Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Abstract

We propose an algorithm to solve polynomial eigenvalue problems via linearization combining several ingredients: a specific choice of linearization, which is constructed using input from tropical algebra and the notion of well-separated tropical roots, an appropriate scaling applied to the linearization and a modified stopping criterion for the QZ iterations that takes advantage of the properties of our scaled linearization. Numerical experiments suggest that our polynomial eigensolver computes all the finite and well-conditioned eigenvalues to high relative accuracy even when they are very different in magnitude.