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 )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          JOBQ, JOBU, JOBV
      INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
     $                   NCYCLE, P
      DOUBLE PRECISION   TOLA, TOLB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
     $                   V( LDV, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DTGSJA computes the generalized singular value decomposition (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 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 orthogonal matrices, Z' denotes the 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
* ( 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.
*
*  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 formed explicitly, or they
*  may be postmultiplied into 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
*                  orthogonal matrix U is returned;
*          = 'N':  U is not computed.
*
*  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
*                  orthogonal matrix V is returned;
*          = 'N':  V is not computed.
*
*  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
*                  orthogonal 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 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).
*
*  B       (input/output) DOUBLE PRECISION 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 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.
*
*  Q       (input/output) DOUBLE PRECISION array, dimension (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 illegal value.
*          = 1:  the procedure does not converge after MAXIT cycles.
*
*  Internal 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.
*
*  =====================================================================