Provided by: liblapack-doc_3.3.1-1_all

**NAME**

LAPACK-3 - 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 dimension is >= (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 dimension is >= (CLEND-CLSTRT+1) The separation from the right neighbor eigenvalue in W. WERR (input) REAL array, dimension dimension is >= (CLEND-CLSTRT+1) WERR contain the semiwidth of the uncertainty interval of the corresponding eigenvalue APPROXIMATION in W SPDIAM (input) REAL estimate of the spectral diameter obtained from the Gerschgorin intervals CLGAPL (input) REAL CLGAPR (input) REAL 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) REAL 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 LAPACK auxiliary routine (version 3.2.2) April 2011 SLARRF(3lapack)