On Asymptotics of Eigenvalues of Seven-Diagonal Toeplitz Matrices

Abstract

Asymptotic formulas are derived that admit a uniform estimate of the remainder for Toeplitz matrices of size \(n\) as \(n \to \infty \) in the case when their symbol \(a(t)\) has the form \(a(t) = (t - 2{{a}_{0}} + {{t}^{{ - 1}}}{{)}^{3}}\). This result is a generalization of the result of Stukopin et al. (2021), who obtained similar asymptotic formulas for a seven-diagonal Toeplitz matrix with a similar symbol in the case \({{a}_{0}} = 1\). The resulting formulas are of high computational efficiency and generalize the classical results of Parter and Widom on asymptotics of extreme eigenvalues.