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

Emil Cătinaş

Authors

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

Abstract views: 328

Abstract

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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