DGGSVD - compute the generalized singular value decomposi­
       tion (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  decomposi­
       tion  (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  struc­
       tures, 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.


       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  orthogo­
       nal 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 con­
       verted to the latter form by taking the nonsingular matrix
       X as

                            X = Q*( I   0    )
                                  ( 0 inv(R) ).


ARGUMENTS


               = '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  con­
               tains  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  con­
               tains  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  general­
               ized 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)

               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/output) 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    (output)INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value.
               >  0:   if  INFO  =  1,  the Jacobi-type procedure
               failed to converge.  For further details, see sub­
               routine 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(P,N)*norm(B)*MAZHEPS.   The  size  of TOLA and
               TOLB may affect the size of backward errors of the
               decomposition.

               Further Details ===============

               2-96  Based  on  modifications by Ming Gu and Huan
               Ren, Computer Science Division, University of Cal­
               ifornia at Berkeley, USA


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