 ZLAR1V(3) sigma I

## 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