# A Matrix-Less Method to Approximate the Spectrum and the Spectral Function of Toeplitz Matrices with Real Eigenvalues

## TL;DR

This paper introduces a matrix-less method to approximate the spectral distribution of Toeplitz matrices with real eigenvalues, validated through numerical experiments and capable of sometimes deriving analytical expressions.

## Contribution

It proposes a novel matrix-less algorithm for spectral approximation of Toeplitz matrices with real eigenvalues, extending previous asymptotic analysis methods.

## Key findings

- Validated the hypothesis that eigenvalues admit an asymptotic expansion with a real first function
- Successfully tested the algorithm on diverse numerical examples
- In some cases, derived analytical expressions for the spectral distribution function

## Abstract

It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of $T_n(f)$ are real for all $n$, then they admit an asymptotic expansion of the same type as considered in previous works [1,10,12,13], where the first function $g$ appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of $T_n(f)$. After validating this working hypothesis through a number of numerical experiments, drawing inspiration from [12], we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $g$. The proposed algorithm is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $g$. Future research directions are outlined at the end of the paper.