Computing

, Volume 45, Issue 4, pp 355–367

Recurrence relations for rational cubic methods II: The Chebyshev method

  • V. Candela
  • A. Marquina
Article

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.

AMS Subject Classification (1980)

Primary 65J15 

Key words

Third order iterative methods a priori error bounds non-linear equations 

Rekursions-Beziehungen für rationale kubische Verfahren II: Die Chebyshev Methode

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

  1. [1]
    Alefeld, G., On the convergence of Halley's method. Amer. Math. Monthly,88, 530–536 (1981).Google Scholar
  2. [2]
    Altman, M., Iterative methods of higher order. Bull. Acad. Pollon., Sci. (Série des sci. math., astr., phys.),IX, 62–68 (1961).Google Scholar
  3. [3]
    Borwein, J. M., Borwein, P. B., On the complexity of familiar functions and numbers. SIAM Rev.30, 4, 589–601 (1988).Google Scholar
  4. [4]
    Brent, R., On the addition of binary numbers. IEEE Trans. on Comp.,C-19, 758–759 (1970).Google Scholar
  5. [5]
    Brent, R., Fast multiple-precision evaluation of elementary functions. J. of the Association for Computing Machinery,23, 2, 242–251 (1976).Google Scholar
  6. [6]
    Brown, G. H., On Halley's variation of Newton's Method. Amer. Math. Monthly,84, 726–727 (1977).Google Scholar
  7. [7]
    Candela, V. F., Marquina, A., Recurrence relations, for rational cubic methods I: the Halley method. Computing44, 169–184 (1990).Google Scholar
  8. [8]
    Döring, B., Einige Sätze über das Verfahren der tangierenden Hyperbeln in Banach-Räumen. Aplikace Mat.,15, 418–464 (1970).Google Scholar
  9. [9]
    Ehrmann, H., Konstruktion und Durchfürung von Iterationsverfahren höherer Ordnung. Arch. Rational Mech. Anal.,4, 65–88 (1959).Google Scholar
  10. [10]
    Gander, W., On Halley's iteration method. Amer. Math. Monthly,92, 131–134 (1985).Google Scholar
  11. [11]
    Hansen, E., Patrick, M., A Family of root finding methods. Numer. Math.,27, 257–269 (1977).Google Scholar
  12. [12]
    Kantorovich, L. V., Akilov, G. P., Functional analysis in normed spaces. Ex. Oxford: Pergamon 1964.Google Scholar
  13. [13]
    Knuth, D., Seminumerical algorithms, 2nd edn. Reading. Mass. Addison Wesley (1981).Google Scholar
  14. [14]
    Krishnamurthy, E. V., On optimal iterative schemes for high-speed division. IEEE Trans. on Comp.,C-19, 3, 227–231 (1970).Google Scholar
  15. [15]
    Miel, G. J. The Kantorovich Theorem with optimal error bounds., Amer. Math. Monthly,86, 212–215 (1979).Google Scholar
  16. [16]
    Miel, G. J., An updated version of the Kantorovich theorem for Newton's method. Computing,27, 237–244 (1981).Google Scholar
  17. [17]
    Ortega, J. M., The Newton-Kantorovich theorem. Amer. Math. Monthly,75, 658–660 (1968).Google Scholar
  18. [18]
    Ortega, J. M., Rheinboldt, W. C., Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.Google Scholar
  19. [19]
    Potra, F. A., Pták, V., Sharp error bounds for Newton's process. Numer. Math.,34, 63–72 (1980).Google Scholar
  20. [20]
    Potra, F. A., Pták, V., Nondiscrete induction and iterative processes. Research Notes in Mathematics,103 Ed. Boston: Pitman 1984.Google Scholar
  21. [21]
    Tapia, R. A., The Kantorovich theorem for Newton's method. Amer. Math. Monthly,78, 389–392 (1971).Google Scholar
  22. [22]
    Taylor, A. Y., Lay, D., Introduction to functional analysis, 2nd edn. New York: J. Wiley, 1980.Google Scholar
  23. [23]
    Yamamoto, T., A method for finding sharp error bounds for Newton's Method under the Kantorovich Assumptions. Numer. Math.,49, 203–220 (1986).Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • V. Candela
    • 1
  • A. Marquina
    • 1
  1. 1.Departamento de Análisis MatemáticoBurjassot, ValenciaSpain

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