A numerical method for polynomial eigenvalue problems using contour integral

Abstract

We propose a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs). The method finds eigenvalues contained in a certain domain which is defined by a surrounding integral path. By evaluating the contour integral numerically along the path, the method reduces the original PEP into a small generalized eigenvalue problem, which has the identical eigenvalues in the domain. When the contour integral is approximated numerically, eigenvalues on the periphery of the path are also obtained. Error analysis shows that, even though condition numbers of those exterior eigenvalues can be huge, the interior eigenvalues are calculated less erroneously. Four numerical examples are presented, which confirm the theoretical predictions.