Quadratic eigenproblems are no problem

Abstract

High-dimensional eigenproblems often arise in the solution of scientific problems involving stability or wave modeling. In this article we present results for a quadratic eigenproblem that we encountered in solving an acoustics problem, specifically in modeling the propagation of waves in a room in which one wall was constructed of sound-absorbing material. Efficient algorithms are known for the standard linear eigenproblem, Ax = x where A is a real or complex-valued square matrix of order n. Generalized eigenproblems of the form Ax = Bx, which occur in nite element formulations, are usually reduced to the standard problem, in a form such as B Ax = x. The reduction requires an expensive inversion operation for one of the matrices involved. Higher-order polynomial eigenproblems are also usually transformed into standard eigenproblems. We discuss here the second-degree (i.e., quadratic) eigenproblem 2C2 + C1 + C0 x = 0 in which the matrices Ci are square matrices.