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, where the first function, called the eigenvalue symbol f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of T n ( f ). This eigenvalue symbol f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of f . Future research directions are outlined at the end of the paper.