Are the Eigenvalues of Banded Symmetric Toeplitz Matrices Known in Almost Closed Form?

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained for the eigenvalues of a Toeplitz matrix, under suitable assumptions on the generating function, the precise asymptotic expansion as the matrix size goes to infinity. In this article we provide numerical evidence that some of these assumptions can be relaxed. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric Toeplitz matrices with a high level of accuracy and a relatively low computational cost.