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       SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,  *> \brief \b DGESVJ
      $                   LDV, WORK, LWORK, INFO )  *
   *  =========== DOCUMENTATION ===========
 *  *
 *  -- LAPACK routine (version 3.3.1)                                  --  * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
 *  *
 *  -- Contributed by Zlatko Drmac of the University of Zagreb and     --  *> \htmlonly
 *  -- Kresimir Veselic of the Fernuniversitaet Hagen                  --  *> Download DGESVJ + dependencies 
 *  -- April 2011                                                      --  *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvj.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvj.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvj.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
   *                          LDV, WORK, LWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N
   *       CHARACTER*1        JOBA, JOBU, JOBV
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), SVA( N ), V( LDV, * ),
   *      $                   WORK( LWORK )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGESVJ computes the singular value decomposition (SVD) of a real
   *> M-by-N matrix A, where M >= N. The SVD of A is written as
   *>                                    [++]   [xx]   [x0]   [xx]
   *>              A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]
   *>                                    [++]   [xx]
   *> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
   *> matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
   *> of SIGMA are the singular values of A. The columns of U and V are the
   *> left and the right singular vectors of A, respectively.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBA
   *> \verbatim
   *>          JOBA is CHARACTER* 1
   *>          Specifies the structure of A.
   *>          = 'L': The input matrix A is lower triangular;
   *>          = 'U': The input matrix A is upper triangular;
   *>          = 'G': The input matrix A is general M-by-N matrix, M >= N.
   *> \endverbatim
   *>
   *> \param[in] JOBU
   *> \verbatim
   *>          JOBU is CHARACTER*1
   *>          Specifies whether to compute the left singular vectors
   *>          (columns of U):
   *>          = 'U': The left singular vectors corresponding to the nonzero
   *>                 singular values are computed and returned in the leading
   *>                 columns of A. See more details in the description of A.
   *>                 The default numerical orthogonality threshold is set to
   *>                 approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
   *>          = 'C': Analogous to JOBU='U', except that user can control the
   *>                 level of numerical orthogonality of the computed left
   *>                 singular vectors. TOL can be set to TOL = CTOL*EPS, where
   *>                 CTOL is given on input in the array WORK.
   *>                 No CTOL smaller than ONE is allowed. CTOL greater
   *>                 than 1 / EPS is meaningless. The option 'C'
   *>                 can be used if M*EPS is satisfactory orthogonality
   *>                 of the computed left singular vectors, so CTOL=M could
   *>                 save few sweeps of Jacobi rotations.
   *>                 See the descriptions of A and WORK(1).
   *>          = 'N': The matrix U is not computed. However, see the
   *>                 description of A.
   *> \endverbatim
   *>
   *> \param[in] JOBV
   *> \verbatim
   *>          JOBV is CHARACTER*1
   *>          Specifies whether to compute the right singular vectors, that
   *>          is, the matrix V:
   *>          = 'V' : the matrix V is computed and returned in the array V
   *>          = 'A' : the Jacobi rotations are applied to the MV-by-N
   *>                  array V. In other words, the right singular vector
   *>                  matrix V is not computed explicitly, instead it is
   *>                  applied to an MV-by-N matrix initially stored in the
   *>                  first MV rows of V.
   *>          = 'N' : the matrix V is not computed and the array V is not
   *>                  referenced
   *> \endverbatim
   *>
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.  
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns of the input matrix A.
   *>          M >= N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, the M-by-N matrix A.
   *>          On exit :
   *>          If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C' :
   *>                 If INFO .EQ. 0 :
   *>                 RANKA orthonormal columns of U are returned in the
   *>                 leading RANKA columns of the array A. Here RANKA <= N
   *>                 is the number of computed singular values of A that are
   *>                 above the underflow threshold DLAMCH('S'). The singular
   *>                 vectors corresponding to underflowed or zero singular
   *>                 values are not computed. The value of RANKA is returned
   *>                 in the array WORK as RANKA=NINT(WORK(2)). Also see the
   *>                 descriptions of SVA and WORK. The computed columns of U
   *>                 are mutually numerically orthogonal up to approximately
   *>                 TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
   *>                 see the description of JOBU.
   *>                 If INFO .GT. 0 :
   *>                 the procedure DGESVJ did not converge in the given number
   *>                 of iterations (sweeps). In that case, the computed
   *>                 columns of U may not be orthogonal up to TOL. The output
   *>                 U (stored in A), SIGMA (given by the computed singular
   *>                 values in SVA(1:N)) and V is still a decomposition of the
   *>                 input matrix A in the sense that the residual
   *>                 ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
   *>
   *>          If JOBU .EQ. 'N' :
   *>                 If INFO .EQ. 0 :
   *>                 Note that the left singular vectors are 'for free' in the
   *>                 one-sided Jacobi SVD algorithm. However, if only the
   *>                 singular values are needed, the level of numerical
   *>                 orthogonality of U is not an issue and iterations are
   *>                 stopped when the columns of the iterated matrix are
   *>                 numerically orthogonal up to approximately M*EPS. Thus,
   *>                 on exit, A contains the columns of U scaled with the
   *>                 corresponding singular values.
   *>                 If INFO .GT. 0 :
   *>                 the procedure DGESVJ did not converge in the given number
   *>                 of iterations (sweeps).
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[out] SVA
   *> \verbatim
   *>          SVA is DOUBLE PRECISION array, dimension (N)
   *>          On exit :
   *>          If INFO .EQ. 0 :
   *>          depending on the value SCALE = WORK(1), we have:
   *>                 If SCALE .EQ. ONE :
   *>                 SVA(1:N) contains the computed singular values of A.
   *>                 During the computation SVA contains the Euclidean column
   *>                 norms of the iterated matrices in the array A.
   *>                 If SCALE .NE. ONE :
   *>                 The singular values of A are SCALE*SVA(1:N), and this
   *>                 factored representation is due to the fact that some of the
   *>                 singular values of A might underflow or overflow.
   *>          If INFO .GT. 0 :
   *>          the procedure DGESVJ did not converge in the given number of
   *>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
   *> \endverbatim
   *>
   *> \param[in] MV
   *> \verbatim
   *>          MV is INTEGER
   *>          If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ
   *>          is applied to the first MV rows of V. See the description of JOBV.
   *> \endverbatim
   *>
   *> \param[in,out] V
   *> \verbatim
   *>          V is DOUBLE PRECISION array, dimension (LDV,N)
   *>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
   *>                         the right singular vectors;
   *>          If JOBV = 'A', then V contains the product of the computed right
   *>                         singular vector matrix and the initial matrix in
   *>                         the array V.
   *>          If JOBV = 'N', then V is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDV
   *> \verbatim
   *>          LDV is INTEGER
   *>          The leading dimension of the array V, LDV .GE. 1.
   *>          If JOBV .EQ. 'V', then LDV .GE. max(1,N).
   *>          If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .
   *> \endverbatim
   *>
   *> \param[in,out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension max(4,M+N).
   *>          On entry :
   *>          If JOBU .EQ. 'C' :
   *>          WORK(1) = CTOL, where CTOL defines the threshold for convergence.
   *>                    The process stops if all columns of A are mutually
   *>                    orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
   *>                    It is required that CTOL >= ONE, i.e. it is not
   *>                    allowed to force the routine to obtain orthogonality
   *>                    below EPS.
   *>          On exit :
   *>          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
   *>                    are the computed singular values of A.
   *>                    (See description of SVA().)
   *>          WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
   *>                    singular values.
   *>          WORK(3) = NINT(WORK(3)) is the number of the computed singular
   *>                    values that are larger than the underflow threshold.
   *>          WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
   *>                    rotations needed for numerical convergence.
   *>          WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
   *>                    This is useful information in cases when DGESVJ did
   *>                    not converge, as it can be used to estimate whether
   *>                    the output is stil useful and for post festum analysis.
   *>          WORK(6) = the largest absolute value over all sines of the
   *>                    Jacobi rotation angles in the last sweep. It can be
   *>                    useful for a post festum analysis.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          length of WORK, WORK >= MAX(6,M+N)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0 : successful exit.
   *>          < 0 : if INFO = -i, then the i-th argument had an illegal value
   *>          > 0 : DGESVJ did not converge in the maximal allowed number (30)
   *>                of sweeps. The output may still be useful. See the
   *>                description of WORK.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleGEcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
   *>  rotations. The rotations are implemented as fast scaled rotations of
   *>  Anda and Park [1]. In the case of underflow of the Jacobi angle, a
   *>  modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
   *>  column interchanges of de Rijk [2]. The relative accuracy of the computed
   *>  singular values and the accuracy of the computed singular vectors (in
   *>  angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
   *>  The condition number that determines the accuracy in the full rank case
   *>  is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
   *>  spectral condition number. The best performance of this Jacobi SVD
   *>  procedure is achieved if used in an  accelerated version of Drmac and
   *>  Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
   *>  Some tunning parameters (marked with [TP]) are available for the
   *>  implementer.
   *>  The computational range for the nonzero singular values is the  machine
   *>  number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
   *>  denormalized singular values can be computed with the corresponding
   *>  gradual loss of accurate digits.
   *> \endverbatim
   *
   *> \par Contributors:
   *  ==================
   *>
   *> \verbatim
   *>
   *>  ============
   *>
   *>  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
   *> \endverbatim
   *
   *> \par References:
   *  ================
   *>
   *> \verbatim
   *>
   *> [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
   *>     SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
   *> [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
   *>     singular value decomposition on a vector computer.
   *>     SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
   *> [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
   *> [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
   *>     value computation in floating point arithmetic.
   *>     SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
   *> [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
   *>     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
   *>     LAPACK Working note 169.
   *> [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
   *>     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
   *>     LAPACK Working note 170.
   *> [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
   *>     QSVD, (H,K)-SVD computations.
   *>     Department of Mathematics, University of Zagreb, 2008.
   *> \endverbatim
   *
   *>  \par Bugs, examples and comments:
   *   =================================
   *>
   *> \verbatim
   *>  ===========================
   *>  Please report all bugs and send interesting test examples and comments to
   *>  drmac@math.hr. Thank you.
   *> \endverbatim
   *>
   *  =====================================================================
         SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
        $                   LDV, WORK, LWORK, INFO )
 *  *
   *  -- 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..--
   *     November 2011
 *  *
 * This routine is also part of SIGMA (version 1.23, October 23. 2008.)  
 * SIGMA is a library of algorithms for highly accurate algorithms for  
 * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the  
 * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.  
 *  
       IMPLICIT           NONE  
 *     ..  
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N        INTEGER            INFO, LDA, LDV, LWORK, M, MV, N
       CHARACTER*1        JOBA, JOBU, JOBV        CHARACTER*1        JOBA, JOBU, JOBV
Line 26 Line 349
      $                   WORK( LWORK )       $                   WORK( LWORK )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGESVJ computes the singular value decomposition (SVD) of a real  
 *  M-by-N matrix A, where M >= N. The SVD of A is written as  
 *                                     [++]   [xx]   [x0]   [xx]  
 *               A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]  
 *                                     [++]   [xx]  
 *  where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal  
 *  matrix, and V is an N-by-N orthogonal matrix. The diagonal elements  
 *  of SIGMA are the singular values of A. The columns of U and V are the  
 *  left and the right singular vectors of A, respectively.  
 *  
 *  Further Details  
 *  ~~~~~~~~~~~~~~~  
 *  The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane  
 *  rotations. The rotations are implemented as fast scaled rotations of  
 *  Anda and Park [1]. In the case of underflow of the Jacobi angle, a  
 *  modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses  
 *  column interchanges of de Rijk [2]. The relative accuracy of the computed  
 *  singular values and the accuracy of the computed singular vectors (in  
 *  angle metric) is as guaranteed by the theory of Demmel and Veselic [3].  
 *  The condition number that determines the accuracy in the full rank case  
 *  is essentially min_{D=diag} kappa(A*D), where kappa(.) is the  
 *  spectral condition number. The best performance of this Jacobi SVD  
 *  procedure is achieved if used in an  accelerated version of Drmac and  
 *  Veselic [5,6], and it is the kernel routine in the SIGMA library [7].  
 *  Some tunning parameters (marked with [TP]) are available for the  
 *  implementer.  
 *  The computational range for the nonzero singular values is the  machine  
 *  number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even  
 *  denormalized singular values can be computed with the corresponding  
 *  gradual loss of accurate digits.  
 *  
 *  Contributors  
 *  ~~~~~~~~~~~~  
 *  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)  
 *  
 *  References  
 *  ~~~~~~~~~~  
 * [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.  
 *     SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.  
 * [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the  
 *     singular value decomposition on a vector computer.  
 *     SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.  
 * [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.  
 * [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular  
 *     value computation in floating point arithmetic.  
 *     SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.  
 * [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.  
 *     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.  
 *     LAPACK Working note 169.  
 * [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.  
 *     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.  
 *     LAPACK Working note 170.  
 * [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,  
 *     QSVD, (H,K)-SVD computations.  
 *     Department of Mathematics, University of Zagreb, 2008.  
 *  
 *  Bugs, Examples and Comments  
 *  ~~~~~~~~~~~~~~~~~~~~~~~~~~~  
 *  Please report all bugs and send interesting test examples and comments to  
 *  drmac@math.hr. Thank you.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBA    (input) CHARACTER* 1  
 *          Specifies the structure of A.  
 *          = 'L': The input matrix A is lower triangular;  
 *          = 'U': The input matrix A is upper triangular;  
 *          = 'G': The input matrix A is general M-by-N matrix, M >= N.  
 *  
 *  JOBU    (input) CHARACTER*1  
 *          Specifies whether to compute the left singular vectors  
 *          (columns of U):  
 *          = 'U': The left singular vectors corresponding to the nonzero  
 *                 singular values are computed and returned in the leading  
 *                 columns of A. See more details in the description of A.  
 *                 The default numerical orthogonality threshold is set to  
 *                 approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').  
 *          = 'C': Analogous to JOBU='U', except that user can control the  
 *                 level of numerical orthogonality of the computed left  
 *                 singular vectors. TOL can be set to TOL = CTOL*EPS, where  
 *                 CTOL is given on input in the array WORK.  
 *                 No CTOL smaller than ONE is allowed. CTOL greater  
 *                 than 1 / EPS is meaningless. The option 'C'  
 *                 can be used if M*EPS is satisfactory orthogonality  
 *                 of the computed left singular vectors, so CTOL=M could  
 *                 save few sweeps of Jacobi rotations.  
 *                 See the descriptions of A and WORK(1).  
 *          = 'N': The matrix U is not computed. However, see the  
 *                 description of A.  
 *  
 *  JOBV    (input) CHARACTER*1  
 *          Specifies whether to compute the right singular vectors, that  
 *          is, the matrix V:  
 *          = 'V' : the matrix V is computed and returned in the array V  
 *          = 'A' : the Jacobi rotations are applied to the MV-by-N  
 *                  array V. In other words, the right singular vector  
 *                  matrix V is not computed explicitly, instead it is  
 *                  applied to an MV-by-N matrix initially stored in the  
 *                  first MV rows of V.  
 *          = 'N' : the matrix V is not computed and the array V is not  
 *                  referenced  
 *  
 *  M       (input) INTEGER  
 *          The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.    
 *  
 *  N       (input) INTEGER  
 *          The number of columns of the input matrix A.  
 *          M >= N >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, the M-by-N matrix A.  
 *          On exit :  
 *          If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C' :  
 *                 If INFO .EQ. 0 :  
 *                 RANKA orthonormal columns of U are returned in the  
 *                 leading RANKA columns of the array A. Here RANKA <= N  
 *                 is the number of computed singular values of A that are  
 *                 above the underflow threshold DLAMCH('S'). The singular  
 *                 vectors corresponding to underflowed or zero singular  
 *                 values are not computed. The value of RANKA is returned  
 *                 in the array WORK as RANKA=NINT(WORK(2)). Also see the  
 *                 descriptions of SVA and WORK. The computed columns of U  
 *                 are mutually numerically orthogonal up to approximately  
 *                 TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),  
 *                 see the description of JOBU.  
 *                 If INFO .GT. 0 :  
 *                 the procedure DGESVJ did not converge in the given number  
 *                 of iterations (sweeps). In that case, the computed  
 *                 columns of U may not be orthogonal up to TOL. The output  
 *                 U (stored in A), SIGMA (given by the computed singular  
 *                 values in SVA(1:N)) and V is still a decomposition of the  
 *                 input matrix A in the sense that the residual  
 *                 ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.  
 *  
 *          If JOBU .EQ. 'N' :  
 *                 If INFO .EQ. 0 :  
 *                 Note that the left singular vectors are 'for free' in the  
 *                 one-sided Jacobi SVD algorithm. However, if only the  
 *                 singular values are needed, the level of numerical  
 *                 orthogonality of U is not an issue and iterations are  
 *                 stopped when the columns of the iterated matrix are  
 *                 numerically orthogonal up to approximately M*EPS. Thus,  
 *                 on exit, A contains the columns of U scaled with the  
 *                 corresponding singular values.  
 *                 If INFO .GT. 0 :  
 *                 the procedure DGESVJ did not converge in the given number  
 *                 of iterations (sweeps).  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,M).  
 *  
 *  SVA     (workspace/output) DOUBLE PRECISION array, dimension (N)  
 *          On exit :  
 *          If INFO .EQ. 0 :  
 *          depending on the value SCALE = WORK(1), we have:  
 *                 If SCALE .EQ. ONE :  
 *                 SVA(1:N) contains the computed singular values of A.  
 *                 During the computation SVA contains the Euclidean column  
 *                 norms of the iterated matrices in the array A.  
 *                 If SCALE .NE. ONE :  
 *                 The singular values of A are SCALE*SVA(1:N), and this  
 *                 factored representation is due to the fact that some of the  
 *                 singular values of A might underflow or overflow.  
 *          If INFO .GT. 0 :  
 *          the procedure DGESVJ did not converge in the given number of  
 *          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.  
 *  
 *  MV      (input) INTEGER  
 *          If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ  
 *          is applied to the first MV rows of V. See the description of JOBV.  
 *  
 *  V       (input/output) DOUBLE PRECISION array, dimension (LDV,N)  
 *          If JOBV = 'V', then V contains on exit the N-by-N matrix of  
 *                         the right singular vectors;  
 *          If JOBV = 'A', then V contains the product of the computed right  
 *                         singular vector matrix and the initial matrix in  
 *                         the array V.  
 *          If JOBV = 'N', then V is not referenced.  
 *  
 *  LDV     (input) INTEGER  
 *          The leading dimension of the array V, LDV .GE. 1.  
 *          If JOBV .EQ. 'V', then LDV .GE. max(1,N).  
 *          If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .  
 *  
 *  WORK    (input/workspace/output) DOUBLE PRECISION array, dimension max(4,M+N).  
 *          On entry :  
 *          If JOBU .EQ. 'C' :  
 *          WORK(1) = CTOL, where CTOL defines the threshold for convergence.  
 *                    The process stops if all columns of A are mutually  
 *                    orthogonal up to CTOL*EPS, EPS=DLAMCH('E').  
 *                    It is required that CTOL >= ONE, i.e. it is not  
 *                    allowed to force the routine to obtain orthogonality  
 *                    below EPS.  
 *          On exit :  
 *          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)  
 *                    are the computed singular values of A.  
 *                    (See description of SVA().)  
 *          WORK(2) = NINT(WORK(2)) is the number of the computed nonzero  
 *                    singular values.  
 *          WORK(3) = NINT(WORK(3)) is the number of the computed singular  
 *                    values that are larger than the underflow threshold.  
 *          WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi  
 *                    rotations needed for numerical convergence.  
 *          WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.  
 *                    This is useful information in cases when DGESVJ did  
 *                    not converge, as it can be used to estimate whether  
 *                    the output is stil useful and for post festum analysis.  
 *          WORK(6) = the largest absolute value over all sines of the  
 *                    Jacobi rotation angles in the last sweep. It can be  
 *                    useful for a post festum analysis.  
 *  
 *  LWORK   (input) INTEGER  
 *          length of WORK, WORK >= MAX(6,M+N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0 : successful exit.  
 *          < 0 : if INFO = -i, then the i-th argument had an illegal value  
 *          > 0 : DGESVJ did not converge in the maximal allowed number (30)  
 *                of sweeps. The output may still be useful. See the  
 *                description of WORK.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Local Parameters ..  *     .. Local Parameters ..
Removed from v.1.6  
changed lines
  Added in v.1.7

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