zggsvd(3F)

NAME
     ZGGSVD - compute the generalized singular value decomposition (GSVD) of
     an M-by-N complex matrix A and P-by-N complex matrix B

SYNOPSIS
     SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB,
                        ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, RWORK,
                        IWORK, INFO )

         CHARACTER      JOBQ, JOBU, JOBV

         INTEGER        INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P

         INTEGER        IWORK( * )

         DOUBLE         PRECISION ALPHA( * ), BETA( * ), RWORK( * )

         COMPLEX*16     A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V(
                        LDV, * ), WORK( * )

PURPOSE
     ZGGSVD computes the generalized singular value decomposition (GSVD) of an
     M-by-N complex matrix A and P-by-N complex matrix B:

           U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )

     where U, V and Q are unitary matrices, and Z' means the conjugate
     transpose of Z.  Let K+L = the effective numerical rank of the matrix
     (A',B')', 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'.
     If ( A',B')' has orthnormal 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'*A x = lambda* B'*B x.
     In some literature, the GSVD of A and B is presented in the form
                      U'*A*X = ( 0 D1 ),   V'*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) )


ARGUMENTS
     JOBU    (input) CHARACTER*1
             = 'U':  Unitary matrix U is computed;
             = 'N':  U is not computed.
     JOBV    (input) CHARACTER*1
             = 'V':  Unitary matrix V is computed;
             = 'N':  V is not computed.

     JOBQ    (input) CHARACTER*1
             = 'Q':  Unitary matrix Q is computed;
             = 'N':  Q is not computed.

     M       (input) INTEGER
             The number of rows of the matrix A.  M >= 0.

     N       (input) INTEGER
             The number of columns of the matrices A and B.  N >= 0.

     P       (input) INTEGER
             The number of rows of the matrix B.  P >= 0.

     K       (output) INTEGER
             L       (output) INTEGER On exit, K and L specify the dimension
             of the subblocks described in Purpose.  K + L = effective
             numerical rank of (A',B')'.

     A       (input/output) 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     (input) INTEGER
             The leading dimension of the array A. LDA >= max(1,M).

     B       (input/output) 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     (input) INTEGER
             The leading dimension of the array B. LDB >= max(1,P).

     ALPHA   (output) DOUBLE PRECISION array, dimension (N)
             BETA    (output) 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       (output) 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     (input) INTEGER
             The leading dimension of the array U. LDU >= max(1,M) if JOBU =
             'U'; LDU >= 1 otherwise.

     V       (output) 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     (input) INTEGER
             The leading dimension of the array V. LDV >= max(1,P) if JOBV =
             'V'; LDV >= 1 otherwise.

     Q       (output) 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     (input) INTEGER
             The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ =
             'Q'; LDQ >= 1 otherwise.

     WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)

     RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)

     IWORK   (workspace) INTEGER array, dimension (N)

     INFO    (output)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.

PARAMETERS
     TOLA    DOUBLE PRECISION
             TOLB    DOUBLE PRECISION TOLA and TOLB are the thresholds to
             determine the effective rank of (A',B')'. 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.