1: *> \brief <b> ZGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGESVDQ + dependencies
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11: *> [TGZ]</a>
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGESVDQ( JOBA, JOBP, JOBR, JOBU, JOBV, M, N, A, LDA,
22: * S, U, LDU, V, LDV, NUMRANK, IWORK, LIWORK,
23: * CWORK, LCWORK, RWORK, LRWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * IMPLICIT NONE
27: * CHARACTER JOBA, JOBP, JOBR, JOBU, JOBV
28: * INTEGER M, N, LDA, LDU, LDV, NUMRANK, LIWORK, LCWORK, LRWORK,
29: * INFO
30: * ..
31: * .. Array Arguments ..
32: * COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( * )
33: * DOUBLE PRECISION S( * ), RWORK( * )
34: * INTEGER IWORK( * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: * ZCGESVDQ computes the singular value decomposition (SVD) of a complex
44: *> M-by-N matrix A, where M >= N. The SVD of A is written as
45: *> [++] [xx] [x0] [xx]
46: *> A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
47: *> [++] [xx]
48: *> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
49: *> matrix, and V is an N-by-N unitary matrix. The diagonal elements
50: *> of SIGMA are the singular values of A. The columns of U and V are the
51: *> left and the right singular vectors of A, respectively.
52: *> \endverbatim
53: *
54: * Arguments
55: * =========
56: *
57: *> \param[in] JOBA
58: *> \verbatim
59: *> JOBA is CHARACTER*1
60: *> Specifies the level of accuracy in the computed SVD
61: *> = 'A' The requested accuracy corresponds to having the backward
62: *> error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,
63: *> where EPS = DLAMCH('Epsilon'). This authorises ZGESVDQ to
64: *> truncate the computed triangular factor in a rank revealing
65: *> QR factorization whenever the truncated part is below the
66: *> threshold of the order of EPS * ||A||_F. This is aggressive
67: *> truncation level.
68: *> = 'M' Similarly as with 'A', but the truncation is more gentle: it
69: *> is allowed only when there is a drop on the diagonal of the
70: *> triangular factor in the QR factorization. This is medium
71: *> truncation level.
72: *> = 'H' High accuracy requested. No numerical rank determination based
73: *> on the rank revealing QR factorization is attempted.
74: *> = 'E' Same as 'H', and in addition the condition number of column
75: *> scaled A is estimated and returned in RWORK(1).
76: *> N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)
77: *> \endverbatim
78: *>
79: *> \param[in] JOBP
80: *> \verbatim
81: *> JOBP is CHARACTER*1
82: *> = 'P' The rows of A are ordered in decreasing order with respect to
83: *> ||A(i,:)||_\infty. This enhances numerical accuracy at the cost
84: *> of extra data movement. Recommended for numerical robustness.
85: *> = 'N' No row pivoting.
86: *> \endverbatim
87: *>
88: *> \param[in] JOBR
89: *> \verbatim
90: *> JOBR is CHARACTER*1
91: *> = 'T' After the initial pivoted QR factorization, ZGESVD is applied to
92: *> the adjoint R**H of the computed triangular factor R. This involves
93: *> some extra data movement (matrix transpositions). Useful for
94: *> experiments, research and development.
95: *> = 'N' The triangular factor R is given as input to CGESVD. This may be
96: *> preferred as it involves less data movement.
97: *> \endverbatim
98: *>
99: *> \param[in] JOBU
100: *> \verbatim
101: *> JOBU is CHARACTER*1
102: *> = 'A' All M left singular vectors are computed and returned in the
103: *> matrix U. See the description of U.
104: *> = 'S' or 'U' N = min(M,N) left singular vectors are computed and returned
105: *> in the matrix U. See the description of U.
106: *> = 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular
107: *> vectors are computed and returned in the matrix U.
108: *> = 'F' The N left singular vectors are returned in factored form as the
109: *> product of the Q factor from the initial QR factorization and the
110: *> N left singular vectors of (R**H , 0)**H. If row pivoting is used,
111: *> then the necessary information on the row pivoting is stored in
112: *> IWORK(N+1:N+M-1).
113: *> = 'N' The left singular vectors are not computed.
114: *> \endverbatim
115: *>
116: *> \param[in] JOBV
117: *> \verbatim
118: *> JOBV is CHARACTER*1
119: *> = 'A', 'V' All N right singular vectors are computed and returned in
120: *> the matrix V.
121: *> = 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular
122: *> vectors are computed and returned in the matrix V. This option is
123: *> allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
124: *> = 'N' The right singular vectors are not computed.
125: *> \endverbatim
126: *>
127: *> \param[in] M
128: *> \verbatim
129: *> M is INTEGER
130: *> The number of rows of the input matrix A. M >= 0.
131: *> \endverbatim
132: *>
133: *> \param[in] N
134: *> \verbatim
135: *> N is INTEGER
136: *> The number of columns of the input matrix A. M >= N >= 0.
137: *> \endverbatim
138: *>
139: *> \param[in,out] A
140: *> \verbatim
141: *> A is COMPLEX*16 array of dimensions LDA x N
142: *> On entry, the input matrix A.
143: *> On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
144: *> the Householder vectors as stored by ZGEQP3. If JOBU = 'F', these Householder
145: *> vectors together with CWORK(1:N) can be used to restore the Q factors from
146: *> the initial pivoted QR factorization of A. See the description of U.
147: *> \endverbatim
148: *>
149: *> \param[in] LDA
150: *> \verbatim
151: *> LDA is INTEGER.
152: *> The leading dimension of the array A. LDA >= max(1,M).
153: *> \endverbatim
154: *>
155: *> \param[out] S
156: *> \verbatim
157: *> S is DOUBLE PRECISION array of dimension N.
158: *> The singular values of A, ordered so that S(i) >= S(i+1).
159: *> \endverbatim
160: *>
161: *> \param[out] U
162: *> \verbatim
163: *> U is COMPLEX*16 array, dimension
164: *> LDU x M if JOBU = 'A'; see the description of LDU. In this case,
165: *> on exit, U contains the M left singular vectors.
166: *> LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this
167: *> case, U contains the leading N or the leading NUMRANK left singular vectors.
168: *> LDU x N if JOBU = 'F' ; see the description of LDU. In this case U
169: *> contains N x N unitary matrix that can be used to form the left
170: *> singular vectors.
171: *> If JOBU = 'N', U is not referenced.
172: *> \endverbatim
173: *>
174: *> \param[in] LDU
175: *> \verbatim
176: *> LDU is INTEGER.
177: *> The leading dimension of the array U.
178: *> If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M).
179: *> If JOBU = 'F', LDU >= max(1,N).
180: *> Otherwise, LDU >= 1.
181: *> \endverbatim
182: *>
183: *> \param[out] V
184: *> \verbatim
185: *> V is COMPLEX*16 array, dimension
186: *> LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .
187: *> If JOBV = 'A', or 'V', V contains the N-by-N unitary matrix V**H;
188: *> If JOBV = 'R', V contains the first NUMRANK rows of V**H (the right
189: *> singular vectors, stored rowwise, of the NUMRANK largest singular values).
190: *> If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
191: *> If JOBV = 'N', and JOBA.NE.'E', V is not referenced.
192: *> \endverbatim
193: *>
194: *> \param[in] LDV
195: *> \verbatim
196: *> LDV is INTEGER
197: *> The leading dimension of the array V.
198: *> If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N).
199: *> Otherwise, LDV >= 1.
200: *> \endverbatim
201: *>
202: *> \param[out] NUMRANK
203: *> \verbatim
204: *> NUMRANK is INTEGER
205: *> NUMRANK is the numerical rank first determined after the rank
206: *> revealing QR factorization, following the strategy specified by the
207: *> value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK
208: *> leading singular values and vectors are then requested in the call
209: *> of CGESVD. The final value of NUMRANK might be further reduced if
210: *> some singular values are computed as zeros.
211: *> \endverbatim
212: *>
213: *> \param[out] IWORK
214: *> \verbatim
215: *> IWORK is INTEGER array, dimension (max(1, LIWORK)).
216: *> On exit, IWORK(1:N) contains column pivoting permutation of the
217: *> rank revealing QR factorization.
218: *> If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence
219: *> of row swaps used in row pivoting. These can be used to restore the
220: *> left singular vectors in the case JOBU = 'F'.
221: *
222: *> If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
223: *> LIWORK(1) returns the minimal LIWORK.
224: *> \endverbatim
225: *>
226: *> \param[in] LIWORK
227: *> \verbatim
228: *> LIWORK is INTEGER
229: *> The dimension of the array IWORK.
230: *> LIWORK >= N + M - 1, if JOBP = 'P';
231: *> LIWORK >= N if JOBP = 'N'.
232: *>
233: *> If LIWORK = -1, then a workspace query is assumed; the routine
234: *> only calculates and returns the optimal and minimal sizes
235: *> for the CWORK, IWORK, and RWORK arrays, and no error
236: *> message related to LCWORK is issued by XERBLA.
237: *> \endverbatim
238: *>
239: *> \param[out] CWORK
240: *> \verbatim
241: *> CWORK is COMPLEX*12 array, dimension (max(2, LCWORK)), used as a workspace.
242: *> On exit, if, on entry, LCWORK.NE.-1, CWORK(1:N) contains parameters
243: *> needed to recover the Q factor from the QR factorization computed by
244: *> ZGEQP3.
245: *
246: *> If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
247: *> CWORK(1) returns the optimal LCWORK, and
248: *> CWORK(2) returns the minimal LCWORK.
249: *> \endverbatim
250: *>
251: *> \param[in,out] LCWORK
252: *> \verbatim
253: *> LCWORK is INTEGER
254: *> The dimension of the array CWORK. It is determined as follows:
255: *> Let LWQP3 = N+1, LWCON = 2*N, and let
256: *> LWUNQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U'
257: *> { MAX( M, 1 ), if JOBU = 'A'
258: *> LWSVD = MAX( 3*N, 1 )
259: *> LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
260: *> LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
261: *> Then the minimal value of LCWORK is:
262: *> = MAX( N + LWQP3, LWSVD ) if only the singular values are needed;
263: *> = MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
264: *> and a scaled condition estimate requested;
265: *>
266: *> = N + MAX( LWQP3, LWSVD, LWUNQ ) if the singular values and the left
267: *> singular vectors are requested;
268: *> = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the singular values and the left
269: *> singular vectors are requested, and also
270: *> a scaled condition estimate requested;
271: *>
272: *> = N + MAX( LWQP3, LWSVD ) if the singular values and the right
273: *> singular vectors are requested;
274: *> = N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
275: *> singular vectors are requested, and also
276: *> a scaled condition etimate requested;
277: *>
278: *> = N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
279: *> independent of JOBR;
280: *> = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
281: *> JOBV = 'R' and, also a scaled condition
282: *> estimate requested; independent of JOBR;
283: *> = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
284: *> N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
285: *> full SVD is requested with JOBV = 'A' or 'V', and
286: *> JOBR ='N'
287: *> = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
288: *> N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
289: *> if the full SVD is requested with JOBV = 'A' or 'V', and
290: *> JOBR ='N', and also a scaled condition number estimate
291: *> requested.
292: *> = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
293: *> N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
294: *> full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
295: *> = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
296: *> N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
297: *> if the full SVD is requested with JOBV = 'A', 'V' and
298: *> JOBR ='T', and also a scaled condition number estimate
299: *> requested.
300: *> Finally, LCWORK must be at least two: LCWORK = MAX( 2, LCWORK ).
301: *>
302: *> If LCWORK = -1, then a workspace query is assumed; the routine
303: *> only calculates and returns the optimal and minimal sizes
304: *> for the CWORK, IWORK, and RWORK arrays, and no error
305: *> message related to LCWORK is issued by XERBLA.
306: *> \endverbatim
307: *>
308: *> \param[out] RWORK
309: *> \verbatim
310: *> RWORK is DOUBLE PRECISION array, dimension (max(1, LRWORK)).
311: *> On exit,
312: *> 1. If JOBA = 'E', RWORK(1) contains an estimate of the condition
313: *> number of column scaled A. If A = C * D where D is diagonal and C
314: *> has unit columns in the Euclidean norm, then, assuming full column rank,
315: *> N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).
316: *> Otherwise, RWORK(1) = -1.
317: *> 2. RWORK(2) contains the number of singular values computed as
318: *> exact zeros in ZGESVD applied to the upper triangular or trapeziodal
319: *> R (from the initial QR factorization). In case of early exit (no call to
320: *> ZGESVD, such as in the case of zero matrix) RWORK(2) = -1.
321: *
322: *> If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
323: *> RWORK(1) returns the minimal LRWORK.
324: *> \endverbatim
325: *>
326: *> \param[in] LRWORK
327: *> \verbatim
328: *> LRWORK is INTEGER.
329: *> The dimension of the array RWORK.
330: *> If JOBP ='P', then LRWORK >= MAX(2, M, 5*N);
331: *> Otherwise, LRWORK >= MAX(2, 5*N).
332: *
333: *> If LRWORK = -1, then a workspace query is assumed; the routine
334: *> only calculates and returns the optimal and minimal sizes
335: *> for the CWORK, IWORK, and RWORK arrays, and no error
336: *> message related to LCWORK is issued by XERBLA.
337: *> \endverbatim
338: *>
339: *> \param[out] INFO
340: *> \verbatim
341: *> INFO is INTEGER
342: *> = 0: successful exit.
343: *> < 0: if INFO = -i, the i-th argument had an illegal value.
344: *> > 0: if ZBDSQR did not converge, INFO specifies how many superdiagonals
345: *> of an intermediate bidiagonal form B (computed in ZGESVD) did not
346: *> converge to zero.
347: *> \endverbatim
348: *
349: *> \par Further Details:
350: * ========================
351: *>
352: *> \verbatim
353: *>
354: *> 1. The data movement (matrix transpose) is coded using simple nested
355: *> DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
356: *> Those DO-loops are easily identified in this source code - by the CONTINUE
357: *> statements labeled with 11**. In an optimized version of this code, the
358: *> nested DO loops should be replaced with calls to an optimized subroutine.
359: *> 2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
360: *> column norm overflow. This is the minial precaution and it is left to the
361: *> SVD routine (CGESVD) to do its own preemptive scaling if potential over-
362: *> or underflows are detected. To avoid repeated scanning of the array A,
363: *> an optimal implementation would do all necessary scaling before calling
364: *> CGESVD and the scaling in CGESVD can be switched off.
365: *> 3. Other comments related to code optimization are given in comments in the
366: *> code, enlosed in [[double brackets]].
367: *> \endverbatim
368: *
369: *> \par Bugs, examples and comments
370: * ===========================
371: *
372: *> \verbatim
373: *> Please report all bugs and send interesting examples and/or comments to
374: *> drmac@math.hr. Thank you.
375: *> \endverbatim
376: *
377: *> \par References
378: * ===============
379: *
380: *> \verbatim
381: *> [1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
382: *> Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
383: *> 44(1): 11:1-11:30 (2017)
384: *>
385: *> SIGMA library, xGESVDQ section updated February 2016.
386: *> Developed and coded by Zlatko Drmac, Department of Mathematics
387: *> University of Zagreb, Croatia, drmac@math.hr
388: *> \endverbatim
389: *
390: *
391: *> \par Contributors:
392: * ==================
393: *>
394: *> \verbatim
395: *> Developed and coded by Zlatko Drmac, Department of Mathematics
396: *> University of Zagreb, Croatia, drmac@math.hr
397: *> \endverbatim
398: *
399: * Authors:
400: * ========
401: *
402: *> \author Univ. of Tennessee
403: *> \author Univ. of California Berkeley
404: *> \author Univ. of Colorado Denver
405: *> \author NAG Ltd.
406: *
407: *> \date November 2018
408: *
409: *> \ingroup complex16GEsing
410: *
411: * =====================================================================
412: SUBROUTINE ZGESVDQ( JOBA, JOBP, JOBR, JOBU, JOBV, M, N, A, LDA,
413: $ S, U, LDU, V, LDV, NUMRANK, IWORK, LIWORK,
414: $ CWORK, LCWORK, RWORK, LRWORK, INFO )
415: * .. Scalar Arguments ..
416: IMPLICIT NONE
417: CHARACTER JOBA, JOBP, JOBR, JOBU, JOBV
418: INTEGER M, N, LDA, LDU, LDV, NUMRANK, LIWORK, LCWORK, LRWORK,
419: $ INFO
420: * ..
421: * .. Array Arguments ..
422: COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( * )
423: DOUBLE PRECISION S( * ), RWORK( * )
424: INTEGER IWORK( * )
425: *
426: * =====================================================================
427: *
428: * .. Parameters ..
429: DOUBLE PRECISION ZERO, ONE
430: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
431: COMPLEX*16 CZERO, CONE
432: PARAMETER ( CZERO = (0.0D0,0.0D0), CONE = (1.0D0,0.0D0) )
433: * ..
434: * .. Local Scalars ..
435: INTEGER IERR, NR, N1, OPTRATIO, p, q
436: INTEGER LWCON, LWQP3, LWRK_ZGELQF, LWRK_ZGESVD, LWRK_ZGESVD2,
437: $ LWRK_ZGEQP3, LWRK_ZGEQRF, LWRK_ZUNMLQ, LWRK_ZUNMQR,
438: $ LWRK_ZUNMQR2, LWLQF, LWQRF, LWSVD, LWSVD2, LWUNQ,
439: $ LWUNQ2, LWUNLQ, MINWRK, MINWRK2, OPTWRK, OPTWRK2,
440: $ IMINWRK, RMINWRK
441: LOGICAL ACCLA, ACCLM, ACCLH, ASCALED, CONDA, DNTWU, DNTWV,
442: $ LQUERY, LSVC0, LSVEC, ROWPRM, RSVEC, RTRANS, WNTUA,
443: $ WNTUF, WNTUR, WNTUS, WNTVA, WNTVR
444: DOUBLE PRECISION BIG, EPSLN, RTMP, SCONDA, SFMIN
445: COMPLEX*16 CTMP
446: * ..
447: * .. Local Arrays
448: COMPLEX*16 CDUMMY(1)
449: DOUBLE PRECISION RDUMMY(1)
450: * ..
451: * .. External Subroutines (BLAS, LAPACK)
452: EXTERNAL ZGELQF, ZGEQP3, ZGEQRF, ZGESVD, ZLACPY, ZLAPMT,
453: $ ZLASCL, ZLASET, ZLASWP, ZDSCAL, DLASET, DLASCL,
454: $ ZPOCON, ZUNMLQ, ZUNMQR, XERBLA
455: * ..
456: * .. External Functions (BLAS, LAPACK)
457: LOGICAL LSAME
458: INTEGER IDAMAX
459: DOUBLE PRECISION ZLANGE, DZNRM2, DLAMCH
460: EXTERNAL LSAME, ZLANGE, IDAMAX, DZNRM2, DLAMCH
461: * ..
462: * .. Intrinsic Functions ..
463: INTRINSIC ABS, CONJG, MAX, MIN, DBLE, SQRT
464: * ..
465: * .. Executable Statements ..
466: *
467: * Test the input arguments
468: *
469: WNTUS = LSAME( JOBU, 'S' ) .OR. LSAME( JOBU, 'U' )
470: WNTUR = LSAME( JOBU, 'R' )
471: WNTUA = LSAME( JOBU, 'A' )
472: WNTUF = LSAME( JOBU, 'F' )
473: LSVC0 = WNTUS .OR. WNTUR .OR. WNTUA
474: LSVEC = LSVC0 .OR. WNTUF
475: DNTWU = LSAME( JOBU, 'N' )
476: *
477: WNTVR = LSAME( JOBV, 'R' )
478: WNTVA = LSAME( JOBV, 'A' ) .OR. LSAME( JOBV, 'V' )
479: RSVEC = WNTVR .OR. WNTVA
480: DNTWV = LSAME( JOBV, 'N' )
481: *
482: ACCLA = LSAME( JOBA, 'A' )
483: ACCLM = LSAME( JOBA, 'M' )
484: CONDA = LSAME( JOBA, 'E' )
485: ACCLH = LSAME( JOBA, 'H' ) .OR. CONDA
486: *
487: ROWPRM = LSAME( JOBP, 'P' )
488: RTRANS = LSAME( JOBR, 'T' )
489: *
490: IF ( ROWPRM ) THEN
491: IMINWRK = MAX( 1, N + M - 1 )
492: RMINWRK = MAX( 2, M, 5*N )
493: ELSE
494: IMINWRK = MAX( 1, N )
495: RMINWRK = MAX( 2, 5*N )
496: END IF
497: LQUERY = (LIWORK .EQ. -1 .OR. LCWORK .EQ. -1 .OR. LRWORK .EQ. -1)
498: INFO = 0
499: IF ( .NOT. ( ACCLA .OR. ACCLM .OR. ACCLH ) ) THEN
500: INFO = -1
501: ELSE IF ( .NOT.( ROWPRM .OR. LSAME( JOBP, 'N' ) ) ) THEN
502: INFO = -2
503: ELSE IF ( .NOT.( RTRANS .OR. LSAME( JOBR, 'N' ) ) ) THEN
504: INFO = -3
505: ELSE IF ( .NOT.( LSVEC .OR. DNTWU ) ) THEN
506: INFO = -4
507: ELSE IF ( WNTUR .AND. WNTVA ) THEN
508: INFO = -5
509: ELSE IF ( .NOT.( RSVEC .OR. DNTWV )) THEN
510: INFO = -5
511: ELSE IF ( M.LT.0 ) THEN
512: INFO = -6
513: ELSE IF ( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
514: INFO = -7
515: ELSE IF ( LDA.LT.MAX( 1, M ) ) THEN
516: INFO = -9
517: ELSE IF ( LDU.LT.1 .OR. ( LSVC0 .AND. LDU.LT.M ) .OR.
518: $ ( WNTUF .AND. LDU.LT.N ) ) THEN
519: INFO = -12
520: ELSE IF ( LDV.LT.1 .OR. ( RSVEC .AND. LDV.LT.N ) .OR.
521: $ ( CONDA .AND. LDV.LT.N ) ) THEN
522: INFO = -14
523: ELSE IF ( LIWORK .LT. IMINWRK .AND. .NOT. LQUERY ) THEN
524: INFO = -17
525: END IF
526: *
527: *
528: IF ( INFO .EQ. 0 ) THEN
529: * .. compute the minimal and the optimal workspace lengths
530: * [[The expressions for computing the minimal and the optimal
531: * values of LCWORK are written with a lot of redundancy and
532: * can be simplified. However, this detailed form is easier for
533: * maintenance and modifications of the code.]]
534: *
535: * .. minimal workspace length for ZGEQP3 of an M x N matrix
536: LWQP3 = N+1
537: * .. minimal workspace length for ZUNMQR to build left singular vectors
538: IF ( WNTUS .OR. WNTUR ) THEN
539: LWUNQ = MAX( N , 1 )
540: ELSE IF ( WNTUA ) THEN
541: LWUNQ = MAX( M , 1 )
542: END IF
543: * .. minimal workspace length for ZPOCON of an N x N matrix
544: LWCON = 2 * N
545: * .. ZGESVD of an N x N matrix
546: LWSVD = MAX( 3 * N, 1 )
547: IF ( LQUERY ) THEN
548: CALL ZGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1,
549: $ RDUMMY, IERR )
550: LWRK_ZGEQP3 = INT( CDUMMY(1) )
551: IF ( WNTUS .OR. WNTUR ) THEN
552: CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
553: $ LDU, CDUMMY, -1, IERR )
554: LWRK_ZUNMQR = INT( CDUMMY(1) )
555: ELSE IF ( WNTUA ) THEN
556: CALL ZUNMQR( 'L', 'N', M, M, N, A, LDA, CDUMMY, U,
557: $ LDU, CDUMMY, -1, IERR )
558: LWRK_ZUNMQR = INT( CDUMMY(1) )
559: ELSE
560: LWRK_ZUNMQR = 0
561: END IF
562: END IF
563: MINWRK = 2
564: OPTWRK = 2
565: IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN
566: * .. minimal and optimal sizes of the complex workspace if
567: * only the singular values are requested
568: IF ( CONDA ) THEN
569: MINWRK = MAX( N+LWQP3, LWCON, LWSVD )
570: ELSE
571: MINWRK = MAX( N+LWQP3, LWSVD )
572: END IF
573: IF ( LQUERY ) THEN
574: CALL ZGESVD( 'N', 'N', N, N, A, LDA, S, U, LDU,
575: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
576: LWRK_ZGESVD = INT( CDUMMY(1) )
577: IF ( CONDA ) THEN
578: OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON, LWRK_ZGESVD )
579: ELSE
580: OPTWRK = MAX( N+LWRK_ZGEQP3, LWRK_ZGESVD )
581: END IF
582: END IF
583: ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN
584: * .. minimal and optimal sizes of the complex workspace if the
585: * singular values and the left singular vectors are requested
586: IF ( CONDA ) THEN
587: MINWRK = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ )
588: ELSE
589: MINWRK = N + MAX( LWQP3, LWSVD, LWUNQ )
590: END IF
591: IF ( LQUERY ) THEN
592: IF ( RTRANS ) THEN
593: CALL ZGESVD( 'N', 'O', N, N, A, LDA, S, U, LDU,
594: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
595: ELSE
596: CALL ZGESVD( 'O', 'N', N, N, A, LDA, S, U, LDU,
597: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
598: END IF
599: LWRK_ZGESVD = INT( CDUMMY(1) )
600: IF ( CONDA ) THEN
601: OPTWRK = N + MAX( LWRK_ZGEQP3, LWCON, LWRK_ZGESVD,
602: $ LWRK_ZUNMQR )
603: ELSE
604: OPTWRK = N + MAX( LWRK_ZGEQP3, LWRK_ZGESVD,
605: $ LWRK_ZUNMQR )
606: END IF
607: END IF
608: ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN
609: * .. minimal and optimal sizes of the complex workspace if the
610: * singular values and the right singular vectors are requested
611: IF ( CONDA ) THEN
612: MINWRK = N + MAX( LWQP3, LWCON, LWSVD )
613: ELSE
614: MINWRK = N + MAX( LWQP3, LWSVD )
615: END IF
616: IF ( LQUERY ) THEN
617: IF ( RTRANS ) THEN
618: CALL ZGESVD( 'O', 'N', N, N, A, LDA, S, U, LDU,
619: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
620: ELSE
621: CALL ZGESVD( 'N', 'O', N, N, A, LDA, S, U, LDU,
622: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
623: END IF
624: LWRK_ZGESVD = INT( CDUMMY(1) )
625: IF ( CONDA ) THEN
626: OPTWRK = N + MAX( LWRK_ZGEQP3, LWCON, LWRK_ZGESVD )
627: ELSE
628: OPTWRK = N + MAX( LWRK_ZGEQP3, LWRK_ZGESVD )
629: END IF
630: END IF
631: ELSE
632: * .. minimal and optimal sizes of the complex workspace if the
633: * full SVD is requested
634: IF ( RTRANS ) THEN
635: MINWRK = MAX( LWQP3, LWSVD, LWUNQ )
636: IF ( CONDA ) MINWRK = MAX( MINWRK, LWCON )
637: MINWRK = MINWRK + N
638: IF ( WNTVA ) THEN
639: * .. minimal workspace length for N x N/2 ZGEQRF
640: LWQRF = MAX( N/2, 1 )
641: * .. minimal workspace lengt for N/2 x N/2 ZGESVD
642: LWSVD2 = MAX( 3 * (N/2), 1 )
643: LWUNQ2 = MAX( N, 1 )
644: MINWRK2 = MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2,
645: $ N/2+LWUNQ2, LWUNQ )
646: IF ( CONDA ) MINWRK2 = MAX( MINWRK2, LWCON )
647: MINWRK2 = N + MINWRK2
648: MINWRK = MAX( MINWRK, MINWRK2 )
649: END IF
650: ELSE
651: MINWRK = MAX( LWQP3, LWSVD, LWUNQ )
652: IF ( CONDA ) MINWRK = MAX( MINWRK, LWCON )
653: MINWRK = MINWRK + N
654: IF ( WNTVA ) THEN
655: * .. minimal workspace length for N/2 x N ZGELQF
656: LWLQF = MAX( N/2, 1 )
657: LWSVD2 = MAX( 3 * (N/2), 1 )
658: LWUNLQ = MAX( N , 1 )
659: MINWRK2 = MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2,
660: $ N/2+LWUNLQ, LWUNQ )
661: IF ( CONDA ) MINWRK2 = MAX( MINWRK2, LWCON )
662: MINWRK2 = N + MINWRK2
663: MINWRK = MAX( MINWRK, MINWRK2 )
664: END IF
665: END IF
666: IF ( LQUERY ) THEN
667: IF ( RTRANS ) THEN
668: CALL ZGESVD( 'O', 'A', N, N, A, LDA, S, U, LDU,
669: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
670: LWRK_ZGESVD = INT( CDUMMY(1) )
671: OPTWRK = MAX(LWRK_ZGEQP3,LWRK_ZGESVD,LWRK_ZUNMQR)
672: IF ( CONDA ) OPTWRK = MAX( OPTWRK, LWCON )
673: OPTWRK = N + OPTWRK
674: IF ( WNTVA ) THEN
675: CALL ZGEQRF(N,N/2,U,LDU,CDUMMY,CDUMMY,-1,IERR)
676: LWRK_ZGEQRF = INT( CDUMMY(1) )
677: CALL ZGESVD( 'S', 'O', N/2,N/2, V,LDV, S, U,LDU,
678: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
679: LWRK_ZGESVD2 = INT( CDUMMY(1) )
680: CALL ZUNMQR( 'R', 'C', N, N, N/2, U, LDU, CDUMMY,
681: $ V, LDV, CDUMMY, -1, IERR )
682: LWRK_ZUNMQR2 = INT( CDUMMY(1) )
683: OPTWRK2 = MAX( LWRK_ZGEQP3, N/2+LWRK_ZGEQRF,
684: $ N/2+LWRK_ZGESVD2, N/2+LWRK_ZUNMQR2 )
685: IF ( CONDA ) OPTWRK2 = MAX( OPTWRK2, LWCON )
686: OPTWRK2 = N + OPTWRK2
687: OPTWRK = MAX( OPTWRK, OPTWRK2 )
688: END IF
689: ELSE
690: CALL ZGESVD( 'S', 'O', N, N, A, LDA, S, U, LDU,
691: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
692: LWRK_ZGESVD = INT( CDUMMY(1) )
693: OPTWRK = MAX(LWRK_ZGEQP3,LWRK_ZGESVD,LWRK_ZUNMQR)
694: IF ( CONDA ) OPTWRK = MAX( OPTWRK, LWCON )
695: OPTWRK = N + OPTWRK
696: IF ( WNTVA ) THEN
697: CALL ZGELQF(N/2,N,U,LDU,CDUMMY,CDUMMY,-1,IERR)
698: LWRK_ZGELQF = INT( CDUMMY(1) )
699: CALL ZGESVD( 'S','O', N/2,N/2, V, LDV, S, U, LDU,
700: $ V, LDV, CDUMMY, -1, RDUMMY, IERR )
701: LWRK_ZGESVD2 = INT( CDUMMY(1) )
702: CALL ZUNMLQ( 'R', 'N', N, N, N/2, U, LDU, CDUMMY,
703: $ V, LDV, CDUMMY,-1,IERR )
704: LWRK_ZUNMLQ = INT( CDUMMY(1) )
705: OPTWRK2 = MAX( LWRK_ZGEQP3, N/2+LWRK_ZGELQF,
706: $ N/2+LWRK_ZGESVD2, N/2+LWRK_ZUNMLQ )
707: IF ( CONDA ) OPTWRK2 = MAX( OPTWRK2, LWCON )
708: OPTWRK2 = N + OPTWRK2
709: OPTWRK = MAX( OPTWRK, OPTWRK2 )
710: END IF
711: END IF
712: END IF
713: END IF
714: *
715: MINWRK = MAX( 2, MINWRK )
716: OPTWRK = MAX( 2, OPTWRK )
717: IF ( LCWORK .LT. MINWRK .AND. (.NOT.LQUERY) ) INFO = -19
718: *
719: END IF
720: *
721: IF (INFO .EQ. 0 .AND. LRWORK .LT. RMINWRK .AND. .NOT. LQUERY) THEN
722: INFO = -21
723: END IF
724: IF( INFO.NE.0 ) THEN
725: CALL XERBLA( 'ZGESVDQ', -INFO )
726: RETURN
727: ELSE IF ( LQUERY ) THEN
728: *
729: * Return optimal workspace
730: *
731: IWORK(1) = IMINWRK
732: CWORK(1) = OPTWRK
733: CWORK(2) = MINWRK
734: RWORK(1) = RMINWRK
735: RETURN
736: END IF
737: *
738: * Quick return if the matrix is void.
739: *
740: IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) ) THEN
741: * .. all output is void.
742: RETURN
743: END IF
744: *
745: BIG = DLAMCH('O')
746: ASCALED = .FALSE.
747: IF ( ROWPRM ) THEN
748: * .. reordering the rows in decreasing sequence in the
749: * ell-infinity norm - this enhances numerical robustness in
750: * the case of differently scaled rows.
751: DO 1904 p = 1, M
752: * RWORK(p) = ABS( A(p,IZAMAX(N,A(p,1),LDA)) )
753: * [[ZLANGE will return NaN if an entry of the p-th row is Nan]]
754: RWORK(p) = ZLANGE( 'M', 1, N, A(p,1), LDA, RDUMMY )
755: * .. check for NaN's and Inf's
756: IF ( ( RWORK(p) .NE. RWORK(p) ) .OR.
757: $ ( (RWORK(p)*ZERO) .NE. ZERO ) ) THEN
758: INFO = -8
759: CALL XERBLA( 'ZGESVDQ', -INFO )
760: RETURN
761: END IF
762: 1904 CONTINUE
763: DO 1952 p = 1, M - 1
764: q = IDAMAX( M-p+1, RWORK(p), 1 ) + p - 1
765: IWORK(N+p) = q
766: IF ( p .NE. q ) THEN
767: RTMP = RWORK(p)
768: RWORK(p) = RWORK(q)
769: RWORK(q) = RTMP
770: END IF
771: 1952 CONTINUE
772: *
773: IF ( RWORK(1) .EQ. ZERO ) THEN
774: * Quick return: A is the M x N zero matrix.
775: NUMRANK = 0
776: CALL DLASET( 'G', N, 1, ZERO, ZERO, S, N )
777: IF ( WNTUS ) CALL ZLASET('G', M, N, CZERO, CONE, U, LDU)
778: IF ( WNTUA ) CALL ZLASET('G', M, M, CZERO, CONE, U, LDU)
779: IF ( WNTVA ) CALL ZLASET('G', N, N, CZERO, CONE, V, LDV)
780: IF ( WNTUF ) THEN
781: CALL ZLASET( 'G', N, 1, CZERO, CZERO, CWORK, N )
782: CALL ZLASET( 'G', M, N, CZERO, CONE, U, LDU )
783: END IF
784: DO 5001 p = 1, N
785: IWORK(p) = p
786: 5001 CONTINUE
787: IF ( ROWPRM ) THEN
788: DO 5002 p = N + 1, N + M - 1
789: IWORK(p) = p - N
790: 5002 CONTINUE
791: END IF
792: IF ( CONDA ) RWORK(1) = -1
793: RWORK(2) = -1
794: RETURN
795: END IF
796: *
797: IF ( RWORK(1) .GT. BIG / SQRT(DBLE(M)) ) THEN
798: * .. to prevent overflow in the QR factorization, scale the
799: * matrix by 1/sqrt(M) if too large entry detected
800: CALL ZLASCL('G',0,0,SQRT(DBLE(M)),ONE, M,N, A,LDA, IERR)
801: ASCALED = .TRUE.
802: END IF
803: CALL ZLASWP( N, A, LDA, 1, M-1, IWORK(N+1), 1 )
804: END IF
805: *
806: * .. At this stage, preemptive scaling is done only to avoid column
807: * norms overflows during the QR factorization. The SVD procedure should
808: * have its own scaling to save the singular values from overflows and
809: * underflows. That depends on the SVD procedure.
810: *
811: IF ( .NOT.ROWPRM ) THEN
812: RTMP = ZLANGE( 'M', M, N, A, LDA, RWORK )
813: IF ( ( RTMP .NE. RTMP ) .OR.
814: $ ( (RTMP*ZERO) .NE. ZERO ) ) THEN
815: INFO = -8
816: CALL XERBLA( 'ZGESVDQ', -INFO )
817: RETURN
818: END IF
819: IF ( RTMP .GT. BIG / SQRT(DBLE(M)) ) THEN
820: * .. to prevent overflow in the QR factorization, scale the
821: * matrix by 1/sqrt(M) if too large entry detected
822: CALL ZLASCL('G',0,0, SQRT(DBLE(M)),ONE, M,N, A,LDA, IERR)
823: ASCALED = .TRUE.
824: END IF
825: END IF
826: *
827: * .. QR factorization with column pivoting
828: *
829: * A * P = Q * [ R ]
830: * [ 0 ]
831: *
832: DO 1963 p = 1, N
833: * .. all columns are free columns
834: IWORK(p) = 0
835: 1963 CONTINUE
836: CALL ZGEQP3( M, N, A, LDA, IWORK, CWORK, CWORK(N+1), LCWORK-N,
837: $ RWORK, IERR )
838: *
839: * If the user requested accuracy level allows truncation in the
840: * computed upper triangular factor, the matrix R is examined and,
841: * if possible, replaced with its leading upper trapezoidal part.
842: *
843: EPSLN = DLAMCH('E')
844: SFMIN = DLAMCH('S')
845: * SMALL = SFMIN / EPSLN
846: NR = N
847: *
848: IF ( ACCLA ) THEN
849: *
850: * Standard absolute error bound suffices. All sigma_i with
851: * sigma_i < N*EPS*||A||_F are flushed to zero. This is an
852: * aggressive enforcement of lower numerical rank by introducing a
853: * backward error of the order of N*EPS*||A||_F.
854: NR = 1
855: RTMP = SQRT(DBLE(N))*EPSLN
856: DO 3001 p = 2, N
857: IF ( ABS(A(p,p)) .LT. (RTMP*ABS(A(1,1))) ) GO TO 3002
858: NR = NR + 1
859: 3001 CONTINUE
860: 3002 CONTINUE
861: *
862: ELSEIF ( ACCLM ) THEN
863: * .. similarly as above, only slightly more gentle (less aggressive).
864: * Sudden drop on the diagonal of R is used as the criterion for being
865: * close-to-rank-deficient. The threshold is set to EPSLN=DLAMCH('E').
866: * [[This can be made more flexible by replacing this hard-coded value
867: * with a user specified threshold.]] Also, the values that underflow
868: * will be truncated.
869: NR = 1
870: DO 3401 p = 2, N
871: IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
872: $ ( ABS(A(p,p)) .LT. SFMIN ) ) GO TO 3402
873: NR = NR + 1
874: 3401 CONTINUE
875: 3402 CONTINUE
876: *
877: ELSE
878: * .. RRQR not authorized to determine numerical rank except in the
879: * obvious case of zero pivots.
880: * .. inspect R for exact zeros on the diagonal;
881: * R(i,i)=0 => R(i:N,i:N)=0.
882: NR = 1
883: DO 3501 p = 2, N
884: IF ( ABS(A(p,p)) .EQ. ZERO ) GO TO 3502
885: NR = NR + 1
886: 3501 CONTINUE
887: 3502 CONTINUE
888: *
889: IF ( CONDA ) THEN
890: * Estimate the scaled condition number of A. Use the fact that it is
891: * the same as the scaled condition number of R.
892: * .. V is used as workspace
893: CALL ZLACPY( 'U', N, N, A, LDA, V, LDV )
894: * Only the leading NR x NR submatrix of the triangular factor
895: * is considered. Only if NR=N will this give a reliable error
896: * bound. However, even for NR < N, this can be used on an
897: * expert level and obtain useful information in the sense of
898: * perturbation theory.
899: DO 3053 p = 1, NR
900: RTMP = DZNRM2( p, V(1,p), 1 )
901: CALL ZDSCAL( p, ONE/RTMP, V(1,p), 1 )
902: 3053 CONTINUE
903: IF ( .NOT. ( LSVEC .OR. RSVEC ) ) THEN
904: CALL ZPOCON( 'U', NR, V, LDV, ONE, RTMP,
905: $ CWORK, RWORK, IERR )
906: ELSE
907: CALL ZPOCON( 'U', NR, V, LDV, ONE, RTMP,
908: $ CWORK(N+1), RWORK, IERR )
909: END IF
910: SCONDA = ONE / SQRT(RTMP)
911: * For NR=N, SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1),
912: * N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
913: * See the reference [1] for more details.
914: END IF
915: *
916: ENDIF
917: *
918: IF ( WNTUR ) THEN
919: N1 = NR
920: ELSE IF ( WNTUS .OR. WNTUF) THEN
921: N1 = N
922: ELSE IF ( WNTUA ) THEN
923: N1 = M
924: END IF
925: *
926: IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
927: *.......................................................................
928: * .. only the singular values are requested
929: *.......................................................................
930: IF ( RTRANS ) THEN
931: *
932: * .. compute the singular values of R**H = [A](1:NR,1:N)**H
933: * .. set the lower triangle of [A] to [A](1:NR,1:N)**H and
934: * the upper triangle of [A] to zero.
935: DO 1146 p = 1, MIN( N, NR )
936: A(p,p) = CONJG(A(p,p))
937: DO 1147 q = p + 1, N
938: A(q,p) = CONJG(A(p,q))
939: IF ( q .LE. NR ) A(p,q) = CZERO
940: 1147 CONTINUE
941: 1146 CONTINUE
942: *
943: CALL ZGESVD( 'N', 'N', N, NR, A, LDA, S, U, LDU,
944: $ V, LDV, CWORK, LCWORK, RWORK, INFO )
945: *
946: ELSE
947: *
948: * .. compute the singular values of R = [A](1:NR,1:N)
949: *
950: IF ( NR .GT. 1 )
951: $ CALL ZLASET( 'L', NR-1,NR-1, CZERO,CZERO, A(2,1), LDA )
952: CALL ZGESVD( 'N', 'N', NR, N, A, LDA, S, U, LDU,
953: $ V, LDV, CWORK, LCWORK, RWORK, INFO )
954: *
955: END IF
956: *
957: ELSE IF ( LSVEC .AND. ( .NOT. RSVEC) ) THEN
958: *.......................................................................
959: * .. the singular values and the left singular vectors requested
960: *.......................................................................""""""""
961: IF ( RTRANS ) THEN
962: * .. apply ZGESVD to R**H
963: * .. copy R**H into [U] and overwrite [U] with the right singular
964: * vectors of R
965: DO 1192 p = 1, NR
966: DO 1193 q = p, N
967: U(q,p) = CONJG(A(p,q))
968: 1193 CONTINUE
969: 1192 CONTINUE
970: IF ( NR .GT. 1 )
971: $ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, U(1,2), LDU )
972: * .. the left singular vectors not computed, the NR right singular
973: * vectors overwrite [U](1:NR,1:NR) as conjugate transposed. These
974: * will be pre-multiplied by Q to build the left singular vectors of A.
975: CALL ZGESVD( 'N', 'O', N, NR, U, LDU, S, U, LDU,
976: $ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
977: *
978: DO 1119 p = 1, NR
979: U(p,p) = CONJG(U(p,p))
980: DO 1120 q = p + 1, NR
981: CTMP = CONJG(U(q,p))
982: U(q,p) = CONJG(U(p,q))
983: U(p,q) = CTMP
984: 1120 CONTINUE
985: 1119 CONTINUE
986: *
987: ELSE
988: * .. apply ZGESVD to R
989: * .. copy R into [U] and overwrite [U] with the left singular vectors
990: CALL ZLACPY( 'U', NR, N, A, LDA, U, LDU )
991: IF ( NR .GT. 1 )
992: $ CALL ZLASET( 'L', NR-1, NR-1, CZERO, CZERO, U(2,1), LDU )
993: * .. the right singular vectors not computed, the NR left singular
994: * vectors overwrite [U](1:NR,1:NR)
995: CALL ZGESVD( 'O', 'N', NR, N, U, LDU, S, U, LDU,
996: $ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
997: * .. now [U](1:NR,1:NR) contains the NR left singular vectors of
998: * R. These will be pre-multiplied by Q to build the left singular
999: * vectors of A.
1000: END IF
1001: *
1002: * .. assemble the left singular vector matrix U of dimensions
1003: * (M x NR) or (M x N) or (M x M).
1004: IF ( ( NR .LT. M ) .AND. ( .NOT.WNTUF ) ) THEN
1005: CALL ZLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
1006: IF ( NR .LT. N1 ) THEN
1007: CALL ZLASET( 'A',NR,N1-NR,CZERO,CZERO,U(1,NR+1), LDU )
1008: CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
1009: $ U(NR+1,NR+1), LDU )
1010: END IF
1011: END IF
1012: *
1013: * The Q matrix from the first QRF is built into the left singular
1014: * vectors matrix U.
1015: *
1016: IF ( .NOT.WNTUF )
1017: $ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
1018: $ LDU, CWORK(N+1), LCWORK-N, IERR )
1019: IF ( ROWPRM .AND. .NOT.WNTUF )
1020: $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(N+1), -1 )
1021: *
1022: ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN
1023: *.......................................................................
1024: * .. the singular values and the right singular vectors requested
1025: *.......................................................................
1026: IF ( RTRANS ) THEN
1027: * .. apply ZGESVD to R**H
1028: * .. copy R**H into V and overwrite V with the left singular vectors
1029: DO 1165 p = 1, NR
1030: DO 1166 q = p, N
1031: V(q,p) = CONJG(A(p,q))
1032: 1166 CONTINUE
1033: 1165 CONTINUE
1034: IF ( NR .GT. 1 )
1035: $ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
1036: * .. the left singular vectors of R**H overwrite V, the right singular
1037: * vectors not computed
1038: IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
1039: CALL ZGESVD( 'O', 'N', N, NR, V, LDV, S, U, LDU,
1040: $ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
1041: *
1042: DO 1121 p = 1, NR
1043: V(p,p) = CONJG(V(p,p))
1044: DO 1122 q = p + 1, NR
1045: CTMP = CONJG(V(q,p))
1046: V(q,p) = CONJG(V(p,q))
1047: V(p,q) = CTMP
1048: 1122 CONTINUE
1049: 1121 CONTINUE
1050: *
1051: IF ( NR .LT. N ) THEN
1052: DO 1103 p = 1, NR
1053: DO 1104 q = NR + 1, N
1054: V(p,q) = CONJG(V(q,p))
1055: 1104 CONTINUE
1056: 1103 CONTINUE
1057: END IF
1058: CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
1059: ELSE
1060: * .. need all N right singular vectors and NR < N
1061: * [!] This is simple implementation that augments [V](1:N,1:NR)
1062: * by padding a zero block. In the case NR << N, a more efficient
1063: * way is to first use the QR factorization. For more details
1064: * how to implement this, see the " FULL SVD " branch.
1065: CALL ZLASET('G', N, N-NR, CZERO, CZERO, V(1,NR+1), LDV)
1066: CALL ZGESVD( 'O', 'N', N, N, V, LDV, S, U, LDU,
1067: $ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
1068: *
1069: DO 1123 p = 1, N
1070: V(p,p) = CONJG(V(p,p))
1071: DO 1124 q = p + 1, N
1072: CTMP = CONJG(V(q,p))
1073: V(q,p) = CONJG(V(p,q))
1074: V(p,q) = CTMP
1075: 1124 CONTINUE
1076: 1123 CONTINUE
1077: CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
1078: END IF
1079: *
1080: ELSE
1081: * .. aply ZGESVD to R
1082: * .. copy R into V and overwrite V with the right singular vectors
1083: CALL ZLACPY( 'U', NR, N, A, LDA, V, LDV )
1084: IF ( NR .GT. 1 )
1085: $ CALL ZLASET( 'L', NR-1, NR-1, CZERO, CZERO, V(2,1), LDV )
1086: * .. the right singular vectors overwrite V, the NR left singular
1087: * vectors stored in U(1:NR,1:NR)
1088: IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
1089: CALL ZGESVD( 'N', 'O', NR, N, V, LDV, S, U, LDU,
1090: $ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
1091: CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
1092: * .. now [V](1:NR,1:N) contains V(1:N,1:NR)**H
1093: ELSE
1094: * .. need all N right singular vectors and NR < N
1095: * [!] This is simple implementation that augments [V](1:NR,1:N)
1096: * by padding a zero block. In the case NR << N, a more efficient
1097: * way is to first use the LQ factorization. For more details
1098: * how to implement this, see the " FULL SVD " branch.
1099: CALL ZLASET('G', N-NR, N, CZERO,CZERO, V(NR+1,1), LDV)
1100: CALL ZGESVD( 'N', 'O', N, N, V, LDV, S, U, LDU,
1101: $ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
1102: CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
1103: END IF
1104: * .. now [V] contains the adjoint of the matrix of the right singular
1105: * vectors of A.
1106: END IF
1107: *
1108: ELSE
1109: *.......................................................................
1110: * .. FULL SVD requested
1111: *.......................................................................
1112: IF ( RTRANS ) THEN
1113: *
1114: * .. apply ZGESVD to R**H [[this option is left for R&D&T]]
1115: *
1116: IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
1117: * .. copy R**H into [V] and overwrite [V] with the left singular
1118: * vectors of R**H
1119: DO 1168 p = 1, NR
1120: DO 1169 q = p, N
1121: V(q,p) = CONJG(A(p,q))
1122: 1169 CONTINUE
1123: 1168 CONTINUE
1124: IF ( NR .GT. 1 )
1125: $ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
1126: *
1127: * .. the left singular vectors of R**H overwrite [V], the NR right
1128: * singular vectors of R**H stored in [U](1:NR,1:NR) as conjugate
1129: * transposed
1130: CALL ZGESVD( 'O', 'A', N, NR, V, LDV, S, V, LDV,
1131: $ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
1132: * .. assemble V
1133: DO 1115 p = 1, NR
1134: V(p,p) = CONJG(V(p,p))
1135: DO 1116 q = p + 1, NR
1136: CTMP = CONJG(V(q,p))
1137: V(q,p) = CONJG(V(p,q))
1138: V(p,q) = CTMP
1139: 1116 CONTINUE
1140: 1115 CONTINUE
1141: IF ( NR .LT. N ) THEN
1142: DO 1101 p = 1, NR
1143: DO 1102 q = NR+1, N
1144: V(p,q) = CONJG(V(q,p))
1145: 1102 CONTINUE
1146: 1101 CONTINUE
1147: END IF
1148: CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
1149: *
1150: DO 1117 p = 1, NR
1151: U(p,p) = CONJG(U(p,p))
1152: DO 1118 q = p + 1, NR
1153: CTMP = CONJG(U(q,p))
1154: U(q,p) = CONJG(U(p,q))
1155: U(p,q) = CTMP
1156: 1118 CONTINUE
1157: 1117 CONTINUE
1158: *
1159: IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
1160: CALL ZLASET('A', M-NR,NR, CZERO,CZERO, U(NR+1,1), LDU)
1161: IF ( NR .LT. N1 ) THEN
1162: CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
1163: CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
1164: $ U(NR+1,NR+1), LDU )
1165: END IF
1166: END IF
1167: *
1168: ELSE
1169: * .. need all N right singular vectors and NR < N
1170: * .. copy R**H into [V] and overwrite [V] with the left singular
1171: * vectors of R**H
1172: * [[The optimal ratio N/NR for using QRF instead of padding
1173: * with zeros. Here hard coded to 2; it must be at least
1174: * two due to work space constraints.]]
1175: * OPTRATIO = ILAENV(6, 'ZGESVD', 'S' // 'O', NR,N,0,0)
1176: * OPTRATIO = MAX( OPTRATIO, 2 )
1177: OPTRATIO = 2
1178: IF ( OPTRATIO*NR .GT. N ) THEN
1179: DO 1198 p = 1, NR
1180: DO 1199 q = p, N
1181: V(q,p) = CONJG(A(p,q))
1182: 1199 CONTINUE
1183: 1198 CONTINUE
1184: IF ( NR .GT. 1 )
1185: $ CALL ZLASET('U',NR-1,NR-1, CZERO,CZERO, V(1,2),LDV)
1186: *
1187: CALL ZLASET('A',N,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
1188: CALL ZGESVD( 'O', 'A', N, N, V, LDV, S, V, LDV,
1189: $ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
1190: *
1191: DO 1113 p = 1, N
1192: V(p,p) = CONJG(V(p,p))
1193: DO 1114 q = p + 1, N
1194: CTMP = CONJG(V(q,p))
1195: V(q,p) = CONJG(V(p,q))
1196: V(p,q) = CTMP
1197: 1114 CONTINUE
1198: 1113 CONTINUE
1199: CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
1200: * .. assemble the left singular vector matrix U of dimensions
1201: * (M x N1), i.e. (M x N) or (M x M).
1202: *
1203: DO 1111 p = 1, N
1204: U(p,p) = CONJG(U(p,p))
1205: DO 1112 q = p + 1, N
1206: CTMP = CONJG(U(q,p))
1207: U(q,p) = CONJG(U(p,q))
1208: U(p,q) = CTMP
1209: 1112 CONTINUE
1210: 1111 CONTINUE
1211: *
1212: IF ( ( N .LT. M ) .AND. .NOT.(WNTUF)) THEN
1213: CALL ZLASET('A',M-N,N,CZERO,CZERO,U(N+1,1),LDU)
1214: IF ( N .LT. N1 ) THEN
1215: CALL ZLASET('A',N,N1-N,CZERO,CZERO,U(1,N+1),LDU)
1216: CALL ZLASET('A',M-N,N1-N,CZERO,CONE,
1217: $ U(N+1,N+1), LDU )
1218: END IF
1219: END IF
1220: ELSE
1221: * .. copy R**H into [U] and overwrite [U] with the right
1222: * singular vectors of R
1223: DO 1196 p = 1, NR
1224: DO 1197 q = p, N
1225: U(q,NR+p) = CONJG(A(p,q))
1226: 1197 CONTINUE
1227: 1196 CONTINUE
1228: IF ( NR .GT. 1 )
1229: $ CALL ZLASET('U',NR-1,NR-1,CZERO,CZERO,U(1,NR+2),LDU)
1230: CALL ZGEQRF( N, NR, U(1,NR+1), LDU, CWORK(N+1),
1231: $ CWORK(N+NR+1), LCWORK-N-NR, IERR )
1232: DO 1143 p = 1, NR
1233: DO 1144 q = 1, N
1234: V(q,p) = CONJG(U(p,NR+q))
1235: 1144 CONTINUE
1236: 1143 CONTINUE
1237: CALL ZLASET('U',NR-1,NR-1,CZERO,CZERO,V(1,2),LDV)
1238: CALL ZGESVD( 'S', 'O', NR, NR, V, LDV, S, U, LDU,
1239: $ V,LDV, CWORK(N+NR+1),LCWORK-N-NR,RWORK, INFO )
1240: CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
1241: CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
1242: CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
1243: CALL ZUNMQR('R','C', N, N, NR, U(1,NR+1), LDU,
1244: $ CWORK(N+1),V,LDV,CWORK(N+NR+1),LCWORK-N-NR,IERR)
1245: CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
1246: * .. assemble the left singular vector matrix U of dimensions
1247: * (M x NR) or (M x N) or (M x M).
1248: IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
1249: CALL ZLASET('A',M-NR,NR,CZERO,CZERO,U(NR+1,1),LDU)
1250: IF ( NR .LT. N1 ) THEN
1251: CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
1252: CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
1253: $ U(NR+1,NR+1),LDU)
1254: END IF
1255: END IF
1256: END IF
1257: END IF
1258: *
1259: ELSE
1260: *
1261: * .. apply ZGESVD to R [[this is the recommended option]]
1262: *
1263: IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
1264: * .. copy R into [V] and overwrite V with the right singular vectors
1265: CALL ZLACPY( 'U', NR, N, A, LDA, V, LDV )
1266: IF ( NR .GT. 1 )
1267: $ CALL ZLASET( 'L', NR-1,NR-1, CZERO,CZERO, V(2,1), LDV )
1268: * .. the right singular vectors of R overwrite [V], the NR left
1269: * singular vectors of R stored in [U](1:NR,1:NR)
1270: CALL ZGESVD( 'S', 'O', NR, N, V, LDV, S, U, LDU,
1271: $ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
1272: CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
1273: * .. now [V](1:NR,1:N) contains V(1:N,1:NR)**H
1274: * .. assemble the left singular vector matrix U of dimensions
1275: * (M x NR) or (M x N) or (M x M).
1276: IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
1277: CALL ZLASET('A', M-NR,NR, CZERO,CZERO, U(NR+1,1), LDU)
1278: IF ( NR .LT. N1 ) THEN
1279: CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
1280: CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
1281: $ U(NR+1,NR+1), LDU )
1282: END IF
1283: END IF
1284: *
1285: ELSE
1286: * .. need all N right singular vectors and NR < N
1287: * .. the requested number of the left singular vectors
1288: * is then N1 (N or M)
1289: * [[The optimal ratio N/NR for using LQ instead of padding
1290: * with zeros. Here hard coded to 2; it must be at least
1291: * two due to work space constraints.]]
1292: * OPTRATIO = ILAENV(6, 'ZGESVD', 'S' // 'O', NR,N,0,0)
1293: * OPTRATIO = MAX( OPTRATIO, 2 )
1294: OPTRATIO = 2
1295: IF ( OPTRATIO * NR .GT. N ) THEN
1296: CALL ZLACPY( 'U', NR, N, A, LDA, V, LDV )
1297: IF ( NR .GT. 1 )
1298: $ CALL ZLASET('L', NR-1,NR-1, CZERO,CZERO, V(2,1),LDV)
1299: * .. the right singular vectors of R overwrite [V], the NR left
1300: * singular vectors of R stored in [U](1:NR,1:NR)
1301: CALL ZLASET('A', N-NR,N, CZERO,CZERO, V(NR+1,1),LDV)
1302: CALL ZGESVD( 'S', 'O', N, N, V, LDV, S, U, LDU,
1303: $ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
1304: CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
1305: * .. now [V] contains the adjoint of the matrix of the right
1306: * singular vectors of A. The leading N left singular vectors
1307: * are in [U](1:N,1:N)
1308: * .. assemble the left singular vector matrix U of dimensions
1309: * (M x N1), i.e. (M x N) or (M x M).
1310: IF ( ( N .LT. M ) .AND. .NOT.(WNTUF)) THEN
1311: CALL ZLASET('A',M-N,N,CZERO,CZERO,U(N+1,1),LDU)
1312: IF ( N .LT. N1 ) THEN
1313: CALL ZLASET('A',N,N1-N,CZERO,CZERO,U(1,N+1),LDU)
1314: CALL ZLASET( 'A',M-N,N1-N,CZERO,CONE,
1315: $ U(N+1,N+1), LDU )
1316: END IF
1317: END IF
1318: ELSE
1319: CALL ZLACPY( 'U', NR, N, A, LDA, U(NR+1,1), LDU )
1320: IF ( NR .GT. 1 )
1321: $ CALL ZLASET('L',NR-1,NR-1,CZERO,CZERO,U(NR+2,1),LDU)
1322: CALL ZGELQF( NR, N, U(NR+1,1), LDU, CWORK(N+1),
1323: $ CWORK(N+NR+1), LCWORK-N-NR, IERR )
1324: CALL ZLACPY('L',NR,NR,U(NR+1,1),LDU,V,LDV)
1325: IF ( NR .GT. 1 )
1326: $ CALL ZLASET('U',NR-1,NR-1,CZERO,CZERO,V(1,2),LDV)
1327: CALL ZGESVD( 'S', 'O', NR, NR, V, LDV, S, U, LDU,
1328: $ V, LDV, CWORK(N+NR+1), LCWORK-N-NR, RWORK, INFO )
1329: CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
1330: CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
1331: CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
1332: CALL ZUNMLQ('R','N',N,N,NR,U(NR+1,1),LDU,CWORK(N+1),
1333: $ V, LDV, CWORK(N+NR+1),LCWORK-N-NR,IERR)
1334: CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
1335: * .. assemble the left singular vector matrix U of dimensions
1336: * (M x NR) or (M x N) or (M x M).
1337: IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
1338: CALL ZLASET('A',M-NR,NR,CZERO,CZERO,U(NR+1,1),LDU)
1339: IF ( NR .LT. N1 ) THEN
1340: CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
1341: CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
1342: $ U(NR+1,NR+1), LDU )
1343: END IF
1344: END IF
1345: END IF
1346: END IF
1347: * .. end of the "R**H or R" branch
1348: END IF
1349: *
1350: * The Q matrix from the first QRF is built into the left singular
1351: * vectors matrix U.
1352: *
1353: IF ( .NOT. WNTUF )
1354: $ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
1355: $ LDU, CWORK(N+1), LCWORK-N, IERR )
1356: IF ( ROWPRM .AND. .NOT.WNTUF )
1357: $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(N+1), -1 )
1358: *
1359: * ... end of the "full SVD" branch
1360: END IF
1361: *
1362: * Check whether some singular values are returned as zeros, e.g.
1363: * due to underflow, and update the numerical rank.
1364: p = NR
1365: DO 4001 q = p, 1, -1
1366: IF ( S(q) .GT. ZERO ) GO TO 4002
1367: NR = NR - 1
1368: 4001 CONTINUE
1369: 4002 CONTINUE
1370: *
1371: * .. if numerical rank deficiency is detected, the truncated
1372: * singular values are set to zero.
1373: IF ( NR .LT. N ) CALL DLASET( 'G', N-NR,1, ZERO,ZERO, S(NR+1), N )
1374: * .. undo scaling; this may cause overflow in the largest singular
1375: * values.
1376: IF ( ASCALED )
1377: $ CALL DLASCL( 'G',0,0, ONE,SQRT(DBLE(M)), NR,1, S, N, IERR )
1378: IF ( CONDA ) RWORK(1) = SCONDA
1379: RWORK(2) = p - NR
1380: * .. p-NR is the number of singular values that are computed as
1381: * exact zeros in ZGESVD() applied to the (possibly truncated)
1382: * full row rank triangular (trapezoidal) factor of A.
1383: NUMRANK = NR
1384: *
1385: RETURN
1386: *
1387: * End of ZGESVDQ
1388: *
1389: END
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