ztgsja(3F)

NAME
     ZTGSJA - compute the generalized singular value decomposition (GSVD) of
     two complex upper triangular (or trapezoidal) 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 decomposition (GSVD) of
     two complex 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 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 nonsingular upper
     triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23
     is (M-K)-by-L upper trapezoidal.

     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 conjugate transpose
     of Z, R is a nonsingular upper triangular 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, V or Q is
     optional.  These matrices may either be formed 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 subblocks 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 necessary, 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- Kogbetliantz 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 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) = 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.

     LDQ     (input) INTEGER
             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 illegal 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 diagonal matrices satisfying

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

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