@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.} }