The tropical scaling for the polynomial eigenvalue problem solved by a contour integral method

Abstract

The contour integral method is a class of efficient methods for computing partial eigenvalues of polynomial eigenvalue problem (PEP). Among them, the recently developed Sakurai–Sugiura method with Rayleigh–Ritz projection (SS‐RR) method has received much attention for its effectiveness. However, the SS‐RR method may suffer from numerical instability when the coefficient matrices of the projected PEP vary widely in norm. To improve the numerical stability, we incorporate the tropical scaling technique into the SS‐RR method and establish upper bounds for the backward error of an approximate eigenpair of the original eigenvalue problem. These bounds shed light on mechanism that the tropical scaling improves the numerical stability of the original SS‐RR method. Numerical experiments show that the actual backward errors can be successfully reduced by scaling and the bounds can predict well the errors occurring before and after scaling.