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On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic

Über eine Verallgemeinerung der Wurzeliterationen für komplexe Polynomnullstellen in Kreis-Intervallarithmetik

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Abstract

Consider a polynomialP (z) of degreen whose zeros are known to lie inn closed disjoint discs, each disc containing one and only one zero. Starting from the known simultaneous interval processes of the third and fourth order, based on Laguerre iterations, two generalised iterative methods in terms of circular regions are derived in this paper. These interval methods make use of the definition of thek-th root of a disc. The order of convergence of the proposed interval methods isk+2 (k≧1). Both procedures are suitable for simultaneous determination of interval approximations containing real or complex zeros of the considered polynomialP. A criterion for the choice of the appropriatek-th root set is also given. For one of the suggested methods a procedure for accelerating the convergence is proposed. Starting from the expression for interval center, the generalised iterative method of the (k+2)-th order in standard arithmetic is derived.

Zusammenfassung

Betrachten wir ein Polynomn-ten Grades, für welches bekannt ist, daß die Nullstellen inn geschlossenen disjunkten Kreisscheiben liegen, wobei jedes Kreisgebiet eine und nur eine Nullstelle enthält. Ausgehend von den bekannten simultanen Intervallverfahren dritter und vierter Ordnung, basierend auf den Laguerreschen Iterationen, werden zwei verallgemeinte Iterationsverfahren für Kreisscheiben formuliert. Diese Intervallmethoden benützen die Definition derk-ten Wurzel einer Kreisscheibe. Die Konvergenzordnung der vorgeschlagenen Intervallverfahren istk+2 (k≧1). Beide Verfahren sind günstig für die simultane Bestimmung von Intervallapproximationen, die die reellen oder komplexen Nullstellen des gegebenen Polynoms enthalten. Ein Kriterium für die Wahl der richtigenk-Wurzelmenge wird angegeben. Für eine der vorgeschlagenen Methoden wird ein Verfahren zur Beschleunigung der Konvergenz angegeben. Ausgehend von Ausdruck für Intervallzentrum ist ein verallgemeintes Iterationsverfahren der Ordnungk+2 für Klassische Arithmetik formuliert.

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  1. Faculty of Electronic Engineering, Beogradska 14, YU-18 000, Niš, Jugoslavia

    M. S. Petković

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  1. M. S. Petković
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Petković, M.S. On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic. Computing 27, 37–55 (1981). https://doi.org/10.1007/BF02243437

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  • DOI: https://doi.org/10.1007/BF02243437

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