338 lines
10 KiB
C
338 lines
10 KiB
C
#include "clapack.h"
|
|
|
|
/* Subroutine */ int dlarrb_(integer *n, doublereal *d__, doublereal *lld,
|
|
integer *ifirst, integer *ilast, doublereal *rtol1, doublereal *rtol2,
|
|
integer *offset, doublereal *w, doublereal *wgap, doublereal *werr,
|
|
doublereal *work, integer *iwork, doublereal *pivmin, doublereal *
|
|
spdiam, integer *twist, integer *info)
|
|
{
|
|
/* System generated locals */
|
|
integer i__1;
|
|
doublereal d__1, d__2;
|
|
|
|
/* Builtin functions */
|
|
double log(doublereal);
|
|
|
|
/* Local variables */
|
|
integer i__, k, r__, i1, ii, ip;
|
|
doublereal gap, mid, tmp, back, lgap, rgap, left;
|
|
integer iter, nint, prev, next;
|
|
doublereal cvrgd, right, width;
|
|
extern integer dlaneg_(integer *, doublereal *, doublereal *, doublereal *
|
|
, doublereal *, integer *);
|
|
integer negcnt;
|
|
doublereal mnwdth;
|
|
integer olnint, maxitr;
|
|
|
|
|
|
/* -- LAPACK auxiliary routine (version 3.1) -- */
|
|
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
|
|
/* November 2006 */
|
|
|
|
/* .. Scalar Arguments .. */
|
|
/* .. */
|
|
/* .. Array Arguments .. */
|
|
/* .. */
|
|
|
|
/* Purpose */
|
|
/* ======= */
|
|
|
|
/* Given the relatively robust representation(RRR) L D L^T, DLARRB */
|
|
/* does "limited" bisection to refine the eigenvalues of L D L^T, */
|
|
/* W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial */
|
|
/* guesses for these eigenvalues are input in W, the corresponding estimate */
|
|
/* of the error in these guesses and their gaps are input in WERR */
|
|
/* and WGAP, respectively. During bisection, intervals */
|
|
/* [left, right] are maintained by storing their mid-points and */
|
|
/* semi-widths in the arrays W and WERR respectively. */
|
|
|
|
/* Arguments */
|
|
/* ========= */
|
|
|
|
/* N (input) INTEGER */
|
|
/* The order of the matrix. */
|
|
|
|
/* D (input) DOUBLE PRECISION array, dimension (N) */
|
|
/* The N diagonal elements of the diagonal matrix D. */
|
|
|
|
/* LLD (input) DOUBLE PRECISION array, dimension (N-1) */
|
|
/* The (N-1) elements L(i)*L(i)*D(i). */
|
|
|
|
/* IFIRST (input) INTEGER */
|
|
/* The index of the first eigenvalue to be computed. */
|
|
|
|
/* ILAST (input) INTEGER */
|
|
/* The index of the last eigenvalue to be computed. */
|
|
|
|
/* RTOL1 (input) DOUBLE PRECISION */
|
|
/* RTOL2 (input) DOUBLE PRECISION */
|
|
/* Tolerance for the convergence of the bisection intervals. */
|
|
/* An interval [LEFT,RIGHT] has converged if */
|
|
/* RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) */
|
|
/* where GAP is the (estimated) distance to the nearest */
|
|
/* eigenvalue. */
|
|
|
|
/* OFFSET (input) INTEGER */
|
|
/* Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET */
|
|
/* through ILAST-OFFSET elements of these arrays are to be used. */
|
|
|
|
/* W (input/output) DOUBLE PRECISION array, dimension (N) */
|
|
/* On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are */
|
|
/* estimates of the eigenvalues of L D L^T indexed IFIRST throug */
|
|
/* ILAST. */
|
|
/* On output, these estimates are refined. */
|
|
|
|
/* WGAP (input/output) DOUBLE PRECISION array, dimension (N-1) */
|
|
/* On input, the (estimated) gaps between consecutive */
|
|
/* eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between */
|
|
/* eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST */
|
|
/* then WGAP(IFIRST-OFFSET) must be set to ZERO. */
|
|
/* On output, these gaps are refined. */
|
|
|
|
/* WERR (input/output) DOUBLE PRECISION array, dimension (N) */
|
|
/* On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are */
|
|
/* the errors in the estimates of the corresponding elements in W. */
|
|
/* On output, these errors are refined. */
|
|
|
|
/* WORK (workspace) DOUBLE PRECISION array, dimension (2*N) */
|
|
/* Workspace. */
|
|
|
|
/* IWORK (workspace) INTEGER array, dimension (2*N) */
|
|
/* Workspace. */
|
|
|
|
/* PIVMIN (input) DOUBLE PRECISION */
|
|
/* The minimum pivot in the Sturm sequence. */
|
|
|
|
/* SPDIAM (input) DOUBLE PRECISION */
|
|
/* The spectral diameter of the matrix. */
|
|
|
|
/* TWIST (input) INTEGER */
|
|
/* The twist index for the twisted factorization that is used */
|
|
/* for the negcount. */
|
|
/* TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T */
|
|
/* TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T */
|
|
/* TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r) */
|
|
|
|
/* INFO (output) INTEGER */
|
|
/* Error flag. */
|
|
|
|
/* Further Details */
|
|
/* =============== */
|
|
|
|
/* Based on contributions by */
|
|
/* Beresford Parlett, University of California, Berkeley, USA */
|
|
/* Jim Demmel, University of California, Berkeley, USA */
|
|
/* Inderjit Dhillon, University of Texas, Austin, USA */
|
|
/* Osni Marques, LBNL/NERSC, USA */
|
|
/* Christof Voemel, University of California, Berkeley, USA */
|
|
|
|
/* ===================================================================== */
|
|
|
|
/* .. Parameters .. */
|
|
/* .. */
|
|
/* .. Local Scalars .. */
|
|
/* .. */
|
|
/* .. External Functions .. */
|
|
|
|
/* .. */
|
|
/* .. Intrinsic Functions .. */
|
|
/* .. */
|
|
/* .. Executable Statements .. */
|
|
|
|
/* Parameter adjustments */
|
|
--iwork;
|
|
--work;
|
|
--werr;
|
|
--wgap;
|
|
--w;
|
|
--lld;
|
|
--d__;
|
|
|
|
/* Function Body */
|
|
*info = 0;
|
|
|
|
maxitr = (integer) ((log(*spdiam + *pivmin) - log(*pivmin)) / log(2.)) +
|
|
2;
|
|
mnwdth = *pivmin * 2.;
|
|
|
|
r__ = *twist;
|
|
if (r__ < 1 || r__ > *n) {
|
|
r__ = *n;
|
|
}
|
|
|
|
/* Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ]. */
|
|
/* The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while */
|
|
/* Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 ) */
|
|
/* for an unconverged interval is set to the index of the next unconverged */
|
|
/* interval, and is -1 or 0 for a converged interval. Thus a linked */
|
|
/* list of unconverged intervals is set up. */
|
|
|
|
i1 = *ifirst;
|
|
/* The number of unconverged intervals */
|
|
nint = 0;
|
|
/* The last unconverged interval found */
|
|
prev = 0;
|
|
rgap = wgap[i1 - *offset];
|
|
i__1 = *ilast;
|
|
for (i__ = i1; i__ <= i__1; ++i__) {
|
|
k = i__ << 1;
|
|
ii = i__ - *offset;
|
|
left = w[ii] - werr[ii];
|
|
right = w[ii] + werr[ii];
|
|
lgap = rgap;
|
|
rgap = wgap[ii];
|
|
gap = min(lgap,rgap);
|
|
/* Make sure that [LEFT,RIGHT] contains the desired eigenvalue */
|
|
/* Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT */
|
|
|
|
/* Do while( NEGCNT(LEFT).GT.I-1 ) */
|
|
|
|
back = werr[ii];
|
|
L20:
|
|
negcnt = dlaneg_(n, &d__[1], &lld[1], &left, pivmin, &r__);
|
|
if (negcnt > i__ - 1) {
|
|
left -= back;
|
|
back *= 2.;
|
|
goto L20;
|
|
}
|
|
|
|
/* Do while( NEGCNT(RIGHT).LT.I ) */
|
|
/* Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT */
|
|
|
|
back = werr[ii];
|
|
L50:
|
|
negcnt = dlaneg_(n, &d__[1], &lld[1], &right, pivmin, &r__);
|
|
if (negcnt < i__) {
|
|
right += back;
|
|
back *= 2.;
|
|
goto L50;
|
|
}
|
|
width = (d__1 = left - right, abs(d__1)) * .5;
|
|
/* Computing MAX */
|
|
d__1 = abs(left), d__2 = abs(right);
|
|
tmp = max(d__1,d__2);
|
|
/* Computing MAX */
|
|
d__1 = *rtol1 * gap, d__2 = *rtol2 * tmp;
|
|
cvrgd = max(d__1,d__2);
|
|
if (width <= cvrgd || width <= mnwdth) {
|
|
/* This interval has already converged and does not need refinement. */
|
|
/* (Note that the gaps might change through refining the */
|
|
/* eigenvalues, however, they can only get bigger.) */
|
|
/* Remove it from the list. */
|
|
iwork[k - 1] = -1;
|
|
/* Make sure that I1 always points to the first unconverged interval */
|
|
if (i__ == i1 && i__ < *ilast) {
|
|
i1 = i__ + 1;
|
|
}
|
|
if (prev >= i1 && i__ <= *ilast) {
|
|
iwork[(prev << 1) - 1] = i__ + 1;
|
|
}
|
|
} else {
|
|
/* unconverged interval found */
|
|
prev = i__;
|
|
++nint;
|
|
iwork[k - 1] = i__ + 1;
|
|
iwork[k] = negcnt;
|
|
}
|
|
work[k - 1] = left;
|
|
work[k] = right;
|
|
/* L75: */
|
|
}
|
|
|
|
/* Do while( NINT.GT.0 ), i.e. there are still unconverged intervals */
|
|
/* and while (ITER.LT.MAXITR) */
|
|
|
|
iter = 0;
|
|
L80:
|
|
prev = i1 - 1;
|
|
i__ = i1;
|
|
olnint = nint;
|
|
i__1 = olnint;
|
|
for (ip = 1; ip <= i__1; ++ip) {
|
|
k = i__ << 1;
|
|
ii = i__ - *offset;
|
|
rgap = wgap[ii];
|
|
lgap = rgap;
|
|
if (ii > 1) {
|
|
lgap = wgap[ii - 1];
|
|
}
|
|
gap = min(lgap,rgap);
|
|
next = iwork[k - 1];
|
|
left = work[k - 1];
|
|
right = work[k];
|
|
mid = (left + right) * .5;
|
|
/* semiwidth of interval */
|
|
width = right - mid;
|
|
/* Computing MAX */
|
|
d__1 = abs(left), d__2 = abs(right);
|
|
tmp = max(d__1,d__2);
|
|
/* Computing MAX */
|
|
d__1 = *rtol1 * gap, d__2 = *rtol2 * tmp;
|
|
cvrgd = max(d__1,d__2);
|
|
if (width <= cvrgd || width <= mnwdth || iter == maxitr) {
|
|
/* reduce number of unconverged intervals */
|
|
--nint;
|
|
/* Mark interval as converged. */
|
|
iwork[k - 1] = 0;
|
|
if (i1 == i__) {
|
|
i1 = next;
|
|
} else {
|
|
/* Prev holds the last unconverged interval previously examined */
|
|
if (prev >= i1) {
|
|
iwork[(prev << 1) - 1] = next;
|
|
}
|
|
}
|
|
i__ = next;
|
|
goto L100;
|
|
}
|
|
prev = i__;
|
|
|
|
/* Perform one bisection step */
|
|
|
|
negcnt = dlaneg_(n, &d__[1], &lld[1], &mid, pivmin, &r__);
|
|
if (negcnt <= i__ - 1) {
|
|
work[k - 1] = mid;
|
|
} else {
|
|
work[k] = mid;
|
|
}
|
|
i__ = next;
|
|
L100:
|
|
;
|
|
}
|
|
++iter;
|
|
/* do another loop if there are still unconverged intervals */
|
|
/* However, in the last iteration, all intervals are accepted */
|
|
/* since this is the best we can do. */
|
|
if (nint > 0 && iter <= maxitr) {
|
|
goto L80;
|
|
}
|
|
|
|
|
|
/* At this point, all the intervals have converged */
|
|
i__1 = *ilast;
|
|
for (i__ = *ifirst; i__ <= i__1; ++i__) {
|
|
k = i__ << 1;
|
|
ii = i__ - *offset;
|
|
/* All intervals marked by '0' have been refined. */
|
|
if (iwork[k - 1] == 0) {
|
|
w[ii] = (work[k - 1] + work[k]) * .5;
|
|
werr[ii] = work[k] - w[ii];
|
|
}
|
|
/* L110: */
|
|
}
|
|
|
|
i__1 = *ilast;
|
|
for (i__ = *ifirst + 1; i__ <= i__1; ++i__) {
|
|
k = i__ << 1;
|
|
ii = i__ - *offset;
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = w[ii] - werr[ii] - w[ii - 1] - werr[ii - 1];
|
|
wgap[ii - 1] = max(d__1,d__2);
|
|
/* L111: */
|
|
}
|
|
return 0;
|
|
|
|
/* End of DLARRB */
|
|
|
|
} /* dlarrb_ */
|