Perturbation, extraction and reﬁnement of invariant pairs for matrix polynomials
Betcke, T. and Kressner, D. (2011) Perturbation, extraction and reﬁnement of invariant pairs for matrix polynomials. Linear Algebra and its Applications, 435 (3). pp. 514-536. ISSN 0024-3795
To link to this article DOI: 10.1016/j.laa.2010.06.029
Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar beneﬁts can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to ﬁll this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a ﬁrst-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe eﬃcient reﬁnement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the eﬀectiveness of our extraction and reﬁnement procedures.