DTGSJA - compute the generalized singular value decomposi­
       tion (GSVD) of two real upper triangular (or  trapezoidal)
       matrices A and B


SYNOPSIS

       SUBROUTINE DTGSJA( JOBU,  JOBV,  JOBQ,  M,  P, N, K, L, A,
                          LDA, B, LDB, TOLA, TOLB,  ALPHA,  BETA,
                          U,  LDU,  V, LDV, Q, LDQ, WORK, NCYCLE,
                          INFO )

           CHARACTER      JOBQ, JOBU, JOBV

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

           DOUBLE         PRECISION TOLA, TOLB

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


PURPOSE

       DTGSJA  computes the generalized singular value decomposi­
       tion (GSVD) of two real upper triangular (or  trapezoidal)
       matrices A and B.  On entry, it is assumed that matrices A
       and B have the following forms, which may be  obtained  by
       the  preprocessing subroutine DGGSVP from a general M-by-N
       matrix A and P-by-N matrix B:

                    N-K-L  K    L
          A =    K ( 0    A12  A13 ) if M-K-L >= 0;
                 L ( 0     0   A23 )
             M-K-L ( 0     0    0  )

                  N-K-L  K    L
          A =  K ( 0    A12  A13 ) if M-K-L < 0;
             M-K ( 0     0   A23 )

                  N-K-L  K    L
          B =  L ( 0     0   B13 )
             P-L ( 0     0    0  )

       where the K-by-K matrix A12 and L-by-L matrix B13 are non­
       singular  upper triangular; A23 is L-by-L upper triangular
       if M-K-L >= 0, otherwise A23 is  (M-K)-by-L  upper  trape­
       zoidal.

       On exit,

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

       where  U,  V and Q are orthogonal matrices, Z' denotes the
       transpose of  Z,  R  is  a  nonsingular  upper  triangular

       of the following structures:

       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 ) K
                   L (  0    0   R22 ) L

       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

                 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.

       R = ( 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  computation of the orthogonal transformation matrices
       U, V or Q is  optional.   These  matrices  may  either  be

       input matrices U1, V1, or Q1.


ARGUMENTS

       JOBU    (input) CHARACTER*1
               = 'U':  U must contain an orthogonal matrix U1  on
               entry, and the product U1*U is returned; = 'I':  U
               is initialized to the unit matrix, and the orthog­
               onal  matrix  U is returned; = 'N':  U is not com­
               puted.

       JOBV    (input) CHARACTER*1
               = 'V':  V must contain an orthogonal matrix V1  on
               entry, and the product V1*V is returned; = 'I':  V
               is initialized to the unit matrix, and the orthog­
               onal  matrix  V is returned; = 'N':  V is not com­
               puted.

       JOBQ    (input) CHARACTER*1
               = 'Q':  Q must contain an orthogonal matrix Q1  on
               entry, and the product Q1*Q is returned; = 'I':  Q
               is initialized to the unit matrix, and the orthog­
               onal  matrix  Q is returned; = 'N':  Q is not com­
               puted.

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

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

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

       K       (input) INTEGER
               L        (input)  INTEGER K and L specify the sub­
               blocks in the input matrices A and B:
               A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-
               L+1:N)  of A and B, whose GSVD is going to be com­
               puted by DTGSJA.  See Further details.

       A       (input/output) DOUBLE PRECISION array, dimension
               (LDA,N)
               On  entry,  the  M-by-N  matrix  A.  On exit, A(N-
               K+1:N,1:MIN(K+L,M)  )  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).

               (LDB,N)
               On entry, the P-by-N matrix B.  On exit, if neces­
               sary, B(M-K+1:L,N+M-K-L+1:N) contains a part of R.
               See Purpose for details.

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

       TOLA    (input) DOUBLE PRECISION
               TOLB    (input) DOUBLE PRECISION TOLA and TOLB are
               the  convergence  criteria  for  the  Jacobi- Kog­
               betliantz iteration procedure. Generally, they are
               the  same  as  used in the preprocessing step, say
               TOLA   =    max(M,N)*norm(A)*MAZHEPS,    TOLB    =
               max(P,N)*norm(B)*MAZHEPS.

       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)
               = diag(C),
               BETA(K+1:K+L)    =  diag(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.   Furthermore,
               if K+L < N, ALPHA(K+L+1:N) = 0 and
               BETA(K+L+1:N)  = 0.

       U       (input/output) DOUBLE PRECISION array, dimension
               (LDU,M)
               On entry, if JOBU = 'U', U must contain  a  matrix
               U1  (usually  the  orthogonal  matrix  returned by
               DGGSVP).  On exit, if JOBU = 'I', U  contains  the
               orthogonal matrix U; if JOBU = 'U', U contains the
               product U1*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       (input/output) DOUBLE PRECISION array, dimension
               (LDV,P)
               On  entry,  if JOBV = 'V', V must contain a matrix
               V1 (usually  the  orthogonal  matrix  returned  by
               DGGSVP).   On  exit, if JOBV = 'I', V contains the
               orthogonal matrix V; if JOBV = 'V', V contains the
               product V1*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.

               (LDQ,N)
               On entry, if JOBQ = 'Q', Q must contain  a  matrix
               Q1  (usually  the  orthogonal  matrix  returned by
               DGGSVP).  On exit, if JOBQ = 'I', Q  contains  the
               orthogonal matrix Q; if JOBQ = 'Q', Q contains the
               product Q1*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
               (2*N)

       NCYCLE  (output) INTEGER
               The number of cycles required for convergence.

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value.
               = 1:  the procedure does not converge after  MAXIT
               cycles.


PARAMETERS

       MAXIT   INTEGER
               MAXIT specifies the total loops that the iterative
               procedure may take. If  after  MAXIT  cycles,  the
               routine fails to converge, we return INFO = 1.

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

               DTGSJA  essentially uses a variant of Kogbetliantz
               algorithm to reduce min(L,M-K)-by-L triangular (or
               trapezoidal)  matrix  A23 and L-by-L matrix B13 to
               the form:

               U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,

               where U1, V1 and Q1 are orthogonal matrix, and  Z'
               is  the  transpose  of  Z.  C1 and S1 are diagonal
               matrices satisfying

               C1**2 + S1**2 = I,

               and R1 is an L-by-L nonsingular  upper  triangular
               matrix.


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