# On some Steffensen-type iterative methods for a class of nonlinear equations

## Abstract

Consider the nonlinear equations $$H(x):=F(x)+G(x)=0$$, with $$F$$ differentiable and $$G$$ continuous, where $$F,G,H:X \rightarrow X$$ are nonlinear operators and $$X$$ is a Banach space.

The Newton method for solving the nonlinear equation $$H(x)=0$$ cannot be applied, and we propose an iterative method for solving this equation by combining the Newton method with the Steffensen method: $x_{k+1} = \big(F^\prime(x_k)+[x_k,\varphi(x_k);G]\big)^{-1}(F(x_k)+G(x_k)),$ where $$\varphi(x)=x-\lambda (F(x)+G(x))$$, $$\lambda >0$$ fixed.

The method is obtained by combining the Newton method for the differentiable part with the Steffensen method for the nondifferentiable part.

We show that the R-convergence order of this method is 2, the same as of the Newton method.

We provide some numerical examples and compare different methods for a nonlinear system in $$\mathbb{R}^2$$.

## Authors

E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

## Keywords

nonlinear equation; Banach space; Newton method; Steffensen method; combined method; nondifferentiable mapping; nonsmooth mapping; r-convergence order.

## Cite this paper as:

E. Cătinaş, On some Steffensen-type iterative methods for a class of nonlinear equations, Rev. Anal. Numér. Théor. Approx., 24 (1995) nos. 1-2, pp. 37-43.

2457-6794

2501-059X

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## References

[1] I.K. Argyros, On the secant method and the Ptak error estimates, Rev. Anal. Numer. Theor. Approx., 24 (1995) nos. 1–2, pp. 3–14.
article on journal website

[2] M. Balazs, A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Numer. Theor. Approx., 21 (1992) no. 2, pp. 111–117.
article on journal website

[3] E. Catinas, On some iterative methods for solving nonlinear equations, Rev. Anal. Numer. Theor. Approx., 23 (1994) no. 1, pp. 47–53.
article on journal website, article on post

[4] G. Goldner, M. Balazs, Asupra metodei coardei si a unei modificari a ei pentru rezolvarea ecuațiilor operationale neliniare, Stud. Cerc. Mat., 20 (1968), pp. 981–990. [English title: On the method of chord and on its modification for solving the nonlinear operator equations]

[5] G. Goldner, M. Balazs, Observații asupra diferențelor divizate și asupra metodei coardei, Revista de Analiza Numerica si Teoria Aproximatiei, 3 (1974) no. 1, pp. 19–30 (in Romanian).
[English title: Remarks on divided differences and method of chords]
article on journal website

[6] L.V. Kantorovici, G.P. Akilov, Functional Analysis, Editura Stiintifica si Enciclopedica, Bucuresti, 1986 (in Romanian).

[7] I. Pavaloiu, On the monotonicity of the sequences of approximations obtained by Steffensen’s method, Mathematica, 35(58) (1993) no. 1, pp. 71–76.
article on post

[8] T. Yamamoto, A note on a posteriori error bound of Zabrejko and Nguen for Zincenko’s iteration, Numer. Funct. Anal. Optimiz., 9 (1987) nos. 9&10, pp. 987–994.
CrossRef

[9] T. Yamamoto, Ball convergence theorems and error estimates for certain iterative methods for nonlinear equations, Japan Journal of Applied Mathematics, 7 (1990) no. 1, pp. 131–143.
CrossRef

[10] X. Chen, T. Yamamoto, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optimiz., 10 (1989) 1&2, pp. 37–48.
CrossRef