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Always convergent methods for nonlinear equations of several variables

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Abstract

We develop always convergent methods for solving nonlinear equations of the form \(f\left (x\right ) =0\) (\(f:\mathbb {R}^{n}\rightarrow \mathbb {R}^{m}\), \(x\in B=\times _{i=1}^{n}\left [ a_{i},b_{i}\right ] \)) under the assumption that f is continuous on B. The suggested methods use continuous space curves lying in the rectangle B and have a kind of monotone convergence to the nearest zero on the given curve, if it exists, or the iterations leave the region in a finite number of steps. The selection of space curves is also investigated. The numerical test results indicate the feasibility and limitations of the suggested methods.

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The author is indebted to the unknown referee whose remarks and suggestions improved the paper.

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  1. Óbuda University, 1034, Budapest, Bécsi u. 96/b, Hungary

    A. Galántai

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  1. A. Galántai
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Galántai, A. Always convergent methods for nonlinear equations of several variables. Numer Algor 78, 625–641 (2018). https://doi.org/10.1007/s11075-017-0392-z

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