A contour integral approach to the computation of invariant pairs

Abstract

We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner [3] and by Beyn and Thümmler [6]. 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 the backward error for invariant pairs and solvents. We then adapt the Sakurai-Sugiura moment method [1] to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues. Numerical refinement via a variant of Newton's method is also studied. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

Eigenvalue embedding problem for quadratic regular matrix polynomials with symmetry structures

“…Consequently, the eigenvalues of Λ are eigenvalues of Q(λ). Thus invariant pair provides a unified perspective on the problem of computing several eigenpairs for a given matrix polynomial [5,22,23,33,50]. Obviously, invariant pair extends the concepts of standard pair [28] and null pair [4].…”