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version 1.8, 2010/12/21 13:53:32 version 1.9, 2011/11/21 20:42:57
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   *> \brief \b DLARRF
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLARRF + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrf.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrf.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrf.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
   *                          W, WGAP, WERR,
   *                          SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
   *                          DPLUS, LPLUS, WORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            CLSTRT, CLEND, INFO, N
   *       DOUBLE PRECISION   CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   D( * ), DPLUS( * ), L( * ), LD( * ),
   *      $          LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> Given the initial representation L D L^T and its cluster of close
   *> eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
   *> W( CLEND ), DLARRF finds a new relatively robust representation
   *> L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
   *> eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix (subblock, if the matrix splitted).
   *> \endverbatim
   *>
   *> \param[in] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>          The N diagonal elements of the diagonal matrix D.
   *> \endverbatim
   *>
   *> \param[in] L
   *> \verbatim
   *>          L is DOUBLE PRECISION array, dimension (N-1)
   *>          The (N-1) subdiagonal elements of the unit bidiagonal
   *>          matrix L.
   *> \endverbatim
   *>
   *> \param[in] LD
   *> \verbatim
   *>          LD is DOUBLE PRECISION array, dimension (N-1)
   *>          The (N-1) elements L(i)*D(i).
   *> \endverbatim
   *>
   *> \param[in] CLSTRT
   *> \verbatim
   *>          CLSTRT is INTEGER
   *>          The index of the first eigenvalue in the cluster.
   *> \endverbatim
   *>
   *> \param[in] CLEND
   *> \verbatim
   *>          CLEND is INTEGER
   *>          The index of the last eigenvalue in the cluster.
   *> \endverbatim
   *>
   *> \param[in] W
   *> \verbatim
   *>          W is DOUBLE PRECISION array, dimension
   *>          dimension is >=  (CLEND-CLSTRT+1)
   *>          The eigenvalue APPROXIMATIONS of L D L^T in ascending order.
   *>          W( CLSTRT ) through W( CLEND ) form the cluster of relatively
   *>          close eigenalues.
   *> \endverbatim
   *>
   *> \param[in,out] WGAP
   *> \verbatim
   *>          WGAP is DOUBLE PRECISION array, dimension
   *>          dimension is >=  (CLEND-CLSTRT+1)
   *>          The separation from the right neighbor eigenvalue in W.
   *> \endverbatim
   *>
   *> \param[in] WERR
   *> \verbatim
   *>          WERR is DOUBLE PRECISION array, dimension
   *>          dimension is  >=  (CLEND-CLSTRT+1)
   *>          WERR contain the semiwidth of the uncertainty
   *>          interval of the corresponding eigenvalue APPROXIMATION in W
   *> \endverbatim
   *>
   *> \param[in] SPDIAM
   *> \verbatim
   *>          SPDIAM is DOUBLE PRECISION
   *>          estimate of the spectral diameter obtained from the
   *>          Gerschgorin intervals
   *> \endverbatim
   *>
   *> \param[in] CLGAPL
   *> \verbatim
   *>          CLGAPL is DOUBLE PRECISION
   *> \endverbatim
   *>
   *> \param[in] CLGAPR
   *> \verbatim
   *>          CLGAPR is DOUBLE PRECISION
   *>          absolute gap on each end of the cluster.
   *>          Set by the calling routine to protect against shifts too close
   *>          to eigenvalues outside the cluster.
   *> \endverbatim
   *>
   *> \param[in] PIVMIN
   *> \verbatim
   *>          PIVMIN is DOUBLE PRECISION
   *>          The minimum pivot allowed in the Sturm sequence.
   *> \endverbatim
   *>
   *> \param[out] SIGMA
   *> \verbatim
   *>          SIGMA is DOUBLE PRECISION
   *>          The shift used to form L(+) D(+) L(+)^T.
   *> \endverbatim
   *>
   *> \param[out] DPLUS
   *> \verbatim
   *>          DPLUS is DOUBLE PRECISION array, dimension (N)
   *>          The N diagonal elements of the diagonal matrix D(+).
   *> \endverbatim
   *>
   *> \param[out] LPLUS
   *> \verbatim
   *>          LPLUS is DOUBLE PRECISION array, dimension (N-1)
   *>          The first (N-1) elements of LPLUS contain the subdiagonal
   *>          elements of the unit bidiagonal matrix L(+).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (2*N)
   *>          Workspace.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          Signals processing OK (=0) or failure (=1)
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup auxOTHERauxiliary
   *
   *> \par Contributors:
   *  ==================
   *>
   *> Beresford Parlett, University of California, Berkeley, USA \n
   *> Jim Demmel, University of California, Berkeley, USA \n
   *> Inderjit Dhillon, University of Texas, Austin, USA \n
   *> Osni Marques, LBNL/NERSC, USA \n
   *> Christof Voemel, University of California, Berkeley, USA
   *
   *  =====================================================================
       SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,        SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
      $                   W, WGAP, WERR,       $                   W, WGAP, WERR,
      $                   SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,       $                   SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
      $                   DPLUS, LPLUS, WORK, INFO )       $                   DPLUS, LPLUS, WORK, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2.2) --  *  -- LAPACK auxiliary 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..--
 *     June 2010  *     November 2011
 **  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            CLSTRT, CLEND, INFO, N        INTEGER            CLSTRT, CLEND, INFO, N
       DOUBLE PRECISION   CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM        DOUBLE PRECISION   CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
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      $          LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )       $          LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  Given the initial representation L D L^T and its cluster of close  
 *  eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...  
 *  W( CLEND ), DLARRF finds a new relatively robust representation  
 *  L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the  
 *  eigenvalues of L(+) D(+) L(+)^T is relatively isolated.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix (subblock, if the matrix splitted).  
 *  
 *  D       (input) DOUBLE PRECISION array, dimension (N)  
 *          The N diagonal elements of the diagonal matrix D.  
 *  
 *  L       (input) DOUBLE PRECISION array, dimension (N-1)  
 *          The (N-1) subdiagonal elements of the unit bidiagonal  
 *          matrix L.  
 *  
 *  LD      (input) DOUBLE PRECISION array, dimension (N-1)  
 *          The (N-1) elements L(i)*D(i).  
 *  
 *  CLSTRT  (input) INTEGER  
 *          The index of the first eigenvalue in the cluster.  
 *  
 *  CLEND   (input) INTEGER  
 *          The index of the last eigenvalue in the cluster.  
 *  
 *  W       (input) DOUBLE PRECISION array, dimension  
 *          dimension is >=  (CLEND-CLSTRT+1)  
 *          The eigenvalue APPROXIMATIONS of L D L^T in ascending order.  
 *          W( CLSTRT ) through W( CLEND ) form the cluster of relatively  
 *          close eigenalues.  
 *  
 *  WGAP    (input/output) DOUBLE PRECISION array, dimension  
 *          dimension is >=  (CLEND-CLSTRT+1)  
 *          The separation from the right neighbor eigenvalue in W.  
 *  
 *  WERR    (input) DOUBLE PRECISION array, dimension  
 *          dimension is  >=  (CLEND-CLSTRT+1)  
 *          WERR contain the semiwidth of the uncertainty  
 *          interval of the corresponding eigenvalue APPROXIMATION in W  
 *  
 *  SPDIAM  (input) DOUBLE PRECISION  
 *          estimate of the spectral diameter obtained from the  
 *          Gerschgorin intervals  
 *  
 *  CLGAPL  (input) DOUBLE PRECISION  
 *  
 *  CLGAPR  (input) DOUBLE PRECISION  
 *          absolute gap on each end of the cluster.  
 *          Set by the calling routine to protect against shifts too close  
 *          to eigenvalues outside the cluster.  
 *  
 *  PIVMIN  (input) DOUBLE PRECISION  
 *          The minimum pivot allowed in the Sturm sequence.  
 *  
 *  SIGMA   (output) DOUBLE PRECISION  
 *          The shift used to form L(+) D(+) L(+)^T.  
 *  
 *  DPLUS   (output) DOUBLE PRECISION array, dimension (N)  
 *          The N diagonal elements of the diagonal matrix D(+).  
 *  
 *  LPLUS   (output) DOUBLE PRECISION array, dimension (N-1)  
 *          The first (N-1) elements of LPLUS contain the subdiagonal  
 *          elements of the unit bidiagonal matrix L(+).  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)  
 *          Workspace.  
 *  
 *  INFO    (output) INTEGER  
 *          Signals processing OK (=0) or failure (=1)  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Beresford Parlett, University of California, Berkeley, USA  
 *     Jim Demmel, University of California, Berkeley, USA  
 *     Inderjit Dhillon, University of Texas, Austin, USA  
 *     Osni Marques, LBNL/NERSC, USA  
 *     Christof Voemel, University of California, Berkeley, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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