?lar1v

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?lar1v

Computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of tridiagonal matrix.

Syntax

call slar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )

call dlar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )

call clar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )

call zlar1v( n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work )

Include Files

mkl.fi

Description

The routine ?lar1v computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix L*D*L^T - λ*I. 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, L*D*L^T - λ*I = L(+)*D(+)*L(+)^T
Progressive qd transform, L*D*L^T - λ*I = U(-)*D(-)*U(-)^T,
Computation of the diagonal elements of the inverse of L*D*L^T - λ*I 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 L*D*L^T.

b1

INTEGER. First index of the submatrix of L*D*L^T.

bn

INTEGER. Last index of the submatrix of L*D*L^T.

lambda

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

The shift. To compute an accurate eigenvector, lambda should be a good approximation to an eigenvalue of L*D*L^T.

l

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

Array, DIMENSION (n-1).

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

d

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

Array, DIMENSION (n).

The n diagonal elements of the diagonal matrix D.

ld

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

Array, DIMENSION (n-1).

The n-1 elements L_i*D_i.

lld

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

Array, DIMENSION (n-1).

The n-1 elements L_i*L_i*D_i.

pivmin

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

The minimum pivot in the Sturm sequence.

gaptol

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

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

z

REAL for slar1v

DOUBLE PRECISION for dlar1v

COMPLEX for clar1v

DOUBLE COMPLEX for zlar1v

Array, DIMENSION (n). 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 (L*D*L^T - lambda*I)^-1 is largest in magnitude. If 1 ≤ r ≤ n, r is unchanged.

work

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

Workspace array, DIMENSION (4*n).

Output Parameters

z

REAL for slar1v

DOUBLE PRECISION for dlar1v

COMPLEX for clar1v

DOUBLE COMPLEX for zlar1v

Array, DIMENSION (n). The (scaled) r-th column of the inverse. z(r) is returned to be 1.

negcnt

INTEGER. If wantnc is .TRUE. then negcnt = the number of pivots < pivmin in the matrix factorization L*D*L^T, and negcnt = -1 otherwise.

ztz

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

The square of the 2-norm of z.

mingma

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

The reciprocal of the largest (in magnitude) diagonal element of the inverse of L*D*L^T - lambda*I.

r

On output, r is the twist index used to compute z. Ideally, r designates the position of the maximum entry in the eigenvector.

isuppz

INTEGER. Array, DIMENSION (2). The support of the vector in Z, that is, the vector z is nonzero only in elements isuppz(1) through isuppz(2).

nrminv

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

Equals 1/sqrt( ztz ).

resid

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

The residual of the FP vector.

resid = ABS( mingma )/sqrt( ztz ).

rqcorr

REAL for slar1v/clar1v

DOUBLE PRECISION for dlar1v/zlar1v

The Rayleigh Quotient correction to lambda.

rqcorr = mingma/ztz.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?lar1v