Eigenvalues and quasi-eigenvalues of banded Toeplitz matrices: some properties and applications

In this paper we consider the spectrum and quasi-eigenvalues of a family of banded Toeplitz matrices and define their extensions to the generalized eigenvalue problem. A diagonal similarity transformation on such matrices that allows a suitable modification of the region containing the quasi-eigenvalues is reported. Two kind of applications have been analyzed: the computation of the eigenvalues and the asymptotic spectra of Toeplitz matrices and the solution of block banded quasi-Toeplitz linear systems that arise after the discretization of an ODE using a boundary value method.