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

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