A contour integral approach to the computation of invariant pairs

TL;DR

This paper introduces a contour integral method for computing invariant pairs of matrix polynomials, extending eigenvalue concepts to nonlinear cases and analyzing their numerical stability.

Contribution

It adapts the Sakurai-Sugiura method for invariant pairs, derives condition numbers, and explores numerical refinement and the relation to matrix triangularization.

Findings

Effective computation of invariant pairs using contour integrals.

Extension of the Sakurai-Sugiura method to multiple eigenvalues.

Numerical refinement improves accuracy of invariant pairs.

Abstract

We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner and by Beyn and Thuemmler. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. We compute formulations for the condition numbers and backward errors of invariant pairs and solvents. We then adapt the Sakurai-Sugiura moment method to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues. Numerical refinement via a variants of Newton's method is also studied. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.