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 LDLT - 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 :

1. Stationary qd transform, LDLT - λI = L+D+L+T,

2. Progressive qd transform, LDLT - λI = U-D-U-T,

3. Computation of the diagonal elements of the inverse of LDLT - 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.

4. 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 LDLT.

b1

First index of the submatrix of LDLT.

bn

Last index of the submatrix of LDLT.

lambda

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

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 rn. If r is input as 0, r is set to the index where (LDLT - σI)-1 is largest in magnitude. If 1 rn, 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 LDLT, 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 LDLT - 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.