Eigenvalue Clusters of Large Tetradiagonal Toeplitz Matrices

Toeplitz matrices are typically non-Hermitian and hence they evade the well-elaborated machinery one can employ in the Hermitian case. In a pioneering paper of 1960, Palle Schmidt and Frank Spitzer showed that the eigenvalues of large banded Toeplitz matrices cluster along a certain limiting set which is the union of finitely many closed analytic arcs. Finding this limiting set nevertheless remains a challenge. We here present an algorithm in the spirit of Richard Beam and Robert Warming that reduces testing O(N2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^2)$$\end{document} points in the plane for membership in the limiting set by testing only O(N) points along a one-dimensional curve. For tetradiagonal Toeplitz matrices, we describe all types of the limiting sets, we classify their exceptional points, and we establish asymptotic formulas for the analytic arcs near their endpoints.