Abdelrahman Said
1106 lines
40 KiB
C
1106 lines
40 KiB
C
/* -- 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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/* Table of constant values */
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static doublereal c_b5 = 0.;
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static integer c__1 = 1;
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static integer c__2 = 2;
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/* > \brief \b DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenv
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alues of L D LT.
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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 DLARRV + dependencies
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> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrv.
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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/dlarrv.
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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/dlarrv.
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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 DLARRV( N, VL, VU, D, L, PIVMIN,
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ISPLIT, M, DOL, DOU, MINRGP,
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RTOL1, RTOL2, W, WERR, WGAP,
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IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
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WORK, IWORK, INFO )
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INTEGER DOL, DOU, INFO, LDZ, M, N
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DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
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INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
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$ ISUPPZ( * ), IWORK( * )
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DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
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$ WGAP( * ), WORK( * )
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DOUBLE PRECISION Z( LDZ, * )
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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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> DLARRV computes the eigenvectors of the tridiagonal matrix
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> T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
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> The input eigenvalues should have been computed by DLARRE.
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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. N >= 0.
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> \endverbatim
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>
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> \param[in] VL
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> \verbatim
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> VL is DOUBLE PRECISION
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> \endverbatim
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>
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> \param[in] VU
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> \verbatim
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> VU is DOUBLE PRECISION
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> Lower and upper bounds of the interval that contains the desired
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> eigenvalues. VL < VU. Needed to compute gaps on the left or right
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> end of the extremal eigenvalues in the desired RANGE.
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> \endverbatim
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>
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> \param[in,out] D
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> \verbatim
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> D is DOUBLE PRECISION array, dimension (N)
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> On entry, the N diagonal elements of the diagonal matrix D.
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> On exit, D may be overwritten.
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> \endverbatim
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>
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> \param[in,out] L
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> \verbatim
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> L is DOUBLE PRECISION array, dimension (N)
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> On entry, the (N-1) subdiagonal elements of the unit
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> bidiagonal matrix L are in elements 1 to N-1 of L
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> (if the matrix is not splitted.) At the end of each block
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> is stored the corresponding shift as given by DLARRE.
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> On exit, L is overwritten.
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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 allowed in the Sturm sequence.
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> \endverbatim
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>
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> \param[in] ISPLIT
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> \verbatim
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> ISPLIT is INTEGER array, dimension (N)
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> The splitting points, at which T breaks up into blocks.
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> The first block consists of rows/columns 1 to
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> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
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> through ISPLIT( 2 ), etc.
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> \endverbatim
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>
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> \param[in] M
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> \verbatim
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> M is INTEGER
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> The total number of input eigenvalues. 0 <= M <= N.
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> \endverbatim
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>
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> \param[in] DOL
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> \verbatim
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> DOL is INTEGER
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> \endverbatim
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>
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> \param[in] DOU
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> \verbatim
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> DOU is INTEGER
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> If the user wants to compute only selected eigenvectors from all
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> the eigenvalues supplied, he can specify an index range DOL:DOU.
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> Or else the setting DOL=1, DOU=M should be applied.
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> Note that DOL and DOU refer to the order in which the eigenvalues
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> are stored in W.
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> If the user wants to compute only selected eigenpairs, then
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> the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
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> computed eigenvectors. All other columns of Z are set to zero.
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> \endverbatim
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>
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> \param[in] MINRGP
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> \verbatim
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> MINRGP is DOUBLE PRECISION
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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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> Parameters for bisection.
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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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> \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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> The first M elements of W contain the APPROXIMATE eigenvalues for
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> which eigenvectors are to be computed. The eigenvalues
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> should be grouped by split-off block and ordered from
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> smallest to largest within the block ( The output array
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> W from DLARRE is expected here ). Furthermore, they are with
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> respect to the shift of the corresponding root representation
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> for their block. On exit, W holds the eigenvalues of the
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> UNshifted matrix.
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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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> The first M elements contain the semiwidth of the uncertainty
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> interval of the corresponding eigenvalue in W
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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)
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> The separation from the right neighbor eigenvalue in W.
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> \endverbatim
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>
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> \param[in] IBLOCK
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> \verbatim
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> IBLOCK is INTEGER array, dimension (N)
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> The indices of the blocks (submatrices) associated with the
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> corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
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> W(i) belongs to the first block from the top, =2 if W(i)
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> belongs to the second block, etc.
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> \endverbatim
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>
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> \param[in] INDEXW
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> \verbatim
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> INDEXW is INTEGER array, dimension (N)
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> The indices of the eigenvalues within each block (submatrix);
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> for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
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> i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.
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> \endverbatim
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>
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> \param[in] GERS
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> \verbatim
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> GERS is DOUBLE PRECISION array, dimension (2*N)
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> The N Gerschgorin intervals (the i-th Gerschgorin interval
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> is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
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> be computed from the original UNshifted matrix.
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> \endverbatim
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>
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> \param[out] Z
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> \verbatim
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> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
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> If INFO = 0, the first M columns of Z contain the
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> orthonormal eigenvectors of the matrix T
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> corresponding to the input eigenvalues, with the i-th
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> column of Z holding the eigenvector associated with W(i).
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> Note: the user must ensure that at least max(1,M) columns are
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> supplied in the array Z.
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> \endverbatim
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>
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> \param[in] LDZ
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> \verbatim
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> LDZ is INTEGER
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> The leading dimension of the array Z. LDZ >= 1, and if
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> JOBZ = 'V', LDZ >= max(1,N).
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> \endverbatim
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>
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> \param[out] ISUPPZ
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> \verbatim
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> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
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> The support of the eigenvectors in Z, i.e., the indices
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> indicating the nonzero elements in Z. The I-th eigenvector
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> is nonzero only in elements ISUPPZ( 2*I-1 ) through
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> ISUPPZ( 2*I ).
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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 (12*N)
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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 (7*N)
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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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> = 0: successful exit
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>
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> > 0: A problem occured in DLARRV.
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> < 0: One of the called subroutines signaled an internal problem.
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> Needs inspection of the corresponding parameter IINFO
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> for further information.
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>
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> =-1: Problem in DLARRB when refining a child's eigenvalues.
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> =-2: Problem in DLARRF when computing the RRR of a child.
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> When a child is inside a tight cluster, it can be difficult
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> to find an RRR. A partial remedy from the user's point of
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> view is to make the parameter MINRGP smaller and recompile.
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> However, as the orthogonality of the computed vectors is
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> proportional to 1/MINRGP, the user should be aware that
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> he might be trading in precision when he decreases MINRGP.
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> =-3: Problem in DLARRB when refining a single eigenvalue
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> after the Rayleigh correction was rejected.
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> = 5: The Rayleigh Quotient Iteration failed to converge to
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> full accuracy in MAXITR steps.
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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 doubleOTHERauxiliary
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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 igraphdlarrv_(integer *n, doublereal *vl, doublereal *vu,
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doublereal *d__, doublereal *l, doublereal *pivmin, integer *isplit,
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integer *m, integer *dol, integer *dou, doublereal *minrgp,
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doublereal *rtol1, doublereal *rtol2, doublereal *w, doublereal *werr,
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doublereal *wgap, integer *iblock, integer *indexw, doublereal *gers,
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doublereal *z__, integer *ldz, integer *isuppz, doublereal *work,
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integer *iwork, integer *info)
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{
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/* System generated locals */
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integer z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5;
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doublereal d__1, d__2;
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logical L__1;
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/* Builtin functions */
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double log(doublereal);
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/* Local variables */
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integer minwsize, i__, j, k, p, q, miniwsize, ii;
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doublereal gl;
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integer im, in;
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doublereal gu, gap, eps, tau, tol, tmp;
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integer zto;
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doublereal ztz;
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integer iend, jblk;
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doublereal lgap;
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integer done;
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doublereal rgap, left;
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integer wend, iter;
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doublereal bstw;
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integer itmp1;
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extern /* Subroutine */ int igraphdscal_(integer *, doublereal *, doublereal *,
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integer *);
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integer indld;
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doublereal fudge;
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integer idone;
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doublereal sigma;
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integer iinfo, iindr;
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doublereal resid;
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logical eskip;
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doublereal right;
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extern /* Subroutine */ int igraphdcopy_(integer *, doublereal *, integer *,
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doublereal *, integer *);
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integer nclus, zfrom;
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doublereal rqtol;
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integer iindc1, iindc2;
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extern /* Subroutine */ int igraphdlar1v_(integer *, integer *, integer *,
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doublereal *, doublereal *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *, logical *,
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integer *, doublereal *, doublereal *, integer *, integer *,
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doublereal *, doublereal *, doublereal *, doublereal *);
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logical stp2ii;
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doublereal lambda;
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extern doublereal igraphdlamch_(char *);
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integer ibegin, indeig;
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logical needbs;
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integer indlld;
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doublereal sgndef, mingma;
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extern /* Subroutine */ int igraphdlarrb_(integer *, doublereal *, doublereal *,
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integer *, integer *, doublereal *, doublereal *, integer *,
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doublereal *, doublereal *, doublereal *, doublereal *, integer *,
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doublereal *, doublereal *, integer *, integer *);
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integer oldien, oldncl, wbegin;
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doublereal spdiam;
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integer negcnt;
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extern /* Subroutine */ int igraphdlarrf_(integer *, doublereal *, doublereal *,
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doublereal *, integer *, integer *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *,
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doublereal *, integer *);
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integer oldcls;
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doublereal savgap;
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integer ndepth;
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doublereal ssigma;
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extern /* Subroutine */ int igraphdlaset_(char *, integer *, integer *,
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doublereal *, doublereal *, doublereal *, integer *);
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logical usedbs;
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integer iindwk, offset;
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doublereal gaptol;
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integer newcls, oldfst, indwrk, windex, oldlst;
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logical usedrq;
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integer newfst, newftt, parity, windmn, windpl, isupmn, newlst, zusedl;
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doublereal bstres;
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integer newsiz, zusedu, zusedw;
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doublereal nrminv, rqcorr;
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logical tryrqc;
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integer isupmx;
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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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The first N entries of WORK are reserved for the eigenvalues
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Parameter adjustments */
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--d__;
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--l;
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--isplit;
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--w;
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--werr;
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--wgap;
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--iblock;
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--indexw;
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--gers;
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z_dim1 = *ldz;
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z_offset = 1 + z_dim1;
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z__ -= z_offset;
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--isuppz;
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--work;
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--iwork;
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/* Function Body */
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indld = *n + 1;
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indlld = (*n << 1) + 1;
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indwrk = *n * 3 + 1;
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minwsize = *n * 12;
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i__1 = minwsize;
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for (i__ = 1; i__ <= i__1; ++i__) {
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work[i__] = 0.;
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/* L5: */
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}
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/* IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
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factorization used to compute the FP vector */
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iindr = 0;
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/* IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
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layer and the one above. */
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iindc1 = *n;
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iindc2 = *n << 1;
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iindwk = *n * 3 + 1;
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miniwsize = *n * 7;
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i__1 = miniwsize;
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for (i__ = 1; i__ <= i__1; ++i__) {
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iwork[i__] = 0;
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/* L10: */
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}
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zusedl = 1;
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if (*dol > 1) {
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/* Set lower bound for use of Z */
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zusedl = *dol - 1;
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}
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zusedu = *m;
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if (*dou < *m) {
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/* Set lower bound for use of Z */
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zusedu = *dou + 1;
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}
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/* The width of the part of Z that is used */
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zusedw = zusedu - zusedl + 1;
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igraphdlaset_("Full", n, &zusedw, &c_b5, &c_b5, &z__[zusedl * z_dim1 + 1], ldz);
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eps = igraphdlamch_("Precision");
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rqtol = eps * 2.;
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/* Set expert flags for standard code. */
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tryrqc = TRUE_;
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if (*dol == 1 && *dou == *m) {
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} else {
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/* Only selected eigenpairs are computed. Since the other evalues
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are not refined by RQ iteration, bisection has to compute to full
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accuracy. */
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*rtol1 = eps * 4.;
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*rtol2 = eps * 4.;
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}
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/* The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
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desired eigenvalues. The support of the nonzero eigenvector
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entries is contained in the interval IBEGIN:IEND.
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Remark that if k eigenpairs are desired, then the eigenvectors
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are stored in k contiguous columns of Z.
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DONE is the number of eigenvectors already computed */
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done = 0;
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ibegin = 1;
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wbegin = 1;
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i__1 = iblock[*m];
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for (jblk = 1; jblk <= i__1; ++jblk) {
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iend = isplit[jblk];
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sigma = l[iend];
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/* Find the eigenvectors of the submatrix indexed IBEGIN
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through IEND. */
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wend = wbegin - 1;
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L15:
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if (wend < *m) {
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if (iblock[wend + 1] == jblk) {
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++wend;
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goto L15;
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}
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}
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if (wend < wbegin) {
|
|
ibegin = iend + 1;
|
|
goto L170;
|
|
} else if (wend < *dol || wbegin > *dou) {
|
|
ibegin = iend + 1;
|
|
wbegin = wend + 1;
|
|
goto L170;
|
|
}
|
|
/* Find local spectral diameter of the block */
|
|
gl = gers[(ibegin << 1) - 1];
|
|
gu = gers[ibegin * 2];
|
|
i__2 = iend;
|
|
for (i__ = ibegin + 1; i__ <= i__2; ++i__) {
|
|
/* Computing MIN */
|
|
d__1 = gers[(i__ << 1) - 1];
|
|
gl = min(d__1,gl);
|
|
/* Computing MAX */
|
|
d__1 = gers[i__ * 2];
|
|
gu = max(d__1,gu);
|
|
/* L20: */
|
|
}
|
|
spdiam = gu - gl;
|
|
/* OLDIEN is the last index of the previous block */
|
|
oldien = ibegin - 1;
|
|
/* Calculate the size of the current block */
|
|
in = iend - ibegin + 1;
|
|
/* The number of eigenvalues in the current block */
|
|
im = wend - wbegin + 1;
|
|
/* This is for a 1x1 block */
|
|
if (ibegin == iend) {
|
|
++done;
|
|
z__[ibegin + wbegin * z_dim1] = 1.;
|
|
isuppz[(wbegin << 1) - 1] = ibegin;
|
|
isuppz[wbegin * 2] = ibegin;
|
|
w[wbegin] += sigma;
|
|
work[wbegin] = w[wbegin];
|
|
ibegin = iend + 1;
|
|
++wbegin;
|
|
goto L170;
|
|
}
|
|
/* The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
|
|
Note that these can be approximations, in this case, the corresp.
|
|
entries of WERR give the size of the uncertainty interval.
|
|
The eigenvalue approximations will be refined when necessary as
|
|
high relative accuracy is required for the computation of the
|
|
corresponding eigenvectors. */
|
|
igraphdcopy_(&im, &w[wbegin], &c__1, &work[wbegin], &c__1);
|
|
/* We store in W the eigenvalue approximations w.r.t. the original
|
|
matrix T. */
|
|
i__2 = im;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
w[wbegin + i__ - 1] += sigma;
|
|
/* L30: */
|
|
}
|
|
/* NDEPTH is the current depth of the representation tree */
|
|
ndepth = 0;
|
|
/* PARITY is either 1 or 0 */
|
|
parity = 1;
|
|
/* NCLUS is the number of clusters for the next level of the
|
|
representation tree, we start with NCLUS = 1 for the root */
|
|
nclus = 1;
|
|
iwork[iindc1 + 1] = 1;
|
|
iwork[iindc1 + 2] = im;
|
|
/* IDONE is the number of eigenvectors already computed in the current
|
|
block */
|
|
idone = 0;
|
|
/* loop while( IDONE.LT.IM )
|
|
generate the representation tree for the current block and
|
|
compute the eigenvectors */
|
|
L40:
|
|
if (idone < im) {
|
|
/* This is a crude protection against infinitely deep trees */
|
|
if (ndepth > *m) {
|
|
*info = -2;
|
|
return 0;
|
|
}
|
|
/* breadth first processing of the current level of the representation
|
|
tree: OLDNCL = number of clusters on current level */
|
|
oldncl = nclus;
|
|
/* reset NCLUS to count the number of child clusters */
|
|
nclus = 0;
|
|
|
|
parity = 1 - parity;
|
|
if (parity == 0) {
|
|
oldcls = iindc1;
|
|
newcls = iindc2;
|
|
} else {
|
|
oldcls = iindc2;
|
|
newcls = iindc1;
|
|
}
|
|
/* Process the clusters on the current level */
|
|
i__2 = oldncl;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
j = oldcls + (i__ << 1);
|
|
/* OLDFST, OLDLST = first, last index of current cluster.
|
|
cluster indices start with 1 and are relative
|
|
to WBEGIN when accessing W, WGAP, WERR, Z */
|
|
oldfst = iwork[j - 1];
|
|
oldlst = iwork[j];
|
|
if (ndepth > 0) {
|
|
/* Retrieve relatively robust representation (RRR) of cluster
|
|
that has been computed at the previous level
|
|
The RRR is stored in Z and overwritten once the eigenvectors
|
|
have been computed or when the cluster is refined */
|
|
if (*dol == 1 && *dou == *m) {
|
|
/* Get representation from location of the leftmost evalue
|
|
of the cluster */
|
|
j = wbegin + oldfst - 1;
|
|
} else {
|
|
if (wbegin + oldfst - 1 < *dol) {
|
|
/* Get representation from the left end of Z array */
|
|
j = *dol - 1;
|
|
} else if (wbegin + oldfst - 1 > *dou) {
|
|
/* Get representation from the right end of Z array */
|
|
j = *dou;
|
|
} else {
|
|
j = wbegin + oldfst - 1;
|
|
}
|
|
}
|
|
igraphdcopy_(&in, &z__[ibegin + j * z_dim1], &c__1, &d__[ibegin]
|
|
, &c__1);
|
|
i__3 = in - 1;
|
|
igraphdcopy_(&i__3, &z__[ibegin + (j + 1) * z_dim1], &c__1, &l[
|
|
ibegin], &c__1);
|
|
sigma = z__[iend + (j + 1) * z_dim1];
|
|
/* Set the corresponding entries in Z to zero */
|
|
igraphdlaset_("Full", &in, &c__2, &c_b5, &c_b5, &z__[ibegin + j
|
|
* z_dim1], ldz);
|
|
}
|
|
/* Compute DL and DLL of current RRR */
|
|
i__3 = iend - 1;
|
|
for (j = ibegin; j <= i__3; ++j) {
|
|
tmp = d__[j] * l[j];
|
|
work[indld - 1 + j] = tmp;
|
|
work[indlld - 1 + j] = tmp * l[j];
|
|
/* L50: */
|
|
}
|
|
if (ndepth > 0) {
|
|
/* P and Q are index of the first and last eigenvalue to compute
|
|
within the current block */
|
|
p = indexw[wbegin - 1 + oldfst];
|
|
q = indexw[wbegin - 1 + oldlst];
|
|
/* Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET
|
|
through the Q-OFFSET elements of these arrays are to be used.
|
|
OFFSET = P-OLDFST */
|
|
offset = indexw[wbegin] - 1;
|
|
/* perform limited bisection (if necessary) to get approximate
|
|
eigenvalues to the precision needed. */
|
|
igraphdlarrb_(&in, &d__[ibegin], &work[indlld + ibegin - 1], &p,
|
|
&q, rtol1, rtol2, &offset, &work[wbegin], &wgap[
|
|
wbegin], &werr[wbegin], &work[indwrk], &iwork[
|
|
iindwk], pivmin, &spdiam, &in, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -1;
|
|
return 0;
|
|
}
|
|
/* We also recompute the extremal gaps. W holds all eigenvalues
|
|
of the unshifted matrix and must be used for computation
|
|
of WGAP, the entries of WORK might stem from RRRs with
|
|
different shifts. The gaps from WBEGIN-1+OLDFST to
|
|
WBEGIN-1+OLDLST are correctly computed in DLARRB.
|
|
However, we only allow the gaps to become greater since
|
|
this is what should happen when we decrease WERR */
|
|
if (oldfst > 1) {
|
|
/* Computing MAX */
|
|
d__1 = wgap[wbegin + oldfst - 2], d__2 = w[wbegin +
|
|
oldfst - 1] - werr[wbegin + oldfst - 1] - w[
|
|
wbegin + oldfst - 2] - werr[wbegin + oldfst -
|
|
2];
|
|
wgap[wbegin + oldfst - 2] = max(d__1,d__2);
|
|
}
|
|
if (wbegin + oldlst - 1 < wend) {
|
|
/* Computing MAX */
|
|
d__1 = wgap[wbegin + oldlst - 1], d__2 = w[wbegin +
|
|
oldlst] - werr[wbegin + oldlst] - w[wbegin +
|
|
oldlst - 1] - werr[wbegin + oldlst - 1];
|
|
wgap[wbegin + oldlst - 1] = max(d__1,d__2);
|
|
}
|
|
/* Each time the eigenvalues in WORK get refined, we store
|
|
the newly found approximation with all shifts applied in W */
|
|
i__3 = oldlst;
|
|
for (j = oldfst; j <= i__3; ++j) {
|
|
w[wbegin + j - 1] = work[wbegin + j - 1] + sigma;
|
|
/* L53: */
|
|
}
|
|
}
|
|
/* Process the current node. */
|
|
newfst = oldfst;
|
|
i__3 = oldlst;
|
|
for (j = oldfst; j <= i__3; ++j) {
|
|
if (j == oldlst) {
|
|
/* we are at the right end of the cluster, this is also the
|
|
boundary of the child cluster */
|
|
newlst = j;
|
|
} else if (wgap[wbegin + j - 1] >= *minrgp * (d__1 = work[
|
|
wbegin + j - 1], abs(d__1))) {
|
|
/* the right relative gap is big enough, the child cluster
|
|
(NEWFST,..,NEWLST) is well separated from the following */
|
|
newlst = j;
|
|
} else {
|
|
/* inside a child cluster, the relative gap is not
|
|
big enough. */
|
|
goto L140;
|
|
}
|
|
/* Compute size of child cluster found */
|
|
newsiz = newlst - newfst + 1;
|
|
/* NEWFTT is the place in Z where the new RRR or the computed
|
|
eigenvector is to be stored */
|
|
if (*dol == 1 && *dou == *m) {
|
|
/* Store representation at location of the leftmost evalue
|
|
of the cluster */
|
|
newftt = wbegin + newfst - 1;
|
|
} else {
|
|
if (wbegin + newfst - 1 < *dol) {
|
|
/* Store representation at the left end of Z array */
|
|
newftt = *dol - 1;
|
|
} else if (wbegin + newfst - 1 > *dou) {
|
|
/* Store representation at the right end of Z array */
|
|
newftt = *dou;
|
|
} else {
|
|
newftt = wbegin + newfst - 1;
|
|
}
|
|
}
|
|
if (newsiz > 1) {
|
|
|
|
/* Current child is not a singleton but a cluster.
|
|
Compute and store new representation of child.
|
|
|
|
|
|
Compute left and right cluster gap.
|
|
|
|
LGAP and RGAP are not computed from WORK because
|
|
the eigenvalue approximations may stem from RRRs
|
|
different shifts. However, W hold all eigenvalues
|
|
of the unshifted matrix. Still, the entries in WGAP
|
|
have to be computed from WORK since the entries
|
|
in W might be of the same order so that gaps are not
|
|
exhibited correctly for very close eigenvalues. */
|
|
if (newfst == 1) {
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = w[wbegin] - werr[wbegin] - *vl;
|
|
lgap = max(d__1,d__2);
|
|
} else {
|
|
lgap = wgap[wbegin + newfst - 2];
|
|
}
|
|
rgap = wgap[wbegin + newlst - 1];
|
|
|
|
/* Compute left- and rightmost eigenvalue of child
|
|
to high precision in order to shift as close
|
|
as possible and obtain as large relative gaps
|
|
as possible */
|
|
|
|
for (k = 1; k <= 2; ++k) {
|
|
if (k == 1) {
|
|
p = indexw[wbegin - 1 + newfst];
|
|
} else {
|
|
p = indexw[wbegin - 1 + newlst];
|
|
}
|
|
offset = indexw[wbegin] - 1;
|
|
igraphdlarrb_(&in, &d__[ibegin], &work[indlld + ibegin
|
|
- 1], &p, &p, &rqtol, &rqtol, &offset, &
|
|
work[wbegin], &wgap[wbegin], &werr[wbegin]
|
|
, &work[indwrk], &iwork[iindwk], pivmin, &
|
|
spdiam, &in, &iinfo);
|
|
/* L55: */
|
|
}
|
|
|
|
if (wbegin + newlst - 1 < *dol || wbegin + newfst - 1
|
|
> *dou) {
|
|
/* if the cluster contains no desired eigenvalues
|
|
skip the computation of that branch of the rep. tree
|
|
|
|
We could skip before the refinement of the extremal
|
|
eigenvalues of the child, but then the representation
|
|
tree could be different from the one when nothing is
|
|
skipped. For this reason we skip at this place. */
|
|
idone = idone + newlst - newfst + 1;
|
|
goto L139;
|
|
}
|
|
|
|
/* Compute RRR of child cluster.
|
|
Note that the new RRR is stored in Z
|
|
|
|
DLARRF needs LWORK = 2*N */
|
|
igraphdlarrf_(&in, &d__[ibegin], &l[ibegin], &work[indld +
|
|
ibegin - 1], &newfst, &newlst, &work[wbegin],
|
|
&wgap[wbegin], &werr[wbegin], &spdiam, &lgap,
|
|
&rgap, pivmin, &tau, &z__[ibegin + newftt *
|
|
z_dim1], &z__[ibegin + (newftt + 1) * z_dim1],
|
|
&work[indwrk], &iinfo);
|
|
if (iinfo == 0) {
|
|
/* a new RRR for the cluster was found by DLARRF
|
|
update shift and store it */
|
|
ssigma = sigma + tau;
|
|
z__[iend + (newftt + 1) * z_dim1] = ssigma;
|
|
/* WORK() are the midpoints and WERR() the semi-width
|
|
Note that the entries in W are unchanged. */
|
|
i__4 = newlst;
|
|
for (k = newfst; k <= i__4; ++k) {
|
|
fudge = eps * 3. * (d__1 = work[wbegin + k -
|
|
1], abs(d__1));
|
|
work[wbegin + k - 1] -= tau;
|
|
fudge += eps * 4. * (d__1 = work[wbegin + k -
|
|
1], abs(d__1));
|
|
/* Fudge errors */
|
|
werr[wbegin + k - 1] += fudge;
|
|
/* Gaps are not fudged. Provided that WERR is small
|
|
when eigenvalues are close, a zero gap indicates
|
|
that a new representation is needed for resolving
|
|
the cluster. A fudge could lead to a wrong decision
|
|
of judging eigenvalues 'separated' which in
|
|
reality are not. This could have a negative impact
|
|
on the orthogonality of the computed eigenvectors.
|
|
L116: */
|
|
}
|
|
++nclus;
|
|
k = newcls + (nclus << 1);
|
|
iwork[k - 1] = newfst;
|
|
iwork[k] = newlst;
|
|
} else {
|
|
*info = -2;
|
|
return 0;
|
|
}
|
|
} else {
|
|
|
|
/* Compute eigenvector of singleton */
|
|
|
|
iter = 0;
|
|
|
|
tol = log((doublereal) in) * 4. * eps;
|
|
|
|
k = newfst;
|
|
windex = wbegin + k - 1;
|
|
/* Computing MAX */
|
|
i__4 = windex - 1;
|
|
windmn = max(i__4,1);
|
|
/* Computing MIN */
|
|
i__4 = windex + 1;
|
|
windpl = min(i__4,*m);
|
|
lambda = work[windex];
|
|
++done;
|
|
/* Check if eigenvector computation is to be skipped */
|
|
if (windex < *dol || windex > *dou) {
|
|
eskip = TRUE_;
|
|
goto L125;
|
|
} else {
|
|
eskip = FALSE_;
|
|
}
|
|
left = work[windex] - werr[windex];
|
|
right = work[windex] + werr[windex];
|
|
indeig = indexw[windex];
|
|
/* Note that since we compute the eigenpairs for a child,
|
|
all eigenvalue approximations are w.r.t the same shift.
|
|
In this case, the entries in WORK should be used for
|
|
computing the gaps since they exhibit even very small
|
|
differences in the eigenvalues, as opposed to the
|
|
entries in W which might "look" the same. */
|
|
if (k == 1) {
|
|
/* In the case RANGE='I' and with not much initial
|
|
accuracy in LAMBDA and VL, the formula
|
|
LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
|
|
can lead to an overestimation of the left gap and
|
|
thus to inadequately early RQI 'convergence'.
|
|
Prevent this by forcing a small left gap.
|
|
Computing MAX */
|
|
d__1 = abs(left), d__2 = abs(right);
|
|
lgap = eps * max(d__1,d__2);
|
|
} else {
|
|
lgap = wgap[windmn];
|
|
}
|
|
if (k == im) {
|
|
/* In the case RANGE='I' and with not much initial
|
|
accuracy in LAMBDA and VU, the formula
|
|
can lead to an overestimation of the right gap and
|
|
thus to inadequately early RQI 'convergence'.
|
|
Prevent this by forcing a small right gap.
|
|
Computing MAX */
|
|
d__1 = abs(left), d__2 = abs(right);
|
|
rgap = eps * max(d__1,d__2);
|
|
} else {
|
|
rgap = wgap[windex];
|
|
}
|
|
gap = min(lgap,rgap);
|
|
if (k == 1 || k == im) {
|
|
/* The eigenvector support can become wrong
|
|
because significant entries could be cut off due to a
|
|
large GAPTOL parameter in LAR1V. Prevent this. */
|
|
gaptol = 0.;
|
|
} else {
|
|
gaptol = gap * eps;
|
|
}
|
|
isupmn = in;
|
|
isupmx = 1;
|
|
/* Update WGAP so that it holds the minimum gap
|
|
to the left or the right. This is crucial in the
|
|
case where bisection is used to ensure that the
|
|
eigenvalue is refined up to the required precision.
|
|
The correct value is restored afterwards. */
|
|
savgap = wgap[windex];
|
|
wgap[windex] = gap;
|
|
/* We want to use the Rayleigh Quotient Correction
|
|
as often as possible since it converges quadratically
|
|
when we are close enough to the desired eigenvalue.
|
|
However, the Rayleigh Quotient can have the wrong sign
|
|
and lead us away from the desired eigenvalue. In this
|
|
case, the best we can do is to use bisection. */
|
|
usedbs = FALSE_;
|
|
usedrq = FALSE_;
|
|
/* Bisection is initially turned off unless it is forced */
|
|
needbs = ! tryrqc;
|
|
L120:
|
|
/* Check if bisection should be used to refine eigenvalue */
|
|
if (needbs) {
|
|
/* Take the bisection as new iterate */
|
|
usedbs = TRUE_;
|
|
itmp1 = iwork[iindr + windex];
|
|
offset = indexw[wbegin] - 1;
|
|
d__1 = eps * 2.;
|
|
igraphdlarrb_(&in, &d__[ibegin], &work[indlld + ibegin
|
|
- 1], &indeig, &indeig, &c_b5, &d__1, &
|
|
offset, &work[wbegin], &wgap[wbegin], &
|
|
werr[wbegin], &work[indwrk], &iwork[
|
|
iindwk], pivmin, &spdiam, &itmp1, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -3;
|
|
return 0;
|
|
}
|
|
lambda = work[windex];
|
|
/* Reset twist index from inaccurate LAMBDA to
|
|
force computation of true MINGMA */
|
|
iwork[iindr + windex] = 0;
|
|
}
|
|
/* Given LAMBDA, compute the eigenvector. */
|
|
L__1 = ! usedbs;
|
|
igraphdlar1v_(&in, &c__1, &in, &lambda, &d__[ibegin], &l[
|
|
ibegin], &work[indld + ibegin - 1], &work[
|
|
indlld + ibegin - 1], pivmin, &gaptol, &z__[
|
|
ibegin + windex * z_dim1], &L__1, &negcnt, &
|
|
ztz, &mingma, &iwork[iindr + windex], &isuppz[
|
|
(windex << 1) - 1], &nrminv, &resid, &rqcorr,
|
|
&work[indwrk]);
|
|
if (iter == 0) {
|
|
bstres = resid;
|
|
bstw = lambda;
|
|
} else if (resid < bstres) {
|
|
bstres = resid;
|
|
bstw = lambda;
|
|
}
|
|
/* Computing MIN */
|
|
i__4 = isupmn, i__5 = isuppz[(windex << 1) - 1];
|
|
isupmn = min(i__4,i__5);
|
|
/* Computing MAX */
|
|
i__4 = isupmx, i__5 = isuppz[windex * 2];
|
|
isupmx = max(i__4,i__5);
|
|
++iter;
|
|
/* sin alpha <= |resid|/gap
|
|
Note that both the residual and the gap are
|
|
proportional to the matrix, so ||T|| doesn't play
|
|
a role in the quotient
|
|
|
|
Convergence test for Rayleigh-Quotient iteration
|
|
(omitted when Bisection has been used) */
|
|
|
|
if (resid > tol * gap && abs(rqcorr) > rqtol * abs(
|
|
lambda) && ! usedbs) {
|
|
/* We need to check that the RQCORR update doesn't
|
|
move the eigenvalue away from the desired one and
|
|
towards a neighbor. -> protection with bisection */
|
|
if (indeig <= negcnt) {
|
|
/* The wanted eigenvalue lies to the left */
|
|
sgndef = -1.;
|
|
} else {
|
|
/* The wanted eigenvalue lies to the right */
|
|
sgndef = 1.;
|
|
}
|
|
/* We only use the RQCORR if it improves the
|
|
the iterate reasonably. */
|
|
if (rqcorr * sgndef >= 0. && lambda + rqcorr <=
|
|
right && lambda + rqcorr >= left) {
|
|
usedrq = TRUE_;
|
|
/* Store new midpoint of bisection interval in WORK */
|
|
if (sgndef == 1.) {
|
|
/* The current LAMBDA is on the left of the true
|
|
eigenvalue */
|
|
left = lambda;
|
|
/* We prefer to assume that the error estimate
|
|
is correct. We could make the interval not
|
|
as a bracket but to be modified if the RQCORR
|
|
chooses to. In this case, the RIGHT side should
|
|
be modified as follows:
|
|
RIGHT = MAX(RIGHT, LAMBDA + RQCORR) */
|
|
} else {
|
|
/* The current LAMBDA is on the right of the true
|
|
eigenvalue */
|
|
right = lambda;
|
|
/* See comment about assuming the error estimate is
|
|
correct above.
|
|
LEFT = MIN(LEFT, LAMBDA + RQCORR) */
|
|
}
|
|
work[windex] = (right + left) * .5;
|
|
/* Take RQCORR since it has the correct sign and
|
|
improves the iterate reasonably */
|
|
lambda += rqcorr;
|
|
/* Update width of error interval */
|
|
werr[windex] = (right - left) * .5;
|
|
} else {
|
|
needbs = TRUE_;
|
|
}
|
|
if (right - left < rqtol * abs(lambda)) {
|
|
/* The eigenvalue is computed to bisection accuracy
|
|
compute eigenvector and stop */
|
|
usedbs = TRUE_;
|
|
goto L120;
|
|
} else if (iter < 10) {
|
|
goto L120;
|
|
} else if (iter == 10) {
|
|
needbs = TRUE_;
|
|
goto L120;
|
|
} else {
|
|
*info = 5;
|
|
return 0;
|
|
}
|
|
} else {
|
|
stp2ii = FALSE_;
|
|
if (usedrq && usedbs && bstres <= resid) {
|
|
lambda = bstw;
|
|
stp2ii = TRUE_;
|
|
}
|
|
if (stp2ii) {
|
|
/* improve error angle by second step */
|
|
L__1 = ! usedbs;
|
|
igraphdlar1v_(&in, &c__1, &in, &lambda, &d__[ibegin]
|
|
, &l[ibegin], &work[indld + ibegin -
|
|
1], &work[indlld + ibegin - 1],
|
|
pivmin, &gaptol, &z__[ibegin + windex
|
|
* z_dim1], &L__1, &negcnt, &ztz, &
|
|
mingma, &iwork[iindr + windex], &
|
|
isuppz[(windex << 1) - 1], &nrminv, &
|
|
resid, &rqcorr, &work[indwrk]);
|
|
}
|
|
work[windex] = lambda;
|
|
}
|
|
|
|
/* Compute FP-vector support w.r.t. whole matrix */
|
|
|
|
isuppz[(windex << 1) - 1] += oldien;
|
|
isuppz[windex * 2] += oldien;
|
|
zfrom = isuppz[(windex << 1) - 1];
|
|
zto = isuppz[windex * 2];
|
|
isupmn += oldien;
|
|
isupmx += oldien;
|
|
/* Ensure vector is ok if support in the RQI has changed */
|
|
if (isupmn < zfrom) {
|
|
i__4 = zfrom - 1;
|
|
for (ii = isupmn; ii <= i__4; ++ii) {
|
|
z__[ii + windex * z_dim1] = 0.;
|
|
/* L122: */
|
|
}
|
|
}
|
|
if (isupmx > zto) {
|
|
i__4 = isupmx;
|
|
for (ii = zto + 1; ii <= i__4; ++ii) {
|
|
z__[ii + windex * z_dim1] = 0.;
|
|
/* L123: */
|
|
}
|
|
}
|
|
i__4 = zto - zfrom + 1;
|
|
igraphdscal_(&i__4, &nrminv, &z__[zfrom + windex * z_dim1],
|
|
&c__1);
|
|
L125:
|
|
/* Update W */
|
|
w[windex] = lambda + sigma;
|
|
/* Recompute the gaps on the left and right
|
|
But only allow them to become larger and not
|
|
smaller (which can only happen through "bad"
|
|
cancellation and doesn't reflect the theory
|
|
where the initial gaps are underestimated due
|
|
to WERR being too crude.) */
|
|
if (! eskip) {
|
|
if (k > 1) {
|
|
/* Computing MAX */
|
|
d__1 = wgap[windmn], d__2 = w[windex] - werr[
|
|
windex] - w[windmn] - werr[windmn];
|
|
wgap[windmn] = max(d__1,d__2);
|
|
}
|
|
if (windex < wend) {
|
|
/* Computing MAX */
|
|
d__1 = savgap, d__2 = w[windpl] - werr[windpl]
|
|
- w[windex] - werr[windex];
|
|
wgap[windex] = max(d__1,d__2);
|
|
}
|
|
}
|
|
++idone;
|
|
}
|
|
/* here ends the code for the current child */
|
|
|
|
L139:
|
|
/* Proceed to any remaining child nodes */
|
|
newfst = j + 1;
|
|
L140:
|
|
;
|
|
}
|
|
/* L150: */
|
|
}
|
|
++ndepth;
|
|
goto L40;
|
|
}
|
|
ibegin = iend + 1;
|
|
wbegin = wend + 1;
|
|
L170:
|
|
;
|
|
}
|
|
|
|
return 0;
|
|
|
|
/* End of DLARRV */
|
|
|
|
} /* igraphdlarrv_ */
|
|
|