# A Framework for Analyzing Nonlinear Eigenproblems and Parametrized Linear Systems

Grammont, Laurence and Higham, Nicholas J. and Tisseur, Françoise (2009) A Framework for Analyzing Nonlinear Eigenproblems and Parametrized Linear Systems. [MIMS Preprint]

Associated with an $n\times n$ matrix polynomial of degree $\ell$, $P(\lambda) = \sum_{j=0}^\ell \lambda^j A_j$, are the eigenvalue problem $P(\lambda)x = 0$ and the linear system problem $P(\omega)x = b$, where in the latter case $x$ is to be computed for many values of the parameter $\omega$. Both problems can be solved by conversion to an equivalent problem $L(\lambda)z = 0$ or $L(\omega)z = c$ that is linear in the parameter $\lambda$ or $\omega$. 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 $N(\lambda)$ to a simpler function $M(\lambda)$, typically a polynomial of degree 1 or 2. Our analysis relates the solutions of the original and linearized problems and in the linear system case indicates how to choose $c$ and recover $x$ from $z$. For the eigenvalue problem this framework includes many special cases studied in the literature, including the vector spaces of pencils $\mathbb{L}_1(P)$ and $\mathbb{L}_2(P)$ 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 $P(\omega)x = b$ 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.