LAPACK  3.6.1
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 *> \date August 2015
337 *
338 *> \ingroup complexOTHERsing
339 *
340 *> \par Contributors:
341 * ==================
342 *>
343 *> Ming Gu and Huan Ren, Computer Science Division, University of
344 *> California at Berkeley, USA
345 *>
346 *
347 *> \par Further Details:
348 * =====================
349 *>
350 *> CGGSVD3 replaces the deprecated subroutine CGGSVD.
351 *>
352 * =====================================================================
353  SUBROUTINE cggsvd3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
354  $ ldb, alpha, beta, u, ldu, v, ldv, q, ldq,
355  $ work, lwork, rwork, iwork, info )
356 *
357 * -- LAPACK driver routine (version 3.6.0) --
358 * -- LAPACK is a software package provided by Univ. of Tennessee, --
359 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
360 * August 2015
361 *
362 * .. Scalar Arguments ..
363  CHARACTER JOBQ, JOBU, JOBV
364  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
365  $ lwork
366 * ..
367 * .. Array Arguments ..
368  INTEGER IWORK( * )
369  REAL ALPHA( * ), BETA( * ), RWORK( * )
370  COMPLEX A( lda, * ), B( ldb, * ), Q( ldq, * ),
371  $ u( ldu, * ), v( ldv, * ), work( * )
372 * ..
373 *
374 * =====================================================================
375 *
376 * .. Local Scalars ..
377  LOGICAL WANTQ, WANTU, WANTV, LQUERY
378  INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
379  REAL ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
380 * ..
381 * .. External Functions ..
382  LOGICAL LSAME
383  REAL CLANGE, SLAMCH
384  EXTERNAL lsame, clange, slamch
385 * ..
386 * .. External Subroutines ..
387  EXTERNAL cggsvp3, ctgsja, scopy, xerbla
388 * ..
389 * .. Intrinsic Functions ..
390  INTRINSIC max, min
391 * ..
392 * .. Executable Statements ..
393 *
394 * Decode and test the input parameters
395 *
396  wantu = lsame( jobu, 'U' )
397  wantv = lsame( jobv, 'V' )
398  wantq = lsame( jobq, 'Q' )
399  lquery = ( lwork.EQ.-1 )
400  lwkopt = 1
401 *
402 * Test the input arguments
403 *
404  info = 0
405  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
406  info = -1
407  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
408  info = -2
409  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
410  info = -3
411  ELSE IF( m.LT.0 ) THEN
412  info = -4
413  ELSE IF( n.LT.0 ) THEN
414  info = -5
415  ELSE IF( p.LT.0 ) THEN
416  info = -6
417  ELSE IF( lda.LT.max( 1, m ) ) THEN
418  info = -10
419  ELSE IF( ldb.LT.max( 1, p ) ) THEN
420  info = -12
421  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
422  info = -16
423  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
424  info = -18
425  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
426  info = -20
427  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
428  info = -24
429  END IF
430 *
431 * Compute workspace
432 *
433  IF( info.EQ.0 ) THEN
434  CALL cggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
435  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
436  $ work, work, -1, info )
437  lwkopt = n + int( work( 1 ) )
438  lwkopt = max( 2*n, lwkopt )
439  lwkopt = max( 1, lwkopt )
440  work( 1 ) = cmplx( lwkopt )
441  END IF
442 *
443  IF( info.NE.0 ) THEN
444  CALL xerbla( 'CGGSVD3', -info )
445  RETURN
446  END IF
447  IF( lquery ) THEN
448  RETURN
449  ENDIF
450 *
451 * Compute the Frobenius norm of matrices A and B
452 *
453  anorm = clange( '1', m, n, a, lda, rwork )
454  bnorm = clange( '1', p, n, b, ldb, rwork )
455 *
456 * Get machine precision and set up threshold for determining
457 * the effective numerical rank of the matrices A and B.
458 *
459  ulp = slamch( 'Precision' )
460  unfl = slamch( 'Safe Minimum' )
461  tola = max( m, n )*max( anorm, unfl )*ulp
462  tolb = max( p, n )*max( bnorm, unfl )*ulp
463 *
464  CALL cggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
465  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
466  $ work, work( n+1 ), lwork-n, info )
467 *
468 * Compute the GSVD of two upper "triangular" matrices
469 *
470  CALL ctgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
471  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
472  $ work, ncycle, info )
473 *
474 * Sort the singular values and store the pivot indices in IWORK
475 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
476 *
477  CALL scopy( n, alpha, 1, rwork, 1 )
478  ibnd = min( l, m-k )
479  DO 20 i = 1, ibnd
480 *
481 * Scan for largest ALPHA(K+I)
482 *
483  isub = i
484  smax = rwork( k+i )
485  DO 10 j = i + 1, ibnd
486  temp = rwork( k+j )
487  IF( temp.GT.smax ) THEN
488  isub = j
489  smax = temp
490  END IF
491  10 CONTINUE
492  IF( isub.NE.i ) THEN
493  rwork( k+isub ) = rwork( k+i )
494  rwork( k+i ) = smax
495  iwork( k+i ) = k + isub
496  ELSE
497  iwork( k+i ) = k + i
498  END IF
499  20 CONTINUE
500 *
501  work( 1 ) = cmplx( lwkopt )
502  RETURN
503 *
504 * End of CGGSVD3
505 *
506  END
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:280
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:356
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:381
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53