cases.
1554 *
1555  CALL sormqr( 'Left', 'No_Tr', m, n1, n, a, lda, work, u,
1556  \$ ldu, work(n+1), lwork-n, ierr )
1557
1558 * The columns of U are normalized. The cost is O(M*N) flops.
1559  temp1 = sqrt(float(m)) * epsln
1560  DO 1973 p = 1, nr
1561  xsc = one / snrm2( m, u(1,p), 1 )
1562  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1563  \$ CALL sscal( m, xsc, u(1,p), 1 )
1564  1973 CONTINUE
1565 *
1566 * If the initial QRF is computed with row pivoting, the left
1567 * singular vectors must be adjusted.
1568 *
1569  IF ( rowpiv )
1570  \$ CALL slaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1571 *
1572  ELSE
1573 *
1574 * .. the initial matrix A has almost orthogonal columns and
1575 * the second QRF is not needed
1576 *
1577  CALL slacpy( 'Upper', n, n, a, lda, work(n+1), n )
1578  IF ( l2pert ) THEN
1579  xsc = sqrt(small)
1580  DO 5970 p = 2, n
1581  temp1 = xsc * work( n + (p-1)*n + p )
1582  DO 5971 q = 1, p - 1
1583  work(n+(q-1)*n+p)=-sign(temp1,work(n+(p-1)*n+q))
1584  5971 CONTINUE
1585  5970 CONTINUE
1586  ELSE
1587  CALL slaset( 'Lower',n-1,n-1,zero,zero,work(n+2),n )
1588  END IF
1589 *
1590  CALL sgesvj( 'Upper', 'U', 'N', n, n, work(n+1), n, sva,
1591  \$ n, u, ldu, work(n+n*n+1), lwork-n-n*n, info )
1592 *
1593  scalem = work(n+n*n+1)
1594  numrank = nint(work(n+n*n+2))
1595  DO 6970 p = 1, n
1596  CALL scopy( n, work(n+(p-1)*n+1), 1, u(1,p), 1 )
1597  CALL sscal( n, sva(p), work(n+(p-1)*n+1), 1 )
1598  6970 CONTINUE
1599 *
1600  CALL strsm( 'Left', 'Upper', 'NoTrans', 'No UD', n, n,
1601  \$ one, a, lda, work(n+1), n )
1602  DO 6972 p = 1, n
1603  CALL scopy( n, work(n+p), n, v(iwork(p),1), ldv )
1604  6972 CONTINUE
1605  temp1 = sqrt(float(n))*epsln
1606  DO 6971 p = 1, n
1607  xsc = one / snrm2( n, v(1,p), 1 )
1608  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1609  \$ CALL sscal( n, xsc, v(1,p), 1 )
1610  6971 CONTINUE
1611 *
1612 * Assemble the left singular vector matrix U (M x N).
1613 *
1614  IF ( n .LT. m ) THEN
1615  CALL slaset( 'A', m-n, n, zero, zero, u(n+1,1), ldu )
1616  IF ( n .LT. n1 ) THEN
1617  CALL slaset( 'A',n, n1-n, zero, zero, u(1,n+1),ldu )
1618  CALL slaset( 'A',m-n,n1-n, zero, one,u(n+1,n+1),ldu )
1619  END IF
1620  END IF
1621  CALL sormqr( 'Left', 'No Tr', m, n1, n, a, lda, work, u,
1622  \$ ldu, work(n+1), lwork-n, ierr )
1623  temp1 = sqrt(float(m))*epsln
1624  DO 6973 p = 1, n1
1625  xsc = one / snrm2( m, u(1,p), 1 )
1626  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1627  \$ CALL sscal( m, xsc, u(1,p), 1 )
1628  6973 CONTINUE
1629 *
1630  IF ( rowpiv )
1631  \$ CALL slaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1632 *
1633  END IF
1634 *
1635 * end of the >> almost orthogonal case << in the full SVD
1636 *
1637  ELSE
1638 *
1639 * This branch deploys a preconditioned Jacobi SVD with explicitly
1640 * accumulated rotations. It is included as optional, mainly for
1641 * experimental purposes. It does perfom well, and can also be used.
1642 * In this implementation, this branch will be automatically activated
1643 * if the condition number sigma_max(A) / sigma_min(A) is predicted
1644 * to be greater than the overflow threshold. This is because the
1645 * a posteriori computation of the singular vectors assumes robust
1646 * implementation of BLAS and some LAPACK procedures, capable of working
1647 * in presence of extreme values. Since that is not always the case, ...
1648 *
1649  DO 7968 p = 1, nr
1650  CALL scopy( n-p+1, a(p,p), lda, v(p,p), 1 )
1651  7968 CONTINUE
1652 *
1653  IF ( l2pert ) THEN
1654  xsc = sqrt(small/epsln)
1655  DO 5969 q = 1, nr
1656  temp1 = xsc*abs( v(q,q) )
1657  DO 5968 p = 1, n
1658  IF ( ( p .GT. q ) .AND. ( abs(v(p,q)) .LE. temp1 )
1659  \$ .OR. ( p .LT. q ) )
1660  \$ v(p,q) = sign( temp1, v(p,q) )
1661  IF ( p .LT. q ) v(p,q) = - v(p,q)
1662  5968 CONTINUE
1663  5969 CONTINUE
1664  ELSE
1665  CALL slaset( 'U', nr-1, nr-1, zero, zero, v(1,2), ldv )
1666  END IF
1667
1668  CALL sgeqrf( n, nr, v, ldv, work(n+1), work(2*n+1),
1669  \$ lwork-2*n, ierr )
1670  CALL slacpy( 'L', n, nr, v, ldv, work(2*n+1), n )
1671 *
1672  DO 7969 p = 1, nr
1673  CALL scopy( nr-p+1, v(p,p), ldv, u(p,p), 1 )
1674  7969 CONTINUE
1675
1676  IF ( l2pert ) THEN
1677  xsc = sqrt(small/epsln)
1678  DO 9970 q = 2, nr
1679  DO 9971 p = 1, q - 1
1680  temp1 = xsc * min(abs(u(p,p)),abs(u(q,q)))
1681  u(p,q) = - sign( temp1, u(q,p) )
1682  9971 CONTINUE
1683  9970 CONTINUE
1684  ELSE
1685  CALL slaset('U', nr-1, nr-1, zero, zero, u(1,2), ldu )
1686  END IF
1687
1688  CALL sgesvj( 'L', 'U', 'V', nr, nr, u, ldu, sva,
1689  \$ n, v, ldv, work(2*n+n*nr+1), lwork-2*n-n*nr, info )
1690  scalem = work(2*n+n*nr+1)
1691  numrank = nint(work(2*n+n*nr+2))
1692
1693  IF ( nr .LT. n ) THEN
1694  CALL slaset( 'A',n-nr,nr,zero,zero,v(nr+1,1),ldv )
1695  CALL slaset( 'A',nr,n-nr,zero,zero,v(1,nr+1),ldv )
1696  CALL slaset( 'A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv )
1697  END IF
1698
1699  CALL sormqr( 'L','N',n,n,nr,work(2*n+1),n,work(n+1),
1700  \$ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
1701 *
1702 * Permute the rows of V using the (column) permutation from the
1703 * first QRF. Also, scale the columns to make them unit in
1704 * Euclidean norm. This applies to all cases.
1705 *
1706  temp1 = sqrt(float(n)) * epsln
1707  DO 7972 q = 1, n
1708  DO 8972 p = 1, n
1709  work(2*n+n*nr+nr+iwork(p)) = v(p,q)
1710  8972 CONTINUE
1711  DO 8973 p = 1, n
1712  v(p,q) = work(2*n+n*nr+nr+p)
1713  8973 CONTINUE
1714  xsc = one / snrm2( n, v(1,q), 1 )
1715  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1716  \$ CALL sscal( n, xsc, v(1,q), 1 )
1717  7972 CONTINUE
1718 *
1719 * At this moment, V contains the right singular vectors of A.
1720 * Next, assemble the left singular vector matrix U (M x N).
1721 *
1722  IF ( nr .LT. m ) THEN
1723  CALL slaset( 'A', m-nr, nr, zero, zero, u(nr+1,1), ldu )
1724  IF ( nr .LT. n1 ) THEN
1725  CALL slaset( 'A',nr, n1-nr, zero, zero, u(1,nr+1),ldu )
1726  CALL slaset( 'A',m-nr,n1-nr, zero, one,u(nr+1,nr+1),ldu )
1727  END IF
1728  END IF
1729 *
1730  CALL sormqr( 'Left', 'No Tr', m, n1, n, a, lda, work, u,
1731  \$ ldu, work(n+1), lwork-n, ierr )
1732 *
1733  IF ( rowpiv )
1734  \$ CALL slaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1735 *
1736 *
1737  END IF
1738  IF ( transp ) THEN
1739 * .. swap U and V because the procedure worked on A^t
1740  DO 6974 p = 1, n
1741  CALL sswap( n, u(1,p), 1, v(1,p), 1 )
1742  6974 CONTINUE
1743  END IF
1744 *
1745  END IF
1746 * end of the full SVD
1747 *
1748 * Undo scaling, if necessary (and possible)
1749 *
1750  IF ( uscal2 .LE. (big/sva(1))*uscal1 ) THEN
1751  CALL slascl( 'G', 0, 0, uscal1, uscal2, nr, 1, sva, n, ierr )
1752  uscal1 = one
1753  uscal2 = one
1754  END IF
1755 *
1756  IF ( nr .LT. n ) THEN
1757  DO 3004 p = nr+1, n
1758  sva(p) = zero
1759  3004 CONTINUE
1760  END IF
1761 *
1762  work(1) = uscal2 * scalem
1763  work(2) = uscal1
1764  IF ( errest ) work(3) = sconda
1765  IF ( lsvec .AND. rsvec ) THEN
1766  work(4) = condr1
1767  work(5) = condr2
1768  END IF
1769  IF ( l2tran ) THEN
1770  work(6) = entra
1771  work(7) = entrat
1772  END IF
1773 *
1774  iwork(1) = nr
1775  iwork(2) = numrank
1776  iwork(3) = warning
1777 *
1778  RETURN
1779 * ..
1780 * .. END OF SGEJSV
1781 * ..
1782  END
1783 *
subroutine slassq(N, X, INCX, SCALE, SUMSQ)
SLASSQ updates a sum of squares represented in scaled form.
Definition: slassq.f:105
subroutine sgesvj(JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, WORK, LWORK, INFO)
SGESVJ
Definition: sgesvj.f:325
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:170
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:183
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:145
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:138
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine slaswp(N, A, LDA, K1, K2, IPIV, INCX)
SLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: slaswp.f:116
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: slaset.f:112
subroutine sgejsv(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
SGEJSV
Definition: sgejsv.f:478
subroutine sormlq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMLQ
Definition: sormlq.f:170
subroutine spocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SPOCON
Definition: spocon.f:123
subroutine sgeqp3(M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
SGEQP3
Definition: sgeqp3.f:153
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:55
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:53
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:130
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
subroutine sgelqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGELQF
Definition: sgelqf.f:137