  This paper presents a method for computing eigenvalues and eigenvectors for
some types of nonlinear eigenvalue problems. The main idea is to approximate
the functions involved in the eigenvalue problem by rational functions and then
apply a form of linearization. Eigenpairs of the expanded form of this
linearization are not extracted directly. Instead, its structure is exploited
to develop a scheme that allows to extract all eigenvalues in a certain region
of the complex plane by solving an eigenvalue problem of much smaller
dimension. Because of its simple implementation and the ability to work
efficiently in large dimensions, the presented method is appealing when solving
challenging engineering problems. A few theoretical results are established to
explain why the new approach works and numerical experiments are presented to
validate the proposed algorithm.
