Approximation of damped quadratic eigenvalue problem by dimension reduction

Abstract

This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem (λ2(v)M+λ(v)D(v)+K)x(v)=0, where M, K are positive definite Hermitian n × n matrices and D(v) is an n × n Hermitian semidefinite matrix which depends on a damping parameter vector v=[v1…vk]∈R+k. With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters vi, which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments.