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