LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
zggsvp.f
Go to the documentation of this file.
1 *> \brief \b ZGGSVP
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download ZGGSVP + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggsvp.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggsvp.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggsvp.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGGSVP( 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, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBQ, JOBU, JOBV
27 * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
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 *> This routine is deprecated and has been replaced by routine ZGGSVP3.
44 *>
45 *> ZGGSVP computes unitary matrices U, V and Q such that
46 *>
47 *> N-K-L K L
48 *> U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
49 *> L ( 0 0 A23 )
50 *> M-K-L ( 0 0 0 )
51 *>
52 *> N-K-L K L
53 *> = K ( 0 A12 A13 ) if M-K-L < 0;
54 *> M-K ( 0 0 A23 )
55 *>
56 *> N-K-L K L
57 *> V**H*B*Q = L ( 0 0 B13 )
58 *> P-L ( 0 0 0 )
59 *>
60 *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
61 *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
62 *> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
63 *> numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H.
64 *>
65 *> This decomposition is the preprocessing step for computing the
66 *> Generalized Singular Value Decomposition (GSVD), see subroutine
67 *> ZGGSVD.
68 *> \endverbatim
69 *
70 * Arguments:
71 * ==========
72 *
73 *> \param[in] JOBU
74 *> \verbatim
75 *> JOBU is CHARACTER*1
76 *> = 'U': Unitary matrix U is computed;
77 *> = 'N': U is not computed.
78 *> \endverbatim
79 *>
80 *> \param[in] JOBV
81 *> \verbatim
82 *> JOBV is CHARACTER*1
83 *> = 'V': Unitary matrix V is computed;
84 *> = 'N': V is not computed.
85 *> \endverbatim
86 *>
87 *> \param[in] JOBQ
88 *> \verbatim
89 *> JOBQ is CHARACTER*1
90 *> = 'Q': Unitary matrix Q is computed;
91 *> = 'N': Q is not computed.
92 *> \endverbatim
93 *>
94 *> \param[in] M
95 *> \verbatim
96 *> M is INTEGER
97 *> The number of rows of the matrix A. M >= 0.
98 *> \endverbatim
99 *>
100 *> \param[in] P
101 *> \verbatim
102 *> P is INTEGER
103 *> The number of rows of the matrix B. P >= 0.
104 *> \endverbatim
105 *>
106 *> \param[in] N
107 *> \verbatim
108 *> N is INTEGER
109 *> The number of columns of the matrices A and B. N >= 0.
110 *> \endverbatim
111 *>
112 *> \param[in,out] A
113 *> \verbatim
114 *> A is COMPLEX*16 array, dimension (LDA,N)
115 *> On entry, the M-by-N matrix A.
116 *> On exit, A contains the triangular (or trapezoidal) matrix
117 *> described in the Purpose section.
118 *> \endverbatim
119 *>
120 *> \param[in] LDA
121 *> \verbatim
122 *> LDA is INTEGER
123 *> The leading dimension of the array A. LDA >= max(1,M).
124 *> \endverbatim
125 *>
126 *> \param[in,out] B
127 *> \verbatim
128 *> B is COMPLEX*16 array, dimension (LDB,N)
129 *> On entry, the P-by-N matrix B.
130 *> On exit, B contains the triangular matrix described in
131 *> the Purpose section.
132 *> \endverbatim
133 *>
134 *> \param[in] LDB
135 *> \verbatim
136 *> LDB is INTEGER
137 *> The leading dimension of the array B. LDB >= max(1,P).
138 *> \endverbatim
139 *>
140 *> \param[in] TOLA
141 *> \verbatim
142 *> TOLA is DOUBLE PRECISION
143 *> \endverbatim
144 *>
145 *> \param[in] TOLB
146 *> \verbatim
147 *> TOLB is DOUBLE PRECISION
148 *>
149 *> TOLA and TOLB are the thresholds to determine the effective
150 *> numerical rank of matrix B and a subblock of A. Generally,
151 *> they are set to
152 *> TOLA = MAX(M,N)*norm(A)*MAZHEPS,
153 *> TOLB = MAX(P,N)*norm(B)*MAZHEPS.
154 *> The size of TOLA and TOLB may affect the size of backward
155 *> errors of the decomposition.
156 *> \endverbatim
157 *>
158 *> \param[out] K
159 *> \verbatim
160 *> K is INTEGER
161 *> \endverbatim
162 *>
163 *> \param[out] L
164 *> \verbatim
165 *> L is INTEGER
166 *>
167 *> On exit, K and L specify the dimension of the subblocks
168 *> described in Purpose section.
169 *> K + L = effective numerical rank of (A**H,B**H)**H.
170 *> \endverbatim
171 *>
172 *> \param[out] U
173 *> \verbatim
174 *> U is COMPLEX*16 array, dimension (LDU,M)
175 *> If JOBU = 'U', U contains the unitary matrix U.
176 *> If JOBU = 'N', U is not referenced.
177 *> \endverbatim
178 *>
179 *> \param[in] LDU
180 *> \verbatim
181 *> LDU is INTEGER
182 *> The leading dimension of the array U. LDU >= max(1,M) if
183 *> JOBU = 'U'; LDU >= 1 otherwise.
184 *> \endverbatim
185 *>
186 *> \param[out] V
187 *> \verbatim
188 *> V is COMPLEX*16 array, dimension (LDV,P)
189 *> If JOBV = 'V', V contains the unitary matrix V.
190 *> If JOBV = 'N', V is not referenced.
191 *> \endverbatim
192 *>
193 *> \param[in] LDV
194 *> \verbatim
195 *> LDV is INTEGER
196 *> The leading dimension of the array V. LDV >= max(1,P) if
197 *> JOBV = 'V'; LDV >= 1 otherwise.
198 *> \endverbatim
199 *>
200 *> \param[out] Q
201 *> \verbatim
202 *> Q is COMPLEX*16 array, dimension (LDQ,N)
203 *> If JOBQ = 'Q', Q contains the unitary matrix Q.
204 *> If JOBQ = 'N', Q is not referenced.
205 *> \endverbatim
206 *>
207 *> \param[in] LDQ
208 *> \verbatim
209 *> LDQ is INTEGER
210 *> The leading dimension of the array Q. LDQ >= max(1,N) if
211 *> JOBQ = 'Q'; LDQ >= 1 otherwise.
212 *> \endverbatim
213 *>
214 *> \param[out] IWORK
215 *> \verbatim
216 *> IWORK is INTEGER array, dimension (N)
217 *> \endverbatim
218 *>
219 *> \param[out] RWORK
220 *> \verbatim
221 *> RWORK is DOUBLE PRECISION array, dimension (2*N)
222 *> \endverbatim
223 *>
224 *> \param[out] TAU
225 *> \verbatim
226 *> TAU is COMPLEX*16 array, dimension (N)
227 *> \endverbatim
228 *>
229 *> \param[out] WORK
230 *> \verbatim
231 *> WORK is COMPLEX*16 array, dimension (max(3*N,M,P))
232 *> \endverbatim
233 *>
234 *> \param[out] INFO
235 *> \verbatim
236 *> INFO is INTEGER
237 *> = 0: successful exit
238 *> < 0: if INFO = -i, the i-th argument had an illegal value.
239 *> \endverbatim
240 *
241 * Authors:
242 * ========
243 *
244 *> \author Univ. of Tennessee
245 *> \author Univ. of California Berkeley
246 *> \author Univ. of Colorado Denver
247 *> \author NAG Ltd.
248 *
249 *> \ingroup complex16OTHERcomputational
250 *
251 *> \par Further Details:
252 * =====================
253 *>
254 *> \verbatim
255 *>
256 *> The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
257 *> with column pivoting to detect the effective numerical rank of the
258 *> a matrix. It may be replaced by a better rank determination strategy.
259 *> \endverbatim
260 *>
261 * =====================================================================
262  SUBROUTINE zggsvp( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
263  $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
264  $ IWORK, RWORK, TAU, WORK, INFO )
265 *
266 * -- LAPACK computational routine --
267 * -- LAPACK is a software package provided by Univ. of Tennessee, --
268 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
269 *
270 * .. Scalar Arguments ..
271  CHARACTER JOBQ, JOBU, JOBV
272  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
273  DOUBLE PRECISION TOLA, TOLB
274 * ..
275 * .. Array Arguments ..
276  INTEGER IWORK( * )
277  DOUBLE PRECISION RWORK( * )
278  COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
279  $ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
280 * ..
281 *
282 * =====================================================================
283 *
284 * .. Parameters ..
285  COMPLEX*16 CZERO, CONE
286  PARAMETER ( CZERO = ( 0.0d+0, 0.0d+0 ),
287  $ cone = ( 1.0d+0, 0.0d+0 ) )
288 * ..
289 * .. Local Scalars ..
290  LOGICAL FORWRD, WANTQ, WANTU, WANTV
291  INTEGER I, J
292  COMPLEX*16 T
293 * ..
294 * .. External Functions ..
295  LOGICAL LSAME
296  EXTERNAL LSAME
297 * ..
298 * .. External Subroutines ..
299  EXTERNAL xerbla, zgeqpf, zgeqr2, zgerq2, zlacpy, zlapmt,
301 * ..
302 * .. Intrinsic Functions ..
303  INTRINSIC abs, dble, dimag, max, min
304 * ..
305 * .. Statement Functions ..
306  DOUBLE PRECISION CABS1
307 * ..
308 * .. Statement Function definitions ..
309  cabs1( t ) = abs( dble( t ) ) + abs( dimag( t ) )
310 * ..
311 * .. Executable Statements ..
312 *
313 * Test the input parameters
314 *
315  wantu = lsame( jobu, 'U' )
316  wantv = lsame( jobv, 'V' )
317  wantq = lsame( jobq, 'Q' )
318  forwrd = .true.
319 *
320  info = 0
321  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
322  info = -1
323  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
324  info = -2
325  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
326  info = -3
327  ELSE IF( m.LT.0 ) THEN
328  info = -4
329  ELSE IF( p.LT.0 ) THEN
330  info = -5
331  ELSE IF( n.LT.0 ) THEN
332  info = -6
333  ELSE IF( lda.LT.max( 1, m ) ) THEN
334  info = -8
335  ELSE IF( ldb.LT.max( 1, p ) ) THEN
336  info = -10
337  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
338  info = -16
339  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
340  info = -18
341  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
342  info = -20
343  END IF
344  IF( info.NE.0 ) THEN
345  CALL xerbla( 'ZGGSVP', -info )
346  RETURN
347  END IF
348 *
349 * QR with column pivoting of B: B*P = V*( S11 S12 )
350 * ( 0 0 )
351 *
352  DO 10 i = 1, n
353  iwork( i ) = 0
354  10 CONTINUE
355  CALL zgeqpf( p, n, b, ldb, iwork, tau, work, rwork, info )
356 *
357 * Update A := A*P
358 *
359  CALL zlapmt( forwrd, m, n, a, lda, iwork )
360 *
361 * Determine the effective rank of matrix B.
362 *
363  l = 0
364  DO 20 i = 1, min( p, n )
365  IF( cabs1( b( i, i ) ).GT.tolb )
366  $ l = l + 1
367  20 CONTINUE
368 *
369  IF( wantv ) THEN
370 *
371 * Copy the details of V, and form V.
372 *
373  CALL zlaset( 'Full', p, p, czero, czero, v, ldv )
374  IF( p.GT.1 )
375  $ CALL zlacpy( 'Lower', p-1, n, b( 2, 1 ), ldb, v( 2, 1 ),
376  $ ldv )
377  CALL zung2r( p, p, min( p, n ), v, ldv, tau, work, info )
378  END IF
379 *
380 * Clean up B
381 *
382  DO 40 j = 1, l - 1
383  DO 30 i = j + 1, l
384  b( i, j ) = czero
385  30 CONTINUE
386  40 CONTINUE
387  IF( p.GT.l )
388  $ CALL zlaset( 'Full', p-l, n, czero, czero, b( l+1, 1 ), ldb )
389 *
390  IF( wantq ) THEN
391 *
392 * Set Q = I and Update Q := Q*P
393 *
394  CALL zlaset( 'Full', n, n, czero, cone, q, ldq )
395  CALL zlapmt( forwrd, n, n, q, ldq, iwork )
396  END IF
397 *
398  IF( p.GE.l .AND. n.NE.l ) THEN
399 *
400 * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
401 *
402  CALL zgerq2( l, n, b, ldb, tau, work, info )
403 *
404 * Update A := A*Z**H
405 *
406  CALL zunmr2( 'Right', 'Conjugate transpose', m, n, l, b, ldb,
407  $ tau, a, lda, work, info )
408  IF( wantq ) THEN
409 *
410 * Update Q := Q*Z**H
411 *
412  CALL zunmr2( 'Right', 'Conjugate transpose', n, n, l, b,
413  $ ldb, tau, q, ldq, work, info )
414  END IF
415 *
416 * Clean up B
417 *
418  CALL zlaset( 'Full', l, n-l, czero, czero, b, ldb )
419  DO 60 j = n - l + 1, n
420  DO 50 i = j - n + l + 1, l
421  b( i, j ) = czero
422  50 CONTINUE
423  60 CONTINUE
424 *
425  END IF
426 *
427 * Let N-L L
428 * A = ( A11 A12 ) M,
429 *
430 * then the following does the complete QR decomposition of A11:
431 *
432 * A11 = U*( 0 T12 )*P1**H
433 * ( 0 0 )
434 *
435  DO 70 i = 1, n - l
436  iwork( i ) = 0
437  70 CONTINUE
438  CALL zgeqpf( m, n-l, a, lda, iwork, tau, work, rwork, info )
439 *
440 * Determine the effective rank of A11
441 *
442  k = 0
443  DO 80 i = 1, min( m, n-l )
444  IF( cabs1( a( i, i ) ).GT.tola )
445  $ k = k + 1
446  80 CONTINUE
447 *
448 * Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N )
449 *
450  CALL zunm2r( 'Left', 'Conjugate transpose', m, l, min( m, n-l ),
451  $ a, lda, tau, a( 1, n-l+1 ), lda, work, info )
452 *
453  IF( wantu ) THEN
454 *
455 * Copy the details of U, and form U
456 *
457  CALL zlaset( 'Full', m, m, czero, czero, u, ldu )
458  IF( m.GT.1 )
459  $ CALL zlacpy( 'Lower', m-1, n-l, a( 2, 1 ), lda, u( 2, 1 ),
460  $ ldu )
461  CALL zung2r( m, m, min( m, n-l ), u, ldu, tau, work, info )
462  END IF
463 *
464  IF( wantq ) THEN
465 *
466 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
467 *
468  CALL zlapmt( forwrd, n, n-l, q, ldq, iwork )
469  END IF
470 *
471 * Clean up A: set the strictly lower triangular part of
472 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
473 *
474  DO 100 j = 1, k - 1
475  DO 90 i = j + 1, k
476  a( i, j ) = czero
477  90 CONTINUE
478  100 CONTINUE
479  IF( m.GT.k )
480  $ CALL zlaset( 'Full', m-k, n-l, czero, czero, a( k+1, 1 ), lda )
481 *
482  IF( n-l.GT.k ) THEN
483 *
484 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
485 *
486  CALL zgerq2( k, n-l, a, lda, tau, work, info )
487 *
488  IF( wantq ) THEN
489 *
490 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H
491 *
492  CALL zunmr2( 'Right', 'Conjugate transpose', n, n-l, k, a,
493  $ lda, tau, q, ldq, work, info )
494  END IF
495 *
496 * Clean up A
497 *
498  CALL zlaset( 'Full', k, n-l-k, czero, czero, a, lda )
499  DO 120 j = n - l - k + 1, n - l
500  DO 110 i = j - n + l + k + 1, k
501  a( i, j ) = czero
502  110 CONTINUE
503  120 CONTINUE
504 *
505  END IF
506 *
507  IF( m.GT.k ) THEN
508 *
509 * QR factorization of A( K+1:M,N-L+1:N )
510 *
511  CALL zgeqr2( m-k, l, a( k+1, n-l+1 ), lda, tau, work, info )
512 *
513  IF( wantu ) THEN
514 *
515 * Update U(:,K+1:M) := U(:,K+1:M)*U1
516 *
517  CALL zunm2r( 'Right', 'No transpose', m, m-k, min( m-k, l ),
518  $ a( k+1, n-l+1 ), lda, tau, u( 1, k+1 ), ldu,
519  $ work, info )
520  END IF
521 *
522 * Clean up
523 *
524  DO 140 j = n - l + 1, n
525  DO 130 i = j - n + k + l + 1, m
526  a( i, j ) = czero
527  130 CONTINUE
528  140 CONTINUE
529 *
530  END IF
531 *
532  RETURN
533 *
534 * End of ZGGSVP
535 *
536  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zgeqpf(M, N, A, LDA, JPVT, TAU, WORK, RWORK, INFO)
ZGEQPF
Definition: zgeqpf.f:148
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 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
subroutine zggsvp(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, INFO)
ZGGSVP
Definition: zggsvp.f:265