Contour Integral Methods for Nonlinear Eigenvalue Problems: A Systems Theoretic Approach

Contour integral methods for eigenvalue problems seek to compute a subset of the spectrum in a bounded region of the complex plane. We briefly survey this class of algorithms, establishing a relationship to system realization and rational interpolation techniques in control theory. This connection casts contour integral methods for linear and nonlinear eigenvalue problems in a general framework that gives perspective on existing methods and suggests a broad class of new algorithms. These methods replace the usual block Hankel matrix pencils (which interpolate at infinity) with Loewner matrix pencils (enabling interpolation at many points in the complex plane). While this framework is novel for linear eigenvalue problems, we focus our presentation on the nonlinear case. The old and new methods share the same intensive computations (the solution of linear systems associated with contour integration), allowing one to explore a vast range of new eigenvalue approximations with little additional work. Numerical examples illustrate the potential of this approach. We also discuss how the concept of filter functions can be employed in this new framework, and we close with a discussion of interpolation point selection.