dggsvd(3F)

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

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

         CHARACTER      JOBQ, JOBU, JOBV

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

         INTEGER        IWORK( * )

         DOUBLE         PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA(
                        * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

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

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

     where U, V and Q are orthogonal matrices, and Z' is the 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 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'.
     If ( A',B')' 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'*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, 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':  Orthogonal matrix U is computed;
             = 'N':  U is not computed.

     JOBV    (input) CHARACTER*1
             = 'V':  Orthogonal matrix V is computed;
             = 'N':  V is not computed.

     JOBQ    (input) CHARACTER*1
             = 'Q':  Orthogonal 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 the Purpose section.  K + L =
             effective numerical rank of (A',B')'.

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

     B       (input/output) 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     (input) INTEGER
             The leading dimension of the array B. LDA >= 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) 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     (input) INTEGER
             The leading dimension of the array U. LDU >= max(1,M) if JOBU =
             'U'; LDU >= 1 otherwise.

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

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

     WORK    (workspace) DOUBLE PRECISION array,
             dimension (max(3*N,M,P)+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 DTGSJA.

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.