  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.

Numerical Analysis