?larrf2

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?larrf2

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

Syntax

void slarrf2(MKL_INT* n, float* d, float* l, float* ld, MKL_INT* clstrt, MKL_INT* clend, MKL_INT* clmid1, MKL_INT* clmid2, float* w, float* wgap, float* werr, MKL_INT* trymid, float* spdiam, float* clgapl, float* clgapr, float* pivmin, float* sigma, float* dplus, float* lplus, float* work, MKL_INT* info);

void dlarrf2(MKL_INT* n, double* d, double* l, double* ld, MKL_INT* clstrt, MKL_INT* clend, MKL_INT* clmid1, MKL_INT* clmid2, double* w, double* wgap, double* werr, MKL_INT* trymid, double* spdiam, double* clgapl, double* clgapr, double* pivmin, double* sigma, double* dplus, double* lplus, double* work, MKL_INT* info);

Include Files

mkl_scalapack.h

Description

Given the initial representation LDL^T and its cluster of close eigenvalues (in a relative measure), defined by the indices of the first and last eigenvalues in the cluster, ?larrf2 finds a new relatively robust representation LDL^T - σ I = L₊D₊L₊^T such that at least one of the eigenvalues of L₊D₊L₊^T is relatively isolated.

This is an enhanced version of ?larrf that also tries shifts in the middle of the cluster, should there be a large gap, in order to break large clusters into at least two pieces.

Input Parameters

n

The order of the matrix (subblock, if the matrix was split).

d

Array of size n

The n diagonal elements of the diagonal matrix D.

l

Array of size n-1

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

ld

Array of size n-1

The (n-1) elements l[i]*d[i].

clstrt

The index of the first eigenvalue in the cluster.

clend

The index of the last eigenvalue in the cluster.

clmid1, clmid2

The index of a middle eigenvalue pair with large gap.

w

Array of size ≥ (clend-clstrt+1)

The eigenvalue approximations of LD L^T in ascending order. w[clstrt - 1] through w[clend - 1] form the cluster of relatively close eigenalues.

wgap

Array of size ≥ (clend-clstrt+1)

The separation from the right neighbor eigenvalue in w.

werr

Array of size ≥ (clend-clstrt+1)

werr contains the semiwidth of the uncertainty interval of the corresponding eigenvalue approximation in w.

spdiam

Estimate of the spectral diameter obtained from the Gerschgorin intervals

clgapl, clgapr

Absolute gap on each end of the cluster.

Set by the calling function to protect against shifts too close to eigenvalues outside the cluster.

pivmin

The minimum pivot allowed in the Sturm sequence.

work

Workspace array of size 2*n

OUTPUT Parameters

wgap

Contains refined values of its input approximations. Very small gaps are unchanged.

sigma

The shift (σ) used to form L₊D₊L₊^T.

dplus

Array of size n

The n diagonal elements of the diagonal matrix D₊.

lplus

Array of size n-1

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

Parent topic: ScaLAPACK Auxiliary Routines

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