# Bilateral approximations of solutions of equations by order three Steffensen-type methods

## Abstract

We study the convergence of a method of Steffensen-type, which is obtained from the Lagrange polynomial of inverse interpolation with controlled nodes. Conditions are given such that the sequences bilaterally approximate the solution of an equation.

## Keywords

nonlinear equations in R; Steffensen type method; monotone iterations.

## References

[1] Costabile, F., Gualtieri, I.M., Luceri, R., A new iterative method for the computation of the solution of nonlinear equations, Numer. Algorithms, 28, pp. 87-100, 2001.

[2] Frontini, M., Hermite interpolation and a new iterative method for the computation of the roots of non-linear equations, Calcolo 40, pp. 109-119, 2003.

[3] Grau, M., An improvement to the computing of nonlinear equation solutions, Numer. Algorithms. 34, pp.1-12, 2003.

[4] Ostrowski, A., Solution of Equations in Euclidian and Banach Spaces, Academic Press New York and London, 1973.

[5] Pavaloiu, I., Optimal efficiency index for iterative methods of interpolatory type, Computer Science Journal of Moldova 1, 5, pp. 20-43,1997. 113

[6] Pavaloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences, Calcolo 32, 1-2, pp.69-82, 1995.

[7] Pavaloiu, I., Optimal problems concerning interpolation methods of solution of equations, Publications de L’Institut Mathematique 52 (66) pp 113-126, 1992.

[8] Pavaloiu, I., Optimal effiency index of a class of Hermite iterative methods, with two steps, Rev. Anal. Numer. Theor. Approx. 29, 1, pp. 83-89, 2000.

[9] Pavaloiu, I., Local convergence of general Steffensen type methods, Rev. Anal. Numer. Theor. Approx. 33,1, 79-86, 2004.

[10] Pavaloiu, I., Pop, N., Interpolation and Applications, Risoprint, Cluj-Napoca, 2005 (in Romanian).

[11] Traub, J.F., Iterative Methods for Solutions of Equations, Pretence-Hall Inc., Englewood Cliffs, New Jersey, 1964.

[12] Turowicz, B.A., Sur les derivees d’ordre superieur d’une function inverse, Ann. Polon. Math. 8 pp. 265-269, 1960.

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##### Cite this paper as:

I. Păvăloiu, Bilateral approximations of solutions of equations by order three Steffensen-type methods, Studia Univ. Babeş-Bolyai, Mathematica, vol. 51 (2006) no. 3, pp. 105-114.

##### Publisher Name

Babes-Bolyai University

0252-1938

2065-961x