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. 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.

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.
    !>
    !>

  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. 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.

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.
    !>
    !>

  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. 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.

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.
    !>
    !>

  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. 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.

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.
    !>
    !>

Author

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