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

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2457-6794

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2501-059X

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References

[1] G. Goldner, M. Balazs, On the method of the chord and on a modification of it for the solution of nonlinear operator equations, Stud. Cerc. Mat., 20 (1968), pp. 981–990 (in Romanian).

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[3] T. Yamamoto, A note on a posteriori error bound of Zabrejko and Nguen for Zincenko’s iteration, Numer. Funct. Anal. Optimiz., 9 (1987) 9&10, pp. 987–994.
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[4] T. Yamamoto, Ball convergence theorems and error estimates for certain iterative methods for nonlinear equations, Japan J. Appl. Math., 7 (1990) no. 1, pp. 131–143.
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[5] X. Chen, T. Yamamoto, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optimiz., 10 (1989) 1&2, pp. 37–48.
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