Aitken-Steffensen-type methods for nonsmooth functions (III)

Ion Păvăloiu
(original), paper

Abstract

We provide sufficient conditions for the convergence of the Steffensen method for solving the scalar equation \(f(x)=0\), without assuming differentiability of \(f\) at other points than the solution \(x^\ast\). We analyze the cases when the Steffensen method generates two sequences which approximate bilaterally the solution.

Author

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

nonlinear equations in R; Aitken-Steffensen method; monotone iterations; bilateral approximations.

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Cite this paper as:

I. Păvăloiu, Aitken-Steffensen-type methods for nonsmooth functions (III), Rev. Anal. Numér. Théor. Approx., 32 (2003) no. 1, pp. 73-77.

About this paper

Journal

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

Publisher Name

Editions de l’Academie Roumaine

DOI

https://doi.org/10.33993/jnaat321-736

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. Numer. Theor.  Approx., 21 no. 2, pp. 111–117, 1992.
[2] Casulli, V. and Trigiante, D. The convergence order for iterative multipoint procedures, Calcolo, 13, no. 1, pp. 25–44, 1977.
[3] Cobzas, S., Mathematical Analysis , Presa Universitara Clujeana, Cluj-Napoca, 1997 (in Romanian).
[4] Ostrowski, A. M., Solution of Equations and Systems of Equations, Academic Press, New York, 1960.
[5] Pavaloiu, I., On the monotonicity of the sequences of approximations obtained by Steffensens’s method, Mathematica (Cluj),35(58), no. 1, pp. 71–76, 1993.
[6] Pavaloiu, I., Bilateral approximations for the solutions of scalar equations, Rev. Anal.Numer. Theor. Approx., 23, no. 1, pp. 95–100, 1994.
[7] Pavaloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences, Calcolo,  32, nos. 1-2, pp. 69–82, 1995.
[8] Pavaloiu, I., Aitken-Steffensen-type methods for nonsmooth functions (I), Rev. Anal. Numer. Theor. Approx., 31, no. 1, pp. 111–116, 2002.
[9] Pavaloiu, I., Aitken–Steffensen type methods for nonsmooth functions (II) , Rev. Anal. Numer. Theor. Approx., 31, no. 2, pp. 203–206, 2002.
[10] Traub, F. J., Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
2003

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