ggsvp3 - ggsvp3: step in ggsvd

Generated automatically by Doxygen for LAPACK from the source code. Version 3.12.0 Thu Aug 7 2025 17:26:25 ggsvp3(3)

subroutinecggsvp3(characterjobu,characterjobv,characterjobq,integerm,integerp,integern,complex,dimension(lda,*)a,integerlda,complex,dimension(ldb,*)b,integerldb,realtola,realtolb,integerk,integerl,complex,dimension(ldu,*)u,integerldu,complex,dimension(ldv,*)v,integerldv,complex,dimension(ldq,*)q,integerldq,integer,dimension(*)iwork,real,dimension(*)rwork,complex,dimension(*)tau,complex,dimension(*)work,integerlwork,integerinfo)CGGSVP3Purpose: CGGSVP3 computes unitary matrices U, V and Q such that N-K-L K L U**H*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**H*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**H,B**H)**H. This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine CGGSVD3. ParametersJOBU JOBU is CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed. JOBV JOBV is CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed. JOBQ JOBQ is CHARACTER*1 = 'Q': Unitary matrix Q is computed; = 'N': Q is not computed. M M is INTEGER The number of rows of the matrix A. M >= 0. P P is INTEGER The number of rows of the matrix B. P >= 0. N N is INTEGER The number of columns of the matrices A and B. N >= 0. A A is COMPLEX 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 LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is COMPLEX 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 LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). TOLA TOLA is REAL TOLB TOLB is REAL 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. K K is INTEGER L 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**H,B**H)**H. U U is COMPLEX array, dimension (LDU,M) If JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N', U is not referenced. LDU LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V V is COMPLEX array, dimension (LDV,P) If JOBV = 'V', V contains the unitary matrix V. If JOBV = 'N', V is not referenced. LDV LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q Q is COMPLEX array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the unitary matrix Q. If JOBQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. IWORK IWORK is INTEGER array, dimension (N) RWORK RWORK is REAL array, dimension (2*N) TAU TAU is COMPLEX array, dimension (N) WORK WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1. 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. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: The subroutine uses LAPACK subroutine CGEQP3 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. CGGSVP3 replaces the deprecated subroutine CGGSVP. subroutinedggsvp3(characterjobu,characterjobv,characterjobq,integerm,integerp,integern,doubleprecision,dimension(lda,*)a,integerlda,doubleprecision,dimension(ldb,*)b,integerldb,doubleprecisiontola,doubleprecisiontolb,integerk,integerl,doubleprecision,dimension(ldu,*)u,integerldu,doubleprecision,dimension(ldv,*)v,integerldv,doubleprecision,dimension(ldq,*)q,integerldq,integer,dimension(*)iwork,doubleprecision,dimension(*)tau,doubleprecision,dimension(*)work,integerlwork,integerinfo)DGGSVP3Purpose: DGGSVP3 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 DGGSVD3. ParametersJOBU JOBU is CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed. JOBV JOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed. JOBQ JOBQ is CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed. M M is INTEGER The number of rows of the matrix A. M >= 0. P P is INTEGER The number of rows of the matrix B. P >= 0. N N is INTEGER The number of columns of the matrices A and B. N >= 0. A 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. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B 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. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). TOLA TOLA is DOUBLE PRECISION TOLB 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. K K is INTEGER L 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. U U is DOUBLE PRECISION array, dimension (LDU,M) If JOBU = 'U', U contains the orthogonal matrix U. If JOBU = 'N', U is not referenced. LDU LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V V is DOUBLE PRECISION array, dimension (LDV,P) If JOBV = 'V', V contains the orthogonal matrix V. If JOBV = 'N', V is not referenced. LDV LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q Q is DOUBLE PRECISION array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the orthogonal matrix Q. If JOBQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. IWORK IWORK is INTEGER array, dimension (N) TAU TAU is DOUBLE PRECISION array, dimension (N) WORK WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1. 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. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: The subroutine uses LAPACK subroutine DGEQP3 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. DGGSVP3 replaces the deprecated subroutine DGGSVP. subroutinesggsvp3(characterjobu,characterjobv,characterjobq,integerm,integerp,integern,real,dimension(lda,*)a,integerlda,real,dimension(ldb,*)b,integerldb,realtola,realtolb,integerk,integerl,real,dimension(ldu,*)u,integerldu,real,dimension(ldv,*)v,integerldv,real,dimension(ldq,*)q,integerldq,integer,dimension(*)iwork,real,dimension(*)tau,real,dimension(*)work,integerlwork,integerinfo)SGGSVP3Purpose: SGGSVP3 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 SGGSVD3. ParametersJOBU JOBU is CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed. JOBV JOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed. JOBQ JOBQ is CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed. M M is INTEGER The number of rows of the matrix A. M >= 0. P P is INTEGER The number of rows of the matrix B. P >= 0. N N is INTEGER The number of columns of the matrices A and B. N >= 0. A A is REAL 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 LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is REAL 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 LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). TOLA TOLA is REAL TOLB TOLB is REAL 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. K K is INTEGER L 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. U U is REAL array, dimension (LDU,M) If JOBU = 'U', U contains the orthogonal matrix U. If JOBU = 'N', U is not referenced. LDU LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V V is REAL array, dimension (LDV,P) If JOBV = 'V', V contains the orthogonal matrix V. If JOBV = 'N', V is not referenced. LDV LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q Q is REAL array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the orthogonal matrix Q. If JOBQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. IWORK IWORK is INTEGER array, dimension (N) TAU TAU is REAL array, dimension (N) WORK WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1. 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. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: The subroutine uses LAPACK subroutine SGEQP3 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. SGGSVP3 replaces the deprecated subroutine SGGSVP. subroutinezggsvp3(characterjobu,characterjobv,characterjobq,integerm,integerp,integern,complex*16,dimension(lda,*)a,integerlda,complex*16,dimension(ldb,*)b,integerldb,doubleprecisiontola,doubleprecisiontolb,integerk,integerl,complex*16,dimension(ldu,*)u,integerldu,complex*16,dimension(ldv,*)v,integerldv,complex*16,dimension(ldq,*)q,integerldq,integer,dimension(*)iwork,doubleprecision,dimension(*)rwork,complex*16,dimension(*)tau,complex*16,dimension(*)work,integerlwork,integerinfo)ZGGSVP3Purpose: ZGGSVP3 computes unitary matrices U, V and Q such that N-K-L K L U**H*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**H*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**H,B**H)**H. This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine ZGGSVD3. ParametersJOBU JOBU is CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed. JOBV JOBV is CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed. JOBQ JOBQ is CHARACTER*1 = 'Q': Unitary matrix Q is computed; = 'N': Q is not computed. M M is INTEGER The number of rows of the matrix A. M >= 0. P P is INTEGER The number of rows of the matrix B. P >= 0. N N is INTEGER The number of columns of the matrices A and B. N >= 0. A A is COMPLEX*16 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 LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is COMPLEX*16 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 LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). TOLA TOLA is DOUBLE PRECISION TOLB 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)*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 K is INTEGER L 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**H,B**H)**H. U U is COMPLEX*16 array, dimension (LDU,M) If JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N', U is not referenced. LDU LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V V is COMPLEX*16 array, dimension (LDV,P) If JOBV = 'V', V contains the unitary matrix V. If JOBV = 'N', V is not referenced. LDV LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q Q is COMPLEX*16 array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the unitary matrix Q. If JOBQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. IWORK IWORK is INTEGER array, dimension (N) RWORK RWORK is DOUBLE PRECISION array, dimension (2*N) TAU TAU is COMPLEX*16 array, dimension (N) WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1. 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. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: The subroutine uses LAPACK subroutine ZGEQP3 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. ZGGSVP3 replaces the deprecated subroutine ZGGSVP.

ggsvp3 - ggsvp3: step in ggsvd

Functions 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 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 subroutine sggsvp3 (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) SGGSVP3 subroutine zggsvp3 (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) ZGGSVP3