Backward error of polynomial eigenvalue problems solved by linearization

It is commonplace in many application domains to utilize polynomial eigenvalue problems to model the behaviour of physical systems. Many techniques exist to compute solutions of these polynomial eigenvalue problems. One of the most frequently used techniques is linearization, in which the polynomial eigenvalue problem is turned into an equivalent linear eigenvalue problem with the same eigenvalues, and with easily recoverable eigenvectors. The eigenvalues and eigenvectors of the linearization are usually computed using a backward stable solver such as the QZ algorithm. Such backward stable algorithms ensure that the computed eigenvalues and eigenvectors of the linearization are exactly those of a nearby linear pencil, where the perturbations are bounded in terms of the machine precision and the norms of the matrices defining the linearization. Although we have solved a nearby linear eigenvalue problem, we are not certain that our computed solution is in fact the exact solution of a nearby polynomial eigenvalue problem. Here, we perform a backward error analysis for the solution of a specific linearization for polynomials expressed in the monomial basis. We use a suitable one-sided factorization of the linearization that allows us to map generic perturbations of the linearization onto structured perturbations of the polynomial coefficients. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)