A numerical method for nonlinear eigenvalue problems using contour integrals

Abstract

A contour integral method is proposed to solve nonlinear eigenvalue problems numerically. The target equation is $F(\lambda)\bm{x}=0$, where the matrix $F(\lambda)$ is an analytic matrix function of $\lambda$. The method can extract only the eigenvalues $\lambda$ in a domain defined by the integral path, by reducing the original problem to a linear eigenvalue problem that has identical eigenvalues in the domain. Theoretical aspects of the method are discussed, and we illustrate how to apply of the method with some numerical examples.