Add graph references
This commit is contained in:
Abdelrahman Said
+439
@@ -0,0 +1,439 @@
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/* -- translated by f2c (version 20240504).
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You must link the resulting object file with libf2c:
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on Microsoft Windows system, link with libf2c.lib;
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on Linux or Unix systems, link with .../path/to/libf2c.a -lm
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or, if you install libf2c.a in a standard place, with -lf2c -lm
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-- in that order, at the end of the command line, as in
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cc *.o -lf2c -lm
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Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
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http://www.netlib.org/f2c/libf2c.zip
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*/
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#include "f2c.h"
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/* > \brief \b DLARRB provides limited bisection to locate eigenvalues for more accuracy.
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=========== DOCUMENTATION ===========
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Online html documentation available at
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http://www.netlib.org/lapack/explore-html/
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> \htmlonly
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> Download DLARRB + dependencies
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> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrb.
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f">
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> [TGZ]</a>
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> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrb.
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f">
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> [ZIP]</a>
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> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrb.
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f">
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> [TXT]</a>
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> \endhtmlonly
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Definition:
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===========
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SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
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RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
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PIVMIN, SPDIAM, TWIST, INFO )
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INTEGER IFIRST, ILAST, INFO, N, OFFSET, TWIST
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DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPDIAM
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INTEGER IWORK( * )
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DOUBLE PRECISION D( * ), LLD( * ), W( * ),
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$ WERR( * ), WGAP( * ), WORK( * )
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> \par Purpose:
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=============
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>
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> \verbatim
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>
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> Given the relatively robust representation(RRR) L D L^T, DLARRB
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> does "limited" bisection to refine the eigenvalues of L D L^T,
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> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
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> guesses for these eigenvalues are input in W, the corresponding estimate
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> of the error in these guesses and their gaps are input in WERR
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> and WGAP, respectively. During bisection, intervals
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> [left, right] are maintained by storing their mid-points and
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> semi-widths in the arrays W and WERR respectively.
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> \endverbatim
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Arguments:
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==========
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> \param[in] N
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> \verbatim
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> N is INTEGER
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> The order of the matrix.
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> \endverbatim
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>
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> \param[in] D
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> \verbatim
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> D is DOUBLE PRECISION array, dimension (N)
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> The N diagonal elements of the diagonal matrix D.
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> \endverbatim
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>
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> \param[in] LLD
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> \verbatim
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> LLD is DOUBLE PRECISION array, dimension (N-1)
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> The (N-1) elements L(i)*L(i)*D(i).
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> \endverbatim
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>
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> \param[in] IFIRST
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> \verbatim
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> IFIRST is INTEGER
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> The index of the first eigenvalue to be computed.
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> \endverbatim
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>
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> \param[in] ILAST
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> \verbatim
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> ILAST is INTEGER
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> The index of the last eigenvalue to be computed.
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> \endverbatim
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>
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> \param[in] RTOL1
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> \verbatim
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> RTOL1 is DOUBLE PRECISION
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> \endverbatim
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>
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> \param[in] RTOL2
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> \verbatim
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> RTOL2 is DOUBLE PRECISION
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> Tolerance for the convergence of the bisection intervals.
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> An interval [LEFT,RIGHT] has converged if
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> RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
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> where GAP is the (estimated) distance to the nearest
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> eigenvalue.
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> \endverbatim
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>
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> \param[in] OFFSET
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> \verbatim
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> OFFSET is INTEGER
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> Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
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> through ILAST-OFFSET elements of these arrays are to be used.
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> \endverbatim
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>
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> \param[in,out] W
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> \verbatim
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> W is DOUBLE PRECISION array, dimension (N)
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> On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
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> estimates of the eigenvalues of L D L^T indexed IFIRST throug
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> ILAST.
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> On output, these estimates are refined.
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> \endverbatim
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>
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> \param[in,out] WGAP
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> \verbatim
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> WGAP is DOUBLE PRECISION array, dimension (N-1)
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> On input, the (estimated) gaps between consecutive
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> eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between
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> eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST
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> then WGAP(IFIRST-OFFSET) must be set to ZERO.
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> On output, these gaps are refined.
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> \endverbatim
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>
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> \param[in,out] WERR
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> \verbatim
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> WERR is DOUBLE PRECISION array, dimension (N)
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> On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
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> the errors in the estimates of the corresponding elements in W.
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> On output, these errors are refined.
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> \endverbatim
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>
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> \param[out] WORK
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> \verbatim
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> WORK is DOUBLE PRECISION array, dimension (2*N)
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> Workspace.
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> \endverbatim
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>
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> \param[out] IWORK
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> \verbatim
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> IWORK is INTEGER array, dimension (2*N)
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> Workspace.
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> \endverbatim
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>
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> \param[in] PIVMIN
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> \verbatim
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> PIVMIN is DOUBLE PRECISION
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> The minimum pivot in the Sturm sequence.
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> \endverbatim
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>
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> \param[in] SPDIAM
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> \verbatim
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> SPDIAM is DOUBLE PRECISION
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> The spectral diameter of the matrix.
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> \endverbatim
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>
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> \param[in] TWIST
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> \verbatim
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> TWIST is INTEGER
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> The twist index for the twisted factorization that is used
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> for the negcount.
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> TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T
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> TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T
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> TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)
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> \endverbatim
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>
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> \param[out] INFO
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> \verbatim
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> INFO is INTEGER
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> Error flag.
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> \endverbatim
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Authors:
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========
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> \author Univ. of Tennessee
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> \author Univ. of California Berkeley
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> \author Univ. of Colorado Denver
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> \author NAG Ltd.
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> \date September 2012
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> \ingroup auxOTHERauxiliary
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> \par Contributors:
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==================
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>
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> Beresford Parlett, University of California, Berkeley, USA \n
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> Jim Demmel, University of California, Berkeley, USA \n
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> Inderjit Dhillon, University of Texas, Austin, USA \n
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> Osni Marques, LBNL/NERSC, USA \n
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> Christof Voemel, University of California, Berkeley, USA
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=====================================================================
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Subroutine */ int igraphdlarrb_(integer *n, doublereal *d__, doublereal *lld,
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integer *ifirst, integer *ilast, doublereal *rtol1, doublereal *rtol2,
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integer *offset, doublereal *w, doublereal *wgap, doublereal *werr,
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doublereal *work, integer *iwork, doublereal *pivmin, doublereal *
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spdiam, integer *twist, integer *info)
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{
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/* System generated locals */
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integer i__1;
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doublereal d__1, d__2;
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/* Builtin functions */
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double log(doublereal);
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/* Local variables */
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integer i__, k, r__, i1, ii, ip;
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doublereal gap, mid, tmp, back, lgap, rgap, left;
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integer iter, nint, prev, next;
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doublereal cvrgd, right, width;
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extern integer igraphdlaneg_(integer *, doublereal *, doublereal *, doublereal *
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, doublereal *, integer *);
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integer negcnt;
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doublereal mnwdth;
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integer olnint, maxitr;
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/* -- LAPACK auxiliary routine (version 3.4.2) --
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-- LAPACK is a software package provided by Univ. of Tennessee, --
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-- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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September 2012
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=====================================================================
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Parameter adjustments */
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--iwork;
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--work;
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--werr;
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--wgap;
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--w;
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--lld;
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--d__;
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/* Function Body */
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*info = 0;
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maxitr = (integer) ((log(*spdiam + *pivmin) - log(*pivmin)) / log(2.)) +
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2;
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mnwdth = *pivmin * 2.;
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r__ = *twist;
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if (r__ < 1 || r__ > *n) {
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r__ = *n;
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}
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/* Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
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The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
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Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
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for an unconverged interval is set to the index of the next unconverged
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interval, and is -1 or 0 for a converged interval. Thus a linked
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list of unconverged intervals is set up. */
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i1 = *ifirst;
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/* The number of unconverged intervals */
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nint = 0;
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/* The last unconverged interval found */
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prev = 0;
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rgap = wgap[i1 - *offset];
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i__1 = *ilast;
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for (i__ = i1; i__ <= i__1; ++i__) {
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k = i__ << 1;
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ii = i__ - *offset;
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left = w[ii] - werr[ii];
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right = w[ii] + werr[ii];
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lgap = rgap;
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rgap = wgap[ii];
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gap = min(lgap,rgap);
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/* Make sure that [LEFT,RIGHT] contains the desired eigenvalue
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Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT
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Do while( NEGCNT(LEFT).GT.I-1 ) */
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back = werr[ii];
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L20:
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negcnt = igraphdlaneg_(n, &d__[1], &lld[1], &left, pivmin, &r__);
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if (negcnt > i__ - 1) {
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left -= back;
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back *= 2.;
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goto L20;
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}
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/* Do while( NEGCNT(RIGHT).LT.I )
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Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT */
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back = werr[ii];
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L50:
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negcnt = igraphdlaneg_(n, &d__[1], &lld[1], &right, pivmin, &r__);
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if (negcnt < i__) {
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right += back;
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back *= 2.;
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goto L50;
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}
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width = (d__1 = left - right, abs(d__1)) * .5;
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/* Computing MAX */
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d__1 = abs(left), d__2 = abs(right);
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tmp = max(d__1,d__2);
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/* Computing MAX */
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d__1 = *rtol1 * gap, d__2 = *rtol2 * tmp;
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cvrgd = max(d__1,d__2);
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if (width <= cvrgd || width <= mnwdth) {
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/* This interval has already converged and does not need refinement.
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(Note that the gaps might change through refining the
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eigenvalues, however, they can only get bigger.)
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Remove it from the list. */
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iwork[k - 1] = -1;
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/* Make sure that I1 always points to the first unconverged interval */
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if (i__ == i1 && i__ < *ilast) {
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i1 = i__ + 1;
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}
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if (prev >= i1 && i__ <= *ilast) {
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iwork[(prev << 1) - 1] = i__ + 1;
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}
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} else {
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/* unconverged interval found */
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prev = i__;
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++nint;
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iwork[k - 1] = i__ + 1;
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iwork[k] = negcnt;
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}
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work[k - 1] = left;
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work[k] = right;
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/* L75: */
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}
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/* Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
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and while (ITER.LT.MAXITR) */
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iter = 0;
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L80:
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prev = i1 - 1;
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i__ = i1;
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olnint = nint;
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i__1 = olnint;
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for (ip = 1; ip <= i__1; ++ip) {
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k = i__ << 1;
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ii = i__ - *offset;
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rgap = wgap[ii];
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lgap = rgap;
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if (ii > 1) {
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lgap = wgap[ii - 1];
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}
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gap = min(lgap,rgap);
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next = iwork[k - 1];
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left = work[k - 1];
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right = work[k];
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mid = (left + right) * .5;
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/* semiwidth of interval */
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width = right - mid;
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/* Computing MAX */
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d__1 = abs(left), d__2 = abs(right);
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tmp = max(d__1,d__2);
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/* Computing MAX */
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d__1 = *rtol1 * gap, d__2 = *rtol2 * tmp;
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cvrgd = max(d__1,d__2);
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if (width <= cvrgd || width <= mnwdth || iter == maxitr) {
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/* reduce number of unconverged intervals */
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--nint;
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/* Mark interval as converged. */
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iwork[k - 1] = 0;
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if (i1 == i__) {
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i1 = next;
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} else {
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/* Prev holds the last unconverged interval previously examined */
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if (prev >= i1) {
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iwork[(prev << 1) - 1] = next;
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}
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}
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i__ = next;
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goto L100;
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}
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prev = i__;
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/* Perform one bisection step */
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negcnt = igraphdlaneg_(n, &d__[1], &lld[1], &mid, pivmin, &r__);
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if (negcnt <= i__ - 1) {
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work[k - 1] = mid;
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} else {
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work[k] = mid;
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}
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i__ = next;
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L100:
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;
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}
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++iter;
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/* do another loop if there are still unconverged intervals
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However, in the last iteration, all intervals are accepted
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since this is the best we can do. */
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if (nint > 0 && iter <= maxitr) {
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goto L80;
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}
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/* At this point, all the intervals have converged */
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i__1 = *ilast;
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for (i__ = *ifirst; i__ <= i__1; ++i__) {
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k = i__ << 1;
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ii = i__ - *offset;
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/* All intervals marked by '0' have been refined. */
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if (iwork[k - 1] == 0) {
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w[ii] = (work[k - 1] + work[k]) * .5;
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werr[ii] = work[k] - w[ii];
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}
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/* L110: */
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}
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i__1 = *ilast;
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for (i__ = *ifirst + 1; i__ <= i__1; ++i__) {
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k = i__ << 1;
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ii = i__ - *offset;
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/* Computing MAX */
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d__1 = 0., d__2 = w[ii] - werr[ii] - w[ii - 1] - werr[ii - 1];
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wgap[ii - 1] = max(d__1,d__2);
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/* L111: */
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}
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return 0;
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/* End of DLARRB */
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} /* igraphdlarrb_ */
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