862 lines
29 KiB
C
862 lines
29 KiB
C
/* dlarre.f -- translated by f2c (version 20061008).
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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 "clapack.h"
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/* Table of constant values */
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static integer c__1 = 1;
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static integer c__2 = 2;
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/* Subroutine */ int dlarre_(char *range, integer *n, doublereal *vl,
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doublereal *vu, integer *il, integer *iu, doublereal *d__, doublereal
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*e, doublereal *e2, doublereal *rtol1, doublereal *rtol2, doublereal *
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spltol, integer *nsplit, integer *isplit, integer *m, doublereal *w,
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doublereal *werr, doublereal *wgap, integer *iblock, integer *indexw,
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doublereal *gers, doublereal *pivmin, doublereal *work, integer *
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iwork, integer *info)
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{
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/* System generated locals */
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integer i__1, i__2;
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doublereal d__1, d__2, d__3;
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/* Builtin functions */
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double sqrt(doublereal), log(doublereal);
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/* Local variables */
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integer i__, j;
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doublereal s1, s2;
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integer mb;
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doublereal gl;
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integer in, mm;
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doublereal gu;
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integer cnt;
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doublereal eps, tau, tmp, rtl;
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integer cnt1, cnt2;
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doublereal tmp1, eabs;
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integer iend, jblk;
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doublereal eold;
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integer indl;
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doublereal dmax__, emax;
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integer wend, idum, indu;
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doublereal rtol;
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integer iseed[4];
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doublereal avgap, sigma;
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extern logical lsame_(char *, char *);
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integer iinfo;
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extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
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doublereal *, integer *);
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logical norep;
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extern /* Subroutine */ int dlasq2_(integer *, doublereal *, integer *);
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extern doublereal dlamch_(char *);
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integer ibegin;
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logical forceb;
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integer irange;
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doublereal sgndef;
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extern /* Subroutine */ int dlarra_(integer *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, integer *, integer *,
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integer *), dlarrb_(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 *), dlarrc_(char *
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, integer *, doublereal *, doublereal *, doublereal *, doublereal
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*, doublereal *, integer *, integer *, integer *, integer *);
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integer wbegin;
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extern /* Subroutine */ int dlarrd_(char *, char *, integer *, doublereal
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*, doublereal *, integer *, integer *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *, integer *
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, integer *, integer *, doublereal *, doublereal *, doublereal *,
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doublereal *, integer *, integer *, doublereal *, integer *,
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integer *);
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doublereal safmin, spdiam;
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extern /* Subroutine */ int dlarrk_(integer *, integer *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, integer *);
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logical usedqd;
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doublereal clwdth, isleft;
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extern /* Subroutine */ int dlarnv_(integer *, integer *, integer *,
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doublereal *);
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doublereal isrght, bsrtol, dpivot;
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/* -- LAPACK auxiliary routine (version 3.2) -- */
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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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/* To find the desired eigenvalues of a given real symmetric */
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/* tridiagonal matrix T, DLARRE sets any "small" off-diagonal */
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/* elements to zero, and for each unreduced block T_i, it finds */
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/* (a) a suitable shift at one end of the block's spectrum, */
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/* (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and */
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/* (c) eigenvalues of each L_i D_i L_i^T. */
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/* The representations and eigenvalues found are then used by */
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/* DSTEMR to compute the eigenvectors of T. */
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/* The accuracy varies depending on whether bisection is used to */
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/* find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to */
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/* conpute all and then discard any unwanted one. */
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/* As an added benefit, DLARRE also outputs the n */
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/* Gerschgorin intervals for the matrices L_i D_i L_i^T. */
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/* Arguments */
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/* ========= */
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/* RANGE (input) CHARACTER */
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/* = 'A': ("All") all eigenvalues will be found. */
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/* = 'V': ("Value") all eigenvalues in the half-open interval */
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/* (VL, VU] will be found. */
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/* = 'I': ("Index") the IL-th through IU-th eigenvalues (of the */
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/* entire matrix) will be found. */
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/* N (input) INTEGER */
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/* The order of the matrix. N > 0. */
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/* VL (input/output) DOUBLE PRECISION */
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/* VU (input/output) DOUBLE PRECISION */
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/* If RANGE='V', the lower and upper bounds for the eigenvalues. */
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/* Eigenvalues less than or equal to VL, or greater than VU, */
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/* will not be returned. VL < VU. */
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/* If RANGE='I' or ='A', DLARRE computes bounds on the desired */
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/* part of the spectrum. */
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/* IL (input) INTEGER */
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/* IU (input) INTEGER */
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/* If RANGE='I', the indices (in ascending order) of the */
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/* smallest and largest eigenvalues to be returned. */
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/* 1 <= IL <= IU <= N. */
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/* D (input/output) DOUBLE PRECISION array, dimension (N) */
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/* On entry, the N diagonal elements of the tridiagonal */
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/* matrix T. */
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/* On exit, the N diagonal elements of the diagonal */
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/* matrices D_i. */
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/* E (input/output) DOUBLE PRECISION array, dimension (N) */
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/* On entry, the first (N-1) entries contain the subdiagonal */
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/* elements of the tridiagonal matrix T; E(N) need not be set. */
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/* On exit, E contains the subdiagonal elements of the unit */
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/* bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), */
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/* 1 <= I <= NSPLIT, contain the base points sigma_i on output. */
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/* E2 (input/output) DOUBLE PRECISION array, dimension (N) */
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/* On entry, the first (N-1) entries contain the SQUARES of the */
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/* subdiagonal elements of the tridiagonal matrix T; */
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/* E2(N) need not be set. */
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/* On exit, the entries E2( ISPLIT( I ) ), */
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/* 1 <= I <= NSPLIT, have been set to zero */
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/* RTOL1 (input) DOUBLE PRECISION */
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/* RTOL2 (input) 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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/* SPLTOL (input) DOUBLE PRECISION */
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/* The threshold for splitting. */
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/* NSPLIT (output) INTEGER */
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/* The number of blocks T splits into. 1 <= NSPLIT <= N. */
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/* ISPLIT (output) 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 ISPLIT(1), */
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/* the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), */
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/* etc., and the NSPLIT-th consists of rows/columns */
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/* ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. */
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/* M (output) INTEGER */
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/* The total number of eigenvalues (of all L_i D_i L_i^T) */
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/* found. */
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/* W (output) DOUBLE PRECISION array, dimension (N) */
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/* The first M elements contain the eigenvalues. The */
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/* eigenvalues of each of the blocks, L_i D_i L_i^T, are */
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/* sorted in ascending order ( DLARRE may use the */
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/* remaining N-M elements as workspace). */
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/* WERR (output) DOUBLE PRECISION array, dimension (N) */
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/* The error bound on the corresponding eigenvalue in W. */
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/* WGAP (output) DOUBLE PRECISION array, dimension (N) */
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/* The separation from the right neighbor eigenvalue in W. */
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/* The gap is only with respect to the eigenvalues of the same block */
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/* as each block has its own representation tree. */
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/* Exception: at the right end of a block we store the left gap */
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/* IBLOCK (output) 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 (output) 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 block 2 */
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/* GERS (output) 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)). */
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/* PIVMIN (output) DOUBLE PRECISION */
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/* The minimum pivot in the Sturm sequence for T. */
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/* WORK (workspace) DOUBLE PRECISION array, dimension (6*N) */
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/* Workspace. */
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/* IWORK (workspace) INTEGER array, dimension (5*N) */
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/* Workspace. */
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/* INFO (output) INTEGER */
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/* = 0: successful exit */
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/* > 0: A problem occured in DLARRE. */
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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 DLARRD. */
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/* = 2: No base representation could be found in MAXTRY iterations. */
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/* Increasing MAXTRY and recompilation might be a remedy. */
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/* =-3: Problem in DLARRB when computing the refined root */
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/* representation for DLASQ2. */
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/* =-4: Problem in DLARRB when preforming bisection on the */
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/* desired part of the spectrum. */
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/* =-5: Problem in DLASQ2. */
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/* =-6: Problem in DLASQ2. */
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/* Further Details */
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/* The base representations are required to suffer very little */
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/* element growth and consequently define all their eigenvalues to */
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/* high relative accuracy. */
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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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/* .. Local Arrays .. */
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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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/* Parameter adjustments */
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--iwork;
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--work;
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--gers;
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--indexw;
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--iblock;
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--wgap;
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--werr;
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--w;
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--isplit;
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--e2;
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--e;
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--d__;
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/* Function Body */
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*info = 0;
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/* Decode RANGE */
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if (lsame_(range, "A")) {
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irange = 1;
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} else if (lsame_(range, "V")) {
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irange = 3;
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} else if (lsame_(range, "I")) {
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irange = 2;
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}
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*m = 0;
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/* Get machine constants */
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safmin = dlamch_("S");
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eps = dlamch_("P");
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/* Set parameters */
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rtl = sqrt(eps);
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bsrtol = sqrt(eps);
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/* Treat case of 1x1 matrix for quick return */
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if (*n == 1) {
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if (irange == 1 || irange == 3 && d__[1] > *vl && d__[1] <= *vu ||
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irange == 2 && *il == 1 && *iu == 1) {
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*m = 1;
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w[1] = d__[1];
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/* The computation error of the eigenvalue is zero */
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werr[1] = 0.;
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wgap[1] = 0.;
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iblock[1] = 1;
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indexw[1] = 1;
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gers[1] = d__[1];
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gers[2] = d__[1];
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}
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/* store the shift for the initial RRR, which is zero in this case */
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e[1] = 0.;
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return 0;
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}
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/* General case: tridiagonal matrix of order > 1 */
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/* Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter. */
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/* Compute maximum off-diagonal entry and pivmin. */
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gl = d__[1];
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gu = d__[1];
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eold = 0.;
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emax = 0.;
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e[*n] = 0.;
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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werr[i__] = 0.;
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wgap[i__] = 0.;
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eabs = (d__1 = e[i__], abs(d__1));
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if (eabs >= emax) {
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emax = eabs;
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}
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tmp1 = eabs + eold;
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gers[(i__ << 1) - 1] = d__[i__] - tmp1;
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/* Computing MIN */
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d__1 = gl, d__2 = gers[(i__ << 1) - 1];
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gl = min(d__1,d__2);
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gers[i__ * 2] = d__[i__] + tmp1;
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/* Computing MAX */
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d__1 = gu, d__2 = gers[i__ * 2];
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gu = max(d__1,d__2);
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eold = eabs;
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/* L5: */
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}
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/* The minimum pivot allowed in the Sturm sequence for T */
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/* Computing MAX */
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/* Computing 2nd power */
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d__3 = emax;
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d__1 = 1., d__2 = d__3 * d__3;
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*pivmin = safmin * max(d__1,d__2);
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/* Compute spectral diameter. The Gerschgorin bounds give an */
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/* estimate that is wrong by at most a factor of SQRT(2) */
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spdiam = gu - gl;
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/* Compute splitting points */
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dlarra_(n, &d__[1], &e[1], &e2[1], spltol, &spdiam, nsplit, &isplit[1], &
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iinfo);
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/* Can force use of bisection instead of faster DQDS. */
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/* Option left in the code for future multisection work. */
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forceb = FALSE_;
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/* Initialize USEDQD, DQDS should be used for ALLRNG unless someone */
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/* explicitly wants bisection. */
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usedqd = irange == 1 && ! forceb;
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if (irange == 1 && ! forceb) {
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/* Set interval [VL,VU] that contains all eigenvalues */
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*vl = gl;
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*vu = gu;
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} else {
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/* We call DLARRD to find crude approximations to the eigenvalues */
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/* in the desired range. In case IRANGE = INDRNG, we also obtain the */
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/* interval (VL,VU] that contains all the wanted eigenvalues. */
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/* An interval [LEFT,RIGHT] has converged if */
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/* RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT)) */
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/* DLARRD needs a WORK of size 4*N, IWORK of size 3*N */
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dlarrd_(range, "B", n, vl, vu, il, iu, &gers[1], &bsrtol, &d__[1], &e[
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1], &e2[1], pivmin, nsplit, &isplit[1], &mm, &w[1], &werr[1],
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vl, vu, &iblock[1], &indexw[1], &work[1], &iwork[1], &iinfo);
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if (iinfo != 0) {
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*info = -1;
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return 0;
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}
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/* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 */
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i__1 = *n;
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for (i__ = mm + 1; i__ <= i__1; ++i__) {
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w[i__] = 0.;
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werr[i__] = 0.;
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iblock[i__] = 0;
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indexw[i__] = 0;
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/* L14: */
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}
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}
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/* ** */
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/* Loop over unreduced blocks */
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ibegin = 1;
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wbegin = 1;
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i__1 = *nsplit;
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for (jblk = 1; jblk <= i__1; ++jblk) {
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iend = isplit[jblk];
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in = iend - ibegin + 1;
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/* 1 X 1 block */
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if (in == 1) {
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if (irange == 1 || irange == 3 && d__[ibegin] > *vl && d__[ibegin]
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<= *vu || irange == 2 && iblock[wbegin] == jblk) {
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++(*m);
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w[*m] = d__[ibegin];
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werr[*m] = 0.;
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/* The gap for a single block doesn't matter for the later */
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/* algorithm and is assigned an arbitrary large value */
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wgap[*m] = 0.;
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iblock[*m] = jblk;
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indexw[*m] = 1;
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++wbegin;
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}
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/* E( IEND ) holds the shift for the initial RRR */
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e[iend] = 0.;
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ibegin = iend + 1;
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goto L170;
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}
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/* Blocks of size larger than 1x1 */
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/* E( IEND ) will hold the shift for the initial RRR, for now set it =0 */
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e[iend] = 0.;
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/* Find local outer bounds GL,GU for the block */
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gl = d__[ibegin];
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gu = d__[ibegin];
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i__2 = iend;
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for (i__ = ibegin; i__ <= i__2; ++i__) {
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/* Computing MIN */
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d__1 = gers[(i__ << 1) - 1];
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gl = min(d__1,gl);
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/* Computing MAX */
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d__1 = gers[i__ * 2];
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gu = max(d__1,gu);
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/* L15: */
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}
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spdiam = gu - gl;
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if (! (irange == 1 && ! forceb)) {
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/* Count the number of eigenvalues in the current block. */
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mb = 0;
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i__2 = mm;
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for (i__ = wbegin; i__ <= i__2; ++i__) {
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if (iblock[i__] == jblk) {
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++mb;
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} else {
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goto L21;
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}
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/* L20: */
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}
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L21:
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if (mb == 0) {
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/* No eigenvalue in the current block lies in the desired range */
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/* E( IEND ) holds the shift for the initial RRR */
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e[iend] = 0.;
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ibegin = iend + 1;
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goto L170;
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|
} else {
|
|
/* Decide whether dqds or bisection is more efficient */
|
|
usedqd = (doublereal) mb > in * .5 && ! forceb;
|
|
wend = wbegin + mb - 1;
|
|
/* Calculate gaps for the current block */
|
|
/* In later stages, when representations for individual */
|
|
/* eigenvalues are different, we use SIGMA = E( IEND ). */
|
|
sigma = 0.;
|
|
i__2 = wend - 1;
|
|
for (i__ = wbegin; i__ <= i__2; ++i__) {
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
|
|
werr[i__]);
|
|
wgap[i__] = max(d__1,d__2);
|
|
/* L30: */
|
|
}
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
|
|
wgap[wend] = max(d__1,d__2);
|
|
/* Find local index of the first and last desired evalue. */
|
|
indl = indexw[wbegin];
|
|
indu = indexw[wend];
|
|
}
|
|
}
|
|
if (irange == 1 && ! forceb || usedqd) {
|
|
/* Case of DQDS */
|
|
/* Find approximations to the extremal eigenvalues of the block */
|
|
dlarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
|
|
rtl, &tmp, &tmp1, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -1;
|
|
return 0;
|
|
}
|
|
/* Computing MAX */
|
|
d__2 = gl, d__3 = tmp - tmp1 - eps * 100. * (d__1 = tmp - tmp1,
|
|
abs(d__1));
|
|
isleft = max(d__2,d__3);
|
|
dlarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
|
|
rtl, &tmp, &tmp1, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -1;
|
|
return 0;
|
|
}
|
|
/* Computing MIN */
|
|
d__2 = gu, d__3 = tmp + tmp1 + eps * 100. * (d__1 = tmp + tmp1,
|
|
abs(d__1));
|
|
isrght = min(d__2,d__3);
|
|
/* Improve the estimate of the spectral diameter */
|
|
spdiam = isrght - isleft;
|
|
} else {
|
|
/* Case of bisection */
|
|
/* Find approximations to the wanted extremal eigenvalues */
|
|
/* Computing MAX */
|
|
d__2 = gl, d__3 = w[wbegin] - werr[wbegin] - eps * 100. * (d__1 =
|
|
w[wbegin] - werr[wbegin], abs(d__1));
|
|
isleft = max(d__2,d__3);
|
|
/* Computing MIN */
|
|
d__2 = gu, d__3 = w[wend] + werr[wend] + eps * 100. * (d__1 = w[
|
|
wend] + werr[wend], abs(d__1));
|
|
isrght = min(d__2,d__3);
|
|
}
|
|
/* Decide whether the base representation for the current block */
|
|
/* L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I */
|
|
/* should be on the left or the right end of the current block. */
|
|
/* The strategy is to shift to the end which is "more populated" */
|
|
/* Furthermore, decide whether to use DQDS for the computation of */
|
|
/* the eigenvalue approximations at the end of DLARRE or bisection. */
|
|
/* dqds is chosen if all eigenvalues are desired or the number of */
|
|
/* eigenvalues to be computed is large compared to the blocksize. */
|
|
if (irange == 1 && ! forceb) {
|
|
/* If all the eigenvalues have to be computed, we use dqd */
|
|
usedqd = TRUE_;
|
|
/* INDL is the local index of the first eigenvalue to compute */
|
|
indl = 1;
|
|
indu = in;
|
|
/* MB = number of eigenvalues to compute */
|
|
mb = in;
|
|
wend = wbegin + mb - 1;
|
|
/* Define 1/4 and 3/4 points of the spectrum */
|
|
s1 = isleft + spdiam * .25;
|
|
s2 = isrght - spdiam * .25;
|
|
} else {
|
|
/* DLARRD has computed IBLOCK and INDEXW for each eigenvalue */
|
|
/* approximation. */
|
|
/* choose sigma */
|
|
if (usedqd) {
|
|
s1 = isleft + spdiam * .25;
|
|
s2 = isrght - spdiam * .25;
|
|
} else {
|
|
tmp = min(isrght,*vu) - max(isleft,*vl);
|
|
s1 = max(isleft,*vl) + tmp * .25;
|
|
s2 = min(isrght,*vu) - tmp * .25;
|
|
}
|
|
}
|
|
/* Compute the negcount at the 1/4 and 3/4 points */
|
|
if (mb > 1) {
|
|
dlarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
|
|
cnt, &cnt1, &cnt2, &iinfo);
|
|
}
|
|
if (mb == 1) {
|
|
sigma = gl;
|
|
sgndef = 1.;
|
|
} else if (cnt1 - indl >= indu - cnt2) {
|
|
if (irange == 1 && ! forceb) {
|
|
sigma = max(isleft,gl);
|
|
} else if (usedqd) {
|
|
/* use Gerschgorin bound as shift to get pos def matrix */
|
|
/* for dqds */
|
|
sigma = isleft;
|
|
} else {
|
|
/* use approximation of the first desired eigenvalue of the */
|
|
/* block as shift */
|
|
sigma = max(isleft,*vl);
|
|
}
|
|
sgndef = 1.;
|
|
} else {
|
|
if (irange == 1 && ! forceb) {
|
|
sigma = min(isrght,gu);
|
|
} else if (usedqd) {
|
|
/* use Gerschgorin bound as shift to get neg def matrix */
|
|
/* for dqds */
|
|
sigma = isrght;
|
|
} else {
|
|
/* use approximation of the first desired eigenvalue of the */
|
|
/* block as shift */
|
|
sigma = min(isrght,*vu);
|
|
}
|
|
sgndef = -1.;
|
|
}
|
|
/* An initial SIGMA has been chosen that will be used for computing */
|
|
/* T - SIGMA I = L D L^T */
|
|
/* Define the increment TAU of the shift in case the initial shift */
|
|
/* needs to be refined to obtain a factorization with not too much */
|
|
/* element growth. */
|
|
if (usedqd) {
|
|
/* The initial SIGMA was to the outer end of the spectrum */
|
|
/* the matrix is definite and we need not retreat. */
|
|
tau = spdiam * eps * *n + *pivmin * 2.;
|
|
} else {
|
|
if (mb > 1) {
|
|
clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
|
|
avgap = (d__1 = clwdth / (doublereal) (wend - wbegin), abs(
|
|
d__1));
|
|
if (sgndef == 1.) {
|
|
/* Computing MAX */
|
|
d__1 = wgap[wbegin];
|
|
tau = max(d__1,avgap) * .5;
|
|
/* Computing MAX */
|
|
d__1 = tau, d__2 = werr[wbegin];
|
|
tau = max(d__1,d__2);
|
|
} else {
|
|
/* Computing MAX */
|
|
d__1 = wgap[wend - 1];
|
|
tau = max(d__1,avgap) * .5;
|
|
/* Computing MAX */
|
|
d__1 = tau, d__2 = werr[wend];
|
|
tau = max(d__1,d__2);
|
|
}
|
|
} else {
|
|
tau = werr[wbegin];
|
|
}
|
|
}
|
|
|
|
for (idum = 1; idum <= 6; ++idum) {
|
|
/* Compute L D L^T factorization of tridiagonal matrix T - sigma I. */
|
|
/* Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of */
|
|
/* pivots in WORK(2*IN+1:3*IN) */
|
|
dpivot = d__[ibegin] - sigma;
|
|
work[1] = dpivot;
|
|
dmax__ = abs(work[1]);
|
|
j = ibegin;
|
|
i__2 = in - 1;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
work[(in << 1) + i__] = 1. / work[i__];
|
|
tmp = e[j] * work[(in << 1) + i__];
|
|
work[in + i__] = tmp;
|
|
dpivot = d__[j + 1] - sigma - tmp * e[j];
|
|
work[i__ + 1] = dpivot;
|
|
/* Computing MAX */
|
|
d__1 = dmax__, d__2 = abs(dpivot);
|
|
dmax__ = max(d__1,d__2);
|
|
++j;
|
|
/* L70: */
|
|
}
|
|
/* check for element growth */
|
|
if (dmax__ > spdiam * 64.) {
|
|
norep = TRUE_;
|
|
} else {
|
|
norep = FALSE_;
|
|
}
|
|
if (usedqd && ! norep) {
|
|
/* Ensure the definiteness of the representation */
|
|
/* All entries of D (of L D L^T) must have the same sign */
|
|
i__2 = in;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
tmp = sgndef * work[i__];
|
|
if (tmp < 0.) {
|
|
norep = TRUE_;
|
|
}
|
|
/* L71: */
|
|
}
|
|
}
|
|
if (norep) {
|
|
/* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin */
|
|
/* shift which makes the matrix definite. So we should end up */
|
|
/* here really only in the case of IRANGE = VALRNG or INDRNG. */
|
|
if (idum == 5) {
|
|
if (sgndef == 1.) {
|
|
/* The fudged Gerschgorin shift should succeed */
|
|
sigma = gl - spdiam * 2. * eps * *n - *pivmin * 4.;
|
|
} else {
|
|
sigma = gu + spdiam * 2. * eps * *n + *pivmin * 4.;
|
|
}
|
|
} else {
|
|
sigma -= sgndef * tau;
|
|
tau *= 2.;
|
|
}
|
|
} else {
|
|
/* an initial RRR is found */
|
|
goto L83;
|
|
}
|
|
/* L80: */
|
|
}
|
|
/* if the program reaches this point, no base representation could be */
|
|
/* found in MAXTRY iterations. */
|
|
*info = 2;
|
|
return 0;
|
|
L83:
|
|
/* At this point, we have found an initial base representation */
|
|
/* T - SIGMA I = L D L^T with not too much element growth. */
|
|
/* Store the shift. */
|
|
e[iend] = sigma;
|
|
/* Store D and L. */
|
|
dcopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
|
|
i__2 = in - 1;
|
|
dcopy_(&i__2, &work[in + 1], &c__1, &e[ibegin], &c__1);
|
|
if (mb > 1) {
|
|
|
|
/* Perturb each entry of the base representation by a small */
|
|
/* (but random) relative amount to overcome difficulties with */
|
|
/* glued matrices. */
|
|
|
|
for (i__ = 1; i__ <= 4; ++i__) {
|
|
iseed[i__ - 1] = 1;
|
|
/* L122: */
|
|
}
|
|
i__2 = (in << 1) - 1;
|
|
dlarnv_(&c__2, iseed, &i__2, &work[1]);
|
|
i__2 = in - 1;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
d__[ibegin + i__ - 1] *= eps * 8. * work[i__] + 1.;
|
|
e[ibegin + i__ - 1] *= eps * 8. * work[in + i__] + 1.;
|
|
/* L125: */
|
|
}
|
|
d__[iend] *= eps * 4. * work[in] + 1.;
|
|
|
|
}
|
|
|
|
/* Don't update the Gerschgorin intervals because keeping track */
|
|
/* of the updates would be too much work in DLARRV. */
|
|
/* We update W instead and use it to locate the proper Gerschgorin */
|
|
/* intervals. */
|
|
/* Compute the required eigenvalues of L D L' by bisection or dqds */
|
|
if (! usedqd) {
|
|
/* If DLARRD has been used, shift the eigenvalue approximations */
|
|
/* according to their representation. This is necessary for */
|
|
/* a uniform DLARRV since dqds computes eigenvalues of the */
|
|
/* shifted representation. In DLARRV, W will always hold the */
|
|
/* UNshifted eigenvalue approximation. */
|
|
i__2 = wend;
|
|
for (j = wbegin; j <= i__2; ++j) {
|
|
w[j] -= sigma;
|
|
werr[j] += (d__1 = w[j], abs(d__1)) * eps;
|
|
/* L134: */
|
|
}
|
|
/* call DLARRB to reduce eigenvalue error of the approximations */
|
|
/* from DLARRD */
|
|
i__2 = iend - 1;
|
|
for (i__ = ibegin; i__ <= i__2; ++i__) {
|
|
/* Computing 2nd power */
|
|
d__1 = e[i__];
|
|
work[i__] = d__[i__] * (d__1 * d__1);
|
|
/* L135: */
|
|
}
|
|
/* use bisection to find EV from INDL to INDU */
|
|
i__2 = indl - 1;
|
|
dlarrb_(&in, &d__[ibegin], &work[ibegin], &indl, &indu, rtol1,
|
|
rtol2, &i__2, &w[wbegin], &wgap[wbegin], &werr[wbegin], &
|
|
work[(*n << 1) + 1], &iwork[1], pivmin, &spdiam, &in, &
|
|
iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -4;
|
|
return 0;
|
|
}
|
|
/* DLARRB computes all gaps correctly except for the last one */
|
|
/* Record distance to VU/GU */
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
|
|
wgap[wend] = max(d__1,d__2);
|
|
i__2 = indu;
|
|
for (i__ = indl; i__ <= i__2; ++i__) {
|
|
++(*m);
|
|
iblock[*m] = jblk;
|
|
indexw[*m] = i__;
|
|
/* L138: */
|
|
}
|
|
} else {
|
|
/* Call dqds to get all eigs (and then possibly delete unwanted */
|
|
/* eigenvalues). */
|
|
/* Note that dqds finds the eigenvalues of the L D L^T representation */
|
|
/* of T to high relative accuracy. High relative accuracy */
|
|
/* might be lost when the shift of the RRR is subtracted to obtain */
|
|
/* the eigenvalues of T. However, T is not guaranteed to define its */
|
|
/* eigenvalues to high relative accuracy anyway. */
|
|
/* Set RTOL to the order of the tolerance used in DLASQ2 */
|
|
/* This is an ESTIMATED error, the worst case bound is 4*N*EPS */
|
|
/* which is usually too large and requires unnecessary work to be */
|
|
/* done by bisection when computing the eigenvectors */
|
|
rtol = log((doublereal) in) * 4. * eps;
|
|
j = ibegin;
|
|
i__2 = in - 1;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
work[(i__ << 1) - 1] = (d__1 = d__[j], abs(d__1));
|
|
work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
|
|
++j;
|
|
/* L140: */
|
|
}
|
|
work[(in << 1) - 1] = (d__1 = d__[iend], abs(d__1));
|
|
work[in * 2] = 0.;
|
|
dlasq2_(&in, &work[1], &iinfo);
|
|
if (iinfo != 0) {
|
|
/* If IINFO = -5 then an index is part of a tight cluster */
|
|
/* and should be changed. The index is in IWORK(1) and the */
|
|
/* gap is in WORK(N+1) */
|
|
*info = -5;
|
|
return 0;
|
|
} else {
|
|
/* Test that all eigenvalues are positive as expected */
|
|
i__2 = in;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
if (work[i__] < 0.) {
|
|
*info = -6;
|
|
return 0;
|
|
}
|
|
/* L149: */
|
|
}
|
|
}
|
|
if (sgndef > 0.) {
|
|
i__2 = indu;
|
|
for (i__ = indl; i__ <= i__2; ++i__) {
|
|
++(*m);
|
|
w[*m] = work[in - i__ + 1];
|
|
iblock[*m] = jblk;
|
|
indexw[*m] = i__;
|
|
/* L150: */
|
|
}
|
|
} else {
|
|
i__2 = indu;
|
|
for (i__ = indl; i__ <= i__2; ++i__) {
|
|
++(*m);
|
|
w[*m] = -work[i__];
|
|
iblock[*m] = jblk;
|
|
indexw[*m] = i__;
|
|
/* L160: */
|
|
}
|
|
}
|
|
i__2 = *m;
|
|
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
|
|
/* the value of RTOL below should be the tolerance in DLASQ2 */
|
|
werr[i__] = rtol * (d__1 = w[i__], abs(d__1));
|
|
/* L165: */
|
|
}
|
|
i__2 = *m - 1;
|
|
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
|
|
/* compute the right gap between the intervals */
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] + werr[
|
|
i__]);
|
|
wgap[i__] = max(d__1,d__2);
|
|
/* L166: */
|
|
}
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = *vu - sigma - (w[*m] + werr[*m]);
|
|
wgap[*m] = max(d__1,d__2);
|
|
}
|
|
/* proceed with next block */
|
|
ibegin = iend + 1;
|
|
wbegin = wend + 1;
|
|
L170:
|
|
;
|
|
}
|
|
|
|
return 0;
|
|
|
|
/* end of DLARRE */
|
|
|
|
} /* dlarre_ */
|