A contour integral approach to the computation of invariant pairs

International audienceWe study some aspects of the invariant pair problem for matrix polynomials,as introduced by Betcke and Kressner and by Beyn and Thuemmler. Invariant pairsextend the notion of eigenvalue-eigenvector pairs, providing a counterpart ofinvariant subspaces for the nonlinear case. Here, a contour integralformulation is applied to compute condition numbers and backward errors forinvariant pairs and solvents. We then adapt the Sakurai-Sugiura moment methodto the computation of invariant pairs, including some classes of problems thathave multiple eigenvalues. Numerical refinement via two variants of Newton'smethod is also studied. Furthermore, we investigate the relation between thematrix solvent problem and the triangularization of matrix polynomials