Alternatives to the Rayleigh Quotient for the Quadratic Eigenvalue Problem

We consider the quadratic eigenvalue problem λ2Ax+ λBx+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 eigenvalue λ. The usual approach is to impose the Galerkin condition r(θ, u) = (θ2A+ θB +C)u ⊥ u, from which it follows that θ must be one of the two solutions to the quadratic equation (u∗Au)θ2+(u∗Bu)θ+(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 λ. We compare the approaches theoretically and by numerical experiments. The methods are extended to approximations from subspaces and to the polynomial eigenvalue problem.