Abstract
In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of “recurrence relations”, analogous to those given for the Newton method by Kantorovich, improving previous results by Döring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.
Zusammenfassung
Wir betrachten ein System von a priori Fehlerabschätzungen für die Konvergenz des Halley-Verfahrens in Banachräumen. Unsere Sätze geben hinreichende Bedingungen an den Startwert, welche die Konvergenz der Halley-Iteration sichern. Sie bestehen aus einem System rekursiver Beziehungen, ähnlich den Bedingungen von Kantorovich für das Newton-Verfahren. Weitere rationale kubische Verfahren werden in einer künftigen Arbeit untersucht.
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This paper is part of the PhD dissertation, realized under the direction of the second named author.
Supported in part by C.A.I.C.Y.T. GR85-0035. University of Valencia (Spain).
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Candela, V., Marquina, A. Recurrence relations for rational cubic methods I: The Halley method. Computing 44, 169–184 (1990). https://doi.org/10.1007/BF02241866
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DOI: https://doi.org/10.1007/BF02241866