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