A framework for analyzing nonlinear eigenproblems and parametrized linear systems

Abstract

Associated with an n × n matrix polynomial of degree ℓ , P ( λ ) = ∑ j = 0 ℓ λ j A j , are the eigenvalue problem P ( λ ) x = 0 and the linear system problem P ( ω ) x = b , where in the latter case x is to be computed for many values of the parameter ω . Both problems can be solved by conversion to an equivalent problem L ( λ ) z = 0 or L ( ω ) z = c that is linear in the parameter λ or ω . This linearization process has received much attention in recent years for the eigenvalue problem, but it is less well understood for the linear system problem. We develop a framework in which more general versions of both problems can be analyzed, based on one-sided factorizations connecting a general nonlinear matrix function N ( λ ) to a simpler function M ( λ ) , typically a polynomial of degree 1 or 2. Our analysis relates the solutions of the original and lower degree problems and in the linear system case indicates how to choose the right-hand side c and recover the solution x from z . For the eigenvalue problem this framework includes many special cases studied in the literature, including the vector spaces of pencils L 1 ( P ) and L 2 ( P ) recently introduced by Mackey, Mackey, Mehl, and Mehrmann and a class of rational problems. We use the framework to investigate the conditioning and stability of the parametrized linear system P ( ω ) x = b and thereby study the effect of scaling, both of the original polynomial and of the pencil L . Our results identify situations in which scaling can potentially greatly improve the conditioning and stability and our numerical results show that dramatic improvements can be achieved in practice.