# On some iterative methods for solving nonlinear equations

## Abstract

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

The Newton method for solving $$H(x)=0$$ cannot be applied, and we propose an iterative method for solving the nonlinear equation, by combining the Newton method (for the differentiable part) with the chord/secant method (for the nondifferentiable part): $x_{k+1} = \big(F^\prime(x_k)+[x_{k-1},x_k;G]\big)^{-1}(F(x_k)+G(x_k)).$

We show that the r-convergence order of the method is the same as of the chord/secant 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; chord method; secant method; combined method; nondifferentiable mapping; nonsmooth mapping; R-convergence order.

## Cite this paper as:

E. Cătinaş, On some iterative methods for solving nonlinear equations, Rev. Anal. Numér. Théor. Approx., 23 (1994) no. 1, pp. 47-53

2457-6794

2501-059X

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