A contour integral approach to the computation of invariant pairs
Résumé
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. Here, a contour integral
formulation is applied to compute condition numbers and backward errors for
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 two variants of Newton's
method is also studied. Furthermore, we investigate the relation between the
matrix solvent problem and the triangularization of matrix polynomials.