# dlarrb (l) - Linux Man Pages

## 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

SUBROUTINE DLARRB(
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

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