A Subspace Approximation Method for the Quadratic Eigenvalue Problem

Abstract

Quadratic eigenvalue problems involving large matrices arise frequently in areas such as the vibration analysis of structures, micro-electro-mechanical systems (MEMS) simulation, and the solution of quadratically constrained least squares problems. The typical approach is to solve the quadratic eigenvalue problem using a mathematically equivalent linearized formulation, resulting in a doubled dimension and, in many cases, a lack of backward stability. This paper introduces an approach to solving the quadratic eigenvalue problem directly without linearizing it. Perturbation subspaces for block eigenvector matrices are used to reduce the modified problem to a sequence of problems of smaller dimension. These perturbation subspaces are shown to be contained in certain generalized Krylov subspaces of the n-dimensional space, where n is the undoubled dimension of the matrices in the quadratic problem. The method converges at least as fast as the corresponding Taylor series, and the convergence can be accelerated further by applying a block generalization of the quadratically convergent Rayleigh quotient iteration. Numerical examples are presented to illustrate the applicability of the method.