Abstract
In this paper we discuss the computational aspects of two algorithms due to E. I. Jury for determining if all the zeros of a polynomial with integer coefficients lie within the unit circle. We show that Jury's original algorithm asymptotically requires an exponential amount of computing time when variable-precision arithmetic is employed. We show that his modified algorithm requires only a polynomially bounded amount of computing time when variable-precision arithmetic is employed. Finally we produce a congruence arithmetic algorithm analogous to Jury's modified algorithm which requires less computing time than Jury's modified algorithm.
Zusammenfassung
In dieser Arbeit diskutieren wir die rechnerischen Gesichtspunkte zweier Algorithmen von E. I. Jury zur Entscheidung ob alle Nullstellen eines Polynoms mit ganzzahligen Koeffizienten im Einheitskreis liegen. Wir zeigen, daß beim ursprünglichen Algorithmus von Jury die Rechenzeit asymptotisch exponentiell mit dem Grad des Polynoms anwächst, wenn mit variable Genauigkeit gerechnet wird. Wir zeigen auch, daß unter denselben Voraussetzungen beim modifizierten Algorithmus von Jury die Rechenzeit durch eine Potenz vom Grad des Polynoms abgeschätzt werden kann. Schließlich geben wir einen auf Kongruenzen beruhenden, zum modifizierten Algorithmus von Jury analogen Algorithmus an, der aber weniger Rechenzeit als dieser benötigt.
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Anderson, P.G., Garey, M.R. & Heindel, L.E. Computational aspects of deciding if all roots of a polynomial lie within the unit circle. Computing 16, 293–304 (1976). https://doi.org/10.1007/BF02252078
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DOI: https://doi.org/10.1007/BF02252078