ZTGSJA - compute the generalized singular value decomposi­
       tion (GSVD) of two complex  upper  triangular  (or  trape­
       zoidal) matrices A and B


SYNOPSIS

       SUBROUTINE ZTGSJA( 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 ALPHA( * ), BETA( * )

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


PURPOSE

       ZTGSJA computes the generalized singular value  decomposi­
       tion  (GSVD)  of  two  complex upper triangular (or trape­
       zoidal) 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  ZGGSVP  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  unitary  matrices,  Z'  denotes  the

       gular matrix, and D1 and  D2  are  ``diagonal''  matrices,
       which are 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 unitary transformation matrices  U,

       explicitly, or  they  may  be  postmultiplied  into  input
       matrices U1, V1, or Q1.


ARGUMENTS

       JOBU    (input) CHARACTER*1
               =  'U':   U  must  contain  a unitary matrix U1 on
               entry, and the product U1*U is returned; = 'I':  U
               is initialized to the unit matrix, and the unitary
               matrix U is returned; = 'N':  U is not computed.

       JOBV    (input) CHARACTER*1
               = 'V':  V must contain  a  unitary  matrix  V1  on
               entry, and the product V1*V is returned; = 'I':  V
               is initialized to the unit matrix, and the unitary
               matrix V is returned; = 'N':  V is not computed.

       JOBQ    (input) CHARACTER*1
               =  'Q':   Q  must  contain  a unitary matrix Q1 on
               entry, and the product Q1*Q is returned; = 'I':  Q
               is initialized to the unit matrix, and the unitary
               matrix Q is returned; = 'N':  Q is not computed.

       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 computed by ZTGSJA.  See Further details.

       A       (input/output) COMPLEX*16 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) COMPLEX*16 array, dimension (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.

               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
               BETA(K+L+1:N)  = 0.

       U       (input/output) COMPLEX*16 array, dimension (LDU,M)
               On entry, if JOBU = 'U', U must contain  a  matrix
               U1   (usually   the  unitary  matrix  returned  by
               ZGGSVP).  On exit, if JOBU = 'I', U  contains  the
               unitary  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) COMPLEX*16 array, dimension (LDV,P)
               On  entry,  if JOBV = 'V', V must contain a matrix
               V1  (usually  the  unitary  matrix   returned   by
               ZGGSVP).   On  exit, if JOBV = 'I', V contains the
               unitary 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) COMPLEX*16 array, dimension (LDQ,N)
               On entry, if JOBQ = 'Q', Q must contain  a  matrix
               Q1   (usually   the  unitary  matrix  returned  by
               ZGGSVP).  On exit, if JOBQ = 'I', Q  contains  the
               unitary  matrix  Q;  if JOBQ = 'Q', Q contains the
               product Q1*Q.  If JOBQ = 'N', Q is not referenced.

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

       WORK    (workspace) COMPLEX*16 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 ===============

               ZTGSJA 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 unitary matrix, and Z' is
               the conjugate transpose of Z.  C1 and S1 are diag­
               onal matrices satisfying

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

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


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