SYNOPSIS

SUBROUTINE ZLAR1V(

N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )

    

LOGICAL WANTNC

    

INTEGER B1, BN, N, NEGCNT, R

    

DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID, RQCORR, ZTZ

    

INTEGER ISUPPZ( * )

    

DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ), WORK( * )

    

COMPLEX*16 Z( * )

PURPOSE

ZLAR1V computes the (scaled) r-th column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix L D L^T - sigma I. When sigma 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 :
(a) Stationary qd transform, L D L^T - sigma I = L(+) D(+) L(+)^T, (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T, (c) Computation of the diagonal elements of the inverse of
    L D L^T - sigma 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.
(d) Computation of the (scaled) r-th column of the inverse using the
    twisted factorization obtained by combining the top part of the
    the stationary and the bottom part of the progressive transform.

ARGUMENTS

N (input) INTEGER

The order of the matrix L D L^T.

B1 (input) INTEGER

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

BN (input) INTEGER

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

LAMBDA (input) DOUBLE PRECISION

The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L^T.

L (input) DOUBLE PRECISION array, dimension (N-1)

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

D (input) DOUBLE PRECISION array, dimension (N)

The n diagonal elements of the diagonal matrix D.

LD (input) DOUBLE PRECISION array, dimension (N-1)

The n-1 elements L(i)*D(i).

LLD (input) DOUBLE PRECISION array, dimension (N-1)

The n-1 elements L(i)*L(i)*D(i).

PIVMIN (input) DOUBLE PRECISION

The minimum pivot in the Sturm sequence.

GAPTOL (input) DOUBLE PRECISION

Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.

Z (input/output) COMPLEX*16 array, dimension (N)

On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1.

WANTNC (input) LOGICAL

Specifies whether NEGCNT has to be computed.

NEGCNT (output) 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 (output) DOUBLE PRECISION

The square of the 2-norm of Z.

MINGMA (output) DOUBLE PRECISION

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

R (input/output) 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 - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged. On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector.

ISUPPZ (output) INTEGER array, dimension (2)

The support of the vector in Z, i.e., the vector Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

NRMINV (output) DOUBLE PRECISION

NRMINV = 1/SQRT( ZTZ )

RESID (output) DOUBLE PRECISION

The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )

RQCORR (output) DOUBLE PRECISION

The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP

WORK (workspace) DOUBLE PRECISION array, dimension (4*N)

FURTHER DETAILS

Based on contributions by

   Beresford Parlett, University of California, Berkeley, USA
   Jim Demmel, University of California, Berkeley, USA

   Inderjit Dhillon, University of Texas, Austin, USA

   Osni Marques, LBNL/NERSC, USA

   Christof Voemel, University of California, Berkeley, USA