Local convergence of general Steffensen type methods

Ion Păvăloiu
(original), paper

Abstract

We study the local convergence of a generalized Steffensen method. We show that this method substantially improves the convergence order of the classical Steffensen method. The convergence order of our method is greater or equal to the number of the controlled nodes used in the Lagrange-type inverse interpolation, which, in its turn, is substantially higher than the convergence orders of the Lagrange type inverse interpolation with uncontrolled nodes (since their convergence order is at most (2)).

Author

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

nonlinear equations in R; Steffensen method.

PDF

PDF-LaTeX file (on the journal website).

Cite this paper as:

I. Păvăloiu, Local convergence of general Steffensen type methods, Rev. Anal. Numér. Théor. Approx., 33 (2004) 1, pp. 79-86.

About this paper

Journal

Revue d’Analyse Numérique et de Théorie de l’Approximation

Publisher Name

Editions de l’Academie Roumaine

Article on the journal website

http://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no1-art10

Print ISSN

1222-9024

Online ISSN

2457-8126

References

[1] Balazs, M., A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Num ́er. Theor. Approx., 21 , no. 2, pp. 111–117, 1992.
[2] Brent, R., Winograd, S. and Walfe, Ph., Optimal iterative processes for root-finding, Numer. Math., 20 , no. 5, pp. 327–341, 1973.
[3] Coman, C., Some practical approximation methods for nonlinear equations, Mathematica – Rev. Anal. Num ́er. Theor. Approx., 11, nos. 1–2, pp. 41–48, 1982.
[4] Cassulli, V. and Trigiante, D., The convergence order for iterative multipoint procedures, Calcolo, 13, no. 1, pp. 25–44, 1977.
[5] Iancu, C., Pavaloiu, I. and Serb, I., Methodes iteratives optimales de type Steffensen obtinues par interpolation inverse , Faculty of Mathematics, “Babes-Bolyai” University, Seminar on Functional analysis and Numerical Methods, Preprint no.1, pp. 81–88, 1983.
[6] Kacewicz, B., An integral-interpolation iterative method for the solution of scalar equations, Numer. Math., 26 , no. 4, pp. 355–365, 1976.
[7] Ostrowski, A., Solution of Equations in Euclidian and Banach Spaces, Academic Press, New York and London, 1973.
[8] Pavaloiu, I., La resolution des equations par interpolation, Mathematica, 23(46), no. 1, pp. 61–72, 1981.
[9] Pavaloiu, I. and Serb, I., Sur des methodes de type interpolatoire `a vitesse de convergence optimale, Rev. Anal. Numer. Theor. Approx., 12 , no. 1, pp. 83–88, 1983.
[10] Pavaloiu, I., Optimal efficiency index for iterative methods of interpolatory type, Computer Science Journal of Moldova, 5 , no. 1 (13), pp. 20–43, 1997.
[11] Traub, J. F., Iterative Methods for the Solution of Equations, Pretince-Hall, Inc. Englewood Clifs, N.J., 1964.
[12] Turowicz, A. B., Sur les deriv ́ees d’ordre superieur d’une fonction inverse, Ann. Polon. Math., 8, pp. 265–269, 1960

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