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version 1.6, 2011/07/22 07:38:05 version 1.9, 2012/07/31 11:06:35
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       SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,  *> \brief \b DGSVJ1
      $                   EPS, SFMIN, TOL, NSWEEP, 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 DGSVJ1 + dependencies 
 *  -- April 2011                                                      --  *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgsvj1.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgsvj1.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgsvj1.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
   *                          EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       DOUBLE PRECISION   EPS, SFMIN, TOL
   *       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
   *       CHARACTER*1        JOBV
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
   *      $                   WORK( LWORK )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGSVJ1 is called from SGESVJ as a pre-processor and that is its main
   *> purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
   *> it targets only particular pivots and it does not check convergence
   *> (stopping criterion). Few tunning parameters (marked by [TP]) are
   *> available for the implementer.
   *>
   *> Further Details
   *> ~~~~~~~~~~~~~~~
   *> DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
   *> the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
   *> off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
   *> block-entries (tiles) of the (1,2) off-diagonal block are marked by the
   *> [x]'s in the following scheme:
   *>
   *>    | *  *  * [x] [x] [x]|
   *>    | *  *  * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
   *>    | *  *  * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
   *>    |[x] [x] [x] *  *  * |
   *>    |[x] [x] [x] *  *  * |
   *>    |[x] [x] [x] *  *  * |
   *>
   *> In terms of the columns of A, the first N1 columns are rotated 'against'
   *> the remaining N-N1 columns, trying to increase the angle between the
   *> corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
   *> tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.
   *> The number of sweeps is given in NSWEEP and the orthogonality threshold
   *> is given in TOL.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBV
   *> \verbatim
   *>          JOBV is CHARACTER*1
   *>          Specifies whether the output from this procedure is used
   *>          to compute the matrix V:
   *>          = 'V': the product of the Jacobi rotations is accumulated
   *>                 by postmulyiplying the N-by-N array V.
   *>                (See the description of V.)
   *>          = 'A': the product of the Jacobi rotations is accumulated
   *>                 by postmulyiplying the MV-by-N array V.
   *>                (See the descriptions of MV and V.)
   *>          = 'N': the Jacobi rotations are not accumulated.
   *> \endverbatim
   *>
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows of the input matrix A.  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] N1
   *> \verbatim
   *>          N1 is INTEGER
   *>          N1 specifies the 2 x 2 block partition, the first N1 columns are
   *>          rotated 'against' the remaining N-N1 columns of A.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, M-by-N matrix A, such that A*diag(D) represents
   *>          the input matrix.
   *>          On exit,
   *>          A_onexit * D_onexit represents the input matrix A*diag(D)
   *>          post-multiplied by a sequence of Jacobi rotations, where the
   *>          rotation threshold and the total number of sweeps are given in
   *>          TOL and NSWEEP, respectively.
   *>          (See the descriptions of N1, D, TOL and NSWEEP.)
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[in,out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>          The array D accumulates the scaling factors from the fast scaled
   *>          Jacobi rotations.
   *>          On entry, A*diag(D) represents the input matrix.
   *>          On exit, A_onexit*diag(D_onexit) represents the input matrix
   *>          post-multiplied by a sequence of Jacobi rotations, where the
   *>          rotation threshold and the total number of sweeps are given in
   *>          TOL and NSWEEP, respectively.
   *>          (See the descriptions of N1, A, TOL and NSWEEP.)
   *> \endverbatim
   *>
   *> \param[in,out] SVA
   *> \verbatim
   *>          SVA is DOUBLE PRECISION array, dimension (N)
   *>          On entry, SVA contains the Euclidean norms of the columns of
   *>          the matrix A*diag(D).
   *>          On exit, SVA contains the Euclidean norms of the columns of
   *>          the matrix onexit*diag(D_onexit).
   *> \endverbatim
   *>
   *> \param[in] MV
   *> \verbatim
   *>          MV is INTEGER
   *>          If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
   *>                           sequence of Jacobi rotations.
   *>          If JOBV = 'N',   then MV is not referenced.
   *> \endverbatim
   *>
   *> \param[in,out] V
   *> \verbatim
   *>          V is DOUBLE PRECISION array, dimension (LDV,N)
   *>          If JOBV .EQ. 'V' then N rows of V are post-multipled by a
   *>                           sequence of Jacobi rotations.
   *>          If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
   *>                           sequence of Jacobi rotations.
   *>          If JOBV = 'N',   then V is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDV
   *> \verbatim
   *>          LDV is INTEGER
   *>          The leading dimension of the array V,  LDV >= 1.
   *>          If JOBV = 'V', LDV .GE. N.
   *>          If JOBV = 'A', LDV .GE. MV.
   *> \endverbatim
   *>
   *> \param[in] EPS
   *> \verbatim
   *>          EPS is DOUBLE PRECISION
   *>          EPS = DLAMCH('Epsilon')
   *> \endverbatim
   *>
   *> \param[in] SFMIN
   *> \verbatim
   *>          SFMIN is DOUBLE PRECISION
   *>          SFMIN = DLAMCH('Safe Minimum')
   *> \endverbatim
   *>
   *> \param[in] TOL
   *> \verbatim
   *>          TOL is DOUBLE PRECISION
   *>          TOL is the threshold for Jacobi rotations. For a pair
   *>          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
   *>          applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
   *> \endverbatim
   *>
   *> \param[in] NSWEEP
   *> \verbatim
   *>          NSWEEP is INTEGER
   *>          NSWEEP is the number of sweeps of Jacobi rotations to be
   *>          performed.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (LWORK)
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          LWORK is the dimension of WORK. LWORK .GE. M.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0 : successful exit.
   *>          < 0 : if INFO = -i, then the i-th argument had an illegal value
   *> \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 Contributors:
   *  ==================
   *>
   *> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
 *  *
   *  =====================================================================
         SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
        $                   EPS, SFMIN, TOL, NSWEEP, 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 ..
       DOUBLE PRECISION   EPS, SFMIN, TOL        DOUBLE PRECISION   EPS, SFMIN, TOL
       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP        INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
Line 27 Line 251
      $                   WORK( LWORK )       $                   WORK( LWORK )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGSVJ1 is called from SGESVJ as a pre-processor and that is its main  
 *  purpose. It applies Jacobi rotations in the same way as SGESVJ does, but  
 *  it targets only particular pivots and it does not check convergence  
 *  (stopping criterion). Few tunning parameters (marked by [TP]) are  
 *  available for the implementer.  
 *  
 *  Further Details  
 *  ~~~~~~~~~~~~~~~  
 *  DGSVJ1 applies few sweeps of Jacobi rotations in the column space of  
 *  the input M-by-N matrix A. The pivot pairs are taken from the (1,2)  
 *  off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The  
 *  block-entries (tiles) of the (1,2) off-diagonal block are marked by the  
 *  [x]'s in the following scheme:  
 *  
 *     | *   *   * [x] [x] [x]|  
 *     | *   *   * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.  
 *     | *   *   * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.  
 *     |[x] [x] [x] *   *   * |  
 *     |[x] [x] [x] *   *   * |  
 *     |[x] [x] [x] *   *   * |  
 *  
 *  In terms of the columns of A, the first N1 columns are rotated 'against'  
 *  the remaining N-N1 columns, trying to increase the angle between the  
 *  corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is  
 *  tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.  
 *  The number of sweeps is given in NSWEEP and the orthogonality threshold  
 *  is given in TOL.  
 *  
 *  Contributors  
 *  ~~~~~~~~~~~~  
 *  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBV    (input) CHARACTER*1  
 *          Specifies whether the output from this procedure is used  
 *          to compute the matrix V:  
 *          = 'V': the product of the Jacobi rotations is accumulated  
 *                 by postmulyiplying the N-by-N array V.  
 *                (See the description of V.)  
 *          = 'A': the product of the Jacobi rotations is accumulated  
 *                 by postmulyiplying the MV-by-N array V.  
 *                (See the descriptions of MV and V.)  
 *          = 'N': the Jacobi rotations are not accumulated.  
 *  
 *  M       (input) INTEGER  
 *          The number of rows of the input matrix A.  M >= 0.  
 *  
 *  N       (input) INTEGER  
 *          The number of columns of the input matrix A.  
 *          M >= N >= 0.  
 *  
 *  N1      (input) INTEGER  
 *          N1 specifies the 2 x 2 block partition, the first N1 columns are  
 *          rotated 'against' the remaining N-N1 columns of A.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, M-by-N matrix A, such that A*diag(D) represents  
 *          the input matrix.  
 *          On exit,  
 *          A_onexit * D_onexit represents the input matrix A*diag(D)  
 *          post-multiplied by a sequence of Jacobi rotations, where the  
 *          rotation threshold and the total number of sweeps are given in  
 *          TOL and NSWEEP, respectively.  
 *          (See the descriptions of N1, D, TOL and NSWEEP.)  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,M).  
 *  
 *  D       (input/workspace/output) DOUBLE PRECISION array, dimension (N)  
 *          The array D accumulates the scaling factors from the fast scaled  
 *          Jacobi rotations.  
 *          On entry, A*diag(D) represents the input matrix.  
 *          On exit, A_onexit*diag(D_onexit) represents the input matrix  
 *          post-multiplied by a sequence of Jacobi rotations, where the  
 *          rotation threshold and the total number of sweeps are given in  
 *          TOL and NSWEEP, respectively.  
 *          (See the descriptions of N1, A, TOL and NSWEEP.)  
 *  
 *  SVA     (input/workspace/output) DOUBLE PRECISION array, dimension (N)  
 *          On entry, SVA contains the Euclidean norms of the columns of  
 *          the matrix A*diag(D).  
 *          On exit, SVA contains the Euclidean norms of the columns of  
 *          the matrix onexit*diag(D_onexit).  
 *  
 *  MV      (input) INTEGER  
 *          If JOBV .EQ. 'A', then MV rows of V are post-multipled by a  
 *                           sequence of Jacobi rotations.  
 *          If JOBV = 'N',   then MV is not referenced.  
 *  
 *  V       (input/output) DOUBLE PRECISION array, dimension (LDV,N)  
 *          If JOBV .EQ. 'V' then N rows of V are post-multipled by a  
 *                           sequence of Jacobi rotations.  
 *          If JOBV .EQ. 'A' then MV rows of V are post-multipled by a  
 *                           sequence of Jacobi rotations.  
 *          If JOBV = 'N',   then V is not referenced.  
 *  
 *  LDV     (input) INTEGER  
 *          The leading dimension of the array V,  LDV >= 1.  
 *          If JOBV = 'V', LDV .GE. N.  
 *          If JOBV = 'A', LDV .GE. MV.  
 *  
 *  EPS     (input) DOUBLE PRECISION  
 *          EPS = DLAMCH('Epsilon')  
 *  
 *  SFMIN   (input) DOUBLE PRECISION  
 *          SFMIN = DLAMCH('Safe Minimum')  
 *  
 *  TOL     (input) DOUBLE PRECISION  
 *          TOL is the threshold for Jacobi rotations. For a pair  
 *          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is  
 *          applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.  
 *  
 *  NSWEEP  (input) INTEGER  
 *          NSWEEP is the number of sweeps of Jacobi rotations to be  
 *          performed.  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK)  
 *  
 *  LWORK   (input) INTEGER  
 *          LWORK is the dimension of WORK. LWORK .GE. M.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0 : successful exit.  
 *          < 0 : if INFO = -i, then the i-th argument had an illegal value  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Local Parameters ..  *     .. Local Parameters ..
       DOUBLE PRECISION   ZERO, HALF, ONE, TWO        DOUBLE PRECISION   ZERO, HALF, ONE
       PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,        PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0 )
      $                   TWO = 2.0D0 )  
 *     ..  *     ..
 *     .. Local Scalars ..  *     .. Local Scalars ..
       DOUBLE PRECISION   AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,        DOUBLE PRECISION   AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
Removed from v.1.6  
changed lines
  Added in v.1.9

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