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.
Scanned paper.
About this paper
Publisher Name
Paper on journal website
Print ISSN
2457-6794
Online ISSN
2501-059X
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