SLARRF(1) LAPACK auxiliary routine (version 3.2) SLARRF(1)NAME
SLARRF - the initial representation L D L^T and its cluster of close
eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ..
SYNOPSIS
SUBROUTINE SLARRF( N, D, L, LD, CLSTRT, CLEND, W, WGAP, WERR, SPDIAM,
CLGAPL, CLGAPR, PIVMIN, SIGMA, DPLUS, LPLUS, WORK,
INFO )
INTEGER CLSTRT, CLEND, INFO, N
REAL CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
REAL D( * ), DPLUS( * ), L( * ), LD( * ), LPLUS( * ), W(
* ), WGAP( * ), WERR( * ), WORK( * )
PURPOSE
Given the initial representation L D L^T and its cluster of close ei‐
genvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W(
CLEND ), SLARRF 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) REAL array, dimension (N)
The N diagonal elements of the diagonal matrix D.
L (input) REAL array, dimension (N-1)
The (N-1) subdiagonal elements of the unit bidiagonal matrix L.
LD (input) REAL 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) REAL 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) REAL array, dimension >= (CLEND-
CLSTRT+1)
The separation from the right neighbor eigenvalue in W.
WERR (input) REAL 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) REAL
The shift used to form L(+) D(+) L(+)^T.
DPLUS (output) REAL array, dimension (N)
The N diagonal elements of the diagonal matrix D(+).
LPLUS (output) REAL array, dimension (N-1)
The first (N-1) elements of LPLUS contain the subdiagonal ele‐
ments of the unit bidiagonal matrix L(+).
WORK (workspace) REAL 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
LAPACK auxiliary routine (versioNovember 2008 SLARRF(1)
