@TechReport{ it:2016-017,
author = {S.-E. Ekstr{\"o}m and S. Serra-Capizzano},
title = {Eigenvalues of Banded Symmetric {T}oeplitz Matrices are
Known Almost in Close Form?},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2016},
number = {2016-017},
month = sep,
abstract = {It is well-known that the eigenvalues of (real) symmetric
banded Toeplitz matrices of size $n$ are approximately
given by an equispaced sampling of the symbol $f(\theta)$,
up to an error which grows at most as $h=(n+1)^{-1}$, where
the symbol is a real-valued cosine polynomial.
Under the condition that $f$ is monotone, we show that
there is hierarchy of symbols so that
\[ \lambda_{j}^{(h)}-f\left(\theta_{j}^{(h)}\right)=\sum_k
c_k\left(\theta_{j}^{(h)}\right)\, h^k,\quad \quad
\theta_j^{(h)}=j\pi h, j=1,\ldots,n, \]
with $c_k(\theta)$ higher order symbols. In the general
case, a more complicate expression holds but still we find
a structural hierarchy of symbols. The latter asymptotic
expansions constitute a starting point for computing the
eigenvalues of large symmetric banded Toeplitz matrices by
using classical extrapolation methods.
Selected numerics are shown in 1D and a similar study is
briefly discussed in the multilevel setting ($d$D, $d\ge
2$) with blocks included, so opening the door to a fast
computation of the spectrum of matrices approximating
partial differential operators.}
}