## 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

## Keywords

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

## 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.Scanned paper.

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## About this paper

##### 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.

##### Publisher Name

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##### Print ISSN

1222-9024

##### Online ISSN

2457-8126