968 lines
36 KiB
C
968 lines
36 KiB
C
#include "clapack.h"
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/* Table of constant values */
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static real c_b5 = 0.f;
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static integer c__1 = 1;
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static integer c__2 = 2;
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/* Subroutine */ int slarrv_(integer *n, real *vl, real *vu, real *d__, real *
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l, real *pivmin, integer *isplit, integer *m, integer *dol, integer *
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dou, real *minrgp, real *rtol1, real *rtol2, real *w, real *werr,
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real *wgap, integer *iblock, integer *indexw, real *gers, real *z__,
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integer *ldz, integer *isuppz, real *work, integer *iwork, integer *
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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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real r__1, r__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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real gl;
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integer im, in;
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real gu, gap, eps, tau, tol, tmp;
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integer zto;
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real ztz;
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integer iend, jblk;
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real lgap;
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integer done;
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real rgap, left;
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integer wend, iter;
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real bstw;
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integer itmp1, indld;
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real fudge;
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integer idone;
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real sigma;
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integer iinfo, iindr;
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real resid;
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extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
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logical eskip;
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real right;
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integer nclus, zfrom;
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extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
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integer *);
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real rqtol;
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integer iindc1, iindc2;
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extern /* Subroutine */ int slar1v_(integer *, integer *, integer *, real
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*, real *, real *, real *, real *, real *, real *, real *,
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logical *, integer *, real *, real *, integer *, integer *, real *
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, real *, real *, real *);
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logical stp2ii;
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real lambda;
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integer ibegin, indeig;
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logical needbs;
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integer indlld;
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real sgndef, mingma;
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extern doublereal slamch_(char *);
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integer oldien, oldncl, wbegin;
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real spdiam;
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integer negcnt, oldcls;
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real savgap;
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integer ndepth;
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real ssigma;
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logical usedbs;
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integer iindwk, offset;
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real gaptol;
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extern /* Subroutine */ int slarrb_(integer *, real *, real *, integer *,
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integer *, real *, real *, integer *, real *, real *, real *,
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real *, integer *, real *, real *, integer *, integer *), slarrf_(
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integer *, real *, real *, real *, integer *, integer *, real *,
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real *, real *, real *, real *, real *, real *, real *, real *,
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real *, real *, integer *);
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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, isupmn, newlst, windpl, zusedl,
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newsiz, zusedu, zusedw;
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real bstres, nrminv;
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logical tryrqc;
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integer isupmx;
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real rqcorr;
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extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *,
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real *, real *, integer *);
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/* -- LAPACK auxiliary routine (version 3.1.1) -- */
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/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
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/* November 2006 */
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/* .. Scalar Arguments .. */
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/* .. */
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/* .. Array Arguments .. */
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/* .. */
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/* Purpose */
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/* ======= */
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/* SLARRV 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 SLARRE. */
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/* Arguments */
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/* ========= */
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/* N (input) INTEGER */
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/* The order of the matrix. N >= 0. */
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/* VL (input) REAL */
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/* VU (input) REAL */
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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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/* D (input/output) REAL 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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/* L (input/output) REAL 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 SLARRE. */
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/* On exit, L is overwritten. */
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/* PIVMIN (in) DOUBLE PRECISION */
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/* The minimum pivot allowed in the Sturm sequence. */
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/* ISPLIT (input) 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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/* M (input) INTEGER */
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/* The total number of input eigenvalues. 0 <= M <= N. */
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/* DOL (input) INTEGER */
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/* DOU (input) 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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/* MINRGP (input) REAL */
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/* RTOL1 (input) REAL */
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/* RTOL2 (input) REAL */
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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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/* W (input/output) REAL 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 SLARRE 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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/* WERR (input/output) REAL 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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/* WGAP (input/output) REAL array, dimension (N) */
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/* The separation from the right neighbor eigenvalue in W. */
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/* IBLOCK (input) 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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/* INDEXW (input) 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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/* GERS (input) REAL 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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/* Z (output) REAL 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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/* LDZ (input) 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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/* ISUPPZ (output) 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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/* WORK (workspace) REAL array, dimension (12*N) */
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/* IWORK (workspace) INTEGER array, dimension (7*N) */
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/* INFO (output) INTEGER */
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/* = 0: successful exit */
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/* > 0: A problem occured in SLARRV. */
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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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/* =-1: Problem in SLARRB when refining a child's eigenvalues. */
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/* =-2: Problem in SLARRF 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 SLARRB 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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/* Further Details */
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/* =============== */
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/* Based on contributions by */
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/* Beresford Parlett, University of California, Berkeley, USA */
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/* Jim Demmel, University of California, Berkeley, USA */
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/* Inderjit Dhillon, University of Texas, Austin, USA */
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/* Osni Marques, LBNL/NERSC, USA */
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/* Christof Voemel, University of California, Berkeley, USA */
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/* ===================================================================== */
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/* .. Parameters .. */
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/* .. */
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/* .. Local Scalars .. */
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/* .. */
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/* .. External Functions .. */
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/* .. */
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/* .. External Subroutines .. */
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/* .. */
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/* .. Intrinsic Functions .. */
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/* .. */
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/* .. Executable Statements .. */
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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.f;
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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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slaset_("Full", n, &zusedw, &c_b5, &c_b5, &z__[zusedl * z_dim1 + 1], ldz);
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eps = slamch_("Precision");
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rqtol = eps * 2.f;
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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.f;
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*rtol2 = eps * 4.f;
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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) {
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ibegin = iend + 1;
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goto L170;
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} else if (wend < *dol || wbegin > *dou) {
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ibegin = iend + 1;
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wbegin = wend + 1;
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goto L170;
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}
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/* Find local spectral diameter of the block */
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gl = gers[(ibegin << 1) - 1];
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gu = gers[ibegin * 2];
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i__2 = iend;
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for (i__ = ibegin + 1; i__ <= i__2; ++i__) {
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/* Computing MIN */
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r__1 = gers[(i__ << 1) - 1];
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gl = dmin(r__1,gl);
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/* Computing MAX */
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r__1 = gers[i__ * 2];
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gu = dmax(r__1,gu);
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/* L20: */
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}
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spdiam = gu - gl;
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/* OLDIEN is the last index of the previous block */
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oldien = ibegin - 1;
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/* Calculate the size of the current block */
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in = iend - ibegin + 1;
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/* The number of eigenvalues in the current block */
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im = wend - wbegin + 1;
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/* This is for a 1x1 block */
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if (ibegin == iend) {
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++done;
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z__[ibegin + wbegin * z_dim1] = 1.f;
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isuppz[(wbegin << 1) - 1] = ibegin;
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isuppz[wbegin * 2] = ibegin;
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w[wbegin] += sigma;
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work[wbegin] = w[wbegin];
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ibegin = iend + 1;
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++wbegin;
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goto L170;
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}
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/* The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND) */
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/* Note that these can be approximations, in this case, the corresp. */
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/* entries of WERR give the size of the uncertainty interval. */
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/* The eigenvalue approximations will be refined when necessary as */
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/* high relative accuracy is required for the computation of the */
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/* corresponding eigenvectors. */
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scopy_(&im, &w[wbegin], &c__1, &work[wbegin], &c__1);
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/* We store in W the eigenvalue approximations w.r.t. the original */
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/* matrix T. */
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i__2 = im;
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for (i__ = 1; i__ <= i__2; ++i__) {
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w[wbegin + i__ - 1] += sigma;
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/* L30: */
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}
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/* NDEPTH is the current depth of the representation tree */
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ndepth = 0;
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/* PARITY is either 1 or 0 */
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parity = 1;
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/* NCLUS is the number of clusters for the next level of the */
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/* representation tree, we start with NCLUS = 1 for the root */
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nclus = 1;
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iwork[iindc1 + 1] = 1;
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iwork[iindc1 + 2] = im;
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/* IDONE is the number of eigenvectors already computed in the current */
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/* block */
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idone = 0;
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/* loop while( IDONE.LT.IM ) */
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/* generate the representation tree for the current block and */
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/* compute the eigenvectors */
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L40:
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if (idone < im) {
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/* This is a crude protection against infinitely deep trees */
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if (ndepth > *m) {
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*info = -2;
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return 0;
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}
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/* breadth first processing of the current level of the representation */
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/* tree: OLDNCL = number of clusters on current level */
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oldncl = nclus;
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/* reset NCLUS to count the number of child clusters */
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nclus = 0;
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parity = 1 - parity;
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if (parity == 0) {
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oldcls = iindc1;
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newcls = iindc2;
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} else {
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oldcls = iindc2;
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newcls = iindc1;
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}
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/* Process the clusters on the current level */
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i__2 = oldncl;
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for (i__ = 1; i__ <= i__2; ++i__) {
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j = oldcls + (i__ << 1);
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/* OLDFST, OLDLST = first, last index of current cluster. */
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/* cluster indices start with 1 and are relative */
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/* to WBEGIN when accessing W, WGAP, WERR, Z */
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oldfst = iwork[j - 1];
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oldlst = iwork[j];
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if (ndepth > 0) {
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/* Retrieve relatively robust representation (RRR) of cluster */
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/* that has been computed at the previous level */
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/* The RRR is stored in Z and overwritten once the eigenvectors */
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/* have been computed or when the cluster is refined */
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if (*dol == 1 && *dou == *m) {
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/* Get representation from location of the leftmost evalue */
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/* of the cluster */
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j = wbegin + oldfst - 1;
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} else {
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if (wbegin + oldfst - 1 < *dol) {
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/* Get representation from the left end of Z array */
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j = *dol - 1;
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} else if (wbegin + oldfst - 1 > *dou) {
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/* Get representation from the right end of Z array */
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j = *dou;
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} else {
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j = wbegin + oldfst - 1;
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}
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}
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scopy_(&in, &z__[ibegin + j * z_dim1], &c__1, &d__[ibegin]
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, &c__1);
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i__3 = in - 1;
|
|
scopy_(&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 */
|
|
slaset_("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., th P-OFFSET */
|
|
/* thru' 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. */
|
|
slarrb_(&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 SLARRB. */
|
|
/* However, we only allow the gaps to become greater since */
|
|
/* this is what should happen when we decrease WERR */
|
|
if (oldfst > 1) {
|
|
/* Computing MAX */
|
|
r__1 = wgap[wbegin + oldfst - 2], r__2 = w[wbegin +
|
|
oldfst - 1] - werr[wbegin + oldfst - 1] - w[
|
|
wbegin + oldfst - 2] - werr[wbegin + oldfst -
|
|
2];
|
|
wgap[wbegin + oldfst - 2] = dmax(r__1,r__2);
|
|
}
|
|
if (wbegin + oldlst - 1 < wend) {
|
|
/* Computing MAX */
|
|
r__1 = wgap[wbegin + oldlst - 1], r__2 = w[wbegin +
|
|
oldlst] - werr[wbegin + oldlst] - w[wbegin +
|
|
oldlst - 1] - werr[wbegin + oldlst - 1];
|
|
wgap[wbegin + oldlst - 1] = dmax(r__1,r__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 * (r__1 = work[
|
|
wbegin + j - 1], dabs(r__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 */
|
|
r__1 = 0.f, r__2 = w[wbegin] - werr[wbegin] - *vl;
|
|
lgap = dmax(r__1,r__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;
|
|
slarrb_(&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 */
|
|
|
|
/* SLARRF needs LWORK = 2*N */
|
|
slarrf_(&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 SLARRF */
|
|
/* 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.f * (r__1 = work[wbegin + k -
|
|
1], dabs(r__1));
|
|
work[wbegin + k - 1] -= tau;
|
|
fudge += eps * 4.f * (r__1 = work[wbegin + k
|
|
- 1], dabs(r__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((real) in) * 4.f * 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 */
|
|
r__1 = dabs(left), r__2 = dabs(right);
|
|
lgap = eps * dmax(r__1,r__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 */
|
|
r__1 = dabs(left), r__2 = dabs(right);
|
|
rgap = eps * dmax(r__1,r__2);
|
|
} else {
|
|
rgap = wgap[windex];
|
|
}
|
|
gap = dmin(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.f;
|
|
} 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;
|
|
r__1 = eps * 2.f;
|
|
slarrb_(&in, &d__[ibegin], &work[indlld + ibegin
|
|
- 1], &indeig, &indeig, &c_b5, &r__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;
|
|
slar1v_(&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 && dabs(rqcorr) > rqtol * dabs(
|
|
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.f;
|
|
} else {
|
|
/* The wanted eigenvalue lies to the right */
|
|
sgndef = 1.f;
|
|
}
|
|
/* We only use the RQCORR if it improves the */
|
|
/* the iterate reasonably. */
|
|
if (rqcorr * sgndef >= 0.f && lambda + rqcorr <=
|
|
right && lambda + rqcorr >= left) {
|
|
usedrq = TRUE_;
|
|
/* Store new midpoint of bisection interval in WORK */
|
|
if (sgndef == 1.f) {
|
|
/* 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) * .5f;
|
|
/* Take RQCORR since it has the correct sign and */
|
|
/* improves the iterate reasonably */
|
|
lambda += rqcorr;
|
|
/* Update width of error interval */
|
|
werr[windex] = (right - left) * .5f;
|
|
} else {
|
|
needbs = TRUE_;
|
|
}
|
|
if (right - left < rqtol * dabs(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;
|
|
slar1v_(&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.f;
|
|
/* L122: */
|
|
}
|
|
}
|
|
if (isupmx > zto) {
|
|
i__4 = isupmx;
|
|
for (ii = zto + 1; ii <= i__4; ++ii) {
|
|
z__[ii + windex * z_dim1] = 0.f;
|
|
/* L123: */
|
|
}
|
|
}
|
|
i__4 = zto - zfrom + 1;
|
|
sscal_(&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 */
|
|
r__1 = wgap[windmn], r__2 = w[windex] - werr[
|
|
windex] - w[windmn] - werr[windmn];
|
|
wgap[windmn] = dmax(r__1,r__2);
|
|
}
|
|
if (windex < wend) {
|
|
/* Computing MAX */
|
|
r__1 = savgap, r__2 = w[windpl] - werr[windpl]
|
|
- w[windex] - werr[windex];
|
|
wgap[windex] = dmax(r__1,r__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 SLARRV */
|
|
|
|
} /* slarrv_ */
|