ggsvp3(3)

NAME
ggsvp3 - ggsvp3: step in ggsvd

SYNOPSIS

  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

Detailed Description

Function Documentation

  subroutine cggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real tola, real tolb, integer k, integer l, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, real, dimension( * ) rwork, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)

CGGSVP3

Purpose:

     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.

Parameters
    JOBU

              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.
              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.

Further Details:

      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.

Definition at line 275 of file cggsvp3.f.

  subroutine dggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, integer k, integer l, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DGGSVP3

Purpose:

     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.

Parameters
    JOBU

              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.
              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.

Further Details:

      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.

Definition at line 269 of file dggsvp3.f.

  subroutine sggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real tola, real tolb, integer k, integer l, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SGGSVP3

Purpose:

     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.

Parameters
    JOBU

              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.
              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.

Further Details:

      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.

Definition at line 269 of file sggsvp3.f.

  subroutine zggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, integer k, integer l, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, double precision, dimension( * ) rwork, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZGGSVP3

Purpose:

     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.

Parameters
    JOBU

              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.
              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.

Further Details:

      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.

Definition at line 275 of file zggsvp3.f.

Author

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