Always convergent methods for solving nonlinear equations

Galántai, Aurél (2015) Always convergent methods for solving nonlinear equations. JOURNAL OF COMPUTATIONAL AND APPLIED MECHANICS, 10 (2). pp. 183-208. ISSN 1586-2070

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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] $) on continuous space curves that are lying in $B$. Under the only assumption that $f$ is continuous these methods 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. Depending on the selection of the curve these methods are always convergent in the previous sense. In the paper we also investigate the selection of curves and also provide numerical test results that indicate the feasibility of the suggested methods.

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