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

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.