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
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 con‐
verged if RIGHT-LEFT.LT.MAX( RTOL1*GAP,
RTOL2*MAX(|LEFT|,|RIGHT|) ) where GAP is the (estimated) dis‐
tance 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-OFF‐
SET) 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
LAPACK auxiliary routine (versioNovember 2008 DLARRB(1)
