Alternatives to the Rayleigh Quotient for the Quadratic Eigenvalue Problem

Michiel E. Hochstenbach and
Henk A. van der Vorst

Mathematical Institute

Utrecht University

P.O. Box 80.010

NL-3508 TA Utrecht, The Netherlands

Abstract
We consider the quadratic eigenvalue problem
*lambda*^{2} Ax + lambda Bx + Cx = 0.
Suppose that *u* is an approximation to an eigenvector *x* (for
instance obtained by a subspace method), and that we want to determine an
approximation to the corresponding *lambda*. The usual approach is to
impose the Galerkin condition
*r(theta; u) = ( theta*^{2} A + theta B + C) u perp u
from which it follows that theta must be one of the two solutions to the
quadratic equation
*( u*^{*}Au ) theta^{2} +
( u^{*}Bu ) + ( u^{*}Cu )= 0.
An unnatural aspect is that if *u = x*, the second solution has in
general no meaning. When *u* is not very accurate, it may not be clear
which solution is the best. Moreover, when the discriminant of the equation
is small, the solutions may be very sensitive to perturbations in *u*.

In this
paper we therefore examine alternative approximations to *lambda*. We
compare the approaches theoretically and by numerical experiments. The
methods are extended to approximations from subspaces and to the polynomial
eigenvalue problem.

Key words. Quadratic eigenvalue problem, Rayleigh quotient, Galerkin, minimum
residual, subspace methods, polynomial eigenvalue problem, backward error,
refined Ritz vector.

AMS subject classifications. 65F15.