LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
 All Classes Files Functions Variables Typedefs Macros
dgejsv.f
Go to the documentation of this file.
1 *> \brief \b DGEJSV
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DGEJSV + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgejsv.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgejsv.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgejsv.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DGEJSV( 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 * DOUBLE PRECISION 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 *> DGEJSV 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 *> \endverbatim
55 *
56 * Arguments:
57 * ==========
58 *
59 *> \param[in] JOBA
60 *> \verbatim
61 *> JOBA is CHARACTER*1
62 *> Specifies the level of accuracy:
63 *> = 'C': This option works well (high relative accuracy) if A = B * D,
64 *> with well-conditioned B and arbitrary diagonal matrix D.
65 *> The accuracy cannot be spoiled by COLUMN scaling. The
66 *> accuracy of the computed output depends on the condition of
67 *> B, and the procedure aims at the best theoretical accuracy.
68 *> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
69 *> bounded by f(M,N)*epsilon* cond(B), independent of D.
70 *> The input matrix is preprocessed with the QRF with column
71 *> pivoting. This initial preprocessing and preconditioning by
72 *> a rank revealing QR factorization is common for all values of
73 *> JOBA. Additional actions are specified as follows:
74 *> = 'E': Computation as with 'C' with an additional estimate of the
75 *> condition number of B. It provides a realistic error bound.
76 *> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
77 *> D1, D2, and well-conditioned matrix C, this option gives
78 *> higher accuracy than the 'C' option. If the structure of the
79 *> input matrix is not known, and relative accuracy is
80 *> desirable, then this option is advisable. The input matrix A
81 *> is preprocessed with QR factorization with FULL (row and
82 *> column) pivoting.
83 *> = 'G' Computation as with 'F' with an additional estimate of the
84 *> condition number of B, where A=D*B. If A has heavily weighted
85 *> rows, then using this condition number gives too pessimistic
86 *> error bound.
87 *> = 'A': Small singular values are the noise and the matrix is treated
88 *> as numerically rank defficient. The error in the computed
89 *> singular values is bounded by f(m,n)*epsilon*||A||.
90 *> The computed SVD A = U * S * V^t restores A up to
91 *> f(m,n)*epsilon*||A||.
92 *> This gives the procedure the licence to discard (set to zero)
93 *> all singular values below N*epsilon*||A||.
94 *> = 'R': Similar as in 'A'. Rank revealing property of the initial
95 *> QR factorization is used do reveal (using triangular factor)
96 *> a gap sigma_{r+1} < epsilon * sigma_r in which case the
97 *> numerical RANK is declared to be r. The SVD is computed with
98 *> absolute error bounds, but more accurately than with 'A'.
99 *> \endverbatim
100 *>
101 *> \param[in] JOBU
102 *> \verbatim
103 *> JOBU is CHARACTER*1
104 *> Specifies whether to compute the columns of U:
105 *> = 'U': N columns of U are returned in the array U.
106 *> = 'F': full set of M left sing. vectors is returned in the array U.
107 *> = 'W': U may be used as workspace of length M*N. See the description
108 *> of U.
109 *> = 'N': U is not computed.
110 *> \endverbatim
111 *>
112 *> \param[in] JOBV
113 *> \verbatim
114 *> JOBV is CHARACTER*1
115 *> Specifies whether to compute the matrix V:
116 *> = 'V': N columns of V are returned in the array V; Jacobi rotations
117 *> are not explicitly accumulated.
118 *> = 'J': N columns of V are returned in the array V, but they are
119 *> computed as the product of Jacobi rotations. This option is
120 *> allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
121 *> = 'W': V may be used as workspace of length N*N. See the description
122 *> of V.
123 *> = 'N': V is not computed.
124 *> \endverbatim
125 *>
126 *> \param[in] JOBR
127 *> \verbatim
128 *> JOBR is CHARACTER*1
129 *> Specifies the RANGE for the singular values. Issues the licence to
130 *> set to zero small positive singular values if they are outside
131 *> specified range. If A .NE. 0 is scaled so that the largest singular
132 *> value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
133 *> the licence to kill columns of A whose norm in c*A is less than
134 *> DSQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
135 *> where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
136 *> = 'N': Do not kill small columns of c*A. This option assumes that
137 *> BLAS and QR factorizations and triangular solvers are
138 *> implemented to work in that range. If the condition of A
139 *> is greater than BIG, use DGESVJ.
140 *> = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]
141 *> (roughly, as described above). This option is recommended.
142 *> ~~~~~~~~~~~~~~~~~~~~~~~~~~~
143 *> For computing the singular values in the FULL range [SFMIN,BIG]
144 *> use DGESVJ.
145 *> \endverbatim
146 *>
147 *> \param[in] JOBT
148 *> \verbatim
149 *> JOBT is CHARACTER*1
150 *> If the matrix is square then the procedure may determine to use
151 *> transposed A if A^t seems to be better with respect to convergence.
152 *> If the matrix is not square, JOBT is ignored. This is subject to
153 *> changes in the future.
154 *> The decision is based on two values of entropy over the adjoint
155 *> orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
156 *> = 'T': transpose if entropy test indicates possibly faster
157 *> convergence of Jacobi process if A^t is taken as input. If A is
158 *> replaced with A^t, then the row pivoting is included automatically.
159 *> = 'N': do not speculate.
160 *> This option can be used to compute only the singular values, or the
161 *> full SVD (U, SIGMA and V). For only one set of singular vectors
162 *> (U or V), the caller should provide both U and V, as one of the
163 *> matrices is used as workspace if the matrix A is transposed.
164 *> The implementer can easily remove this constraint and make the
165 *> code more complicated. See the descriptions of U and V.
166 *> \endverbatim
167 *>
168 *> \param[in] JOBP
169 *> \verbatim
170 *> JOBP is CHARACTER*1
171 *> Issues the licence to introduce structured perturbations to drown
172 *> denormalized numbers. This licence should be active if the
173 *> denormals are poorly implemented, causing slow computation,
174 *> especially in cases of fast convergence (!). For details see [1,2].
175 *> For the sake of simplicity, this perturbations are included only
176 *> when the full SVD or only the singular values are requested. The
177 *> implementer/user can easily add the perturbation for the cases of
178 *> computing one set of singular vectors.
179 *> = 'P': introduce perturbation
180 *> = 'N': do not perturb
181 *> \endverbatim
182 *>
183 *> \param[in] M
184 *> \verbatim
185 *> M is INTEGER
186 *> The number of rows of the input matrix A. M >= 0.
187 *> \endverbatim
188 *>
189 *> \param[in] N
190 *> \verbatim
191 *> N is INTEGER
192 *> The number of columns of the input matrix A. M >= N >= 0.
193 *> \endverbatim
194 *>
195 *> \param[in,out] A
196 *> \verbatim
197 *> A is DOUBLE PRECISION array, dimension (LDA,N)
198 *> On entry, the M-by-N matrix A.
199 *> \endverbatim
200 *>
201 *> \param[in] LDA
202 *> \verbatim
203 *> LDA is INTEGER
204 *> The leading dimension of the array A. LDA >= max(1,M).
205 *> \endverbatim
206 *>
207 *> \param[out] SVA
208 *> \verbatim
209 *> SVA is DOUBLE PRECISION array, dimension (N)
210 *> On exit,
211 *> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
212 *> computation SVA contains Euclidean column norms of the
213 *> iterated matrices in the array A.
214 *> - For WORK(1) .NE. WORK(2): The singular values of A are
215 *> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
216 *> sigma_max(A) overflows or if small singular values have been
217 *> saved from underflow by scaling the input matrix A.
218 *> - If JOBR='R' then some of the singular values may be returned
219 *> as exact zeros obtained by "set to zero" because they are
220 *> below the numerical rank threshold or are denormalized numbers.
221 *> \endverbatim
222 *>
223 *> \param[out] U
224 *> \verbatim
225 *> U is DOUBLE PRECISION array, dimension ( LDU, N )
226 *> If JOBU = 'U', then U contains on exit the M-by-N matrix of
227 *> the left singular vectors.
228 *> If JOBU = 'F', then U contains on exit the M-by-M matrix of
229 *> the left singular vectors, including an ONB
230 *> of the orthogonal complement of the Range(A).
231 *> If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
232 *> then U is used as workspace if the procedure
233 *> replaces A with A^t. In that case, [V] is computed
234 *> in U as left singular vectors of A^t and then
235 *> copied back to the V array. This 'W' option is just
236 *> a reminder to the caller that in this case U is
237 *> reserved as workspace of length N*N.
238 *> If JOBU = 'N' U is not referenced.
239 *> \endverbatim
240 *>
241 *> \param[in] LDU
242 *> \verbatim
243 *> LDU is INTEGER
244 *> The leading dimension of the array U, LDU >= 1.
245 *> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
246 *> \endverbatim
247 *>
248 *> \param[out] V
249 *> \verbatim
250 *> V is DOUBLE PRECISION array, dimension ( LDV, N )
251 *> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
252 *> the right singular vectors;
253 *> If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
254 *> then V is used as workspace if the pprocedure
255 *> replaces A with A^t. In that case, [U] is computed
256 *> in V as right singular vectors of A^t and then
257 *> copied back to the U array. This 'W' option is just
258 *> a reminder to the caller that in this case V is
259 *> reserved as workspace of length N*N.
260 *> If JOBV = 'N' V is not referenced.
261 *> \endverbatim
262 *>
263 *> \param[in] LDV
264 *> \verbatim
265 *> LDV is INTEGER
266 *> The leading dimension of the array V, LDV >= 1.
267 *> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
268 *> \endverbatim
269 *>
270 *> \param[out] WORK
271 *> \verbatim
272 *> WORK is DOUBLE PRECISION array, dimension at least LWORK.
273 *> On exit, if N.GT.0 .AND. M.GT.0 (else not referenced),
274 *> WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
275 *> that SCALE*SVA(1:N) are the computed singular values
276 *> of A. (See the description of SVA().)
277 *> WORK(2) = See the description of WORK(1).
278 *> WORK(3) = SCONDA is an estimate for the condition number of
279 *> column equilibrated A. (If JOBA .EQ. 'E' or 'G')
280 *> SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
281 *> It is computed using DPOCON. It holds
282 *> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
283 *> where R is the triangular factor from the QRF of A.
284 *> However, if R is truncated and the numerical rank is
285 *> determined to be strictly smaller than N, SCONDA is
286 *> returned as -1, thus indicating that the smallest
287 *> singular values might be lost.
288 *>
289 *> If full SVD is needed, the following two condition numbers are
290 *> useful for the analysis of the algorithm. They are provied for
291 *> a developer/implementer who is familiar with the details of
292 *> the method.
293 *>
294 *> WORK(4) = an estimate of the scaled condition number of the
295 *> triangular factor in the first QR factorization.
296 *> WORK(5) = an estimate of the scaled condition number of the
297 *> triangular factor in the second QR factorization.
298 *> The following two parameters are computed if JOBT .EQ. 'T'.
299 *> They are provided for a developer/implementer who is familiar
300 *> with the details of the method.
301 *>
302 *> WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
303 *> of diag(A^t*A) / Trace(A^t*A) taken as point in the
304 *> probability simplex.
305 *> WORK(7) = the entropy of A*A^t.
306 *> \endverbatim
307 *>
308 *> \param[in] LWORK
309 *> \verbatim
310 *> LWORK is INTEGER
311 *> Length of WORK to confirm proper allocation of work space.
312 *> LWORK depends on the job:
313 *>
314 *> If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
315 *> -> .. no scaled condition estimate required (JOBE.EQ.'N'):
316 *> LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
317 *> ->> For optimal performance (blocked code) the optimal value
318 *> is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
319 *> block size for DGEQP3 and DGEQRF.
320 *> In general, optimal LWORK is computed as
321 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
322 *> -> .. an estimate of the scaled condition number of A is
323 *> required (JOBA='E', 'G'). In this case, LWORK is the maximum
324 *> of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
325 *> ->> For optimal performance (blocked code) the optimal value
326 *> is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
327 *> In general, the optimal length LWORK is computed as
328 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
329 *> N+N*N+LWORK(DPOCON),7).
330 *>
331 *> If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
332 *> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
333 *> -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
334 *> where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
335 *> DORMLQ. In general, the optimal length LWORK is computed as
336 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
337 *> N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
338 *>
339 *> If SIGMA and the left singular vectors are needed
340 *> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
341 *> -> For optimal performance:
342 *> if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
343 *> if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
344 *> where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
345 *> In general, the optimal length LWORK is computed as
346 *> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
347 *> 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
348 *> Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or
349 *> M*NB (for JOBU.EQ.'F').
350 *>
351 *> If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
352 *> -> if JOBV.EQ.'V'
353 *> the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
354 *> -> if JOBV.EQ.'J' the minimal requirement is
355 *> LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
356 *> -> For optimal performance, LWORK should be additionally
357 *> larger than N+M*NB, where NB is the optimal block size
358 *> for DORMQR.
359 *> \endverbatim
360 *>
361 *> \param[out] IWORK
362 *> \verbatim
363 *> IWORK is INTEGER array, dimension M+3*N.
364 *> On exit,
365 *> IWORK(1) = the numerical rank determined after the initial
366 *> QR factorization with pivoting. See the descriptions
367 *> of JOBA and JOBR.
368 *> IWORK(2) = the number of the computed nonzero singular values
369 *> IWORK(3) = if nonzero, a warning message:
370 *> If IWORK(3).EQ.1 then some of the column norms of A
371 *> were denormalized floats. The requested high accuracy
372 *> is not warranted by the data.
373 *> \endverbatim
374 *>
375 *> \param[out] INFO
376 *> \verbatim
377 *> INFO is INTEGER
378 *> < 0 : if INFO = -i, then the i-th argument had an illegal value.
379 *> = 0 : successfull exit;
380 *> > 0 : DGEJSV did not converge in the maximal allowed number
381 *> of sweeps. The computed values may be inaccurate.
382 *> \endverbatim
383 *
384 * Authors:
385 * ========
386 *
387 *> \author Univ. of Tennessee
388 *> \author Univ. of California Berkeley
389 *> \author Univ. of Colorado Denver
390 *> \author NAG Ltd.
391 *
392 *> \date September 2012
393 *
394 *> \ingroup doubleGEsing
395 *
396 *> \par Further Details:
397 * =====================
398 *>
399 *> \verbatim
400 *>
401 *> DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3,
402 *> DGEQRF, and DGELQF as preprocessors and preconditioners. Optionally, an
403 *> additional row pivoting can be used as a preprocessor, which in some
404 *> cases results in much higher accuracy. An example is matrix A with the
405 *> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
406 *> diagonal matrices and C is well-conditioned matrix. In that case, complete
407 *> pivoting in the first QR factorizations provides accuracy dependent on the
408 *> condition number of C, and independent of D1, D2. Such higher accuracy is
409 *> not completely understood theoretically, but it works well in practice.
410 *> Further, if A can be written as A = B*D, with well-conditioned B and some
411 *> diagonal D, then the high accuracy is guaranteed, both theoretically and
412 *> in software, independent of D. For more details see [1], [2].
413 *> The computational range for the singular values can be the full range
414 *> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
415 *> & LAPACK routines called by DGEJSV are implemented to work in that range.
416 *> If that is not the case, then the restriction for safe computation with
417 *> the singular values in the range of normalized IEEE numbers is that the
418 *> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
419 *> overflow. This code (DGEJSV) is best used in this restricted range,
420 *> meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
421 *> returned as zeros. See JOBR for details on this.
422 *> Further, this implementation is somewhat slower than the one described
423 *> in [1,2] due to replacement of some non-LAPACK components, and because
424 *> the choice of some tuning parameters in the iterative part (DGESVJ) is
425 *> left to the implementer on a particular machine.
426 *> The rank revealing QR factorization (in this code: DGEQP3) should be
427 *> implemented as in [3]. We have a new version of DGEQP3 under development
428 *> that is more robust than the current one in LAPACK, with a cleaner cut in
429 *> rank defficient cases. It will be available in the SIGMA library [4].
430 *> If M is much larger than N, it is obvious that the inital QRF with
431 *> column pivoting can be preprocessed by the QRF without pivoting. That
432 *> well known trick is not used in DGEJSV because in some cases heavy row
433 *> weighting can be treated with complete pivoting. The overhead in cases
434 *> M much larger than N is then only due to pivoting, but the benefits in
435 *> terms of accuracy have prevailed. The implementer/user can incorporate
436 *> this extra QRF step easily. The implementer can also improve data movement
437 *> (matrix transpose, matrix copy, matrix transposed copy) - this
438 *> implementation of DGEJSV uses only the simplest, naive data movement.
439 *> \endverbatim
440 *
441 *> \par Contributors:
442 * ==================
443 *>
444 *> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
445 *
446 *> \par References:
447 * ================
448 *>
449 *> \verbatim
450 *>
451 *> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
452 *> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
453 *> LAPACK Working note 169.
454 *> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
455 *> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
456 *> LAPACK Working note 170.
457 *> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
458 *> factorization software - a case study.
459 *> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
460 *> LAPACK Working note 176.
461 *> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
462 *> QSVD, (H,K)-SVD computations.
463 *> Department of Mathematics, University of Zagreb, 2008.
464 *> \endverbatim
465 *
466 *> \par Bugs, examples and comments:
467 * =================================
468 *>
469 *> Please report all bugs and send interesting examples and/or comments to
470 *> drmac@math.hr. Thank you.
471 *>
472 * =====================================================================
473  SUBROUTINE dgejsv( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
474  $ m, n, a, lda, sva, u, ldu, v, ldv,
475  $ work, lwork, iwork, info )
476 *
477 * -- LAPACK computational routine (version 3.4.2) --
478 * -- LAPACK is a software package provided by Univ. of Tennessee, --
479 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
480 * September 2012
481 *
482 * .. Scalar Arguments ..
483  IMPLICIT NONE
484  INTEGER info, lda, ldu, ldv, lwork, m, n
485 * ..
486 * .. Array Arguments ..
487  DOUBLE PRECISION a( lda, * ), sva( n ), u( ldu, * ), v( ldv, * ),
488  $ work( lwork )
489  INTEGER iwork( * )
490  CHARACTER*1 joba, jobp, jobr, jobt, jobu, jobv
491 * ..
492 *
493 * ===========================================================================
494 *
495 * .. Local Parameters ..
496  DOUBLE PRECISION zero, one
497  parameter( zero = 0.0d0, one = 1.0d0 )
498 * ..
499 * .. Local Scalars ..
500  DOUBLE PRECISION aapp, aaqq, aatmax, aatmin, big, big1, cond_ok,
501  $ condr1, condr2, entra, entrat, epsln, maxprj, scalem,
502  $ sconda, sfmin, small, temp1, uscal1, uscal2, xsc
503  INTEGER ierr, n1, nr, numrank, p, q, warning
504  LOGICAL almort, defr, errest, goscal, jracc, kill, lsvec,
505  $ l2aber, l2kill, l2pert, l2rank, l2tran,
506  $ noscal, rowpiv, rsvec, transp
507 * ..
508 * .. Intrinsic Functions ..
509  INTRINSIC dabs, dlog, dmax1, dmin1, dble,
510  $ max0, min0, idnint, dsign, dsqrt
511 * ..
512 * .. External Functions ..
513  DOUBLE PRECISION dlamch, dnrm2
514  INTEGER idamax
515  LOGICAL lsame
516  EXTERNAL idamax, lsame, dlamch, dnrm2
517 * ..
518 * .. External Subroutines ..
519  EXTERNAL dcopy, dgelqf, dgeqp3, dgeqrf, dlacpy, dlascl,
522 *
523  EXTERNAL dgesvj
524 * ..
525 *
526 * Test the input arguments
527 *
528  lsvec = lsame( jobu, 'U' ) .OR. lsame( jobu, 'F' )
529  jracc = lsame( jobv, 'J' )
530  rsvec = lsame( jobv, 'V' ) .OR. jracc
531  rowpiv = lsame( joba, 'F' ) .OR. lsame( joba, 'G' )
532  l2rank = lsame( joba, 'R' )
533  l2aber = lsame( joba, 'A' )
534  errest = lsame( joba, 'E' ) .OR. lsame( joba, 'G' )
535  l2tran = lsame( jobt, 'T' )
536  l2kill = lsame( jobr, 'R' )
537  defr = lsame( jobr, 'N' )
538  l2pert = lsame( jobp, 'P' )
539 *
540  IF ( .NOT.(rowpiv .OR. l2rank .OR. l2aber .OR.
541  $ errest .OR. lsame( joba, 'C' ) )) THEN
542  info = - 1
543  ELSE IF ( .NOT.( lsvec .OR. lsame( jobu, 'N' ) .OR.
544  $ lsame( jobu, 'W' )) ) THEN
545  info = - 2
546  ELSE IF ( .NOT.( rsvec .OR. lsame( jobv, 'N' ) .OR.
547  $ lsame( jobv, 'W' )) .OR. ( jracc .AND. (.NOT.lsvec) ) ) THEN
548  info = - 3
549  ELSE IF ( .NOT. ( l2kill .OR. defr ) ) THEN
550  info = - 4
551  ELSE IF ( .NOT. ( l2tran .OR. lsame( jobt, 'N' ) ) ) THEN
552  info = - 5
553  ELSE IF ( .NOT. ( l2pert .OR. lsame( jobp, 'N' ) ) ) THEN
554  info = - 6
555  ELSE IF ( m .LT. 0 ) THEN
556  info = - 7
557  ELSE IF ( ( n .LT. 0 ) .OR. ( n .GT. m ) ) THEN
558  info = - 8
559  ELSE IF ( lda .LT. m ) THEN
560  info = - 10
561  ELSE IF ( lsvec .AND. ( ldu .LT. m ) ) THEN
562  info = - 13
563  ELSE IF ( rsvec .AND. ( ldv .LT. n ) ) THEN
564  info = - 14
565  ELSE IF ( (.NOT.(lsvec .OR. rsvec .OR. errest).AND.
566  & (lwork .LT. max0(7,4*n+1,2*m+n))) .OR.
567  & (.NOT.(lsvec .OR. rsvec) .AND. errest .AND.
568  & (lwork .LT. max0(7,4*n+n*n,2*m+n))) .OR.
569  & (lsvec .AND. (.NOT.rsvec) .AND. (lwork .LT. max0(7,2*m+n,4*n+1)))
570  & .OR.
571  & (rsvec .AND. (.NOT.lsvec) .AND. (lwork .LT. max0(7,2*m+n,4*n+1)))
572  & .OR.
573  & (lsvec .AND. rsvec .AND. (.NOT.jracc) .AND.
574  & (lwork.LT.max0(2*m+n,6*n+2*n*n)))
575  & .OR. (lsvec .AND. rsvec .AND. jracc .AND.
576  & lwork.LT.max0(2*m+n,4*n+n*n,2*n+n*n+6)))
577  & THEN
578  info = - 17
579  ELSE
580 * #:)
581  info = 0
582  END IF
583 *
584  IF ( info .NE. 0 ) THEN
585 * #:(
586  CALL xerbla( 'DGEJSV', - info )
587  RETURN
588  END IF
589 *
590 * Quick return for void matrix (Y3K safe)
591 * #:)
592  IF ( ( m .EQ. 0 ) .OR. ( n .EQ. 0 ) ) RETURN
593 *
594 * Determine whether the matrix U should be M x N or M x M
595 *
596  IF ( lsvec ) THEN
597  n1 = n
598  IF ( lsame( jobu, 'F' ) ) n1 = m
599  END IF
600 *
601 * Set numerical parameters
602 *
603 *! NOTE: Make sure DLAMCH() does not fail on the target architecture.
604 *
605  epsln = dlamch('Epsilon')
606  sfmin = dlamch('SafeMinimum')
607  small = sfmin / epsln
608  big = dlamch('O')
609 * BIG = ONE / SFMIN
610 *
611 * Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
612 *
613 *(!) If necessary, scale SVA() to protect the largest norm from
614 * overflow. It is possible that this scaling pushes the smallest
615 * column norm left from the underflow threshold (extreme case).
616 *
617  scalem = one / dsqrt(dble(m)*dble(n))
618  noscal = .true.
619  goscal = .true.
620  DO 1874 p = 1, n
621  aapp = zero
622  aaqq = one
623  CALL dlassq( m, a(1,p), 1, aapp, aaqq )
624  IF ( aapp .GT. big ) THEN
625  info = - 9
626  CALL xerbla( 'DGEJSV', -info )
627  RETURN
628  END IF
629  aaqq = dsqrt(aaqq)
630  IF ( ( aapp .LT. (big / aaqq) ) .AND. noscal ) THEN
631  sva(p) = aapp * aaqq
632  ELSE
633  noscal = .false.
634  sva(p) = aapp * ( aaqq * scalem )
635  IF ( goscal ) THEN
636  goscal = .false.
637  CALL dscal( p-1, scalem, sva, 1 )
638  END IF
639  END IF
640  1874 CONTINUE
641 *
642  IF ( noscal ) scalem = one
643 *
644  aapp = zero
645  aaqq = big
646  DO 4781 p = 1, n
647  aapp = dmax1( aapp, sva(p) )
648  IF ( sva(p) .NE. zero ) aaqq = dmin1( aaqq, sva(p) )
649  4781 CONTINUE
650 *
651 * Quick return for zero M x N matrix
652 * #:)
653  IF ( aapp .EQ. zero ) THEN
654  IF ( lsvec ) CALL dlaset( 'G', m, n1, zero, one, u, ldu )
655  IF ( rsvec ) CALL dlaset( 'G', n, n, zero, one, v, ldv )
656  work(1) = one
657  work(2) = one
658  IF ( errest ) work(3) = one
659  IF ( lsvec .AND. rsvec ) THEN
660  work(4) = one
661  work(5) = one
662  END IF
663  IF ( l2tran ) THEN
664  work(6) = zero
665  work(7) = zero
666  END IF
667  iwork(1) = 0
668  iwork(2) = 0
669  iwork(3) = 0
670  RETURN
671  END IF
672 *
673 * Issue warning if denormalized column norms detected. Override the
674 * high relative accuracy request. Issue licence to kill columns
675 * (set them to zero) whose norm is less than sigma_max / BIG (roughly).
676 * #:(
677  warning = 0
678  IF ( aaqq .LE. sfmin ) THEN
679  l2rank = .true.
680  l2kill = .true.
681  warning = 1
682  END IF
683 *
684 * Quick return for one-column matrix
685 * #:)
686  IF ( n .EQ. 1 ) THEN
687 *
688  IF ( lsvec ) THEN
689  CALL dlascl( 'G',0,0,sva(1),scalem, m,1,a(1,1),lda,ierr )
690  CALL dlacpy( 'A', m, 1, a, lda, u, ldu )
691 * computing all M left singular vectors of the M x 1 matrix
692  IF ( n1 .NE. n ) THEN
693  CALL dgeqrf( m, n, u,ldu, work, work(n+1),lwork-n,ierr )
694  CALL dorgqr( m,n1,1, u,ldu,work,work(n+1),lwork-n,ierr )
695  CALL dcopy( m, a(1,1), 1, u(1,1), 1 )
696  END IF
697  END IF
698  IF ( rsvec ) THEN
699  v(1,1) = one
700  END IF
701  IF ( sva(1) .LT. (big*scalem) ) THEN
702  sva(1) = sva(1) / scalem
703  scalem = one
704  END IF
705  work(1) = one / scalem
706  work(2) = one
707  IF ( sva(1) .NE. zero ) THEN
708  iwork(1) = 1
709  IF ( ( sva(1) / scalem) .GE. sfmin ) THEN
710  iwork(2) = 1
711  ELSE
712  iwork(2) = 0
713  END IF
714  ELSE
715  iwork(1) = 0
716  iwork(2) = 0
717  END IF
718  IF ( errest ) work(3) = one
719  IF ( lsvec .AND. rsvec ) THEN
720  work(4) = one
721  work(5) = one
722  END IF
723  IF ( l2tran ) THEN
724  work(6) = zero
725  work(7) = zero
726  END IF
727  RETURN
728 *
729  END IF
730 *
731  transp = .false.
732  l2tran = l2tran .AND. ( m .EQ. n )
733 *
734  aatmax = -one
735  aatmin = big
736  IF ( rowpiv .OR. l2tran ) THEN
737 *
738 * Compute the row norms, needed to determine row pivoting sequence
739 * (in the case of heavily row weighted A, row pivoting is strongly
740 * advised) and to collect information needed to compare the
741 * structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).
742 *
743  IF ( l2tran ) THEN
744  DO 1950 p = 1, m
745  xsc = zero
746  temp1 = one
747  CALL dlassq( n, a(p,1), lda, xsc, temp1 )
748 * DLASSQ gets both the ell_2 and the ell_infinity norm
749 * in one pass through the vector
750  work(m+n+p) = xsc * scalem
751  work(n+p) = xsc * (scalem*dsqrt(temp1))
752  aatmax = dmax1( aatmax, work(n+p) )
753  IF (work(n+p) .NE. zero) aatmin = dmin1(aatmin,work(n+p))
754  1950 CONTINUE
755  ELSE
756  DO 1904 p = 1, m
757  work(m+n+p) = scalem*dabs( a(p,idamax(n,a(p,1),lda)) )
758  aatmax = dmax1( aatmax, work(m+n+p) )
759  aatmin = dmin1( aatmin, work(m+n+p) )
760  1904 CONTINUE
761  END IF
762 *
763  END IF
764 *
765 * For square matrix A try to determine whether A^t would be better
766 * input for the preconditioned Jacobi SVD, with faster convergence.
767 * The decision is based on an O(N) function of the vector of column
768 * and row norms of A, based on the Shannon entropy. This should give
769 * the right choice in most cases when the difference actually matters.
770 * It may fail and pick the slower converging side.
771 *
772  entra = zero
773  entrat = zero
774  IF ( l2tran ) THEN
775 *
776  xsc = zero
777  temp1 = one
778  CALL dlassq( n, sva, 1, xsc, temp1 )
779  temp1 = one / temp1
780 *
781  entra = zero
782  DO 1113 p = 1, n
783  big1 = ( ( sva(p) / xsc )**2 ) * temp1
784  IF ( big1 .NE. zero ) entra = entra + big1 * dlog(big1)
785  1113 CONTINUE
786  entra = - entra / dlog(dble(n))
787 *
788 * Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.
789 * It is derived from the diagonal of A^t * A. Do the same with the
790 * diagonal of A * A^t, compute the entropy of the corresponding
791 * probability distribution. Note that A * A^t and A^t * A have the
792 * same trace.
793 *
794  entrat = zero
795  DO 1114 p = n+1, n+m
796  big1 = ( ( work(p) / xsc )**2 ) * temp1
797  IF ( big1 .NE. zero ) entrat = entrat + big1 * dlog(big1)
798  1114 CONTINUE
799  entrat = - entrat / dlog(dble(m))
800 *
801 * Analyze the entropies and decide A or A^t. Smaller entropy
802 * usually means better input for the algorithm.
803 *
804  transp = ( entrat .LT. entra )
805 *
806 * If A^t is better than A, transpose A.
807 *
808  IF ( transp ) THEN
809 * In an optimal implementation, this trivial transpose
810 * should be replaced with faster transpose.
811  DO 1115 p = 1, n - 1
812  DO 1116 q = p + 1, n
813  temp1 = a(q,p)
814  a(q,p) = a(p,q)
815  a(p,q) = temp1
816  1116 CONTINUE
817  1115 CONTINUE
818  DO 1117 p = 1, n
819  work(m+n+p) = sva(p)
820  sva(p) = work(n+p)
821  1117 CONTINUE
822  temp1 = aapp
823  aapp = aatmax
824  aatmax = temp1
825  temp1 = aaqq
826  aaqq = aatmin
827  aatmin = temp1
828  kill = lsvec
829  lsvec = rsvec
830  rsvec = kill
831  IF ( lsvec ) n1 = n
832 *
833  rowpiv = .true.
834  END IF
835 *
836  END IF
837 * END IF L2TRAN
838 *
839 * Scale the matrix so that its maximal singular value remains less
840 * than DSQRT(BIG) -- the matrix is scaled so that its maximal column
841 * has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep
842 * DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and
843 * BLAS routines that, in some implementations, are not capable of
844 * working in the full interval [SFMIN,BIG] and that they may provoke
845 * overflows in the intermediate results. If the singular values spread
846 * from SFMIN to BIG, then DGESVJ will compute them. So, in that case,
847 * one should use DGESVJ instead of DGEJSV.
848 *
849  big1 = dsqrt( big )
850  temp1 = dsqrt( big / dble(n) )
851 *
852  CALL dlascl( 'G', 0, 0, aapp, temp1, n, 1, sva, n, ierr )
853  IF ( aaqq .GT. (aapp * sfmin) ) THEN
854  aaqq = ( aaqq / aapp ) * temp1
855  ELSE
856  aaqq = ( aaqq * temp1 ) / aapp
857  END IF
858  temp1 = temp1 * scalem
859  CALL dlascl( 'G', 0, 0, aapp, temp1, m, n, a, lda, ierr )
860 *
861 * To undo scaling at the end of this procedure, multiply the
862 * computed singular values with USCAL2 / USCAL1.
863 *
864  uscal1 = temp1
865  uscal2 = aapp
866 *
867  IF ( l2kill ) THEN
868 * L2KILL enforces computation of nonzero singular values in
869 * the restricted range of condition number of the initial A,
870 * sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN).
871  xsc = dsqrt( sfmin )
872  ELSE
873  xsc = small
874 *
875 * Now, if the condition number of A is too big,
876 * sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN,
877 * as a precaution measure, the full SVD is computed using DGESVJ
878 * with accumulated Jacobi rotations. This provides numerically
879 * more robust computation, at the cost of slightly increased run
880 * time. Depending on the concrete implementation of BLAS and LAPACK
881 * (i.e. how they behave in presence of extreme ill-conditioning) the
882 * implementor may decide to remove this switch.
883  IF ( ( aaqq.LT.dsqrt(sfmin) ) .AND. lsvec .AND. rsvec ) THEN
884  jracc = .true.
885  END IF
886 *
887  END IF
888  IF ( aaqq .LT. xsc ) THEN
889  DO 700 p = 1, n
890  IF ( sva(p) .LT. xsc ) THEN
891  CALL dlaset( 'A', m, 1, zero, zero, a(1,p), lda )
892  sva(p) = zero
893  END IF
894  700 CONTINUE
895  END IF
896 *
897 * Preconditioning using QR factorization with pivoting
898 *
899  IF ( rowpiv ) THEN
900 * Optional row permutation (Bjoerck row pivoting):
901 * A result by Cox and Higham shows that the Bjoerck's
902 * row pivoting combined with standard column pivoting
903 * has similar effect as Powell-Reid complete pivoting.
904 * The ell-infinity norms of A are made nonincreasing.
905  DO 1952 p = 1, m - 1
906  q = idamax( m-p+1, work(m+n+p), 1 ) + p - 1
907  iwork(2*n+p) = q
908  IF ( p .NE. q ) THEN
909  temp1 = work(m+n+p)
910  work(m+n+p) = work(m+n+q)
911  work(m+n+q) = temp1
912  END IF
913  1952 CONTINUE
914  CALL dlaswp( n, a, lda, 1, m-1, iwork(2*n+1), 1 )
915  END IF
916 *
917 * End of the preparation phase (scaling, optional sorting and
918 * transposing, optional flushing of small columns).
919 *
920 * Preconditioning
921 *
922 * If the full SVD is needed, the right singular vectors are computed
923 * from a matrix equation, and for that we need theoretical analysis
924 * of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF.
925 * In all other cases the first RR QRF can be chosen by other criteria
926 * (eg speed by replacing global with restricted window pivoting, such
927 * as in SGEQPX from TOMS # 782). Good results will be obtained using
928 * SGEQPX with properly (!) chosen numerical parameters.
929 * Any improvement of DGEQP3 improves overal performance of DGEJSV.
930 *
931 * A * P1 = Q1 * [ R1^t 0]^t:
932  DO 1963 p = 1, n
933 * .. all columns are free columns
934  iwork(p) = 0
935  1963 CONTINUE
936  CALL dgeqp3( m,n,a,lda, iwork,work, work(n+1),lwork-n, ierr )
937 *
938 * The upper triangular matrix R1 from the first QRF is inspected for
939 * rank deficiency and possibilities for deflation, or possible
940 * ill-conditioning. Depending on the user specified flag L2RANK,
941 * the procedure explores possibilities to reduce the numerical
942 * rank by inspecting the computed upper triangular factor. If
943 * L2RANK or L2ABER are up, then DGEJSV will compute the SVD of
944 * A + dA, where ||dA|| <= f(M,N)*EPSLN.
945 *
946  nr = 1
947  IF ( l2aber ) THEN
948 * Standard absolute error bound suffices. All sigma_i with
949 * sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
950 * agressive enforcement of lower numerical rank by introducing a
951 * backward error of the order of N*EPSLN*||A||.
952  temp1 = dsqrt(dble(n))*epsln
953  DO 3001 p = 2, n
954  IF ( dabs(a(p,p)) .GE. (temp1*dabs(a(1,1))) ) THEN
955  nr = nr + 1
956  ELSE
957  go to 3002
958  END IF
959  3001 CONTINUE
960  3002 CONTINUE
961  ELSE IF ( l2rank ) THEN
962 * .. similarly as above, only slightly more gentle (less agressive).
963 * Sudden drop on the diagonal of R1 is used as the criterion for
964 * close-to-rank-defficient.
965  temp1 = dsqrt(sfmin)
966  DO 3401 p = 2, n
967  IF ( ( dabs(a(p,p)) .LT. (epsln*dabs(a(p-1,p-1))) ) .OR.
968  $ ( dabs(a(p,p)) .LT. small ) .OR.
969  $ ( l2kill .AND. (dabs(a(p,p)) .LT. temp1) ) ) go to 3402
970  nr = nr + 1
971  3401 CONTINUE
972  3402 CONTINUE
973 *
974  ELSE
975 * The goal is high relative accuracy. However, if the matrix
976 * has high scaled condition number the relative accuracy is in
977 * general not feasible. Later on, a condition number estimator
978 * will be deployed to estimate the scaled condition number.
979 * Here we just remove the underflowed part of the triangular
980 * factor. This prevents the situation in which the code is
981 * working hard to get the accuracy not warranted by the data.
982  temp1 = dsqrt(sfmin)
983  DO 3301 p = 2, n
984  IF ( ( dabs(a(p,p)) .LT. small ) .OR.
985  $ ( l2kill .AND. (dabs(a(p,p)) .LT. temp1) ) ) go to 3302
986  nr = nr + 1
987  3301 CONTINUE
988  3302 CONTINUE
989 *
990  END IF
991 *
992  almort = .false.
993  IF ( nr .EQ. n ) THEN
994  maxprj = one
995  DO 3051 p = 2, n
996  temp1 = dabs(a(p,p)) / sva(iwork(p))
997  maxprj = dmin1( maxprj, temp1 )
998  3051 CONTINUE
999  IF ( maxprj**2 .GE. one - dble(n)*epsln ) almort = .true.
1000  END IF
1001 *
1002 *
1003  sconda = - one
1004  condr1 = - one
1005  condr2 = - one
1006 *
1007  IF ( errest ) THEN
1008  IF ( n .EQ. nr ) THEN
1009  IF ( rsvec ) THEN
1010 * .. V is available as workspace
1011  CALL dlacpy( 'U', n, n, a, lda, v, ldv )
1012  DO 3053 p = 1, n
1013  temp1 = sva(iwork(p))
1014  CALL dscal( p, one/temp1, v(1,p), 1 )
1015  3053 CONTINUE
1016  CALL dpocon( 'U', n, v, ldv, one, temp1,
1017  $ work(n+1), iwork(2*n+m+1), ierr )
1018  ELSE IF ( lsvec ) THEN
1019 * .. U is available as workspace
1020  CALL dlacpy( 'U', n, n, a, lda, u, ldu )
1021  DO 3054 p = 1, n
1022  temp1 = sva(iwork(p))
1023  CALL dscal( p, one/temp1, u(1,p), 1 )
1024  3054 CONTINUE
1025  CALL dpocon( 'U', n, u, ldu, one, temp1,
1026  $ work(n+1), iwork(2*n+m+1), ierr )
1027  ELSE
1028  CALL dlacpy( 'U', n, n, a, lda, work(n+1), n )
1029  DO 3052 p = 1, n
1030  temp1 = sva(iwork(p))
1031  CALL dscal( p, one/temp1, work(n+(p-1)*n+1), 1 )
1032  3052 CONTINUE
1033 * .. the columns of R are scaled to have unit Euclidean lengths.
1034  CALL dpocon( 'U', n, work(n+1), n, one, temp1,
1035  $ work(n+n*n+1), iwork(2*n+m+1), ierr )
1036  END IF
1037  sconda = one / dsqrt(temp1)
1038 * SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
1039 * N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
1040  ELSE
1041  sconda = - one
1042  END IF
1043  END IF
1044 *
1045  l2pert = l2pert .AND. ( dabs( a(1,1)/a(nr,nr) ) .GT. dsqrt(big1) )
1046 * If there is no violent scaling, artificial perturbation is not needed.
1047 *
1048 * Phase 3:
1049 *
1050  IF ( .NOT. ( rsvec .OR. lsvec ) ) THEN
1051 *
1052 * Singular Values only
1053 *
1054 * .. transpose A(1:NR,1:N)
1055  DO 1946 p = 1, min0( n-1, nr )
1056  CALL dcopy( n-p, a(p,p+1), lda, a(p+1,p), 1 )
1057  1946 CONTINUE
1058 *
1059 * The following two DO-loops introduce small relative perturbation
1060 * into the strict upper triangle of the lower triangular matrix.
1061 * Small entries below the main diagonal are also changed.
1062 * This modification is useful if the computing environment does not
1063 * provide/allow FLUSH TO ZERO underflow, for it prevents many
1064 * annoying denormalized numbers in case of strongly scaled matrices.
1065 * The perturbation is structured so that it does not introduce any
1066 * new perturbation of the singular values, and it does not destroy
1067 * the job done by the preconditioner.
1068 * The licence for this perturbation is in the variable L2PERT, which
1069 * should be .FALSE. if FLUSH TO ZERO underflow is active.
1070 *
1071  IF ( .NOT. almort ) THEN
1072 *
1073  IF ( l2pert ) THEN
1074 * XSC = DSQRT(SMALL)
1075  xsc = epsln / dble(n)
1076  DO 4947 q = 1, nr
1077  temp1 = xsc*dabs(a(q,q))
1078  DO 4949 p = 1, n
1079  IF ( ( (p.GT.q) .AND. (dabs(a(p,q)).LE.temp1) )
1080  $ .OR. ( p .LT. q ) )
1081  $ a(p,q) = dsign( temp1, a(p,q) )
1082  4949 CONTINUE
1083  4947 CONTINUE
1084  ELSE
1085  CALL dlaset( 'U', nr-1,nr-1, zero,zero, a(1,2),lda )
1086  END IF
1087 *
1088 * .. second preconditioning using the QR factorization
1089 *
1090  CALL dgeqrf( n,nr, a,lda, work, work(n+1),lwork-n, ierr )
1091 *
1092 * .. and transpose upper to lower triangular
1093  DO 1948 p = 1, nr - 1
1094  CALL dcopy( nr-p, a(p,p+1), lda, a(p+1,p), 1 )
1095  1948 CONTINUE
1096 *
1097  END IF
1098 *
1099 * Row-cyclic Jacobi SVD algorithm with column pivoting
1100 *
1101 * .. again some perturbation (a "background noise") is added
1102 * to drown denormals
1103  IF ( l2pert ) THEN
1104 * XSC = DSQRT(SMALL)
1105  xsc = epsln / dble(n)
1106  DO 1947 q = 1, nr
1107  temp1 = xsc*dabs(a(q,q))
1108  DO 1949 p = 1, nr
1109  IF ( ( (p.GT.q) .AND. (dabs(a(p,q)).LE.temp1) )
1110  $ .OR. ( p .LT. q ) )
1111  $ a(p,q) = dsign( temp1, a(p,q) )
1112  1949 CONTINUE
1113  1947 CONTINUE
1114  ELSE
1115  CALL dlaset( 'U', nr-1, nr-1, zero, zero, a(1,2), lda )
1116  END IF
1117 *
1118 * .. and one-sided Jacobi rotations are started on a lower
1119 * triangular matrix (plus perturbation which is ignored in
1120 * the part which destroys triangular form (confusing?!))
1121 *
1122  CALL dgesvj( 'L', 'NoU', 'NoV', nr, nr, a, lda, sva,
1123  $ n, v, ldv, work, lwork, info )
1124 *
1125  scalem = work(1)
1126  numrank = idnint(work(2))
1127 *
1128 *
1129  ELSE IF ( rsvec .AND. ( .NOT. lsvec ) ) THEN
1130 *
1131 * -> Singular Values and Right Singular Vectors <-
1132 *
1133  IF ( almort ) THEN
1134 *
1135 * .. in this case NR equals N
1136  DO 1998 p = 1, nr
1137  CALL dcopy( n-p+1, a(p,p), lda, v(p,p), 1 )
1138  1998 CONTINUE
1139  CALL dlaset( 'Upper', nr-1, nr-1, zero, zero, v(1,2), ldv )
1140 *
1141  CALL dgesvj( 'L','U','N', n, nr, v,ldv, sva, nr, a,lda,
1142  $ work, lwork, info )
1143  scalem = work(1)
1144  numrank = idnint(work(2))
1145 
1146  ELSE
1147 *
1148 * .. two more QR factorizations ( one QRF is not enough, two require
1149 * accumulated product of Jacobi rotations, three are perfect )
1150 *
1151  CALL dlaset( 'Lower', nr-1, nr-1, zero, zero, a(2,1), lda )
1152  CALL dgelqf( nr, n, a, lda, work, work(n+1), lwork-n, ierr)
1153  CALL dlacpy( 'Lower', nr, nr, a, lda, v, ldv )
1154  CALL dlaset( 'Upper', nr-1, nr-1, zero, zero, v(1,2), ldv )
1155  CALL dgeqrf( nr, nr, v, ldv, work(n+1), work(2*n+1),
1156  $ lwork-2*n, ierr )
1157  DO 8998 p = 1, nr
1158  CALL dcopy( nr-p+1, v(p,p), ldv, v(p,p), 1 )
1159  8998 CONTINUE
1160  CALL dlaset( 'Upper', nr-1, nr-1, zero, zero, v(1,2), ldv )
1161 *
1162  CALL dgesvj( 'Lower', 'U','N', nr, nr, v,ldv, sva, nr, u,
1163  $ ldu, work(n+1), lwork, info )
1164  scalem = work(n+1)
1165  numrank = idnint(work(n+2))
1166  IF ( nr .LT. n ) THEN
1167  CALL dlaset( 'A',n-nr, nr, zero,zero, v(nr+1,1), ldv )
1168  CALL dlaset( 'A',nr, n-nr, zero,zero, v(1,nr+1), ldv )
1169  CALL dlaset( 'A',n-nr,n-nr,zero,one, v(nr+1,nr+1), ldv )
1170  END IF
1171 *
1172  CALL dormlq( 'Left', 'Transpose', n, n, nr, a, lda, work,
1173  $ v, ldv, work(n+1), lwork-n, ierr )
1174 *
1175  END IF
1176 *
1177  DO 8991 p = 1, n
1178  CALL dcopy( n, v(p,1), ldv, a(iwork(p),1), lda )
1179  8991 CONTINUE
1180  CALL dlacpy( 'All', n, n, a, lda, v, ldv )
1181 *
1182  IF ( transp ) THEN
1183  CALL dlacpy( 'All', n, n, v, ldv, u, ldu )
1184  END IF
1185 *
1186  ELSE IF ( lsvec .AND. ( .NOT. rsvec ) ) THEN
1187 *
1188 * .. Singular Values and Left Singular Vectors ..
1189 *
1190 * .. second preconditioning step to avoid need to accumulate
1191 * Jacobi rotations in the Jacobi iterations.
1192  DO 1965 p = 1, nr
1193  CALL dcopy( n-p+1, a(p,p), lda, u(p,p), 1 )
1194  1965 CONTINUE
1195  CALL dlaset( 'Upper', nr-1, nr-1, zero, zero, u(1,2), ldu )
1196 *
1197  CALL dgeqrf( n, nr, u, ldu, work(n+1), work(2*n+1),
1198  $ lwork-2*n, ierr )
1199 *
1200  DO 1967 p = 1, nr - 1
1201  CALL dcopy( nr-p, u(p,p+1), ldu, u(p+1,p), 1 )
1202  1967 CONTINUE
1203  CALL dlaset( 'Upper', nr-1, nr-1, zero, zero, u(1,2), ldu )
1204 *
1205  CALL dgesvj( 'Lower', 'U', 'N', nr,nr, u, ldu, sva, nr, a,
1206  $ lda, work(n+1), lwork-n, info )
1207  scalem = work(n+1)
1208  numrank = idnint(work(n+2))
1209 *
1210  IF ( nr .LT. m ) THEN
1211  CALL dlaset( 'A', m-nr, nr,zero, zero, u(nr+1,1), ldu )
1212  IF ( nr .LT. n1 ) THEN
1213  CALL dlaset( 'A',nr, n1-nr, zero, zero, u(1,nr+1), ldu )
1214  CALL dlaset( 'A',m-nr,n1-nr,zero,one,u(nr+1,nr+1), ldu )
1215  END IF
1216  END IF
1217 *
1218  CALL dormqr( 'Left', 'No Tr', m, n1, n, a, lda, work, u,
1219  $ ldu, work(n+1), lwork-n, ierr )
1220 *
1221  IF ( rowpiv )
1222  $ CALL dlaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1223 *
1224  DO 1974 p = 1, n1
1225  xsc = one / dnrm2( m, u(1,p), 1 )
1226  CALL dscal( m, xsc, u(1,p), 1 )
1227  1974 CONTINUE
1228 *
1229  IF ( transp ) THEN
1230  CALL dlacpy( 'All', n, n, u, ldu, v, ldv )
1231  END IF
1232 *
1233  ELSE
1234 *
1235 * .. Full SVD ..
1236 *
1237  IF ( .NOT. jracc ) THEN
1238 *
1239  IF ( .NOT. almort ) THEN
1240 *
1241 * Second Preconditioning Step (QRF [with pivoting])
1242 * Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
1243 * equivalent to an LQF CALL. Since in many libraries the QRF
1244 * seems to be better optimized than the LQF, we do explicit
1245 * transpose and use the QRF. This is subject to changes in an
1246 * optimized implementation of DGEJSV.
1247 *
1248  DO 1968 p = 1, nr
1249  CALL dcopy( n-p+1, a(p,p), lda, v(p,p), 1 )
1250  1968 CONTINUE
1251 *
1252 * .. the following two loops perturb small entries to avoid
1253 * denormals in the second QR factorization, where they are
1254 * as good as zeros. This is done to avoid painfully slow
1255 * computation with denormals. The relative size of the perturbation
1256 * is a parameter that can be changed by the implementer.
1257 * This perturbation device will be obsolete on machines with
1258 * properly implemented arithmetic.
1259 * To switch it off, set L2PERT=.FALSE. To remove it from the
1260 * code, remove the action under L2PERT=.TRUE., leave the ELSE part.
1261 * The following two loops should be blocked and fused with the
1262 * transposed copy above.
1263 *
1264  IF ( l2pert ) THEN
1265  xsc = dsqrt(small)
1266  DO 2969 q = 1, nr
1267  temp1 = xsc*dabs( v(q,q) )
1268  DO 2968 p = 1, n
1269  IF ( ( p .GT. q ) .AND. ( dabs(v(p,q)) .LE. temp1 )
1270  $ .OR. ( p .LT. q ) )
1271  $ v(p,q) = dsign( temp1, v(p,q) )
1272  IF ( p .LT. q ) v(p,q) = - v(p,q)
1273  2968 CONTINUE
1274  2969 CONTINUE
1275  ELSE
1276  CALL dlaset( 'U', nr-1, nr-1, zero, zero, v(1,2), ldv )
1277  END IF
1278 *
1279 * Estimate the row scaled condition number of R1
1280 * (If R1 is rectangular, N > NR, then the condition number
1281 * of the leading NR x NR submatrix is estimated.)
1282 *
1283  CALL dlacpy( 'L', nr, nr, v, ldv, work(2*n+1), nr )
1284  DO 3950 p = 1, nr
1285  temp1 = dnrm2(nr-p+1,work(2*n+(p-1)*nr+p),1)
1286  CALL dscal(nr-p+1,one/temp1,work(2*n+(p-1)*nr+p),1)
1287  3950 CONTINUE
1288  CALL dpocon('Lower',nr,work(2*n+1),nr,one,temp1,
1289  $ work(2*n+nr*nr+1),iwork(m+2*n+1),ierr)
1290  condr1 = one / dsqrt(temp1)
1291 * .. here need a second oppinion on the condition number
1292 * .. then assume worst case scenario
1293 * R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)
1294 * more conservative <=> CONDR1 .LT. DSQRT(DBLE(N))
1295 *
1296  cond_ok = dsqrt(dble(nr))
1297 *[TP] COND_OK is a tuning parameter.
1298 
1299  IF ( condr1 .LT. cond_ok ) THEN
1300 * .. the second QRF without pivoting. Note: in an optimized
1301 * implementation, this QRF should be implemented as the QRF
1302 * of a lower triangular matrix.
1303 * R1^t = Q2 * R2
1304  CALL dgeqrf( n, nr, v, ldv, work(n+1), work(2*n+1),
1305  $ lwork-2*n, ierr )
1306 *
1307  IF ( l2pert ) THEN
1308  xsc = dsqrt(small)/epsln
1309  DO 3959 p = 2, nr
1310  DO 3958 q = 1, p - 1
1311  temp1 = xsc * dmin1(dabs(v(p,p)),dabs(v(q,q)))
1312  IF ( dabs(v(q,p)) .LE. temp1 )
1313  $ v(q,p) = dsign( temp1, v(q,p) )
1314  3958 CONTINUE
1315  3959 CONTINUE
1316  END IF
1317 *
1318  IF ( nr .NE. n )
1319  $ CALL dlacpy( 'A', n, nr, v, ldv, work(2*n+1), n )
1320 * .. save ...
1321 *
1322 * .. this transposed copy should be better than naive
1323  DO 1969 p = 1, nr - 1
1324  CALL dcopy( nr-p, v(p,p+1), ldv, v(p+1,p), 1 )
1325  1969 CONTINUE
1326 *
1327  condr2 = condr1
1328 *
1329  ELSE
1330 *
1331 * .. ill-conditioned case: second QRF with pivoting
1332 * Note that windowed pivoting would be equaly good
1333 * numerically, and more run-time efficient. So, in
1334 * an optimal implementation, the next call to DGEQP3
1335 * should be replaced with eg. CALL SGEQPX (ACM TOMS #782)
1336 * with properly (carefully) chosen parameters.
1337 *
1338 * R1^t * P2 = Q2 * R2
1339  DO 3003 p = 1, nr
1340  iwork(n+p) = 0
1341  3003 CONTINUE
1342  CALL dgeqp3( n, nr, v, ldv, iwork(n+1), work(n+1),
1343  $ work(2*n+1), lwork-2*n, ierr )
1344 ** CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
1345 ** $ LWORK-2*N, IERR )
1346  IF ( l2pert ) THEN
1347  xsc = dsqrt(small)
1348  DO 3969 p = 2, nr
1349  DO 3968 q = 1, p - 1
1350  temp1 = xsc * dmin1(dabs(v(p,p)),dabs(v(q,q)))
1351  IF ( dabs(v(q,p)) .LE. temp1 )
1352  $ v(q,p) = dsign( temp1, v(q,p) )
1353  3968 CONTINUE
1354  3969 CONTINUE
1355  END IF
1356 *
1357  CALL dlacpy( 'A', n, nr, v, ldv, work(2*n+1), n )
1358 *
1359  IF ( l2pert ) THEN
1360  xsc = dsqrt(small)
1361  DO 8970 p = 2, nr
1362  DO 8971 q = 1, p - 1
1363  temp1 = xsc * dmin1(dabs(v(p,p)),dabs(v(q,q)))
1364  v(p,q) = - dsign( temp1, v(q,p) )
1365  8971 CONTINUE
1366  8970 CONTINUE
1367  ELSE
1368  CALL dlaset( 'L',nr-1,nr-1,zero,zero,v(2,1),ldv )
1369  END IF
1370 * Now, compute R2 = L3 * Q3, the LQ factorization.
1371  CALL dgelqf( nr, nr, v, ldv, work(2*n+n*nr+1),
1372  $ work(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, ierr )
1373 * .. and estimate the condition number
1374  CALL dlacpy( 'L',nr,nr,v,ldv,work(2*n+n*nr+nr+1),nr )
1375  DO 4950 p = 1, nr
1376  temp1 = dnrm2( p, work(2*n+n*nr+nr+p), nr )
1377  CALL dscal( p, one/temp1, work(2*n+n*nr+nr+p), nr )
1378  4950 CONTINUE
1379  CALL dpocon( 'L',nr,work(2*n+n*nr+nr+1),nr,one,temp1,
1380  $ work(2*n+n*nr+nr+nr*nr+1),iwork(m+2*n+1),ierr )
1381  condr2 = one / dsqrt(temp1)
1382 *
1383  IF ( condr2 .GE. cond_ok ) THEN
1384 * .. save the Householder vectors used for Q3
1385 * (this overwrittes the copy of R2, as it will not be
1386 * needed in this branch, but it does not overwritte the
1387 * Huseholder vectors of Q2.).
1388  CALL dlacpy( 'U', nr, nr, v, ldv, work(2*n+1), n )
1389 * .. and the rest of the information on Q3 is in
1390 * WORK(2*N+N*NR+1:2*N+N*NR+N)
1391  END IF
1392 *
1393  END IF
1394 *
1395  IF ( l2pert ) THEN
1396  xsc = dsqrt(small)
1397  DO 4968 q = 2, nr
1398  temp1 = xsc * v(q,q)
1399  DO 4969 p = 1, q - 1
1400 * V(p,q) = - DSIGN( TEMP1, V(q,p) )
1401  v(p,q) = - dsign( temp1, v(p,q) )
1402  4969 CONTINUE
1403  4968 CONTINUE
1404  ELSE
1405  CALL dlaset( 'U', nr-1,nr-1, zero,zero, v(1,2), ldv )
1406  END IF
1407 *
1408 * Second preconditioning finished; continue with Jacobi SVD
1409 * The input matrix is lower trinagular.
1410 *
1411 * Recover the right singular vectors as solution of a well
1412 * conditioned triangular matrix equation.
1413 *
1414  IF ( condr1 .LT. cond_ok ) THEN
1415 *
1416  CALL dgesvj( 'L','U','N',nr,nr,v,ldv,sva,nr,u,
1417  $ ldu,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,info )
1418  scalem = work(2*n+n*nr+nr+1)
1419  numrank = idnint(work(2*n+n*nr+nr+2))
1420  DO 3970 p = 1, nr
1421  CALL dcopy( nr, v(1,p), 1, u(1,p), 1 )
1422  CALL dscal( nr, sva(p), v(1,p), 1 )
1423  3970 CONTINUE
1424 
1425 * .. pick the right matrix equation and solve it
1426 *
1427  IF ( nr .EQ. n ) THEN
1428 * :)) .. best case, R1 is inverted. The solution of this matrix
1429 * equation is Q2*V2 = the product of the Jacobi rotations
1430 * used in DGESVJ, premultiplied with the orthogonal matrix
1431 * from the second QR factorization.
1432  CALL dtrsm( 'L','U','N','N', nr,nr,one, a,lda, v,ldv )
1433  ELSE
1434 * .. R1 is well conditioned, but non-square. Transpose(R2)
1435 * is inverted to get the product of the Jacobi rotations
1436 * used in DGESVJ. The Q-factor from the second QR
1437 * factorization is then built in explicitly.
1438  CALL dtrsm('L','U','T','N',nr,nr,one,work(2*n+1),
1439  $ n,v,ldv)
1440  IF ( nr .LT. n ) THEN
1441  CALL dlaset('A',n-nr,nr,zero,zero,v(nr+1,1),ldv)
1442  CALL dlaset('A',nr,n-nr,zero,zero,v(1,nr+1),ldv)
1443  CALL dlaset('A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv)
1444  END IF
1445  CALL dormqr('L','N',n,n,nr,work(2*n+1),n,work(n+1),
1446  $ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr)
1447  END IF
1448 *
1449  ELSE IF ( condr2 .LT. cond_ok ) THEN
1450 *
1451 * :) .. the input matrix A is very likely a relative of
1452 * the Kahan matrix :)
1453 * The matrix R2 is inverted. The solution of the matrix equation
1454 * is Q3^T*V3 = the product of the Jacobi rotations (appplied to
1455 * the lower triangular L3 from the LQ factorization of
1456 * R2=L3*Q3), pre-multiplied with the transposed Q3.
1457  CALL dgesvj( 'L', 'U', 'N', nr, nr, v, ldv, sva, nr, u,
1458  $ ldu, work(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, info )
1459  scalem = work(2*n+n*nr+nr+1)
1460  numrank = idnint(work(2*n+n*nr+nr+2))
1461  DO 3870 p = 1, nr
1462  CALL dcopy( nr, v(1,p), 1, u(1,p), 1 )
1463  CALL dscal( nr, sva(p), u(1,p), 1 )
1464  3870 CONTINUE
1465  CALL dtrsm('L','U','N','N',nr,nr,one,work(2*n+1),n,u,ldu)
1466 * .. apply the permutation from the second QR factorization
1467  DO 873 q = 1, nr
1468  DO 872 p = 1, nr
1469  work(2*n+n*nr+nr+iwork(n+p)) = u(p,q)
1470  872 CONTINUE
1471  DO 874 p = 1, nr
1472  u(p,q) = work(2*n+n*nr+nr+p)
1473  874 CONTINUE
1474  873 CONTINUE
1475  IF ( nr .LT. n ) THEN
1476  CALL dlaset( 'A',n-nr,nr,zero,zero,v(nr+1,1),ldv )
1477  CALL dlaset( 'A',nr,n-nr,zero,zero,v(1,nr+1),ldv )
1478  CALL dlaset( 'A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv )
1479  END IF
1480  CALL dormqr( 'L','N',n,n,nr,work(2*n+1),n,work(n+1),
1481  $ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
1482  ELSE
1483 * Last line of defense.
1484 * #:( This is a rather pathological case: no scaled condition
1485 * improvement after two pivoted QR factorizations. Other
1486 * possibility is that the rank revealing QR factorization
1487 * or the condition estimator has failed, or the COND_OK
1488 * is set very close to ONE (which is unnecessary). Normally,
1489 * this branch should never be executed, but in rare cases of
1490 * failure of the RRQR or condition estimator, the last line of
1491 * defense ensures that DGEJSV completes the task.
1492 * Compute the full SVD of L3 using DGESVJ with explicit
1493 * accumulation of Jacobi rotations.
1494  CALL dgesvj( 'L', 'U', 'V', nr, nr, v, ldv, sva, nr, u,
1495  $ ldu, work(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, info )
1496  scalem = work(2*n+n*nr+nr+1)
1497  numrank = idnint(work(2*n+n*nr+nr+2))
1498  IF ( nr .LT. n ) THEN
1499  CALL dlaset( 'A',n-nr,nr,zero,zero,v(nr+1,1),ldv )
1500  CALL dlaset( 'A',nr,n-nr,zero,zero,v(1,nr+1),ldv )
1501  CALL dlaset( 'A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv )
1502  END IF
1503  CALL dormqr( 'L','N',n,n,nr,work(2*n+1),n,work(n+1),
1504  $ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
1505 *
1506  CALL dormlq( 'L', 'T', nr, nr, nr, work(2*n+1), n,
1507  $ work(2*n+n*nr+1), u, ldu, work(2*n+n*nr+nr+1),
1508  $ lwork-2*n-n*nr-nr, ierr )
1509  DO 773 q = 1, nr
1510  DO 772 p = 1, nr
1511  work(2*n+n*nr+nr+iwork(n+p)) = u(p,q)
1512  772 CONTINUE
1513  DO 774 p = 1, nr
1514  u(p,q) = work(2*n+n*nr+nr+p)
1515  774 CONTINUE
1516  773 CONTINUE
1517 *
1518  END IF
1519 *
1520 * Permute the rows of V using the (column) permutation from the
1521 * first QRF. Also, scale the columns to make them unit in
1522 * Euclidean norm. This applies to all cases.
1523 *
1524  temp1 = dsqrt(dble(n)) * epsln
1525  DO 1972 q = 1, n
1526  DO 972 p = 1, n
1527  work(2*n+n*nr+nr+iwork(p)) = v(p,q)
1528  972 CONTINUE
1529  DO 973 p = 1, n
1530  v(p,q) = work(2*n+n*nr+nr+p)
1531  973 CONTINUE
1532  xsc = one / dnrm2( n, v(1,q), 1 )
1533  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1534  $ CALL dscal( n, xsc, v(1,q), 1 )
1535  1972 CONTINUE
1536 * At this moment, V contains the right singular vectors of A.
1537 * Next, assemble the left singular vector matrix U (M x N).
1538  IF ( nr .LT. m ) THEN
1539  CALL dlaset( 'A', m-nr, nr, zero, zero, u(nr+1,1), ldu )
1540  IF ( nr .LT. n1 ) THEN
1541  CALL dlaset('A',nr,n1-nr,zero,zero,u(1,nr+1),ldu)
1542  CALL dlaset('A',m-nr,n1-nr,zero,one,u(nr+1,nr+1),ldu)
1543  END IF
1544  END IF
1545 *
1546 * The Q matrix from the first QRF is built into the left singular
1547 * matrix U. This applies to all cases.
1548 *
1549  CALL dormqr( 'Left', 'No_Tr', m, n1, n, a, lda, work, u,
1550  $ ldu, work(n+1), lwork-n, ierr )
1551 
1552 * The columns of U are normalized. The cost is O(M*N) flops.
1553  temp1 = dsqrt(dble(m)) * epsln
1554  DO 1973 p = 1, nr
1555  xsc = one / dnrm2( m, u(1,p), 1 )
1556  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1557  $ CALL dscal( m, xsc, u(1,p), 1 )
1558  1973 CONTINUE
1559 *
1560 * If the initial QRF is computed with row pivoting, the left
1561 * singular vectors must be adjusted.
1562 *
1563  IF ( rowpiv )
1564  $ CALL dlaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1565 *
1566  ELSE
1567 *
1568 * .. the initial matrix A has almost orthogonal columns and
1569 * the second QRF is not needed
1570 *
1571  CALL dlacpy( 'Upper', n, n, a, lda, work(n+1), n )
1572  IF ( l2pert ) THEN
1573  xsc = dsqrt(small)
1574  DO 5970 p = 2, n
1575  temp1 = xsc * work( n + (p-1)*n + p )
1576  DO 5971 q = 1, p - 1
1577  work(n+(q-1)*n+p)=-dsign(temp1,work(n+(p-1)*n+q))
1578  5971 CONTINUE
1579  5970 CONTINUE
1580  ELSE
1581  CALL dlaset( 'Lower',n-1,n-1,zero,zero,work(n+2),n )
1582  END IF
1583 *
1584  CALL dgesvj( 'Upper', 'U', 'N', n, n, work(n+1), n, sva,
1585  $ n, u, ldu, work(n+n*n+1), lwork-n-n*n, info )
1586 *
1587  scalem = work(n+n*n+1)
1588  numrank = idnint(work(n+n*n+2))
1589  DO 6970 p = 1, n
1590  CALL dcopy( n, work(n+(p-1)*n+1), 1, u(1,p), 1 )
1591  CALL dscal( n, sva(p), work(n+(p-1)*n+1), 1 )
1592  6970 CONTINUE
1593 *
1594  CALL dtrsm( 'Left', 'Upper', 'NoTrans', 'No UD', n, n,
1595  $ one, a, lda, work(n+1), n )
1596  DO 6972 p = 1, n
1597  CALL dcopy( n, work(n+p), n, v(iwork(p),1), ldv )
1598  6972 CONTINUE
1599  temp1 = dsqrt(dble(n))*epsln
1600  DO 6971 p = 1, n
1601  xsc = one / dnrm2( n, v(1,p), 1 )
1602  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1603  $ CALL dscal( n, xsc, v(1,p), 1 )
1604  6971 CONTINUE
1605 *
1606 * Assemble the left singular vector matrix U (M x N).
1607 *
1608  IF ( n .LT. m ) THEN
1609  CALL dlaset( 'A', m-n, n, zero, zero, u(n+1,1), ldu )
1610  IF ( n .LT. n1 ) THEN
1611  CALL dlaset( 'A',n, n1-n, zero, zero, u(1,n+1),ldu )
1612  CALL dlaset( 'A',m-n,n1-n, zero, one,u(n+1,n+1),ldu )
1613  END IF
1614  END IF
1615  CALL dormqr( 'Left', 'No Tr', m, n1, n, a, lda, work, u,
1616  $ ldu, work(n+1), lwork-n, ierr )
1617  temp1 = dsqrt(dble(m))*epsln
1618  DO 6973 p = 1, n1
1619  xsc = one / dnrm2( m, u(1,p), 1 )
1620  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1621  $ CALL dscal( m, xsc, u(1,p), 1 )
1622  6973 CONTINUE
1623 *
1624  IF ( rowpiv )
1625  $ CALL dlaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1626 *
1627  END IF
1628 *
1629 * end of the >> almost orthogonal case << in the full SVD
1630 *
1631  ELSE
1632 *
1633 * This branch deploys a preconditioned Jacobi SVD with explicitly
1634 * accumulated rotations. It is included as optional, mainly for
1635 * experimental purposes. It does perfom well, and can also be used.
1636 * In this implementation, this branch will be automatically activated
1637 * if the condition number sigma_max(A) / sigma_min(A) is predicted
1638 * to be greater than the overflow threshold. This is because the
1639 * a posteriori computation of the singular vectors assumes robust
1640 * implementation of BLAS and some LAPACK procedures, capable of working
1641 * in presence of extreme values. Since that is not always the case, ...
1642 *
1643  DO 7968 p = 1, nr
1644  CALL dcopy( n-p+1, a(p,p), lda, v(p,p), 1 )
1645  7968 CONTINUE
1646 *
1647  IF ( l2pert ) THEN
1648  xsc = dsqrt(small/epsln)
1649  DO 5969 q = 1, nr
1650  temp1 = xsc*dabs( v(q,q) )
1651  DO 5968 p = 1, n
1652  IF ( ( p .GT. q ) .AND. ( dabs(v(p,q)) .LE. temp1 )
1653  $ .OR. ( p .LT. q ) )
1654  $ v(p,q) = dsign( temp1, v(p,q) )
1655  IF ( p .LT. q ) v(p,q) = - v(p,q)
1656  5968 CONTINUE
1657  5969 CONTINUE
1658  ELSE
1659  CALL dlaset( 'U', nr-1, nr-1, zero, zero, v(1,2), ldv )
1660  END IF
1661 
1662  CALL dgeqrf( n, nr, v, ldv, work(n+1), work(2*n+1),
1663  $ lwork-2*n, ierr )
1664  CALL dlacpy( 'L', n, nr, v, ldv, work(2*n+1), n )
1665 *
1666  DO 7969 p = 1, nr
1667  CALL dcopy( nr-p+1, v(p,p), ldv, u(p,p), 1 )
1668  7969 CONTINUE
1669 
1670  IF ( l2pert ) THEN
1671  xsc = dsqrt(small/epsln)
1672  DO 9970 q = 2, nr
1673  DO 9971 p = 1, q - 1
1674  temp1 = xsc * dmin1(dabs(u(p,p)),dabs(u(q,q)))
1675  u(p,q) = - dsign( temp1, u(q,p) )
1676  9971 CONTINUE
1677  9970 CONTINUE
1678  ELSE
1679  CALL dlaset('U', nr-1, nr-1, zero, zero, u(1,2), ldu )
1680  END IF
1681 
1682  CALL dgesvj( 'G', 'U', 'V', nr, nr, u, ldu, sva,
1683  $ n, v, ldv, work(2*n+n*nr+1), lwork-2*n-n*nr, info )
1684  scalem = work(2*n+n*nr+1)
1685  numrank = idnint(work(2*n+n*nr+2))
1686 
1687  IF ( nr .LT. n ) THEN
1688  CALL dlaset( 'A',n-nr,nr,zero,zero,v(nr+1,1),ldv )
1689  CALL dlaset( 'A',nr,n-nr,zero,zero,v(1,nr+1),ldv )
1690  CALL dlaset( 'A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv )
1691  END IF
1692 
1693  CALL dormqr( 'L','N',n,n,nr,work(2*n+1),n,work(n+1),
1694  $ v,ldv,work(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
1695 *
1696 * Permute the rows of V using the (column) permutation from the
1697 * first QRF. Also, scale the columns to make them unit in
1698 * Euclidean norm. This applies to all cases.
1699 *
1700  temp1 = dsqrt(dble(n)) * epsln
1701  DO 7972 q = 1, n
1702  DO 8972 p = 1, n
1703  work(2*n+n*nr+nr+iwork(p)) = v(p,q)
1704  8972 CONTINUE
1705  DO 8973 p = 1, n
1706  v(p,q) = work(2*n+n*nr+nr+p)
1707  8973 CONTINUE
1708  xsc = one / dnrm2( n, v(1,q), 1 )
1709  IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
1710  $ CALL dscal( n, xsc, v(1,q), 1 )
1711  7972 CONTINUE
1712 *
1713 * At this moment, V contains the right singular vectors of A.
1714 * Next, assemble the left singular vector matrix U (M x N).
1715 *
1716  IF ( nr .LT. m ) THEN
1717  CALL dlaset( 'A', m-nr, nr, zero, zero, u(nr+1,1), ldu )
1718  IF ( nr .LT. n1 ) THEN
1719  CALL dlaset( 'A',nr, n1-nr, zero, zero, u(1,nr+1),ldu )
1720  CALL dlaset( 'A',m-nr,n1-nr, zero, one,u(nr+1,nr+1),ldu )
1721  END IF
1722  END IF
1723 *
1724  CALL dormqr( 'Left', 'No Tr', m, n1, n, a, lda, work, u,
1725  $ ldu, work(n+1), lwork-n, ierr )
1726 *
1727  IF ( rowpiv )
1728  $ CALL dlaswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
1729 *
1730 *
1731  END IF
1732  IF ( transp ) THEN
1733 * .. swap U and V because the procedure worked on A^t
1734  DO 6974 p = 1, n
1735  CALL dswap( n, u(1,p), 1, v(1,p), 1 )
1736  6974 CONTINUE
1737  END IF
1738 *
1739  END IF
1740 * end of the full SVD
1741 *
1742 * Undo scaling, if necessary (and possible)
1743 *
1744  IF ( uscal2 .LE. (big/sva(1))*uscal1 ) THEN
1745  CALL dlascl( 'G', 0, 0, uscal1, uscal2, nr, 1, sva, n, ierr )
1746  uscal1 = one
1747  uscal2 = one
1748  END IF
1749 *
1750  IF ( nr .LT. n ) THEN
1751  DO 3004 p = nr+1, n
1752  sva(p) = zero
1753  3004 CONTINUE
1754  END IF
1755 *
1756  work(1) = uscal2 * scalem
1757  work(2) = uscal1
1758  IF ( errest ) work(3) = sconda
1759  IF ( lsvec .AND. rsvec ) THEN
1760  work(4) = condr1
1761  work(5) = condr2
1762  END IF
1763  IF ( l2tran ) THEN
1764  work(6) = entra
1765  work(7) = entrat
1766  END IF
1767 *
1768  iwork(1) = nr
1769  iwork(2) = numrank
1770  iwork(3) = warning
1771 *
1772  RETURN
1773 * ..
1774 * .. END OF DGEJSV
1775 * ..
1776  END
1777 *