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845 lines
28 KiB
845 lines
28 KiB
#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 3.1) -- */ |
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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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|
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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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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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|
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/* Blocks of size larger than 1x1 */ |
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|
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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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|
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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 { |
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/* Decide whether dqds or bisection is more efficient */ |
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usedqd = (doublereal) mb > in * .5 && ! forceb; |
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wend = wbegin + mb - 1; |
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/* Calculate gaps for the current block */ |
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/* In later stages, when representations for individual */ |
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/* eigenvalues are different, we use SIGMA = E( IEND ). */ |
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sigma = 0.; |
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i__2 = wend - 1; |
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for (i__ = wbegin; i__ <= i__2; ++i__) { |
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/* Computing MAX */ |
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d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] + |
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werr[i__]); |
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wgap[i__] = max(d__1,d__2); |
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/* L30: */ |
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} |
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/* Computing MAX */ |
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d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]); |
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wgap[wend] = max(d__1,d__2); |
|
/* Find local index of the first and last desired evalue. */ |
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indl = indexw[wbegin]; |
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indu = indexw[wend]; |
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} |
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} |
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if (irange == 1 && ! forceb || usedqd) { |
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/* Case of DQDS */ |
|
/* Find approximations to the extremal eigenvalues of the block */ |
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dlarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, & |
|
rtl, &tmp, &tmp1, &iinfo); |
|
if (iinfo != 0) { |
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*info = -1; |
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return 0; |
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} |
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/* Computing MAX */ |
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d__2 = gl, d__3 = tmp - tmp1 - eps * 100. * (d__1 = tmp - tmp1, |
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abs(d__1)); |
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isleft = max(d__2,d__3); |
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dlarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, & |
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rtl, &tmp, &tmp1, &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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/* Computing MIN */ |
|
d__2 = gu, d__3 = tmp + tmp1 + eps * 100. * (d__1 = tmp + tmp1, |
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abs(d__1)); |
|
isrght = min(d__2,d__3); |
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/* Improve the estimate of the spectral diameter */ |
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spdiam = isrght - isleft; |
|
} else { |
|
/* Case of bisection */ |
|
/* Find approximations to the wanted extremal eigenvalues */ |
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/* Computing MAX */ |
|
d__2 = gl, d__3 = w[wbegin] - werr[wbegin] - eps * 100. * (d__1 = |
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w[wbegin] - werr[wbegin], abs(d__1)); |
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isleft = max(d__2,d__3); |
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/* Computing MIN */ |
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d__2 = gu, d__3 = w[wend] + werr[wend] + eps * 100. * (d__1 = w[ |
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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_ */
|
|
|