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 eigenvalues (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 elements 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