zlar1v (l) - Linux Manuals

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

Command to display zlar1v manual in Linux: $ man l zlar1v

NAME

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

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