Provided by: liblapack-doc_3.12.0-3build1.1_all bug

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) 'diagonal'
            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.

                     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) 'diagonal' 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.

                     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) 'diagonal' 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.

                     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) 'diagonal'
            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.

                     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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