  Contour integral methods for nonlinear 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 techniques in control theory. This connection motivates a new
general framework for contour integral methods (for linear and nonlinear
eigenvalue problems), building on recent developments in multi-point rational
interpolation of dynamical systems. These new techniques, which replace the
usual Hankel matrices with Loewner matrix pencils, incorporate general
interpolation schemes and permit ready recovery of eigenvectors. Because the
main computations (the solution of linear systems associated with contour
integration) are identical for these Loewner methods and the traditional Hankel
approach, a variety of new eigenvalue approximations can be explored with
modest 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 show how contour methods enable a data-driven modal
truncation method for model reduction.
