DLARRB(3f)

NAME
DLARRB - 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