Local convergence of some Newton-type methods for nonlinear systems

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Ion Păvăloiu

Abstract

In order to approximate the solutions of nonlinear systems\[F(x)=0,\]with \(F:D\subseteq {\mathbb R}^{n}\rightarrow {\mathbb R}^{n}\),\(n\in {\Bbb N}\), we consider the method\begin{align*}x_{k+1} & =x_{k}-A_{k}F(x_{k})\label{f1.4}\\A_{k+1} & =A_{k}(2I-F^{\prime}(x_{k+1})A_{k}),\;k=0,1,..., \,A_{0}\in M_{n}({\Bbb R}), x_0 \in D,\end{align*}and we study its local convergence.

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Keywords
nonlinear systems of equations
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