LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
dggsvd3.f
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1 *> \brief <b> DGGSVD3 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 DGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
22 * LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
23 * LWORK, 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 * DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
32 * $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
33 * $ V( LDV, * ), WORK( * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> DGGSVD3 computes the generalized singular value decomposition (GSVD)
43 *> of an M-by-N real matrix A and P-by-N real matrix B:
44 *>
45 *> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
46 *>
47 *> where U, V and Q are orthogonal matrices.
48 *> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
49 *> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
50 *> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
51 *> 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 orthogonal
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**T.
108 *> If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is
109 *> also equal to the CS decomposition of A and B. Furthermore, the GSVD
110 *> can be used to derive the solution of the eigenvalue problem:
111 *> A**T*A x = lambda* B**T*B x.
112 *> In some literature, the GSVD of A and B is presented in the form
113 *> U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
114 *> where U and V are orthogonal and X is nonsingular, 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': Orthogonal 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': Orthogonal 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': Orthogonal 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**T,B**T)**T.
176 *> \endverbatim
177 *>
178 *> \param[in,out] A
179 *> \verbatim
180 *> A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB,N)
195 *> On entry, the P-by-N matrix B.
196 *> On exit, B contains the triangular matrix R if M-K-L < 0.
197 *> 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 DOUBLE PRECISION array, dimension (N)
209 *> \endverbatim
210 *>
211 *> \param[out] BETA
212 *> \verbatim
213 *> BETA is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDU,M)
233 *> If JOBU = 'U', U contains the M-by-M orthogonal 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 DOUBLE PRECISION array, dimension (LDV,P)
247 *> If JOBV = 'V', V contains the P-by-P orthogonal 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 DOUBLE PRECISION array, dimension (LDQ,N)
261 *> If JOBQ = 'Q', Q contains the N-by-N orthogonal 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 DOUBLE PRECISION 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] IWORK
290 *> \verbatim
291 *> IWORK is INTEGER array, dimension (N)
292 *> On exit, IWORK stores the sorting information. More
293 *> precisely, the following loop will sort ALPHA
294 *> for I = K+1, min(M,K+L)
295 *> swap ALPHA(I) and ALPHA(IWORK(I))
296 *> endfor
297 *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
298 *> \endverbatim
299 *>
300 *> \param[out] INFO
301 *> \verbatim
302 *> INFO is INTEGER
303 *> = 0: successful exit.
304 *> < 0: if INFO = -i, the i-th argument had an illegal value.
305 *> > 0: if INFO = 1, the Jacobi-type procedure failed to
306 *> converge. For further details, see subroutine DTGSJA.
307 *> \endverbatim
308 *
309 *> \par Internal Parameters:
310 * =========================
311 *>
312 *> \verbatim
313 *> TOLA DOUBLE PRECISION
314 *> TOLB DOUBLE PRECISION
315 *> TOLA and TOLB are the thresholds to determine the effective
316 *> rank of (A**T,B**T)**T. Generally, they are set to
317 *> TOLA = MAX(M,N)*norm(A)*MACHEPS,
318 *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
319 *> The size of TOLA and TOLB may affect the size of backward
320 *> errors of the decomposition.
321 *> \endverbatim
322 *
323 * Authors:
324 * ========
325 *
326 *> \author Univ. of Tennessee
327 *> \author Univ. of California Berkeley
328 *> \author Univ. of Colorado Denver
329 *> \author NAG Ltd.
330 *
331 *> \ingroup doubleGEsing
332 *
333 *> \par Contributors:
334 * ==================
335 *>
336 *> Ming Gu and Huan Ren, Computer Science Division, University of
337 *> California at Berkeley, USA
338 *>
339 *
340 *> \par Further Details:
341 * =====================
342 *>
343 *> DGGSVD3 replaces the deprecated subroutine DGGSVD.
344 *>
345 * =====================================================================
346  SUBROUTINE dggsvd3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
347  $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
348  $ WORK, LWORK, IWORK, INFO )
349 *
350 * -- LAPACK driver routine --
351 * -- LAPACK is a software package provided by Univ. of Tennessee, --
352 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
353 *
354 * .. Scalar Arguments ..
355  CHARACTER JOBQ, JOBU, JOBV
356  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
357  $ lwork
358 * ..
359 * .. Array Arguments ..
360  INTEGER IWORK( * )
361  DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
362  $ beta( * ), q( ldq, * ), u( ldu, * ),
363  $ v( ldv, * ), work( * )
364 * ..
365 *
366 * =====================================================================
367 *
368 * .. Local Scalars ..
369  LOGICAL WANTQ, WANTU, WANTV, LQUERY
370  INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
371  DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
372 * ..
373 * .. External Functions ..
374  LOGICAL LSAME
375  DOUBLE PRECISION DLAMCH, DLANGE
376  EXTERNAL lsame, dlamch, dlange
377 * ..
378 * .. External Subroutines ..
379  EXTERNAL dcopy, dggsvp3, dtgsja, xerbla
380 * ..
381 * .. Intrinsic Functions ..
382  INTRINSIC max, min
383 * ..
384 * .. Executable Statements ..
385 *
386 * Decode and test the input parameters
387 *
388  wantu = lsame( jobu, 'U' )
389  wantv = lsame( jobv, 'V' )
390  wantq = lsame( jobq, 'Q' )
391  lquery = ( lwork.EQ.-1 )
392  lwkopt = 1
393 *
394 * Test the input arguments
395 *
396  info = 0
397  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
398  info = -1
399  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
400  info = -2
401  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
402  info = -3
403  ELSE IF( m.LT.0 ) THEN
404  info = -4
405  ELSE IF( n.LT.0 ) THEN
406  info = -5
407  ELSE IF( p.LT.0 ) THEN
408  info = -6
409  ELSE IF( lda.LT.max( 1, m ) ) THEN
410  info = -10
411  ELSE IF( ldb.LT.max( 1, p ) ) THEN
412  info = -12
413  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
414  info = -16
415  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
416  info = -18
417  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
418  info = -20
419  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
420  info = -24
421  END IF
422 *
423 * Compute workspace
424 *
425  IF( info.EQ.0 ) THEN
426  CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
427  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
428  $ work, -1, info )
429  lwkopt = n + int( work( 1 ) )
430  lwkopt = max( 2*n, lwkopt )
431  lwkopt = max( 1, lwkopt )
432  work( 1 ) = dble( lwkopt )
433  END IF
434 *
435  IF( info.NE.0 ) THEN
436  CALL xerbla( 'DGGSVD3', -info )
437  RETURN
438  END IF
439  IF( lquery ) THEN
440  RETURN
441  ENDIF
442 *
443 * Compute the Frobenius norm of matrices A and B
444 *
445  anorm = dlange( '1', m, n, a, lda, work )
446  bnorm = dlange( '1', p, n, b, ldb, work )
447 *
448 * Get machine precision and set up threshold for determining
449 * the effective numerical rank of the matrices A and B.
450 *
451  ulp = dlamch( 'Precision' )
452  unfl = dlamch( 'Safe Minimum' )
453  tola = max( m, n )*max( anorm, unfl )*ulp
454  tolb = max( p, n )*max( bnorm, unfl )*ulp
455 *
456 * Preprocessing
457 *
458  CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
459  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
460  $ work( n+1 ), lwork-n, info )
461 *
462 * Compute the GSVD of two upper "triangular" matrices
463 *
464  CALL dtgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
465  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
466  $ work, ncycle, info )
467 *
468 * Sort the singular values and store the pivot indices in IWORK
469 * Copy ALPHA to WORK, then sort ALPHA in WORK
470 *
471  CALL dcopy( n, alpha, 1, work, 1 )
472  ibnd = min( l, m-k )
473  DO 20 i = 1, ibnd
474 *
475 * Scan for largest ALPHA(K+I)
476 *
477  isub = i
478  smax = work( k+i )
479  DO 10 j = i + 1, ibnd
480  temp = work( k+j )
481  IF( temp.GT.smax ) THEN
482  isub = j
483  smax = temp
484  END IF
485  10 CONTINUE
486  IF( isub.NE.i ) THEN
487  work( k+isub ) = work( k+i )
488  work( k+i ) = smax
489  iwork( k+i ) = k + isub
490  ELSE
491  iwork( k+i ) = k + i
492  END IF
493  20 CONTINUE
494 *
495  work( 1 ) = dble( lwkopt )
496  RETURN
497 *
498 * End of DGGSVD3
499 *
500  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dggsvd3(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, IWORK, INFO)
DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
Definition: dggsvd3.f:349
subroutine dtgsja(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)
DTGSJA
Definition: dtgsja.f:378
subroutine dggsvp3(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
DGGSVP3
Definition: dggsvp3.f:272