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cggsvp.f
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1 *> \brief \b CGGSVP
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CGGSVP + 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/cggsvp.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggsvp.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CGGSVP( 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 * REAL TOLA, TOLB
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IWORK( * )
32 * REAL RWORK( * )
33 * COMPLEX 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 *> CGGSVP 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 *> CGGSVD.
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 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 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 REAL
141 *> \endverbatim
142 *>
143 *> \param[in] TOLB
144 *> \verbatim
145 *> TOLB is REAL
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)*MACHEPS,
151 *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
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 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 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 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 REAL array, dimension (2*N)
220 *> \endverbatim
221 *>
222 *> \param[out] TAU
223 *> \verbatim
224 *> TAU is COMPLEX array, dimension (N)
225 *> \endverbatim
226 *>
227 *> \param[out] WORK
228 *> \verbatim
229 *> WORK is COMPLEX array, dimension (max(3*N,M,P))
230 *> \endverbatim
231 *>
232 *> \param[out] INFO
233 *> \verbatim
234 *> INFO is INTEGER
235 *> = 0: successful exit
236 *> < 0: if INFO = -i, the i-th argument had an illegal value.
237 *> \endverbatim
238 *
239 * Authors:
240 * ========
241 *
242 *> \author Univ. of Tennessee
243 *> \author Univ. of California Berkeley
244 *> \author Univ. of Colorado Denver
245 *> \author NAG Ltd.
246 *
247 *> \date November 2011
248 *
249 *> \ingroup complexOTHERcomputational
250 *
251 *> \par Further Details:
252 * =====================
253 *>
254 *> The subroutine uses LAPACK subroutine CGEQPF for the QR factorization
255 *> with column pivoting to detect the effective numerical rank of the
256 *> a matrix. It may be replaced by a better rank determination strategy.
257 *>
258 * =====================================================================
259  SUBROUTINE cggsvp( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
260  $ tola, tolb, k, l, u, ldu, v, ldv, q, ldq,
261  $ iwork, rwork, tau, work, info )
262 *
263 * -- LAPACK computational routine (version 3.4.0) --
264 * -- LAPACK is a software package provided by Univ. of Tennessee, --
265 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266 * November 2011
267 *
268 * .. Scalar Arguments ..
269  CHARACTER jobq, jobu, jobv
270  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p
271  REAL tola, tolb
272 * ..
273 * .. Array Arguments ..
274  INTEGER iwork( * )
275  REAL rwork( * )
276  COMPLEX a( lda, * ), b( ldb, * ), q( ldq, * ),
277  $ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
278 * ..
279 *
280 * =====================================================================
281 *
282 * .. Parameters ..
283  COMPLEX czero, cone
284  parameter( czero = ( 0.0e+0, 0.0e+0 ),
285  $ cone = ( 1.0e+0, 0.0e+0 ) )
286 * ..
287 * .. Local Scalars ..
288  LOGICAL forwrd, wantq, wantu, wantv
289  INTEGER i, j
290  COMPLEX t
291 * ..
292 * .. External Functions ..
293  LOGICAL lsame
294  EXTERNAL lsame
295 * ..
296 * .. External Subroutines ..
297  EXTERNAL cgeqpf, cgeqr2, cgerq2, clacpy, clapmt, claset,
298  $ cung2r, cunm2r, cunmr2, xerbla
299 * ..
300 * .. Intrinsic Functions ..
301  INTRINSIC abs, aimag, max, min, real
302 * ..
303 * .. Statement Functions ..
304  REAL cabs1
305 * ..
306 * .. Statement Function definitions ..
307  cabs1( t ) = abs( REAL( T ) ) + abs( aimag( t ) )
308 * ..
309 * .. Executable Statements ..
310 *
311 * Test the input parameters
312 *
313  wantu = lsame( jobu, 'U' )
314  wantv = lsame( jobv, 'V' )
315  wantq = lsame( jobq, 'Q' )
316  forwrd = .true.
317 *
318  info = 0
319  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
320  info = -1
321  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
322  info = -2
323  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
324  info = -3
325  ELSE IF( m.LT.0 ) THEN
326  info = -4
327  ELSE IF( p.LT.0 ) THEN
328  info = -5
329  ELSE IF( n.LT.0 ) THEN
330  info = -6
331  ELSE IF( lda.LT.max( 1, m ) ) THEN
332  info = -8
333  ELSE IF( ldb.LT.max( 1, p ) ) THEN
334  info = -10
335  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
336  info = -16
337  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
338  info = -18
339  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
340  info = -20
341  END IF
342  IF( info.NE.0 ) THEN
343  CALL xerbla( 'CGGSVP', -info )
344  return
345  END IF
346 *
347 * QR with column pivoting of B: B*P = V*( S11 S12 )
348 * ( 0 0 )
349 *
350  DO 10 i = 1, n
351  iwork( i ) = 0
352  10 continue
353  CALL cgeqpf( p, n, b, ldb, iwork, tau, work, rwork, info )
354 *
355 * Update A := A*P
356 *
357  CALL clapmt( forwrd, m, n, a, lda, iwork )
358 *
359 * Determine the effective rank of matrix B.
360 *
361  l = 0
362  DO 20 i = 1, min( p, n )
363  IF( cabs1( b( i, i ) ).GT.tolb )
364  $ l = l + 1
365  20 continue
366 *
367  IF( wantv ) THEN
368 *
369 * Copy the details of V, and form V.
370 *
371  CALL claset( 'Full', p, p, czero, czero, v, ldv )
372  IF( p.GT.1 )
373  $ CALL clacpy( 'Lower', p-1, n, b( 2, 1 ), ldb, v( 2, 1 ),
374  $ ldv )
375  CALL cung2r( p, p, min( p, n ), v, ldv, tau, work, info )
376  END IF
377 *
378 * Clean up B
379 *
380  DO 40 j = 1, l - 1
381  DO 30 i = j + 1, l
382  b( i, j ) = czero
383  30 continue
384  40 continue
385  IF( p.GT.l )
386  $ CALL claset( 'Full', p-l, n, czero, czero, b( l+1, 1 ), ldb )
387 *
388  IF( wantq ) THEN
389 *
390 * Set Q = I and Update Q := Q*P
391 *
392  CALL claset( 'Full', n, n, czero, cone, q, ldq )
393  CALL clapmt( forwrd, n, n, q, ldq, iwork )
394  END IF
395 *
396  IF( p.GE.l .AND. n.NE.l ) THEN
397 *
398 * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
399 *
400  CALL cgerq2( l, n, b, ldb, tau, work, info )
401 *
402 * Update A := A*Z**H
403 *
404  CALL cunmr2( 'Right', 'Conjugate transpose', m, n, l, b, ldb,
405  $ tau, a, lda, work, info )
406  IF( wantq ) THEN
407 *
408 * Update Q := Q*Z**H
409 *
410  CALL cunmr2( 'Right', 'Conjugate transpose', n, n, l, b,
411  $ ldb, tau, q, ldq, work, info )
412  END IF
413 *
414 * Clean up B
415 *
416  CALL claset( 'Full', l, n-l, czero, czero, b, ldb )
417  DO 60 j = n - l + 1, n
418  DO 50 i = j - n + l + 1, l
419  b( i, j ) = czero
420  50 continue
421  60 continue
422 *
423  END IF
424 *
425 * Let N-L L
426 * A = ( A11 A12 ) M,
427 *
428 * then the following does the complete QR decomposition of A11:
429 *
430 * A11 = U*( 0 T12 )*P1**H
431 * ( 0 0 )
432 *
433  DO 70 i = 1, n - l
434  iwork( i ) = 0
435  70 continue
436  CALL cgeqpf( m, n-l, a, lda, iwork, tau, work, rwork, info )
437 *
438 * Determine the effective rank of A11
439 *
440  k = 0
441  DO 80 i = 1, min( m, n-l )
442  IF( cabs1( a( i, i ) ).GT.tola )
443  $ k = k + 1
444  80 continue
445 *
446 * Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N )
447 *
448  CALL cunm2r( 'Left', 'Conjugate transpose', m, l, min( m, n-l ),
449  $ a, lda, tau, a( 1, n-l+1 ), lda, work, info )
450 *
451  IF( wantu ) THEN
452 *
453 * Copy the details of U, and form U
454 *
455  CALL claset( 'Full', m, m, czero, czero, u, ldu )
456  IF( m.GT.1 )
457  $ CALL clacpy( 'Lower', m-1, n-l, a( 2, 1 ), lda, u( 2, 1 ),
458  $ ldu )
459  CALL cung2r( m, m, min( m, n-l ), u, ldu, tau, work, info )
460  END IF
461 *
462  IF( wantq ) THEN
463 *
464 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
465 *
466  CALL clapmt( forwrd, n, n-l, q, ldq, iwork )
467  END IF
468 *
469 * Clean up A: set the strictly lower triangular part of
470 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
471 *
472  DO 100 j = 1, k - 1
473  DO 90 i = j + 1, k
474  a( i, j ) = czero
475  90 continue
476  100 continue
477  IF( m.GT.k )
478  $ CALL claset( 'Full', m-k, n-l, czero, czero, a( k+1, 1 ), lda )
479 *
480  IF( n-l.GT.k ) THEN
481 *
482 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
483 *
484  CALL cgerq2( k, n-l, a, lda, tau, work, info )
485 *
486  IF( wantq ) THEN
487 *
488 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H
489 *
490  CALL cunmr2( 'Right', 'Conjugate transpose', n, n-l, k, a,
491  $ lda, tau, q, ldq, work, info )
492  END IF
493 *
494 * Clean up A
495 *
496  CALL claset( 'Full', k, n-l-k, czero, czero, a, lda )
497  DO 120 j = n - l - k + 1, n - l
498  DO 110 i = j - n + l + k + 1, k
499  a( i, j ) = czero
500  110 continue
501  120 continue
502 *
503  END IF
504 *
505  IF( m.GT.k ) THEN
506 *
507 * QR factorization of A( K+1:M,N-L+1:N )
508 *
509  CALL cgeqr2( m-k, l, a( k+1, n-l+1 ), lda, tau, work, info )
510 *
511  IF( wantu ) THEN
512 *
513 * Update U(:,K+1:M) := U(:,K+1:M)*U1
514 *
515  CALL cunm2r( 'Right', 'No transpose', m, m-k, min( m-k, l ),
516  $ a( k+1, n-l+1 ), lda, tau, u( 1, k+1 ), ldu,
517  $ work, info )
518  END IF
519 *
520 * Clean up
521 *
522  DO 140 j = n - l + 1, n
523  DO 130 i = j - n + k + l + 1, m
524  a( i, j ) = czero
525  130 continue
526  140 continue
527 *
528  END IF
529 *
530  return
531 *
532 * End of CGGSVP
533 *
534  END