LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
cggsvd3.f
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1 *> \brief <b> CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
22 * LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
23 * LWORK, RWORK, IWORK, 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 * ..
29 * .. Array Arguments ..
30 * INTEGER IWORK( * )
31 * REAL ALPHA( * ), BETA( * ), RWORK( * )
32 * COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
33 * $ U( LDU, * ), V( LDV, * ), WORK( * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> CGGSVD3 computes the generalized singular value decomposition (GSVD)
43 *> of an M-by-N complex matrix A and P-by-N complex matrix B:
44 *>
45 *> U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R )
46 *>
47 *> where U, V and Q are unitary matrices.
48 *> Let K+L = the effective numerical rank of the
49 *> matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
50 *> triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
51 *> matrices and of the following structures, respectively:
52 *>
53 *> If M-K-L >= 0,
54 *>
55 *> K L
56 *> D1 = K ( I 0 )
57 *> L ( 0 C )
58 *> M-K-L ( 0 0 )
59 *>
60 *> K L
61 *> D2 = L ( 0 S )
62 *> P-L ( 0 0 )
63 *>
64 *> N-K-L K L
65 *> ( 0 R ) = K ( 0 R11 R12 )
66 *> L ( 0 0 R22 )
67 *>
68 *> where
69 *>
70 *> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
71 *> S = diag( BETA(K+1), ... , BETA(K+L) ),
72 *> C**2 + S**2 = I.
73 *>
74 *> R is stored in A(1:K+L,N-K-L+1:N) on exit.
75 *>
76 *> If M-K-L < 0,
77 *>
78 *> K M-K K+L-M
79 *> D1 = K ( I 0 0 )
80 *> M-K ( 0 C 0 )
81 *>
82 *> K M-K K+L-M
83 *> D2 = M-K ( 0 S 0 )
84 *> K+L-M ( 0 0 I )
85 *> P-L ( 0 0 0 )
86 *>
87 *> N-K-L K M-K K+L-M
88 *> ( 0 R ) = K ( 0 R11 R12 R13 )
89 *> M-K ( 0 0 R22 R23 )
90 *> K+L-M ( 0 0 0 R33 )
91 *>
92 *> where
93 *>
94 *> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
95 *> S = diag( BETA(K+1), ... , BETA(M) ),
96 *> C**2 + S**2 = I.
97 *>
98 *> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
99 *> ( 0 R22 R23 )
100 *> in B(M-K+1:L,N+M-K-L+1:N) on exit.
101 *>
102 *> The routine computes C, S, R, and optionally the unitary
103 *> transformation matrices U, V and Q.
104 *>
105 *> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
106 *> A and B implicitly gives the SVD of A*inv(B):
107 *> A*inv(B) = U*(D1*inv(D2))*V**H.
108 *> If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
109 *> equal to the CS decomposition of A and B. Furthermore, the GSVD can
110 *> be used to derive the solution of the eigenvalue problem:
111 *> A**H*A x = lambda* B**H*B x.
112 *> In some literature, the GSVD of A and B is presented in the form
113 *> U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 )
114 *> where U and V are orthogonal and X is nonsingular, and D1 and D2 are
115 *> ``diagonal''. The former GSVD form can be converted to the latter
116 *> form by taking the nonsingular matrix X as
117 *>
118 *> X = Q*( I 0 )
119 *> ( 0 inv(R) )
120 *> \endverbatim
121 *
122 * Arguments:
123 * ==========
124 *
125 *> \param[in] JOBU
126 *> \verbatim
127 *> JOBU is CHARACTER*1
128 *> = 'U': Unitary matrix U is computed;
129 *> = 'N': U is not computed.
130 *> \endverbatim
131 *>
132 *> \param[in] JOBV
133 *> \verbatim
134 *> JOBV is CHARACTER*1
135 *> = 'V': Unitary matrix V is computed;
136 *> = 'N': V is not computed.
137 *> \endverbatim
138 *>
139 *> \param[in] JOBQ
140 *> \verbatim
141 *> JOBQ is CHARACTER*1
142 *> = 'Q': Unitary matrix Q is computed;
143 *> = 'N': Q is not computed.
144 *> \endverbatim
145 *>
146 *> \param[in] M
147 *> \verbatim
148 *> M is INTEGER
149 *> The number of rows of the matrix A. M >= 0.
150 *> \endverbatim
151 *>
152 *> \param[in] N
153 *> \verbatim
154 *> N is INTEGER
155 *> The number of columns of the matrices A and B. N >= 0.
156 *> \endverbatim
157 *>
158 *> \param[in] P
159 *> \verbatim
160 *> P is INTEGER
161 *> The number of rows of the matrix B. P >= 0.
162 *> \endverbatim
163 *>
164 *> \param[out] K
165 *> \verbatim
166 *> K is INTEGER
167 *> \endverbatim
168 *>
169 *> \param[out] L
170 *> \verbatim
171 *> L is INTEGER
172 *>
173 *> On exit, K and L specify the dimension of the subblocks
174 *> described in Purpose.
175 *> K + L = effective numerical rank of (A**H,B**H)**H.
176 *> \endverbatim
177 *>
178 *> \param[in,out] A
179 *> \verbatim
180 *> A is COMPLEX array, dimension (LDA,N)
181 *> On entry, the M-by-N matrix A.
182 *> On exit, A contains the triangular matrix R, or part of R.
183 *> See Purpose for details.
184 *> \endverbatim
185 *>
186 *> \param[in] LDA
187 *> \verbatim
188 *> LDA is INTEGER
189 *> The leading dimension of the array A. LDA >= max(1,M).
190 *> \endverbatim
191 *>
192 *> \param[in,out] B
193 *> \verbatim
194 *> B is COMPLEX array, dimension (LDB,N)
195 *> On entry, the P-by-N matrix B.
196 *> On exit, B contains part of the triangular matrix R if
197 *> M-K-L < 0. See Purpose for details.
198 *> \endverbatim
199 *>
200 *> \param[in] LDB
201 *> \verbatim
202 *> LDB is INTEGER
203 *> The leading dimension of the array B. LDB >= max(1,P).
204 *> \endverbatim
205 *>
206 *> \param[out] ALPHA
207 *> \verbatim
208 *> ALPHA is REAL array, dimension (N)
209 *> \endverbatim
210 *>
211 *> \param[out] BETA
212 *> \verbatim
213 *> BETA is REAL array, dimension (N)
214 *>
215 *> On exit, ALPHA and BETA contain the generalized singular
216 *> value pairs of A and B;
217 *> ALPHA(1:K) = 1,
218 *> BETA(1:K) = 0,
219 *> and if M-K-L >= 0,
220 *> ALPHA(K+1:K+L) = C,
221 *> BETA(K+1:K+L) = S,
222 *> or if M-K-L < 0,
223 *> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
224 *> BETA(K+1:M) =S, BETA(M+1:K+L) =1
225 *> and
226 *> ALPHA(K+L+1:N) = 0
227 *> BETA(K+L+1:N) = 0
228 *> \endverbatim
229 *>
230 *> \param[out] U
231 *> \verbatim
232 *> U is COMPLEX array, dimension (LDU,M)
233 *> If JOBU = 'U', U contains the M-by-M unitary matrix U.
234 *> If JOBU = 'N', U is not referenced.
235 *> \endverbatim
236 *>
237 *> \param[in] LDU
238 *> \verbatim
239 *> LDU is INTEGER
240 *> The leading dimension of the array U. LDU >= max(1,M) if
241 *> JOBU = 'U'; LDU >= 1 otherwise.
242 *> \endverbatim
243 *>
244 *> \param[out] V
245 *> \verbatim
246 *> V is COMPLEX array, dimension (LDV,P)
247 *> If JOBV = 'V', V contains the P-by-P unitary matrix V.
248 *> If JOBV = 'N', V is not referenced.
249 *> \endverbatim
250 *>
251 *> \param[in] LDV
252 *> \verbatim
253 *> LDV is INTEGER
254 *> The leading dimension of the array V. LDV >= max(1,P) if
255 *> JOBV = 'V'; LDV >= 1 otherwise.
256 *> \endverbatim
257 *>
258 *> \param[out] Q
259 *> \verbatim
260 *> Q is COMPLEX array, dimension (LDQ,N)
261 *> If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
262 *> If JOBQ = 'N', Q is not referenced.
263 *> \endverbatim
264 *>
265 *> \param[in] LDQ
266 *> \verbatim
267 *> LDQ is INTEGER
268 *> The leading dimension of the array Q. LDQ >= max(1,N) if
269 *> JOBQ = 'Q'; LDQ >= 1 otherwise.
270 *> \endverbatim
271 *>
272 *> \param[out] WORK
273 *> \verbatim
274 *> WORK is COMPLEX array, dimension (MAX(1,LWORK))
275 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
276 *> \endverbatim
277 *>
278 *> \param[in] LWORK
279 *> \verbatim
280 *> LWORK is INTEGER
281 *> The dimension of the array WORK.
282 *>
283 *> If LWORK = -1, then a workspace query is assumed; the routine
284 *> only calculates the optimal size of the WORK array, returns
285 *> this value as the first entry of the WORK array, and no error
286 *> message related to LWORK is issued by XERBLA.
287 *> \endverbatim
288 *>
289 *> \param[out] RWORK
290 *> \verbatim
291 *> RWORK is REAL array, dimension (2*N)
292 *> \endverbatim
293 *>
294 *> \param[out] IWORK
295 *> \verbatim
296 *> IWORK is INTEGER array, dimension (N)
297 *> On exit, IWORK stores the sorting information. More
298 *> precisely, the following loop will sort ALPHA
299 *> for I = K+1, min(M,K+L)
300 *> swap ALPHA(I) and ALPHA(IWORK(I))
301 *> endfor
302 *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
303 *> \endverbatim
304 *>
305 *> \param[out] INFO
306 *> \verbatim
307 *> INFO is INTEGER
308 *> = 0: successful exit.
309 *> < 0: if INFO = -i, the i-th argument had an illegal value.
310 *> > 0: if INFO = 1, the Jacobi-type procedure failed to
311 *> converge. For further details, see subroutine CTGSJA.
312 *> \endverbatim
313 *
314 *> \par Internal Parameters:
315 * =========================
316 *>
317 *> \verbatim
318 *> TOLA REAL
319 *> TOLB REAL
320 *> TOLA and TOLB are the thresholds to determine the effective
321 *> rank of (A**H,B**H)**H. Generally, they are set to
322 *> TOLA = MAX(M,N)*norm(A)*MACHEPS,
323 *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
324 *> The size of TOLA and TOLB may affect the size of backward
325 *> errors of the decomposition.
326 *> \endverbatim
327 *
328 * Authors:
329 * ========
330 *
331 *> \author Univ. of Tennessee
332 *> \author Univ. of California Berkeley
333 *> \author Univ. of Colorado Denver
334 *> \author NAG Ltd.
335 *
336 *> \ingroup complexGEsing
337 *
338 *> \par Contributors:
339 * ==================
340 *>
341 *> Ming Gu and Huan Ren, Computer Science Division, University of
342 *> California at Berkeley, USA
343 *>
344 *
345 *> \par Further Details:
346 * =====================
347 *>
348 *> CGGSVD3 replaces the deprecated subroutine CGGSVD.
349 *>
350 * =====================================================================
351  SUBROUTINE cggsvd3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
352  $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
353  $ WORK, LWORK, RWORK, IWORK, INFO )
354 *
355 * -- LAPACK driver routine --
356 * -- LAPACK is a software package provided by Univ. of Tennessee, --
357 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
358 *
359 * .. Scalar Arguments ..
360  CHARACTER JOBQ, JOBU, JOBV
361  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
362  $ lwork
363 * ..
364 * .. Array Arguments ..
365  INTEGER IWORK( * )
366  REAL ALPHA( * ), BETA( * ), RWORK( * )
367  COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
368  $ u( ldu, * ), v( ldv, * ), work( * )
369 * ..
370 *
371 * =====================================================================
372 *
373 * .. Local Scalars ..
374  LOGICAL WANTQ, WANTU, WANTV, LQUERY
375  INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
376  REAL ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
377 * ..
378 * .. External Functions ..
379  LOGICAL LSAME
380  REAL CLANGE, SLAMCH
381  EXTERNAL lsame, clange, slamch
382 * ..
383 * .. External Subroutines ..
384  EXTERNAL cggsvp3, ctgsja, scopy, xerbla
385 * ..
386 * .. Intrinsic Functions ..
387  INTRINSIC max, min
388 * ..
389 * .. Executable Statements ..
390 *
391 * Decode and test the input parameters
392 *
393  wantu = lsame( jobu, 'U' )
394  wantv = lsame( jobv, 'V' )
395  wantq = lsame( jobq, 'Q' )
396  lquery = ( lwork.EQ.-1 )
397  lwkopt = 1
398 *
399 * Test the input arguments
400 *
401  info = 0
402  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
403  info = -1
404  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
405  info = -2
406  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
407  info = -3
408  ELSE IF( m.LT.0 ) THEN
409  info = -4
410  ELSE IF( n.LT.0 ) THEN
411  info = -5
412  ELSE IF( p.LT.0 ) THEN
413  info = -6
414  ELSE IF( lda.LT.max( 1, m ) ) THEN
415  info = -10
416  ELSE IF( ldb.LT.max( 1, p ) ) THEN
417  info = -12
418  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
419  info = -16
420  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
421  info = -18
422  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
423  info = -20
424  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
425  info = -24
426  END IF
427 *
428 * Compute workspace
429 *
430  IF( info.EQ.0 ) THEN
431  CALL cggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
432  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
433  $ work, work, -1, info )
434  lwkopt = n + int( work( 1 ) )
435  lwkopt = max( 2*n, lwkopt )
436  lwkopt = max( 1, lwkopt )
437  work( 1 ) = cmplx( lwkopt )
438  END IF
439 *
440  IF( info.NE.0 ) THEN
441  CALL xerbla( 'CGGSVD3', -info )
442  RETURN
443  END IF
444  IF( lquery ) THEN
445  RETURN
446  ENDIF
447 *
448 * Compute the Frobenius norm of matrices A and B
449 *
450  anorm = clange( '1', m, n, a, lda, rwork )
451  bnorm = clange( '1', p, n, b, ldb, rwork )
452 *
453 * Get machine precision and set up threshold for determining
454 * the effective numerical rank of the matrices A and B.
455 *
456  ulp = slamch( 'Precision' )
457  unfl = slamch( 'Safe Minimum' )
458  tola = max( m, n )*max( anorm, unfl )*ulp
459  tolb = max( p, n )*max( bnorm, unfl )*ulp
460 *
461  CALL cggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
462  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
463  $ work, work( n+1 ), lwork-n, info )
464 *
465 * Compute the GSVD of two upper "triangular" matrices
466 *
467  CALL ctgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
468  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
469  $ work, ncycle, info )
470 *
471 * Sort the singular values and store the pivot indices in IWORK
472 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
473 *
474  CALL scopy( n, alpha, 1, rwork, 1 )
475  ibnd = min( l, m-k )
476  DO 20 i = 1, ibnd
477 *
478 * Scan for largest ALPHA(K+I)
479 *
480  isub = i
481  smax = rwork( k+i )
482  DO 10 j = i + 1, ibnd
483  temp = rwork( k+j )
484  IF( temp.GT.smax ) THEN
485  isub = j
486  smax = temp
487  END IF
488  10 CONTINUE
489  IF( isub.NE.i ) THEN
490  rwork( k+isub ) = rwork( k+i )
491  rwork( k+i ) = smax
492  iwork( k+i ) = k + isub
493  ELSE
494  iwork( k+i ) = k + i
495  END IF
496  20 CONTINUE
497 *
498  work( 1 ) = cmplx( lwkopt )
499  RETURN
500 *
501 * End of CGGSVD3
502 *
503  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cggsvd3(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, RWORK, IWORK, INFO)
CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
Definition: cggsvd3.f:354
subroutine cggsvp3(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)
CGGSVP3
Definition: cggsvp3.f:278
subroutine ctgsja(JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
CTGSJA
Definition: ctgsja.f:379
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82