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It is known that the generating function $f$ of a sequence of Toeplitzmatrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of theeigenvalues of $T_n(f)$ if $f$ is not real. In a recent paper, we assume as aworking hypothesis that, if the eigenvalues of $T_n(f)$ are real for all $n$,then they admit an asymptotic expansion where the first function $g$ appearingin this expansion is real and describes the asymptotic distribution of theeigenvalues of $T_n(f)$. In this paper we extend this idea to Toeplitz matriceswith complex eigenvalues. The paper is predominantly a numerical exploration ofdifferent typical cases, and presents several avenues of possible futureresearch.

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