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


  • Emil Cătinaş Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy



Let \(H(x):=F(x)+G(x)=0\), 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 cannot be applied for solving the nonlinear equation \(H(x)=0\), 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 \(\mathbb{R}^2\).


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Argyros, I.K., On the Secant Method and the Pták Error Estimates. Revue d'Analyse Numérique et de Théorie de l'Approximation, (to appear).

Balázs, M., A Bilateral Approximating Method for Finding the Real Roots of Real Equations. Revue d'analyse Numérique et de Théorie de l'Approximation 21, 2 (1992), pp. 111-117.

Cătinaş, E., On some Iterative Methods for Solving Nonlinear Equations, Revue d'Analyse Numérique et de Théorie de l'Approximation, 23, 1 (1994), pp. 47-53.

Goldner, G., Balázs, M., On the Method of Chord and on Its Modification for Solving the Nonlinear Operatorial Equaitons, Studii şi cercetări matematice, 20 (1968).

Goldner, G. Balázs, M., Remarks on Divided Differences and Method of Chords, Revista de Analiză Numerică şi Teoria Aproximaţiei, 3, 1, (1974), pp. 19-30.

Kantorovici, L.V., Akilov, G.P., Analiza Funcţională, Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1986.

Păvăloiu, I., On the Monotonicity of the Sequences of Approximations Obtained by Steffensen's Method, Mathematica, 35(58), 1 (1993), pp. 71-76.

Tetsuro, Y., A Note on a Posteriori Error Bound of Zabrejko and Nguen for Zincenko's Iteration, Numer. Funct. Anal. and Optimiz., 9, 9 & 10, (1987), pp. 987-994, https://doi.org/10.1080/01630568708816270.

Tetsuro, Y., Ball Convergence Theorems and Error Estimates for Certain Iterative Methods for Nonlinear Equations, Japan Journal of Applied Mathematics, 7, 1 (1990), pp. 131-143, https://doi.org/10.1007/BF03167895.

Xiaojun, C., Tetsuro, Y., Convergence Domains of Certain Iterative Methods for Solving Nonlinear Equations, Numer. Funct. Anal. and Optimz., 10 (1 & 2), 1989, pp. 37-48, https://doi.org/10.1080/01630568908816289.




How to Cite

Cătinaş, E. (1995). On some Steffensen-type iterative methods for a class of nonlinear equations. Rev. Anal. Numér. Théor. Approx., 24(1), 37–43. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art4