A Matrix-Less Method to Approximate the Spectrum and the Spectral Function of Toeplitz Matrices with Real Eigenvalues
It addresses the problem of approximating eigenvalue distributions for Toeplitz matrices with non-real generating functions, but the approach is incremental and relies on a working hypothesis.
The paper proposes a matrix-less algorithm to approximate the eigenvalue distribution function of Toeplitz matrices with real eigenvalues, validated through numerical experiments where analytical expressions are sometimes recovered.
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.