Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials

Abstract

For each square matrix polynomial $P(λ)$ of odd degree, a block-symmetric block-tridiagonal pencil $\mathcal{T}_{P}(λ)$ was introduced by Antoniou and Vologiannidis in 2004, and a variation $\mathcal{R}_P(λ)$ was introduced by Mackey et al. in 2010. These two pencils have several appealing properties, namely they are always strong linearizations of $P(λ)$, they are easy to construct from the coefficients of $P(λ)$, the eigenvectors of $P(λ)$ can be recovered easily from those of $\mathcal{T}_P(λ)$ and $\mathcal{R}_P(λ)$, the two pencils are symmetric (resp. Hermitian) when $P(λ)$ is, and they preserve the sign characteristic of $P(λ)$ when $P(λ)$ is Hermitian. In this paper we study the numerical behavior of $\mathcal{T}_{P}(λ)$ and $\mathcal{R}_P(λ)$. We compare the conditioning of a finite, nonzero, simple eigenvalue $δ$ of $P(λ)$, when considered an eigenvalue of $P(λ)$ and an eigenvalue of $\mathcal{T}_{P}(λ)$. We also compare the backward error of an approximate eigenpair $(z,δ)$ of $\mathcal{T}_{P}(λ)$ with the backward error of an approximate eigenpair $(x,δ)$ of $P(λ)$, where $x$ was recovered from $z$ in an appropriate way. When the matrix coefficients of $P(λ)$ have similar norms and $P(λ)$ is scaled so that the largest norm of the matrix coefficients of $P(λ)$ is one, we conclude that $\mathcal{T}_{P}(λ)$ and $\mathcal{R}_P(λ)$ have good numerical properties in terms of eigenvalue conditioning and backward error. Moreover, we compare the numerical behavior of $\mathcal{T}_{P}(λ)$ with that of other well-studied linearizations in the literature, and conclude that $\mathcal{T}_{P}(λ)$ performs better than these linearizations when $P(λ)$ has odd degree and has been scaled.