Quadratic Eigenproblems Are No Problem

High-dimensional eigenproblems often arise in the solution of scientiic problems involving stability or wave modeling. In this article we present results for a quadratic eigenproblem that we encountered in solving an acoustics problem, speciically in modeling the propagation of waves in a room in which one wall was constructed of sound-absorbing material. EEcient 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 ?1 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 2 C 2 + C 1 + C 0 x = 0 in which the matrices C i are square matrices. In a discussion of the quadratic eigenvalue problem, Saad 3] refers to a lack of solution methods: \There seems to be a dichotomy between the need of users, mostly in nite elements modeling, and the numerical methods that numerical analysts develop." Commenting on general, higher-order eigenvalue problems, Bai 1] says, \Besides transforming such an eigenvalue 1