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version 1.8, 2011/07/22 07:38:12 version 1.9, 2011/11/21 20:43:06
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   *> \brief \b DTGSJA
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DTGSJA + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsja.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsja.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsja.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       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 )
   * 
   *       .. 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( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),
   *>
   *> where U, V and Q are orthogonal matrices.
   *> 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBU
   *> \verbatim
   *>          JOBU is 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.
   *> \endverbatim
   *>
   *> \param[in] JOBV
   *> \verbatim
   *>          JOBV is 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.
   *> \endverbatim
   *>
   *> \param[in] JOBQ
   *> \verbatim
   *>          JOBQ is 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.
   *> \endverbatim
   *>
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows of the matrix A.  M >= 0.
   *> \endverbatim
   *>
   *> \param[in] P
   *> \verbatim
   *>          P is INTEGER
   *>          The number of rows of the matrix B.  P >= 0.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns of the matrices A and B.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] K
   *> \verbatim
   *>          K is INTEGER
   *> \endverbatim
   *>
   *> \param[in] L
   *> \verbatim
   *>          L is 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.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is 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.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A. LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is 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.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,P).
   *> \endverbatim
   *>
   *> \param[in] TOLA
   *> \verbatim
   *>          TOLA is DOUBLE PRECISION
   *> \endverbatim
   *>
   *> \param[in] TOLB
   *> \verbatim
   *>          TOLB is 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.
   *> \endverbatim
   *>
   *> \param[out] ALPHA
   *> \verbatim
   *>          ALPHA is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] BETA
   *> \verbatim
   *>          BETA is 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.
   *> \endverbatim
   *>
   *> \param[in,out] U
   *> \verbatim
   *>          U is 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.
   *> \endverbatim
   *>
   *> \param[in] LDU
   *> \verbatim
   *>          LDU is INTEGER
   *>          The leading dimension of the array U. LDU >= max(1,M) if
   *>          JOBU = 'U'; LDU >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[in,out] V
   *> \verbatim
   *>          V is 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.
   *> \endverbatim
   *>
   *> \param[in] LDV
   *> \verbatim
   *>          LDV is INTEGER
   *>          The leading dimension of the array V. LDV >= max(1,P) if
   *>          JOBV = 'V'; LDV >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[in,out] Q
   *> \verbatim
   *>          Q is 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.
   *> \endverbatim
   *>
   *> \param[in] LDQ
   *> \verbatim
   *>          LDQ is INTEGER
   *>          The leading dimension of the array Q. LDQ >= max(1,N) if
   *>          JOBQ = 'Q'; LDQ >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (2*N)
   *> \endverbatim
   *>
   *> \param[out] NCYCLE
   *> \verbatim
   *>          NCYCLE is INTEGER
   *>          The number of cycles required for convergence.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is 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.
   *> \endverbatim
   *>
   *> \verbatim
   *>  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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  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**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
   *>
   *>  where U1, V1 and Q1 are orthogonal matrix, and Z**T 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.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,        SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
      $                   LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,       $                   LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
      $                   Q, LDQ, WORK, NCYCLE, INFO )       $                   Q, LDQ, WORK, NCYCLE, INFO )
 *  *
 *  -- LAPACK routine (version 3.3.1)                                  --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2009                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBQ, JOBU, JOBV        CHARACTER          JOBQ, JOBU, JOBV
Line 19 Line 395
      $                   V( LDV, * ), WORK( * )       $                   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**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),  
 *  
 *  where U, V and Q are orthogonal matrices.  
 *  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**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,  
 *  
 *  where U1, V1 and Q1 are orthogonal matrix, and Z**T 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.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
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  Added in v.1.9

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