--- rpl/lapack/lapack/dggsvp.f 2010/08/13 21:03:46 1.6 +++ rpl/lapack/lapack/dggsvp.f 2023/08/07 08:38:51 1.19 @@ -1,11 +1,262 @@ +*> \brief \b DGGSVP +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DGGSVP + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, +* TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, +* IWORK, TAU, WORK, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER JOBQ, JOBU, JOBV +* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P +* DOUBLE PRECISION TOLA, TOLB +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ) +* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), +* $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> This routine is deprecated and has been replaced by routine DGGSVP3. +*> +*> DGGSVP computes orthogonal matrices U, V and Q such that +*> +*> N-K-L K L +*> U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; +*> L ( 0 0 A23 ) +*> M-K-L ( 0 0 0 ) +*> +*> N-K-L K L +*> = K ( 0 A12 A13 ) if M-K-L < 0; +*> M-K ( 0 0 A23 ) +*> +*> N-K-L K L +*> V**T*B*Q = L ( 0 0 B13 ) +*> P-L ( 0 0 0 ) +*> +*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular +*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, +*> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective +*> numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T. +*> +*> This decomposition is the preprocessing step for computing the +*> Generalized Singular Value Decomposition (GSVD), see subroutine +*> DGGSVD. +*> \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] P +*> \verbatim +*> P is INTEGER +*> The number of rows of the matrix B. P >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrices A and B. N >= 0. +*> \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 (or trapezoidal) matrix +*> described in the Purpose section. +*> \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 described in +*> the Purpose section. +*> \endverbatim +*> +*> \param[in] LDB +*> \verbatim +*> LDB is INTEGER +*> The leading dimension of the array B. LDB >= max(1,P). +*> \endverbatim +*> +*> \param[in] TOLA +*> \verbatim +*> TOLA is DOUBLE PRECISION +*> \endverbatim +*> +*> \param[in] TOLB +*> \verbatim +*> TOLB is DOUBLE PRECISION +*> +*> TOLA and TOLB are the thresholds to determine the effective +*> numerical rank of matrix B and a subblock of A. 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 +*> +*> \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 section. +*> K + L = effective numerical rank of (A**T,B**T)**T. +*> \endverbatim +*> +*> \param[out] U +*> \verbatim +*> U is DOUBLE PRECISION array, dimension (LDU,M) +*> If JOBU = 'U', U contains the 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 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 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] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N) +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is DOUBLE PRECISION array, dimension (N) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is DOUBLE PRECISION array, dimension (max(3*N,M,P)) +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -i, the i-th argument had an illegal value. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup doubleOTHERcomputational +* +*> \par Further Details: +* ===================== +*> +*> The subroutine uses LAPACK subroutine DGEQPF for the QR factorization +*> with column pivoting to detect the effective numerical rank of the +*> a matrix. It may be replaced by a better rank determination strategy. +*> +* ===================================================================== SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, $ IWORK, TAU, WORK, INFO ) * -* -- LAPACK routine (version 3.2) -- +* -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 * * .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV @@ -18,132 +269,6 @@ $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) * .. * -* Purpose -* ======= -* -* DGGSVP computes orthogonal matrices U, V and Q such that -* -* N-K-L K L -* U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; -* L ( 0 0 A23 ) -* M-K-L ( 0 0 0 ) -* -* N-K-L K L -* = K ( 0 A12 A13 ) if M-K-L < 0; -* M-K ( 0 0 A23 ) -* -* N-K-L K L -* V'*B*Q = L ( 0 0 B13 ) -* P-L ( 0 0 0 ) -* -* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular -* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, -* otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective -* numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the -* transpose of Z. -* -* This decomposition is the preprocessing step for computing the -* Generalized Singular Value Decomposition (GSVD), see subroutine -* DGGSVD. -* -* Arguments -* ========= -* -* JOBU (input) CHARACTER*1 -* = 'U': Orthogonal matrix U is computed; -* = 'N': U is not computed. -* -* JOBV (input) CHARACTER*1 -* = 'V': Orthogonal matrix V is computed; -* = 'N': V is not computed. -* -* JOBQ (input) CHARACTER*1 -* = 'Q': Orthogonal matrix Q is computed; -* = 'N': Q is not computed. -* -* M (input) INTEGER -* The number of rows of the matrix A. M >= 0. -* -* P (input) INTEGER -* The number of rows of the matrix B. P >= 0. -* -* N (input) INTEGER -* The number of columns of the matrices A and B. N >= 0. -* -* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) -* On entry, the M-by-N matrix A. -* On exit, A contains the triangular (or trapezoidal) matrix -* described in the Purpose section. -* -* LDA (input) INTEGER -* The leading dimension of the array A. LDA >= max(1,M). -* -* B (input/output) DOUBLE PRECISION array, dimension (LDB,N) -* On entry, the P-by-N matrix B. -* On exit, B contains the triangular matrix described in -* the Purpose section. -* -* LDB (input) INTEGER -* The leading dimension of the array B. LDB >= max(1,P). -* -* TOLA (input) DOUBLE PRECISION -* TOLB (input) DOUBLE PRECISION -* TOLA and TOLB are the thresholds to determine the effective -* numerical rank of matrix B and a subblock of A. Generally, -* they are set to -* TOLA = MAX(M,N)*norm(A)*MAZHEPS, -* TOLB = MAX(P,N)*norm(B)*MAZHEPS. -* The size of TOLA and TOLB may affect the size of backward -* errors of the decomposition. -* -* K (output) INTEGER -* L (output) INTEGER -* On exit, K and L specify the dimension of the subblocks -* described in Purpose. -* K + L = effective numerical rank of (A',B')'. -* -* U (output) DOUBLE PRECISION array, dimension (LDU,M) -* If JOBU = 'U', U contains the orthogonal matrix U. -* If JOBU = 'N', U is not referenced. -* -* LDU (input) INTEGER -* The leading dimension of the array U. LDU >= max(1,M) if -* JOBU = 'U'; LDU >= 1 otherwise. -* -* V (output) DOUBLE PRECISION array, dimension (LDV,P) -* If JOBV = 'V', V contains the orthogonal matrix V. -* If JOBV = 'N', V is not referenced. -* -* LDV (input) INTEGER -* The leading dimension of the array V. LDV >= max(1,P) if -* JOBV = 'V'; LDV >= 1 otherwise. -* -* Q (output) DOUBLE PRECISION array, dimension (LDQ,N) -* If JOBQ = 'Q', Q contains the orthogonal matrix Q. -* If JOBQ = 'N', Q is not referenced. -* -* LDQ (input) INTEGER -* The leading dimension of the array Q. LDQ >= max(1,N) if -* JOBQ = 'Q'; LDQ >= 1 otherwise. -* -* IWORK (workspace) INTEGER array, dimension (N) -* -* TAU (workspace) DOUBLE PRECISION array, dimension (N) -* -* WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P)) -* -* INFO (output) INTEGER -* = 0: successful exit -* < 0: if INFO = -i, the i-th argument had an illegal value. -* -* -* Further Details -* =============== -* -* The subroutine uses LAPACK subroutine DGEQPF for the QR factorization -* with column pivoting to detect the effective numerical rank of the -* a matrix. It may be replaced by a better rank determination strategy. -* * ===================================================================== * * .. Parameters .. @@ -258,14 +383,14 @@ * CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO ) * -* Update A := A*Z' +* Update A := A*Z**T * CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A, $ LDA, WORK, INFO ) * IF( WANTQ ) THEN * -* Update Q := Q*Z' +* Update Q := Q*Z**T * CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q, $ LDQ, WORK, INFO ) @@ -287,7 +412,7 @@ * * then the following does the complete QR decomposition of A11: * -* A11 = U*( 0 T12 )*P1' +* A11 = U*( 0 T12 )*P1**T * ( 0 0 ) * DO 70 I = 1, N - L @@ -303,7 +428,7 @@ $ K = K + 1 80 CONTINUE * -* Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N ) +* Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N ) * CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA, $ TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) @@ -345,7 +470,7 @@ * IF( WANTQ ) THEN * -* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' +* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T * CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU, $ Q, LDQ, WORK, INFO )