On some iterative methods for solving nonlinear equations

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 \(\mathbb{R}^2\).

E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

nonlinear equation; Banach space; Newton method; chord method; secant method; combined method; nondifferentiable mapping; nonsmooth mapping; R-convergence order.

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

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