# Backward Error of Polynomial Eigenproblems Solved by Linearization

Higham, Nicholas J. and Li, Ren-Cang and Tisseur, Françoise (2007) Backward Error of Polynomial Eigenproblems Solved by Linearization. SIAM Journal on Matrix Analysis and Applications, 29 (4). pp. 1218-1241. ISSN 0895-4798

The most widely used approach for solving the polynomial eigenvalue problem $P(\lambda)x = \bigl(\sum_{i=0}^m \l^i A_i\bigr) x = 0$ in $n\times n$ matrices $A_i$ is to linearize to produce a larger order pencil $L(\lambda) = \lambda X + Y$, whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial $P$, infinitely many linearizations $L$ exist and approximate eigenpairs of $P$ computed via linearization can have widely varying backward errors. Two main factors affect the backward error. First, because $L$ is usually highly structured, perturbations to $L$ cannot directly be interpreted as equivalent perturbations to $P$. Second, the short'' eigenvectors of $P$ can be recovered from the long'' eigenvectors of $L$ in potentially many ways, with differing implications on the backward error for $P$. We show that if a certain one-sided factorization relating $L$ to $P$ can be found then a simple formula permits recovery of right eigenvectors of $P$ from those of $L$, and the backward error of an approximate eigenpair of $P$ can be bounded in terms of the backward error for the corresponding approximate eigenpair of $L$. A similar factorization has the same implications for left eigenvectors. We use this technique to derive backward error bounds depending only on the norms of the $A_i$ for the companion pencils and for the vector space $\DL(P)$ of pencils recently identified by Mackey, Mackey, Mehl, and Mehrmann. In all cases, sufficient conditions are identified for an optimal backward error for $P$. These results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of $P$. Other contributions of this work are a block scaling of the companion pencils that yields improved backward error bounds; a demonstration that the bounds are applicable to certain structured linearizations of structured polynomials; and backward error bounds specialized to the quadratic case, including analysis of the benefits of a scaling recently proposed by Fan, Lin, and Van Dooren. The results herein make no assumptions on the stability of the method applied to $L$ or whether the method is direct or iterative.