A Matrix-Less Method to Approximate the Spectrum and the Spectral Function of Toeplitz Matrices with Real Eigenvalues
Abstract
It is known that the generating function f of a sequence of Toeplitz matrices \Tn(f)\n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(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 Tn(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.