On the convergence order of some Aitken-Steffensen type methods

Abstract

In this note we make a comparative study of the convergence orders for the Steffensen, Aitken and Aitken-Steffensen methods. We provide some conditions ensuring their local convergence. We study the case when the auxiliary operators used have convergence orders \(r_1,r_2 \in \mathbb {N}\) respectively. We show that the Steffensen, Aitken and Aitken-Steffensen methods have the convergence orders \(r_1+1\), \(r_1+r_2\) and \(r_1r_2+r_1\) respectively.

Author

Keywords

nonlinear equations in R; Steffensen, Aitken and Aitken-Steffensen methods.

References

[1] ARGYROS, I. K., On the solution of nonlinear equotions with a nonilifferentiable term, Revue d’analyse numerique et de theorie de l’approximation,22,no.2, pp.  125-135,  1993.
[2] ARGYROS, I. K., An error analysis for the Steffensen method under generalized, Zabrejlco-Nguen-type assumptions, Revue d’analyse numerique et de theorie de L’approximatìon, 25, nos. 7-2, pp.11-22, 1996.
[3] ARGYROS, I. K., Polynomial Operator Equations in Abstract Spaces and Applications, CRC PressLLC, Boca Raton, Florida, 1998.
[4] ARGYROS, L K. and SZIDAROVZKY, F., The theory and Applications of lteration Methods, C.R.C. Press, Boca Raton, Florida, 1993.
[5] BALAzs, M. and GOLDNER, G., On the approrimate solution of equations in Hilbert space by a Steffensen-type method, Revue d’analyse numerique et de theorie de l’aproximation, L7, no. 1, pp. 19-23,1gB8
[6] CATINAS, E., On Some Steffensen-type iterative methods for a class of nonlinear equations, Revue d’analyse numerique et de theorie de l’approximation, 24, nos. 1-2, pp.37-43, 1995.
[7] GRAVES, L, M., Riemann integration and Taylor’s theorem in general analysis, Trans. Amer. Math. Soc., 29, pp.
163-1 77, 1927.
[8] KANTOROVICH, L. V. and AKILOV, G. P., Functional Analysis, Pergamon Press, Oxford, 1982.
[9] PAVALOIU, I.,  Sur Ia méthode de Steffensen pour la résolution des équations-opérationnelles non lineaires, Rev. Roum. Math. Pure et Appl., XIII, no. 6, pp. 857-861, 1968.
[10] PAVALOIU, I., Sur une généralisation de la methode de Steffensen, Revue d’analyse numerique et de
theorie de l’approximation, 21, no. 1, pp. 59-65, 1992.
[11] PAVALOIU, I., Bilateral approx mations for the solutions of scalar equations, Revue d’analyse numerique et de theorie de l’approximation, 23, no. 1, pp. 95-100, 1994.

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Cite this paper as:

I. Păvăloiu, On the convergence order of some Aitken-Steffensen type methods, Rev. Anal. Numér. Théor. Approx., 32(2003) no. 2, pp. 193-202.

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1222-9024

Online ISSN

2457-8126

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