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Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials | | M. I. Bueno
; F. M. Dopico
; S. Furtado
; L. Medina
; | | Date: |
13 Jun 2017 | | Abstract: | 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. | | Source: | arXiv, 1706.4150 | | Services: | Forum | Review | PDF | Favorites | |
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