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 lambda2 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) = ( theta2 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 ) theta2 + ( 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.