# dlarrf (l) - Linux Man Pages

## NAME

DLARRF - the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ..

## SYNOPSIS

SUBROUTINE DLARRF(
N, D, L, LD, CLSTRT, CLEND, W, WGAP, WERR, SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA, DPLUS, LPLUS, WORK, INFO )

INTEGER CLSTRT, CLEND, INFO, N

DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM

DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ), LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )

## PURPOSE

Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), DLARRF finds a new relatively robust representation L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.

## ARGUMENTS

N (input) INTEGER
The order of the matrix (subblock, if the matrix splitted).
D (input) DOUBLE PRECISION array, dimension (N)
The N diagonal elements of the diagonal matrix D.
L (input) DOUBLE PRECISION array, dimension (N-1)
The (N-1) subdiagonal elements of the unit bidiagonal matrix L.
LD (input) DOUBLE PRECISION array, dimension (N-1)
The (N-1) elements L(i)*D(i).
CLSTRT (input) INTEGER
The index of the first eigenvalue in the cluster.
CLEND (input) INTEGER
The index of the last eigenvalue in the cluster.
W (input) DOUBLE PRECISION array, dimension >= (CLEND-CLSTRT+1)
The eigenvalue APPROXIMATIONS of L D L^T in ascending order. W( CLSTRT ) through W( CLEND ) form the cluster of relatively close eigenalues.
WGAP (input/output) DOUBLE PRECISION array, dimension >= (CLEND-CLSTRT+1)
The separation from the right neighbor eigenvalue in W.
WERR (input) DOUBLE PRECISION array, dimension >= (CLEND-CLSTRT+1)
WERR contain the semiwidth of the uncertainty interval of the corresponding eigenvalue APPROXIMATION in W SPDIAM (input) estimate of the spectral diameter obtained from the Gerschgorin intervals CLGAPL, CLGAPR (input) absolute gap on each end of the cluster. Set by the calling routine to protect against shifts too close to eigenvalues outside the cluster.
PIVMIN (input) DOUBLE PRECISION
The minimum pivot allowed in the Sturm sequence.
SIGMA (output) DOUBLE PRECISION
The shift used to form L(+) D(+) L(+)^T.
DPLUS (output) DOUBLE PRECISION array, dimension (N)
The N diagonal elements of the diagonal matrix D(+).
LPLUS (output) DOUBLE PRECISION array, dimension (N-1)
The first (N-1) elements of LPLUS contain the subdiagonal elements of the unit bidiagonal matrix L(+).
WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
Workspace.

## 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