?larrf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?larrf

Finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.

Syntax

call slarrf( n, d, l, ld, clstrt, clend, w, wgap, werr, spdiam, clgapl, clgapr, pivmin, sigma, dplus, lplus, work, info )

call dlarrf( n, d, l, ld, clstrt, clend, w, wgap, werr, spdiam, clgapl, clgapr, pivmin, sigma, dplus, lplus, work, info )

Include Files

mkl.fi

Description

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), the routine ?larrf finds a new relatively robust representation

L*D*L^T - σ_i*I = L(+)*D(+)*L(+)^T

such that at least one of the eigenvalues of L(+)*D*(+)*L(+)^T is relatively isolated.

Input Parameters

n

INTEGER. The order of the matrix (subblock, if the matrix is splitted).

d

REAL for slarrf

DOUBLE PRECISION for dlarrf

Array, DIMENSION (n). The n diagonal elements of the diagonal matrix D.

l

REAL for slarrf

DOUBLE PRECISION for dlarrf

Array, DIMENSION (n-1).

The (n-1) subdiagonal elements of the unit bidiagonal matrix L.

ld

REAL for slarrf

DOUBLE PRECISION for dlarrf

Array, DIMENSION (n-1).

The n-1 elements L_i*D_i.

clstrt

INTEGER. The index of the first eigenvalue in the cluster.

clend

INTEGER. The index of the last eigenvalue in the cluster.

w

REAL for slarrf

DOUBLE PRECISION for dlarrf

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 eigenvalues.

wgap

REAL for slarrf

DOUBLE PRECISION for dlarrf

Array, DIMENSION ≥ (clend -clstrt+1). The separation from the right neighbor eigenvalue in w.

werr

REAL for slarrf

DOUBLE PRECISION for dlarrf

Array, DIMENSION ≥ (clend -clstrt+1). On input, werr contains the semiwidth of the uncertainty interval of the corresponding eigenvalue approximation in w.

spdiam

REAL for slarrf

DOUBLE PRECISION for dlarrf

Estimate of the spectral diameter obtained from the Gerschgorin intervals.

clgapl, clgapr

REAL for slarrf

DOUBLE PRECISION for dlarrf

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

REAL for slarrf

DOUBLE PRECISION for dlarrf

The minimum pivot allowed in the Sturm sequence.

work

REAL for slarrf

DOUBLE PRECISION for dlarrf

Workspace array, DIMENSION (2*n).

Output Parameters

wgap

On output, the gaps are refined.

sigma

REAL for slarrf

DOUBLE PRECISION for dlarrf

The shift used to form L(+)*D*(+)*L(+)^T.

dplus

REAL for slarrf

DOUBLE PRECISION for dlarrf

Array, DIMENSION (n). The n diagonal elements of the diagonal matrix D(+).

lplus

REAL for slarrf

DOUBLE PRECISION for dlarrf

Array, DIMENSION (n). The first (n-1) elements of lplus contain the subdiagonal elements of the unit bidiagonal matrix L(+).

Parent topic: LAPACK Auxiliary Routines

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?larrf