?lar1va

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?lar1va

Computes scaled eigenvector corresponding to given eigenvalue.

Syntax

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

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

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

INTEGER

The order of the matrix LDL^T.

b1

INTEGER

First index of the submatrix of LDL^T.

bn

INTEGER

Last index of the submatrix of LDL^T.

lambda

REAL for slar1va

DOUBLE PRECISION for dlar1va

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

l

REAL for slar1va

DOUBLE PRECISION for dlar1va

Array of size n-1

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

d

REAL for slar1va

DOUBLE PRECISION for dlar1va

Array of size n

The n diagonal elements of the diagonal matrix D.

ld

REAL for slar1va

DOUBLE PRECISION for dlar1va

Array of size n-1

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

lld

REAL for slar1va

DOUBLE PRECISION for dlar1va

Array of size n-1

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

pivmin

REAL for slar1va

DOUBLE PRECISION for dlar1va

The minimum pivot in the Sturm sequence.

gaptol

REAL for slar1va

DOUBLE PRECISION for dlar1va

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

z

REAL for slar1va

DOUBLE PRECISION for dlar1va

Array of size n

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

wantnc

LOGICAL

Specifies whether negcnt has to be computed.

r

INTEGER

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

REAL for slar1va

DOUBLE PRECISION for dlar1va

(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) equals 1.

negcnt

INTEGER

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

ztz

REAL for slar1va

DOUBLE PRECISION for dlar1va

The square of the 2-norm of z.

mingma

REAL for slar1va

DOUBLE PRECISION for dlar1va

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

INTEGER array of size 2

The support of the vector in z, i.e., the vector z is non-zero only in elements isuppz(1) through isuppz(2).

nrminv

REAL for slar1va

DOUBLE PRECISION for dlar1va

nrminv = 1/SQRT( ztz )

resid

REAL for slar1va

DOUBLE PRECISION for dlar1va

The residual of the FP vector.

resid = ABS( mingma )/SQRT( ztz )

rqcorr

REAL for slar1va

DOUBLE PRECISION for dlar1va

The Rayleigh Quotient correction to lambda.

Parent topic: ScaLAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?lar1va

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