1 *> \brief <b> DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices</b>
3 * =========== DOCUMENTATION ===========
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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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 )
25 * .. Scalar Arguments ..
26 * CHARACTER JOBQ, JOBU, JOBV
27 * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK
29 * .. Array Arguments ..
31 * DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
32 * $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
33 * $ V( LDV, * ), WORK( * )
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:
45 *> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
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:
65 *> ( 0 R ) = K ( 0 R11 R12 )
70 *> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
71 *> S = diag( BETA(K+1), ... , BETA(K+L) ),
74 *> R is stored in A(1:K+L,N-K-L+1:N) on exit.
88 *> ( 0 R ) = K ( 0 R11 R12 R13 )
89 *> M-K ( 0 0 R22 R23 )
90 *> K+L-M ( 0 0 0 R33 )
94 *> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
95 *> S = diag( BETA(K+1), ... , BETA(M) ),
98 *> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
100 *> in B(M-K+1:L,N+M-K-L+1:N) on exit.
102 *> The routine computes C, S, R, and optionally the orthogonal
103 *> transformation matrices U, V and Q.
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
127 *> JOBU is CHARACTER*1
128 *> = 'U': Orthogonal matrix U is computed;
129 *> = 'N': U is not computed.
134 *> JOBV is CHARACTER*1
135 *> = 'V': Orthogonal matrix V is computed;
136 *> = 'N': V is not computed.
141 *> JOBQ is CHARACTER*1
142 *> = 'Q': Orthogonal matrix Q is computed;
143 *> = 'N': Q is not computed.
149 *> The number of rows of the matrix A. M >= 0.
155 *> The number of columns of the matrices A and B. N >= 0.
161 *> The number of rows of the matrix B. P >= 0.
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.
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.
189 *> The leading dimension of the array A. LDA >= max(1,M).
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.
203 *> The leading dimension of the array B. LDB >= max(1,P).
208 *> ALPHA is DOUBLE PRECISION array, dimension (N)
213 *> BETA is DOUBLE PRECISION array, dimension (N)
215 *> On exit, ALPHA and BETA contain the generalized singular
216 *> value pairs of A and B;
219 *> and if M-K-L >= 0,
220 *> ALPHA(K+1:K+L) = C,
221 *> BETA(K+1:K+L) = S,
223 *> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
224 *> BETA(K+1:M) =S, BETA(M+1:K+L) =1
226 *> ALPHA(K+L+1:N) = 0
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.
240 *> The leading dimension of the array U. LDU >= max(1,M) if
241 *> JOBU = 'U'; LDU >= 1 otherwise.
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.
254 *> The leading dimension of the array V. LDV >= max(1,P) if
255 *> JOBV = 'V'; LDV >= 1 otherwise.
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.
268 *> The leading dimension of the array Q. LDQ >= max(1,N) if
269 *> JOBQ = 'Q'; LDQ >= 1 otherwise.
274 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
275 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
281 *> The dimension of the array WORK.
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.
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))
297 *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
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.
309 *> \par Internal Parameters:
310 * =========================
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.
326 *> \author Univ. of Tennessee
327 *> \author Univ. of California Berkeley
328 *> \author Univ. of Colorado Denver
333 *> \ingroup doubleGEsing
335 *> \par Contributors:
338 *> Ming Gu and Huan Ren, Computer Science Division, University of
339 *> California at Berkeley, USA
342 *> \par Further Details:
343 * =====================
345 *> DGGSVD3 replaces the deprecated subroutine DGGSVD.
347 * =====================================================================
348 SUBROUTINE DGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
349 $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
350 $ WORK, LWORK, IWORK, INFO )
352 * -- LAPACK driver routine (version 3.6.0) --
353 * -- LAPACK is a software package provided by Univ. of Tennessee, --
354 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
357 * .. Scalar Arguments ..
358 CHARACTER JOBQ, JOBU, JOBV
359 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
362 * .. Array Arguments ..
364 DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
365 $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
366 $ V( LDV, * ), WORK( * )
369 * =====================================================================
371 * .. Local Scalars ..
372 LOGICAL WANTQ, WANTU, WANTV, LQUERY
373 INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
374 DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
376 * .. External Functions ..
378 DOUBLE PRECISION DLAMCH, DLANGE
379 EXTERNAL LSAME, DLAMCH, DLANGE
381 * .. External Subroutines ..
382 EXTERNAL DCOPY, DGGSVP3, DTGSJA, XERBLA
384 * .. Intrinsic Functions ..
387 * .. Executable Statements ..
389 * Decode and test the input parameters
391 WANTU = LSAME( JOBU, 'U' )
392 WANTV = LSAME( JOBV, 'V' )
393 WANTQ = LSAME( JOBQ, 'Q' )
394 LQUERY = ( LWORK.EQ.-1 )
397 * Test the input arguments
400 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
402 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
404 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
406 ELSE IF( M.LT.0 ) THEN
408 ELSE IF( N.LT.0 ) THEN
410 ELSE IF( P.LT.0 ) THEN
412 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
414 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
416 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
418 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
420 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
422 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
429 CALL DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
430 $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
432 LWKOPT = N + INT( WORK( 1 ) )
433 LWKOPT = MAX( 2*N, LWKOPT )
434 LWKOPT = MAX( 1, LWKOPT )
435 WORK( 1 ) = DBLE( LWKOPT )
439 CALL XERBLA( 'DGGSVD3', -INFO )
446 * Compute the Frobenius norm of matrices A and B
448 ANORM = DLANGE( '1', M, N, A, LDA, WORK )
449 BNORM = DLANGE( '1', P, N, B, LDB, WORK )
451 * Get machine precision and set up threshold for determining
452 * the effective numerical rank of the matrices A and B.
454 ULP = DLAMCH( 'Precision' )
455 UNFL = DLAMCH( 'Safe Minimum' )
456 TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
457 TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
461 CALL DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
462 $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
463 $ WORK( N+1 ), LWORK-N, INFO )
465 * Compute the GSVD of two upper "triangular" matrices
467 CALL DTGSJA( 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 )
471 * Sort the singular values and store the pivot indices in IWORK
472 * Copy ALPHA to WORK, then sort ALPHA in WORK
474 CALL DCOPY( N, ALPHA, 1, WORK, 1 )
478 * Scan for largest ALPHA(K+I)
482 DO 10 J = I + 1, IBND
484 IF( TEMP.GT.SMAX ) THEN
490 WORK( K+ISUB ) = WORK( K+I )
492 IWORK( K+I ) = K + ISUB
498 WORK( 1 ) = DBLE( LWKOPT )