ggsvd3(3)

NAME
ggsvd3 - ggsvd3: SVD, QR iteration

SYNOPSIS

  Functions

subroutine cggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork, info)
CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
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)
DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
subroutine sggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)
SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
subroutine zggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork, info)
ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Detailed Description

Function Documentation

  subroutine cggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alpha, real, dimension( * ) beta, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

    !>
    !> CGGSVD3 computes the generalized singular value decomposition (GSVD)
    !> of an M-by-N complex matrix A and P-by-N complex matrix B:
    !>
    !>       U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )
    !>
    !> where U, V and Q are unitary matrices.
    !> Let K+L = the effective numerical rank of the
    !> matrix (A**H,B**H)**H, 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)
    !> 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 unitary
    !> 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**H.
    !> If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x.
    !> In some literature, the GSVD of A and B is presented in the form
    !>                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
    !> where U and V are orthogonal and X is nonsingular, and 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) )
    !>

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.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrices A and B.  N >= 0.
    !>

    P

    !>          P is INTEGER
    !>          The number of rows of the matrix B.  P >= 0.
    !>

    K

    !>          K is INTEGER
    !>

    L

    !>          L is INTEGER
    !>
    !>          On exit, K and L specify the dimension of the subblocks
    !>          described in Purpose.
    !>          K + L = effective numerical rank of (A**H,B**H)**H.
    !>

    A

    !>          A is COMPLEX 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.
    !>

    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 part of the triangular matrix R if
    !>          M-K-L < 0.  See Purpose for details.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,P).
    !>

    ALPHA

    !>          ALPHA is REAL array, dimension (N)
    !>

    BETA

    !>          BETA is REAL 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
    !>

    U

    !>          U is COMPLEX array, dimension (LDU,M)
    !>          If JOBU = 'U', U contains the M-by-M 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 P-by-P 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 N-by-N 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.
    !>

    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.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (2*N)
    !>

    IWORK

    !>          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).
    !>

    INFO

    !>          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 CTGSJA.
    !>

Internal Parameters:

    !>  TOLA    REAL
    !>  TOLB    REAL
    !>          TOLA and TOLB are the thresholds to determine the effective
    !>          rank of (A**H,B**H)**H. 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:
    CGGSVD3 replaces the deprecated subroutine CGGSVD.

  subroutine dggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

    !>
    !> 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)  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) ).
    !>

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.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrices A and B.  N >= 0.
    !>

    P

    !>          P is INTEGER
    !>          The number of rows of the matrix B.  P >= 0.
    !>

    K

    !>          K is INTEGER
    !>

    L

    !>          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.
    !>

    A

    !>          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.
    !>

    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 R if M-K-L < 0.
    !>          See Purpose for details.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,P).
    !>

    ALPHA

    !>          ALPHA is DOUBLE PRECISION array, dimension (N)
    !>

    BETA

    !>          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
    !>

    U

    !>          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.
    !>

    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 P-by-P 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 N-by-N 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.
    !>

    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.
    !>

    IWORK

    !>          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).
    !>

    INFO

    !>          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.
    !>

Internal Parameters:

    !>  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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:
    DGGSVD3 replaces the deprecated subroutine DGGSVD.

  subroutine sggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alpha, real, dimension( * ) beta, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

    !>
    !> SGGSVD3 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)  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) ).
    !>

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.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrices A and B.  N >= 0.
    !>

    P

    !>          P is INTEGER
    !>          The number of rows of the matrix B.  P >= 0.
    !>

    K

    !>          K is INTEGER
    !>

    L

    !>          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.
    !>

    A

    !>          A is REAL 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.
    !>

    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 R if M-K-L < 0.
    !>          See Purpose for details.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,P).
    !>

    ALPHA

    !>          ALPHA is REAL array, dimension (N)
    !>

    BETA

    !>          BETA is REAL 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
    !>

    U

    !>          U is REAL array, dimension (LDU,M)
    !>          If JOBU = 'U', U contains the M-by-M 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 P-by-P 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 N-by-N 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.
    !>

    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.
    !>

    IWORK

    !>          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).
    !>

    INFO

    !>          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 STGSJA.
    !>

Internal Parameters:

    !>  TOLA    REAL
    !>  TOLB    REAL
    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:
    SGGSVD3 replaces the deprecated subroutine SGGSVD.

  subroutine zggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

    !>
    !> ZGGSVD3 computes the generalized singular value decomposition (GSVD)
    !> of an M-by-N complex matrix A and P-by-N complex matrix B:
    !>
    !>       U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )
    !>
    !> where U, V and Q are unitary matrices.
    !> Let K+L = the effective numerical rank of the
    !> matrix (A**H,B**H)**H, 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)
    !> 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 unitary
    !> 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**H.
    !> If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x.
    !> In some literature, the GSVD of A and B is presented in the form
    !>                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
    !> where U and V are orthogonal and X is nonsingular, and 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) )
    !>

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.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrices A and B.  N >= 0.
    !>

    P

    !>          P is INTEGER
    !>          The number of rows of the matrix B.  P >= 0.
    !>

    K

    !>          K is INTEGER
    !>

    L

    !>          L is INTEGER
    !>
    !>          On exit, K and L specify the dimension of the subblocks
    !>          described in Purpose.
    !>          K + L = effective numerical rank of (A**H,B**H)**H.
    !>

    A

    !>          A is COMPLEX*16 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.
    !>

    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 part of the triangular matrix R if
    !>          M-K-L < 0.  See Purpose for details.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,P).
    !>

    ALPHA

    !>          ALPHA is DOUBLE PRECISION array, dimension (N)
    !>

    BETA

    !>          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
    !>

    U

    !>          U is COMPLEX*16 array, dimension (LDU,M)
    !>          If JOBU = 'U', U contains the M-by-M 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 P-by-P 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 N-by-N 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.
    !>

    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.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (2*N)
    !>

    IWORK

    !>          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).
    !>

    INFO

    !>          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 ZTGSJA.
    !>

Internal Parameters:

    !>  TOLA    DOUBLE PRECISION
    !>  TOLB    DOUBLE PRECISION
    !>          TOLA and TOLB are the thresholds to determine the effective
    !>          rank of (A**H,B**H)**H. 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:
    ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

Author

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