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In this paper, we assume as a working hypothesis that, if the eigenvalues of T<\/jats:italic>n<\/jats:italic><\/jats:sub>(f<\/jats:italic>) are real for all n<\/jats:italic>, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\\mathfrak {f}$<\/jats:tex-math>\n f<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of T<\/jats:italic>n<\/jats:italic><\/jats:sub>(f<\/jats:italic>). This eigenvalue symbol $\\mathfrak {f}$<\/jats:tex-math>\n f<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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 $\\mathfrak {f}$<\/jats:tex-math>\n f<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The proposed algorithm, which opposed to previous versions, does not need any information about neither f<\/jats:italic> nor $\\mathfrak {f}$<\/jats:tex-math>\n f<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\\mathfrak {f}$<\/jats:tex-math>\n f<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Future research directions are outlined at the end of the paper.<\/jats:p>","DOI":"10.1007\/s11075-021-01130-9","type":"journal-article","created":{"date-parts":[[2021,6,23]],"date-time":"2021-06-23T23:04:53Z","timestamp":1624489493000},"page":"701-720","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues"],"prefix":"10.1007","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7875-7543","authenticated-orcid":false,"given":"Sven-Erik","family":"Ekstr\u00f6m","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2131-7643","authenticated-orcid":false,"given":"Paris","family":"Vassalos","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,24]]},"reference":[{"key":"1130_CR1","unstructured":"Generic Linear Algebra.jl. https:\/\/github.com\/JuliaLinearAlgebra\/GenericLinearAlgebra.jl"},{"key":"1130_CR2","doi-asserted-by":"publisher","first-page":"867","DOI":"10.1007\/s11075-017-0404-z","volume":"78","author":"F Ahmad","year":"2017","unstructured":"Ahmad, F., Al-Aidarous, E. 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