?lar1va

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

Version

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?lar1va

Computes scaled eigenvector corresponding to given eigenvalue.

Syntax

void slar1va(MKL_INT* n, MKL_INT* b1, MKL_INT* bn, float* lambda, float* d, float* l, float* ld, float* lld, float* pivmin, float* gaptol, float* z, MKL_INT* wantnc, MKL_INT* negcnt, float* ztz, float* mingma, MKL_INT* r, MKL_INT* isuppz, float* nrminv, float* resid, float* rqcorr, float* work);

void dlar1va(MKL_INT* n, MKL_INT* b1, MKL_INT* bn, double* lambda, double* d, double* l, double* ld, double* lld, double* pivmin, double* gaptol, double* z, MKL_INT* wantnc, MKL_INT* negcnt, double* ztz, double* mingma, MKL_INT* r, MKL_INT* isuppz, double* nrminv, double* resid, double* rqcorr, double* work);

Include Files

mkl_scalapack.h

Description

?slar1va computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDL^T - λI. When λ is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. The following steps accomplish this computation :

Stationary qd transform, LDL^T - λI = L₊D₊L₊^T,
Progressive qd transform, LDL^T - λI = U_-D_-U_-^T,
Computation of the diagonal elements of the inverse of LDL^T - λI by combining the above transforms, and choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude.
Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the stationary and the bottom part of the progressive transform.

Input Parameters

n

The order of the matrix LDL^T.

b1

First index of the submatrix of LDL^T.

bn

Last index of the submatrix of LDL^T.

lambda

The shift λ. In order to compute an accurate eigenvector, lambda should be a good approximation to an eigenvalue of LDL^T.

l

Array of size n-1

The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 0 to n-2.

d

Array of size n

The n diagonal elements of the diagonal matrix D.

ld

Array of size n-1

The n-1 elements l[i]*d[i], i=0,...,n-2.

lld

Array of size n-1

The n-1 elements l[i]*l]i]*d[i], i=0,...,n-2.

pivmin

The minimum pivot in the Sturm sequence.

gaptol

Tolerance that indicates when eigenvector entries are negligible with respect to their contribution to the residual.

z

Array of size n

On input, all entries of z must be set to 0.

wantnc

Specifies whether negcnt has to be computed.

r

The twist index for the twisted factorization used to compute z.

On input, 0 ≤r≤n. If r is input as 0, r is set to the index where (LDL^T - σI)^-1 is largest in magnitude. If 1 ≤r≤n, r is unchanged.

Ideally, r designates the position of the maximum entry in the eigenvector.

work

(Workspace) array of size 4*n

OUTPUT Parameters

z

On output, z contains the (scaled) r-th column of the inverse. The scaling is such that z[r-1] equals 1.

negcnt

If wantncis non-zero then negcnt = the number of pivots < pivmin in the matrix factorization LDL^T, and negcnt = -1 otherwise.

ztz

The square of the 2-norm of z.

mingma

The reciprocal of the largest (in magnitude) diagonal element of the inverse of LDL^T - σI.

r

On output, r contains the twist index used to compute z.

isuppz

array of size 2

The support of the vector in z, i.e., the vector z is non-zero only in elements isuppz[0] and isuppz[1].

nrminv

nrminv = 1/SQRT( ztz )

resid

The residual of the FP vector.

resid = ABS( mingma )/SQRT( ztz )

rqcorr

The Rayleigh Quotient correction to lambda.

Parent topic: ScaLAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

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