On Asymptotics of Eigenvalues of Seven-Diagonal Toeplitz Matrices

Pub Date : 2024-07-18 DOI:10.1134/s0965542524700404

I. V. Voronin

引用次数: 0

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.

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论七条对角线托普利兹矩阵特征值的渐近性

摘要本文导出了一个渐近公式，当符号\(a(t)\)的形式为\(a(t) = (t - 2{{a}_{0}} + {{t}^{{ - 1}}}{)}^{3}}\) 时，可以对大小为 \(n\) 的托普利兹矩阵的余数进行统一估计。这一结果是对 Stukopin 等人（2021 年）结果的推广，他们在 \({{a}_{0}} = 1\) 的情况下，为具有类似符号的七对角托普利兹矩阵获得了类似的渐近公式。所得到的公式具有很高的计算效率，并推广了帕特和维多姆关于极值特征值渐近的经典结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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