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


SYNOPSIS

       SUBROUTINE STGSJA( 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

           REAL           TOLA, TOLB

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


PURPOSE

       STGSJA  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 SGGSVP 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 STGSJA.  See Further details.

       A       (input/output) REAL 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).

       B       (input/output) REAL array, dimension (LDB,N)
               On  entry,  the  P-by-N  matrix  B.   On  exit, if

               of R.  See Purpose for details.

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

       TOLA    (input) REAL
               TOLB    (input) REAL TOLA and TOLB are the conver­
               gence criteria for the Jacobi- Kogbetliantz itera­
               tion procedure. Generally, they are  the  same  as
               used   in  the  preprocessing  step,  say  TOLA  =
               max(M,N)*norm(A)*MACHEPS,          TOLB          =
               max(P,N)*norm(B)*MACHEPS.

       ALPHA   (output) REAL array, dimension (N)
               BETA     (output)  REAL  array,  dimension  (N) On
               exit, ALPHA and BETA contain the generalized  sin­
               gular 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) REAL array, dimension (LDU,M)
               On entry, if JOBU = 'U', U must contain  a  matrix
               U1  (usually  the  orthogonal  matrix  returned by
               SGGSVP).  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) REAL array, dimension (LDV,P)
               On entry, if JOBV = 'V', V must contain  a  matrix
               V1  (usually  the  orthogonal  matrix  returned by
               SGGSVP).  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.

       Q       (input/output) REAL array, dimension (LDQ,N)
               On entry, if JOBQ = 'Q', Q must contain  a  matrix
               Q1  (usually  the  orthogonal  matrix  returned by
               SGGSVP).  On exit, if JOBQ = 'I', Q  contains  the
               orthogonal matrix Q; if JOBQ = 'Q', Q contains the


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

       WORK    (workspace) REAL 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 ===============

               STGSJA 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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