*> \brief DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGGSVD3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * 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 ) * * .. Scalar Arguments .. * CHARACTER JOBQ, JOBU, JOBV * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK * .. * .. Array Arguments .. * INTEGER IWORK( * ) * DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), * \$ BETA( * ), Q( LDQ, * ), U( LDU, * ), * \$ V( LDV, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DGGSVD3 computes the generalized singular value decomposition (GSVD) *> of an M-by-N real matrix A and P-by-N real matrix B: *> *> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) *> *> where U, V and Q are orthogonal matrices. *> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T, *> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and *> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the *> following structures, respectively: *> *> If M-K-L >= 0, *> *> K L *> D1 = K ( I 0 ) *> L ( 0 C ) *> M-K-L ( 0 0 ) *> *> K L *> D2 = L ( 0 S ) *> P-L ( 0 0 ) *> *> N-K-L K L *> ( 0 R ) = K ( 0 R11 R12 ) *> L ( 0 0 R22 ) *> *> where *> *> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), *> S = diag( BETA(K+1), ... , BETA(K+L) ), *> C**2 + S**2 = I. *> *> R is stored in A(1:K+L,N-K-L+1:N) on exit. *> *> If M-K-L < 0, *> *> K M-K K+L-M *> D1 = K ( I 0 0 ) *> M-K ( 0 C 0 ) *> *> K M-K K+L-M *> D2 = M-K ( 0 S 0 ) *> K+L-M ( 0 0 I ) *> P-L ( 0 0 0 ) *> *> N-K-L K M-K K+L-M *> ( 0 R ) = K ( 0 R11 R12 R13 ) *> M-K ( 0 0 R22 R23 ) *> K+L-M ( 0 0 0 R33 ) *> *> where *> *> C = diag( ALPHA(K+1), ... , ALPHA(M) ), *> S = diag( BETA(K+1), ... , BETA(M) ), *> C**2 + S**2 = I. *> *> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored *> ( 0 R22 R23 ) *> in B(M-K+1:L,N+M-K-L+1:N) on exit. *> *> The routine computes C, S, R, and optionally the orthogonal *> transformation matrices U, V and Q. *> *> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of *> A and B implicitly gives the SVD of A*inv(B): *> A*inv(B) = U*(D1*inv(D2))*V**T. *> If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is *> also equal to the CS decomposition of A and B. Furthermore, the GSVD *> can be used to derive the solution of the eigenvalue problem: *> A**T*A x = lambda* B**T*B x. *> In some literature, the GSVD of A and B is presented in the form *> U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 ) *> where U and V are orthogonal and X is nonsingular, D1 and D2 are *> ``diagonal''. The former GSVD form can be converted to the latter *> form by taking the nonsingular matrix X as *> *> X = Q*( I 0 ) *> ( 0 inv(R) ). *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBU *> \verbatim *> JOBU is CHARACTER*1 *> = 'U': Orthogonal matrix U is computed; *> = 'N': U is not computed. *> \endverbatim *> *> \param[in] JOBV *> \verbatim *> JOBV is CHARACTER*1 *> = 'V': Orthogonal matrix V is computed; *> = 'N': V is not computed. *> \endverbatim *> *> \param[in] JOBQ *> \verbatim *> JOBQ is CHARACTER*1 *> = 'Q': Orthogonal matrix Q is computed; *> = 'N': Q is not computed. *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in] P *> \verbatim *> P is INTEGER *> The number of rows of the matrix B. P >= 0. *> \endverbatim *> *> \param[out] K *> \verbatim *> K is INTEGER *> \endverbatim *> *> \param[out] L *> \verbatim *> L is INTEGER *> *> On exit, K and L specify the dimension of the subblocks *> described in Purpose. *> K + L = effective numerical rank of (A**T,B**T)**T. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is DOUBLE PRECISION array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, A contains the triangular matrix R, or part of R. *> See Purpose for details. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB,N) *> On entry, the P-by-N matrix B. *> On exit, B contains the triangular matrix R if M-K-L < 0. *> See Purpose for details. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,P). *> \endverbatim *> *> \param[out] ALPHA *> \verbatim *> ALPHA is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is DOUBLE PRECISION array, dimension (N) *> *> On exit, ALPHA and BETA contain the generalized singular *> value pairs of A and B; *> ALPHA(1:K) = 1, *> BETA(1:K) = 0, *> and if M-K-L >= 0, *> ALPHA(K+1:K+L) = C, *> BETA(K+1:K+L) = S, *> or if M-K-L < 0, *> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 *> BETA(K+1:M) =S, BETA(M+1:K+L) =1 *> and *> ALPHA(K+L+1:N) = 0 *> BETA(K+L+1:N) = 0 *> \endverbatim *> *> \param[out] U *> \verbatim *> U is DOUBLE PRECISION array, dimension (LDU,M) *> If JOBU = 'U', U contains the M-by-M orthogonal matrix U. *> If JOBU = 'N', U is not referenced. *> \endverbatim *> *> \param[in] LDU *> \verbatim *> LDU is INTEGER *> The leading dimension of the array U. LDU >= max(1,M) if *> JOBU = 'U'; LDU >= 1 otherwise. *> \endverbatim *> *> \param[out] V *> \verbatim *> V is DOUBLE PRECISION array, dimension (LDV,P) *> If JOBV = 'V', V contains the P-by-P orthogonal matrix V. *> If JOBV = 'N', V is not referenced. *> \endverbatim *> *> \param[in] LDV *> \verbatim *> LDV is INTEGER *> The leading dimension of the array V. LDV >= max(1,P) if *> JOBV = 'V'; LDV >= 1 otherwise. *> \endverbatim *> *> \param[out] Q *> \verbatim *> Q is DOUBLE PRECISION array, dimension (LDQ,N) *> If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. *> If JOBQ = 'N', Q is not referenced. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. LDQ >= max(1,N) if *> JOBQ = 'Q'; LDQ >= 1 otherwise. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal size of the WORK array, returns *> this value as the first entry of the WORK array, and no error *> message related to LWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (N) *> On exit, IWORK stores the sorting information. More *> precisely, the following loop will sort ALPHA *> for I = K+1, min(M,K+L) *> swap ALPHA(I) and ALPHA(IWORK(I)) *> endfor *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, the i-th argument had an illegal value. *> > 0: if INFO = 1, the Jacobi-type procedure failed to *> converge. For further details, see subroutine DTGSJA. *> \endverbatim * *> \par Internal Parameters: * ========================= *> *> \verbatim *> TOLA DOUBLE PRECISION *> TOLB DOUBLE PRECISION *> TOLA and TOLB are the thresholds to determine the effective *> rank of (A**T,B**T)**T. Generally, they are set to *> TOLA = MAX(M,N)*norm(A)*MACHEPS, *> TOLB = MAX(P,N)*norm(B)*MACHEPS. *> The size of TOLA and TOLB may affect the size of backward *> errors of the decomposition. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date August 2015 * *> \ingroup doubleGEsing * *> \par Contributors: * ================== *> *> Ming Gu and Huan Ren, Computer Science Division, University of *> California at Berkeley, USA *> * *> \par Further Details: * ===================== *> *> DGGSVD3 replaces the deprecated subroutine DGGSVD. *> * ===================================================================== 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 ) * * -- LAPACK driver routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * August 2015 * * .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, \$ LWORK * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), \$ BETA( * ), Q( LDQ, * ), U( LDU, * ), \$ V( LDV, * ), WORK( * ) * .. * * ===================================================================== * * .. Local Scalars .. LOGICAL WANTQ, WANTU, WANTV, LQUERY INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, DLAMCH, DLANGE * .. * .. External Subroutines .. EXTERNAL DCOPY, DGGSVP3, DTGSJA, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Decode and test the input parameters * WANTU = LSAME( JOBU, 'U' ) WANTV = LSAME( JOBV, 'V' ) WANTQ = LSAME( JOBQ, 'Q' ) LQUERY = ( LWORK.EQ.-1 ) LWKOPT = 1 * * Test the input arguments * INFO = 0 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( P.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -12 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN INFO = -16 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN INFO = -18 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -20 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN INFO = -24 END IF * * Compute workspace * IF( INFO.EQ.0 ) THEN CALL DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, \$ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK, \$ WORK, -1, INFO ) LWKOPT = N + INT( WORK( 1 ) ) LWKOPT = MAX( 2*N, LWKOPT ) LWKOPT = MAX( 1, LWKOPT ) WORK( 1 ) = DBLE( LWKOPT ) END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGSVD3', -INFO ) RETURN END IF IF( LQUERY ) THEN RETURN ENDIF * * Compute the Frobenius norm of matrices A and B * ANORM = DLANGE( '1', M, N, A, LDA, WORK ) BNORM = DLANGE( '1', P, N, B, LDB, WORK ) * * Get machine precision and set up threshold for determining * the effective numerical rank of the matrices A and B. * ULP = DLAMCH( 'Precision' ) UNFL = DLAMCH( 'Safe Minimum' ) TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP * * Preprocessing * CALL DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, \$ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK, \$ WORK( N+1 ), LWORK-N, INFO ) * * Compute the GSVD of two upper "triangular" matrices * CALL 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 ) * * Sort the singular values and store the pivot indices in IWORK * Copy ALPHA to WORK, then sort ALPHA in WORK * CALL DCOPY( N, ALPHA, 1, WORK, 1 ) IBND = MIN( L, M-K ) DO 20 I = 1, IBND * * Scan for largest ALPHA(K+I) * ISUB = I SMAX = WORK( K+I ) DO 10 J = I + 1, IBND TEMP = WORK( K+J ) IF( TEMP.GT.SMAX ) THEN ISUB = J SMAX = TEMP END IF 10 CONTINUE IF( ISUB.NE.I ) THEN WORK( K+ISUB ) = WORK( K+I ) WORK( K+I ) = SMAX IWORK( K+I ) = K + ISUB ELSE IWORK( K+I ) = K + I END IF 20 CONTINUE * WORK( 1 ) = DBLE( LWKOPT ) RETURN * * End of DGGSVD3 * END