Abstract
This paper discusses a set of algorithms which, given a polynomial equation with integer coefficients and without any multiple roots, uses exact (infinite precision) integer arithmetic and the Vincent-Uspensky-Akritas theorem to compute intervals containing the real roots of the polynomial equation. Theoretical computing time bounds are developed for these algorithms which are proven to be the fastest existing; this fact is also verified by the empirical results which are included in this article.
Zusammenfassung
Es werden einige Algorithmen diskutiert, die unter Verwendung exakter ganzzahliger Arithmetik und des Vincent-Uspensky-Akritas-Theorems für ein gegebenes Polynom mit ganzzahligen Koeffizienten und ohne mehrfache Wurzeln Intervalle berechnen, die die reellen Nullstellen des Polynoms enthalten. Für diese Algorithmen werden theoretische Rechenzeitschranken entwickelt, und es wird bewiesen und durch empirische Resultate belegt, daß diese Algorithmen die schnellsten unter den bisher existierenden sind.
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Akritas, A.G. The fastest exact algorithms for the isolation of the real roots of a polynomial equation. Computing 24, 299–313 (1980). https://doi.org/10.1007/BF02237816
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DOI: https://doi.org/10.1007/BF02237816