@TechReport{ it:2017-015,
author = {Sven-Erik Ekstr{\"o}m and Carlo Garoni},
title = {An Interpolation-Extrapolation Algorithm for Computing the
Eigenvalues of Preconditioned Banded Symmetric Toeplitz
Matrices},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2017},
number = {2017-015},
month = aug,
abstract = {In the past few years, Bogoya, B{\"o}ttcher, Grudsky, and
Maximenko obtained for the eigenvalues of a Toeplitz matrix
$T_n(f)$, under suitable assumptions on the generating
function $f$, the precise asymptotic expansion as the
matrix size $n$ goes to infinity. On the basis of several
numerical experiments, it was conjectured by
Serra-Capizzano that a completely analogous expansion also
holds for the eigenvalues of the preconditioned Toeplitz
matrix $T_n(u)^{-1}T_n(v)$, provided $f=v/u$ is monotone
and further conditions on $u$ and $v$ are satisfied. Based
on this expansion, we here propose and analyze an
interpolation--extrapolation algorithm for computing the
eigenvalues of $T_n(u)^{-1}T_n(v)$. We illustrate the
performance of the algorithm through numerical experiments
and we also present its generalization to the case where
$f=v/u$ is non-monotone.}
}