2. EL 6 ELS
  3. lapack
  4. dlarrb(l)

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DLARRB(1) LAPACK auxiliary routine (version 3.2) DLARRB(1)

NAME

DLARRB - the relatively robust representation(RRR) L D L^T, DLARRB does "limited" bisection to refine the eigenvalues of L D L^T,

SYNOPSIS

N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO )

INTEGER IFIRST, ILAST, INFO, N, OFFSET, TWIST DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPDIAM INTEGER IWORK( * ) DOUBLE PRECISION D( * ), LLD( * ), W( * ), WERR( * ), WGAP( * ), WORK( * )

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

The order of the matrix.
The N diagonal elements of the diagonal matrix D.
The (N-1) elements L(i)*L(i)*D(i).
The index of the first eigenvalue to be computed.
The index of the last eigenvalue to be computed.
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 for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.
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.
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.
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.
Workspace.
Workspace.
The minimum pivot in the Sturm sequence.
The spectral diameter of the matrix.
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)
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

November 2008 LAPACK auxiliary routine (version 3.2)