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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References
Alefeld, G.: On the Convergence of Halley's Method. Amer. Math. Monthly,88, 530–536 (1981)
Brown, G. H.: On Halley's variation of Newton's Method. Amer. Math. Monthly,84, 726–727 (1977)
Davies, M. and Dawson, B.: On the global convergence of Halley'siteration formula. Num. Math.,24, 133–135 (1975)
Döring, B.: Einige sätze über das Verfahren der tangierenden Hyperbeln in Banach-Räumen. Aplikace Mat.15, 418–464 (1970)
Ehrmann, H.: Konstruktion und Durchfürung von Iterationsverfahren höherer Ordnung. Arch. Rational Mech. Anal.,4, 65–88 (1959)
Gander, W.: On Halley's iteration method. Amer. Math. Monthly,92, 131–134 (1985)
Kantorovich, L. V. and Akilov, G. P.: Functional Analysis in normed spaces. Ed. Oxford: Pergamon 1964
Miel, G. J.: The Kantorovich Theorem with optimal error bounds. Amer. Math. Monthly,86, 212–215 (1979)
Miel, G. J.: An updated version of the Kantorovich theorem for Newton's method. Computing,27, 237–244 (1981)
Ortega, J. M.: The Newton-Kantorovich theorem. Amer. Math. Monthly,75, 658–660 (1968)
Ortega, J. M. and Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. Ed. New York: Academic Press 1970
Potra, F. A. and Pták, V.: Sharp error bounds for Newton's process. Numer. Math.,34, 63–72 (1980)
Potra, F. A. and Pták, V.: Nondiscrete induction and iterative processes. Research Notes in Mathematics,103, Ed. Boston: Pitman 1984
Tapia, R. A.: The Kantorovich theorem for Newton's method. Amer. Math. Monthly,78, 389–392 (1971)
Taylor, A. Y. and Lay, D.: Introduction to Functional Analysis. 2nd. Ed. New York: J. Wiley, 1980
Yamamoto, T.: A method for finding sharp error bounds for Newton's Method under the Kantorovich Assumptions. Numer. Math.49, 203–220 (1986)
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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