LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
zggsvp3.f
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1 *> \brief \b ZGGSVP3
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download ZGGSVP3 + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggsvp3.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggsvp3.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
22 * TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
23 * IWORK, RWORK, TAU, WORK, LWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBQ, JOBU, JOBV
27 * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK
28 * DOUBLE PRECISION TOLA, TOLB
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IWORK( * )
32 * DOUBLE PRECISION RWORK( * )
33 * COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
34 * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> ZGGSVP3 computes unitary matrices U, V and Q such that
44 *>
45 *> N-K-L K L
46 *> U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
47 *> L ( 0 0 A23 )
48 *> M-K-L ( 0 0 0 )
49 *>
50 *> N-K-L K L
51 *> = K ( 0 A12 A13 ) if M-K-L < 0;
52 *> M-K ( 0 0 A23 )
53 *>
54 *> N-K-L K L
55 *> V**H*B*Q = L ( 0 0 B13 )
56 *> P-L ( 0 0 0 )
57 *>
58 *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
59 *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
60 *> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
61 *> numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H.
62 *>
63 *> This decomposition is the preprocessing step for computing the
64 *> Generalized Singular Value Decomposition (GSVD), see subroutine
65 *> ZGGSVD3.
66 *> \endverbatim
67 *
68 * Arguments:
69 * ==========
70 *
71 *> \param[in] JOBU
72 *> \verbatim
73 *> JOBU is CHARACTER*1
74 *> = 'U': Unitary matrix U is computed;
75 *> = 'N': U is not computed.
76 *> \endverbatim
77 *>
78 *> \param[in] JOBV
79 *> \verbatim
80 *> JOBV is CHARACTER*1
81 *> = 'V': Unitary matrix V is computed;
82 *> = 'N': V is not computed.
83 *> \endverbatim
84 *>
85 *> \param[in] JOBQ
86 *> \verbatim
87 *> JOBQ is CHARACTER*1
88 *> = 'Q': Unitary matrix Q is computed;
89 *> = 'N': Q is not computed.
90 *> \endverbatim
91 *>
92 *> \param[in] M
93 *> \verbatim
94 *> M is INTEGER
95 *> The number of rows of the matrix A. M >= 0.
96 *> \endverbatim
97 *>
98 *> \param[in] P
99 *> \verbatim
100 *> P is INTEGER
101 *> The number of rows of the matrix B. P >= 0.
102 *> \endverbatim
103 *>
104 *> \param[in] N
105 *> \verbatim
106 *> N is INTEGER
107 *> The number of columns of the matrices A and B. N >= 0.
108 *> \endverbatim
109 *>
110 *> \param[in,out] A
111 *> \verbatim
112 *> A is COMPLEX*16 array, dimension (LDA,N)
113 *> On entry, the M-by-N matrix A.
114 *> On exit, A contains the triangular (or trapezoidal) matrix
115 *> described in the Purpose section.
116 *> \endverbatim
117 *>
118 *> \param[in] LDA
119 *> \verbatim
120 *> LDA is INTEGER
121 *> The leading dimension of the array A. LDA >= max(1,M).
122 *> \endverbatim
123 *>
124 *> \param[in,out] B
125 *> \verbatim
126 *> B is COMPLEX*16 array, dimension (LDB,N)
127 *> On entry, the P-by-N matrix B.
128 *> On exit, B contains the triangular matrix described in
129 *> the Purpose section.
130 *> \endverbatim
131 *>
132 *> \param[in] LDB
133 *> \verbatim
134 *> LDB is INTEGER
135 *> The leading dimension of the array B. LDB >= max(1,P).
136 *> \endverbatim
137 *>
138 *> \param[in] TOLA
139 *> \verbatim
140 *> TOLA is DOUBLE PRECISION
141 *> \endverbatim
142 *>
143 *> \param[in] TOLB
144 *> \verbatim
145 *> TOLB is DOUBLE PRECISION
146 *>
147 *> TOLA and TOLB are the thresholds to determine the effective
148 *> numerical rank of matrix B and a subblock of A. Generally,
149 *> they are set to
150 *> TOLA = MAX(M,N)*norm(A)*MAZHEPS,
151 *> TOLB = MAX(P,N)*norm(B)*MAZHEPS.
152 *> The size of TOLA and TOLB may affect the size of backward
153 *> errors of the decomposition.
154 *> \endverbatim
155 *>
156 *> \param[out] K
157 *> \verbatim
158 *> K is INTEGER
159 *> \endverbatim
160 *>
161 *> \param[out] L
162 *> \verbatim
163 *> L is INTEGER
164 *>
165 *> On exit, K and L specify the dimension of the subblocks
166 *> described in Purpose section.
167 *> K + L = effective numerical rank of (A**H,B**H)**H.
168 *> \endverbatim
169 *>
170 *> \param[out] U
171 *> \verbatim
172 *> U is COMPLEX*16 array, dimension (LDU,M)
173 *> If JOBU = 'U', U contains the unitary matrix U.
174 *> If JOBU = 'N', U is not referenced.
175 *> \endverbatim
176 *>
177 *> \param[in] LDU
178 *> \verbatim
179 *> LDU is INTEGER
180 *> The leading dimension of the array U. LDU >= max(1,M) if
181 *> JOBU = 'U'; LDU >= 1 otherwise.
182 *> \endverbatim
183 *>
184 *> \param[out] V
185 *> \verbatim
186 *> V is COMPLEX*16 array, dimension (LDV,P)
187 *> If JOBV = 'V', V contains the unitary matrix V.
188 *> If JOBV = 'N', V is not referenced.
189 *> \endverbatim
190 *>
191 *> \param[in] LDV
192 *> \verbatim
193 *> LDV is INTEGER
194 *> The leading dimension of the array V. LDV >= max(1,P) if
195 *> JOBV = 'V'; LDV >= 1 otherwise.
196 *> \endverbatim
197 *>
198 *> \param[out] Q
199 *> \verbatim
200 *> Q is COMPLEX*16 array, dimension (LDQ,N)
201 *> If JOBQ = 'Q', Q contains the unitary matrix Q.
202 *> If JOBQ = 'N', Q is not referenced.
203 *> \endverbatim
204 *>
205 *> \param[in] LDQ
206 *> \verbatim
207 *> LDQ is INTEGER
208 *> The leading dimension of the array Q. LDQ >= max(1,N) if
209 *> JOBQ = 'Q'; LDQ >= 1 otherwise.
210 *> \endverbatim
211 *>
212 *> \param[out] IWORK
213 *> \verbatim
214 *> IWORK is INTEGER array, dimension (N)
215 *> \endverbatim
216 *>
217 *> \param[out] RWORK
218 *> \verbatim
219 *> RWORK is DOUBLE PRECISION array, dimension (2*N)
220 *> \endverbatim
221 *>
222 *> \param[out] TAU
223 *> \verbatim
224 *> TAU is COMPLEX*16 array, dimension (N)
225 *> \endverbatim
226 *>
227 *> \param[out] WORK
228 *> \verbatim
229 *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
230 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
231 *> \endverbatim
232 *>
233 *> \param[in] LWORK
234 *> \verbatim
235 *> LWORK is INTEGER
236 *> The dimension of the array WORK.
237 *>
238 *> If LWORK = -1, then a workspace query is assumed; the routine
239 *> only calculates the optimal size of the WORK array, returns
240 *> this value as the first entry of the WORK array, and no error
241 *> message related to LWORK is issued by XERBLA.
242 *> \endverbatim
243 *>
244 *> \param[out] INFO
245 *> \verbatim
246 *> INFO is INTEGER
247 *> = 0: successful exit
248 *> < 0: if INFO = -i, the i-th argument had an illegal value.
249 *> \endverbatim
250 *
251 * Authors:
252 * ========
253 *
254 *> \author Univ. of Tennessee
255 *> \author Univ. of California Berkeley
256 *> \author Univ. of Colorado Denver
257 *> \author NAG Ltd.
258 *
259 *> \ingroup complex16OTHERcomputational
260 *
261 *> \par Further Details:
262 * =====================
263 *
264 *> \verbatim
265 *>
266 *> The subroutine uses LAPACK subroutine ZGEQP3 for the QR factorization
267 *> with column pivoting to detect the effective numerical rank of the
268 *> a matrix. It may be replaced by a better rank determination strategy.
269 *>
270 *> ZGGSVP3 replaces the deprecated subroutine ZGGSVP.
271 *>
272 *> \endverbatim
273 *>
274 * =====================================================================
275  SUBROUTINE zggsvp3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
276  $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
277  $ IWORK, RWORK, TAU, WORK, LWORK, INFO )
278 *
279 * -- LAPACK computational routine --
280 * -- LAPACK is a software package provided by Univ. of Tennessee, --
281 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
282 *
283  IMPLICIT NONE
284 *
285 * .. Scalar Arguments ..
286  CHARACTER JOBQ, JOBU, JOBV
287  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
288  $ lwork
289  DOUBLE PRECISION TOLA, TOLB
290 * ..
291 * .. Array Arguments ..
292  INTEGER IWORK( * )
293  DOUBLE PRECISION RWORK( * )
294  COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
295  $ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
296 * ..
297 *
298 * =====================================================================
299 *
300 * .. Parameters ..
301  COMPLEX*16 CZERO, CONE
302  PARAMETER ( CZERO = ( 0.0d+0, 0.0d+0 ),
303  $ cone = ( 1.0d+0, 0.0d+0 ) )
304 * ..
305 * .. Local Scalars ..
306  LOGICAL FORWRD, WANTQ, WANTU, WANTV, LQUERY
307  INTEGER I, J, LWKOPT
308 * ..
309 * .. External Functions ..
310  LOGICAL LSAME
311  EXTERNAL LSAME
312 * ..
313 * .. External Subroutines ..
314  EXTERNAL xerbla, zgeqp3, zgeqr2, zgerq2, zlacpy, zlapmt,
316 * ..
317 * .. Intrinsic Functions ..
318  INTRINSIC abs, dble, dimag, max, min
319 * ..
320 * .. Executable Statements ..
321 *
322 * Test the input parameters
323 *
324  wantu = lsame( jobu, 'U' )
325  wantv = lsame( jobv, 'V' )
326  wantq = lsame( jobq, 'Q' )
327  forwrd = .true.
328  lquery = ( lwork.EQ.-1 )
329  lwkopt = 1
330 *
331 * Test the input arguments
332 *
333  info = 0
334  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
335  info = -1
336  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
337  info = -2
338  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
339  info = -3
340  ELSE IF( m.LT.0 ) THEN
341  info = -4
342  ELSE IF( p.LT.0 ) THEN
343  info = -5
344  ELSE IF( n.LT.0 ) THEN
345  info = -6
346  ELSE IF( lda.LT.max( 1, m ) ) THEN
347  info = -8
348  ELSE IF( ldb.LT.max( 1, p ) ) THEN
349  info = -10
350  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
351  info = -16
352  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
353  info = -18
354  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
355  info = -20
356  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
357  info = -24
358  END IF
359 *
360 * Compute workspace
361 *
362  IF( info.EQ.0 ) THEN
363  CALL zgeqp3( p, n, b, ldb, iwork, tau, work, -1, rwork, info )
364  lwkopt = int( work( 1 ) )
365  IF( wantv ) THEN
366  lwkopt = max( lwkopt, p )
367  END IF
368  lwkopt = max( lwkopt, min( n, p ) )
369  lwkopt = max( lwkopt, m )
370  IF( wantq ) THEN
371  lwkopt = max( lwkopt, n )
372  END IF
373  CALL zgeqp3( m, n, a, lda, iwork, tau, work, -1, rwork, info )
374  lwkopt = max( lwkopt, int( work( 1 ) ) )
375  lwkopt = max( 1, lwkopt )
376  work( 1 ) = dcmplx( lwkopt )
377  END IF
378 *
379  IF( info.NE.0 ) THEN
380  CALL xerbla( 'ZGGSVP3', -info )
381  RETURN
382  END IF
383  IF( lquery ) THEN
384  RETURN
385  ENDIF
386 *
387 * QR with column pivoting of B: B*P = V*( S11 S12 )
388 * ( 0 0 )
389 *
390  DO 10 i = 1, n
391  iwork( i ) = 0
392  10 CONTINUE
393  CALL zgeqp3( p, n, b, ldb, iwork, tau, work, lwork, rwork, info )
394 *
395 * Update A := A*P
396 *
397  CALL zlapmt( forwrd, m, n, a, lda, iwork )
398 *
399 * Determine the effective rank of matrix B.
400 *
401  l = 0
402  DO 20 i = 1, min( p, n )
403  IF( abs( b( i, i ) ).GT.tolb )
404  $ l = l + 1
405  20 CONTINUE
406 *
407  IF( wantv ) THEN
408 *
409 * Copy the details of V, and form V.
410 *
411  CALL zlaset( 'Full', p, p, czero, czero, v, ldv )
412  IF( p.GT.1 )
413  $ CALL zlacpy( 'Lower', p-1, n, b( 2, 1 ), ldb, v( 2, 1 ),
414  $ ldv )
415  CALL zung2r( p, p, min( p, n ), v, ldv, tau, work, info )
416  END IF
417 *
418 * Clean up B
419 *
420  DO 40 j = 1, l - 1
421  DO 30 i = j + 1, l
422  b( i, j ) = czero
423  30 CONTINUE
424  40 CONTINUE
425  IF( p.GT.l )
426  $ CALL zlaset( 'Full', p-l, n, czero, czero, b( l+1, 1 ), ldb )
427 *
428  IF( wantq ) THEN
429 *
430 * Set Q = I and Update Q := Q*P
431 *
432  CALL zlaset( 'Full', n, n, czero, cone, q, ldq )
433  CALL zlapmt( forwrd, n, n, q, ldq, iwork )
434  END IF
435 *
436  IF( p.GE.l .AND. n.NE.l ) THEN
437 *
438 * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
439 *
440  CALL zgerq2( l, n, b, ldb, tau, work, info )
441 *
442 * Update A := A*Z**H
443 *
444  CALL zunmr2( 'Right', 'Conjugate transpose', m, n, l, b, ldb,
445  $ tau, a, lda, work, info )
446  IF( wantq ) THEN
447 *
448 * Update Q := Q*Z**H
449 *
450  CALL zunmr2( 'Right', 'Conjugate transpose', n, n, l, b,
451  $ ldb, tau, q, ldq, work, info )
452  END IF
453 *
454 * Clean up B
455 *
456  CALL zlaset( 'Full', l, n-l, czero, czero, b, ldb )
457  DO 60 j = n - l + 1, n
458  DO 50 i = j - n + l + 1, l
459  b( i, j ) = czero
460  50 CONTINUE
461  60 CONTINUE
462 *
463  END IF
464 *
465 * Let N-L L
466 * A = ( A11 A12 ) M,
467 *
468 * then the following does the complete QR decomposition of A11:
469 *
470 * A11 = U*( 0 T12 )*P1**H
471 * ( 0 0 )
472 *
473  DO 70 i = 1, n - l
474  iwork( i ) = 0
475  70 CONTINUE
476  CALL zgeqp3( m, n-l, a, lda, iwork, tau, work, lwork, rwork,
477  $ info )
478 *
479 * Determine the effective rank of A11
480 *
481  k = 0
482  DO 80 i = 1, min( m, n-l )
483  IF( abs( a( i, i ) ).GT.tola )
484  $ k = k + 1
485  80 CONTINUE
486 *
487 * Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N )
488 *
489  CALL zunm2r( 'Left', 'Conjugate transpose', m, l, min( m, n-l ),
490  $ a, lda, tau, a( 1, n-l+1 ), lda, work, info )
491 *
492  IF( wantu ) THEN
493 *
494 * Copy the details of U, and form U
495 *
496  CALL zlaset( 'Full', m, m, czero, czero, u, ldu )
497  IF( m.GT.1 )
498  $ CALL zlacpy( 'Lower', m-1, n-l, a( 2, 1 ), lda, u( 2, 1 ),
499  $ ldu )
500  CALL zung2r( m, m, min( m, n-l ), u, ldu, tau, work, info )
501  END IF
502 *
503  IF( wantq ) THEN
504 *
505 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
506 *
507  CALL zlapmt( forwrd, n, n-l, q, ldq, iwork )
508  END IF
509 *
510 * Clean up A: set the strictly lower triangular part of
511 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
512 *
513  DO 100 j = 1, k - 1
514  DO 90 i = j + 1, k
515  a( i, j ) = czero
516  90 CONTINUE
517  100 CONTINUE
518  IF( m.GT.k )
519  $ CALL zlaset( 'Full', m-k, n-l, czero, czero, a( k+1, 1 ), lda )
520 *
521  IF( n-l.GT.k ) THEN
522 *
523 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
524 *
525  CALL zgerq2( k, n-l, a, lda, tau, work, info )
526 *
527  IF( wantq ) THEN
528 *
529 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H
530 *
531  CALL zunmr2( 'Right', 'Conjugate transpose', n, n-l, k, a,
532  $ lda, tau, q, ldq, work, info )
533  END IF
534 *
535 * Clean up A
536 *
537  CALL zlaset( 'Full', k, n-l-k, czero, czero, a, lda )
538  DO 120 j = n - l - k + 1, n - l
539  DO 110 i = j - n + l + k + 1, k
540  a( i, j ) = czero
541  110 CONTINUE
542  120 CONTINUE
543 *
544  END IF
545 *
546  IF( m.GT.k ) THEN
547 *
548 * QR factorization of A( K+1:M,N-L+1:N )
549 *
550  CALL zgeqr2( m-k, l, a( k+1, n-l+1 ), lda, tau, work, info )
551 *
552  IF( wantu ) THEN
553 *
554 * Update U(:,K+1:M) := U(:,K+1:M)*U1
555 *
556  CALL zunm2r( 'Right', 'No transpose', m, m-k, min( m-k, l ),
557  $ a( k+1, n-l+1 ), lda, tau, u( 1, k+1 ), ldu,
558  $ work, info )
559  END IF
560 *
561 * Clean up
562 *
563  DO 140 j = n - l + 1, n
564  DO 130 i = j - n + k + l + 1, m
565  a( i, j ) = czero
566  130 CONTINUE
567  140 CONTINUE
568 *
569  END IF
570 *
571  work( 1 ) = dcmplx( lwkopt )
572  RETURN
573 *
574 * End of ZGGSVP3
575 *
576  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zgeqp3(M, N, A, LDA, JPVT, TAU, WORK, LWORK, RWORK, INFO)
ZGEQP3
Definition: zgeqp3.f:159
subroutine zgeqr2(M, N, A, LDA, TAU, WORK, INFO)
ZGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Definition: zgeqr2.f:130
subroutine zgerq2(M, N, A, LDA, TAU, WORK, INFO)
ZGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
Definition: zgerq2.f:123
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlapmt(FORWRD, M, N, X, LDX, K)
ZLAPMT performs a forward or backward permutation of the columns of a matrix.
Definition: zlapmt.f:104
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine zung2r(M, N, K, A, LDA, TAU, WORK, INFO)
ZUNG2R
Definition: zung2r.f:114
subroutine zggsvp3(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, LWORK, INFO)
ZGGSVP3
Definition: zggsvp3.f:278
subroutine zunmr2(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
ZUNMR2 multiplies a general matrix by the unitary matrix from a RQ factorization determined by cgerqf...
Definition: zunmr2.f:159
subroutine zunm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
ZUNM2R multiplies a general matrix by the unitary matrix from a QR factorization determined by cgeqrf...
Definition: zunm2r.f:159