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version 1.6, 2010/08/13 21:03:51 version 1.20, 2023/08/07 08:38:57
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   *> \brief \b DLARRB provides limited bisection to locate eigenvalues for more accuracy.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DLARRB + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrb.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrb.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrb.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
   *                          RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
   *                          PIVMIN, SPDIAM, TWIST, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
   *       DOUBLE PRECISION   PIVMIN, RTOL1, RTOL2, SPDIAM
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   D( * ), LLD( * ), W( * ),
   *      $                   WERR( * ), WGAP( * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> Given the relatively robust representation(RRR) L D L^T, DLARRB
   *> does "limited" bisection to refine the eigenvalues of L D L^T,
   *> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
   *> guesses for these eigenvalues are input in W, the corresponding estimate
   *> of the error in these guesses and their gaps are input in WERR
   *> and WGAP, respectively. During bisection, intervals
   *> [left, right] are maintained by storing their mid-points and
   *> semi-widths in the arrays W and WERR respectively.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix.
   *> \endverbatim
   *>
   *> \param[in] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>          The N diagonal elements of the diagonal matrix D.
   *> \endverbatim
   *>
   *> \param[in] LLD
   *> \verbatim
   *>          LLD is DOUBLE PRECISION array, dimension (N-1)
   *>          The (N-1) elements L(i)*L(i)*D(i).
   *> \endverbatim
   *>
   *> \param[in] IFIRST
   *> \verbatim
   *>          IFIRST is INTEGER
   *>          The index of the first eigenvalue to be computed.
   *> \endverbatim
   *>
   *> \param[in] ILAST
   *> \verbatim
   *>          ILAST is INTEGER
   *>          The index of the last eigenvalue to be computed.
   *> \endverbatim
   *>
   *> \param[in] RTOL1
   *> \verbatim
   *>          RTOL1 is DOUBLE PRECISION
   *> \endverbatim
   *>
   *> \param[in] RTOL2
   *> \verbatim
   *>          RTOL2 is DOUBLE PRECISION
   *>          Tolerance for the convergence of the bisection intervals.
   *>          An interval [LEFT,RIGHT] has converged if
   *>          RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
   *>          where GAP is the (estimated) distance to the nearest
   *>          eigenvalue.
   *> \endverbatim
   *>
   *> \param[in] OFFSET
   *> \verbatim
   *>          OFFSET is INTEGER
   *>          Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
   *>          through ILAST-OFFSET elements of these arrays are to be used.
   *> \endverbatim
   *>
   *> \param[in,out] W
   *> \verbatim
   *>          W is DOUBLE PRECISION array, dimension (N)
   *>          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
   *>          estimates of the eigenvalues of L D L^T indexed IFIRST through
   *>          ILAST.
   *>          On output, these estimates are refined.
   *> \endverbatim
   *>
   *> \param[in,out] WGAP
   *> \verbatim
   *>          WGAP is DOUBLE PRECISION array, dimension (N-1)
   *>          On input, the (estimated) gaps between consecutive
   *>          eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between
   *>          eigenvalues I and I+1. Note that if IFIRST = ILAST
   *>          then WGAP(IFIRST-OFFSET) must be set to ZERO.
   *>          On output, these gaps are refined.
   *> \endverbatim
   *>
   *> \param[in,out] WERR
   *> \verbatim
   *>          WERR is DOUBLE PRECISION array, dimension (N)
   *>          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
   *>          the errors in the estimates of the corresponding elements in W.
   *>          On output, these errors are refined.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (2*N)
   *>          Workspace.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (2*N)
   *>          Workspace.
   *> \endverbatim
   *>
   *> \param[in] PIVMIN
   *> \verbatim
   *>          PIVMIN is DOUBLE PRECISION
   *>          The minimum pivot in the Sturm sequence.
   *> \endverbatim
   *>
   *> \param[in] SPDIAM
   *> \verbatim
   *>          SPDIAM is DOUBLE PRECISION
   *>          The spectral diameter of the matrix.
   *> \endverbatim
   *>
   *> \param[in] TWIST
   *> \verbatim
   *>          TWIST is INTEGER
   *>          The twist index for the twisted factorization that is used
   *>          for the negcount.
   *>          TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T
   *>          TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T
   *>          TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          Error flag.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup OTHERauxiliary
   *
   *> \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 DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,        SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
      $                   RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,       $                   RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
      $                   PIVMIN, SPDIAM, TWIST, INFO )       $                   PIVMIN, SPDIAM, TWIST, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine --
 *  -- 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 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST        INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
Line 17 Line 208
      $                   WERR( * ), WGAP( * ), WORK( * )       $                   WERR( * ), WGAP( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  Given the relatively robust representation(RRR) L D L^T, DLARRB  
 *  does "limited" bisection to refine the eigenvalues of L D L^T,  
 *  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial  
 *  guesses for these eigenvalues are input in W, the corresponding estimate  
 *  of the error in these guesses and their gaps are input in WERR  
 *  and WGAP, respectively. During bisection, intervals  
 *  [left, right] are maintained by storing their mid-points and  
 *  semi-widths in the arrays W and WERR respectively.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix.  
 *  
 *  D       (input) DOUBLE PRECISION array, dimension (N)  
 *          The N diagonal elements of the diagonal matrix D.  
 *  
 *  LLD     (input) DOUBLE PRECISION array, dimension (N-1)  
 *          The (N-1) elements L(i)*L(i)*D(i).  
 *  
 *  IFIRST  (input) INTEGER  
 *          The index of the first eigenvalue to be computed.  
 *  
 *  ILAST   (input) INTEGER  
 *          The index of the last eigenvalue to be computed.  
 *  
 *  RTOL1   (input) DOUBLE PRECISION  
 *  RTOL2   (input) DOUBLE PRECISION  
 *          Tolerance for the convergence of the bisection intervals.  
 *          An interval [LEFT,RIGHT] has converged if  
 *          RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )  
 *          where GAP is the (estimated) distance to the nearest  
 *          eigenvalue.  
 *  
 *  OFFSET  (input) INTEGER  
 *          Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET  
 *          through ILAST-OFFSET elements of these arrays are to be used.  
 *  
 *  W       (input/output) DOUBLE PRECISION array, dimension (N)  
 *          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are  
 *          estimates of the eigenvalues of L D L^T indexed IFIRST throug  
 *          ILAST.  
 *          On output, these estimates are refined.  
 *  
 *  WGAP    (input/output) DOUBLE PRECISION array, dimension (N-1)  
 *          On input, the (estimated) gaps between consecutive  
 *          eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between  
 *          eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST  
 *          then WGAP(IFIRST-OFFSET) must be set to ZERO.  
 *          On output, these gaps are refined.  
 *  
 *  WERR    (input/output) DOUBLE PRECISION array, dimension (N)  
 *          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are  
 *          the errors in the estimates of the corresponding elements in W.  
 *          On output, these errors are refined.  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)  
 *          Workspace.  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (2*N)  
 *          Workspace.  
 *  
 *  PIVMIN  (input) DOUBLE PRECISION  
 *          The minimum pivot in the Sturm sequence.  
 *  
 *  SPDIAM  (input) DOUBLE PRECISION  
 *          The spectral diameter of the matrix.  
 *  
 *  TWIST   (input) INTEGER  
 *          The twist index for the twisted factorization that is used  
 *          for the negcount.  
 *          TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T  
 *          TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T  
 *          TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)  
 *  
 *  INFO    (output) INTEGER  
 *          Error flag.  
 *  
 *  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 ..
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 *  *
       INFO = 0        INFO = 0
 *  *
   *     Quick return if possible
   *
         IF( N.LE.0 ) THEN
            RETURN
         END IF
   *
       MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /        MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
      $           LOG( TWO ) ) + 2       $           LOG( TWO ) ) + 2
       MNWDTH = TWO * PIVMIN        MNWDTH = TWO * PIVMIN
Removed from v.1.6  
changed lines
  Added in v.1.20

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