@TechReport{ it:2017-012,
author = {Fayyaz Ahmad and Eman Salem Al-Aidarous and Dina Abdullah
Alrehaili and Sven-Erik Ekstr{\"o}m and Isabella Furci and
Stefano Serra-Capizzano},
title = {Are the Eigenvalues of Preconditioned Banded Symmetric
{T}oeplitz Matrices Known in Almost Closed Form?},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2017},
number = {2017-012},
month = jun,
abstract = {Bogoya, B{\"o}ttcher, Grudsky, and Maximenko have recently
obtained the precise asymptotic expansion for the
eigenvalues of a sequence of Toeplitz matrices
$\{T_n(f)\}$, under suitable assumptions on the associated
generating function $f$. In this paper we provide numerical
evidence that some of these assumptions can be relaxed and
extended to the case of a sequence of preconditioned
Toeplitz matrices $\{T_n^{-1}(g)T_n(f)\}$, for $f$
trigonometric polynomial, $g$ nonnegative, not identically
zero trigonometric polynomial, $r=f/g$, and where the ratio
$r(\cdot)$ plays the same role as $f(\cdot)$ in the
nonpreconditioned case. Moreover, based on the eigenvalue
asymptotics, we devise an extrapolation algorithm for
computing the eigenvalues of preconditioned banded
symmetric Toeplitz matrices with a high level of accuracy,
with a relatively low computational cost, and with
potential application to the computation of the spectrum of
differential operators.}
}