Abstract
We continue the analysis of rational cubic methods, initiated in [7]. In this paper, we obtain a system of a priori error bounds for the Chebyshev method in Banach spaces through a local convergence theorem that provides sufficient conditions on the initial point in order to ensure the convergence of Chebyshev iterates. The error estimates are exact for second degree polynomials. We also discuss some applications.
Zusammenfassung
Wir betrachten ein System von a priori Fehlerabschätzungen für die Konvergenz des Chebyshev-Verfahrens in, Banachräumen. Unsere Sätze geben hinreichende Bedingungen an, den Startwert, welche die Konvergenz der Chebyshev-Iteration sichern. Sie bestehen aus einem System rekursiver Beziehungen, ähnlich den Bedingungen von Kantorvich für das Newton-Verfahren.
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References
Alefeld, G., On the convergence of Halley's method. Amer. Math. Monthly,88, 530–536 (1981).
Altman, M., Iterative methods of higher order. Bull. Acad. Pollon., Sci. (Série des sci. math., astr., phys.),IX, 62–68 (1961).
Borwein, J. M., Borwein, P. B., On the complexity of familiar functions and numbers. SIAM Rev.30, 4, 589–601 (1988).
Brent, R., On the addition of binary numbers. IEEE Trans. on Comp.,C-19, 758–759 (1970).
Brent, R., Fast multiple-precision evaluation of elementary functions. J. of the Association for Computing Machinery,23, 2, 242–251 (1976).
Brown, G. H., On Halley's variation of Newton's Method. Amer. Math. Monthly,84, 726–727 (1977).
Candela, V. F., Marquina, A., Recurrence relations, for rational cubic methods I: the Halley method. Computing44, 169–184 (1990).
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).
Hansen, E., Patrick, M., A Family of root finding methods. Numer. Math.,27, 257–269 (1977).
Kantorovich, L. V., Akilov, G. P., Functional analysis in normed spaces. Ex. Oxford: Pergamon 1964.
Knuth, D., Seminumerical algorithms, 2nd edn. Reading. Mass. Addison Wesley (1981).
Krishnamurthy, E. V., On optimal iterative schemes for high-speed division. IEEE Trans. on Comp.,C-19, 3, 227–231 (1970).
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., Rheinboldt, W. C., Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.
Potra, F. A., Pták, V., Sharp error bounds for Newton's process. Numer. Math.,34, 63–72 (1980).
Potra, F. A., 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., Lay, D., Introduction to functional analysis, 2nd edn. 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 II: The Chebyshev method. Computing 45, 355–367 (1990). https://doi.org/10.1007/BF02238803
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DOI: https://doi.org/10.1007/BF02238803