Local convergence of some Newton type methods for nonlinear systems

In order to approximate the solutions of nonlinear systems \[F(x)=0,\] with \(F:D\subseteq {\mathbb R}^n \rightarrow {\mathbb R}^n\), \(n\in {\mathbb N}\), we consider the method

\begin{align*} x_{k+1} & =x_{k}-A_{k}F(x_{k})\\A_{k+1} & =A_{k}\big(2I-F^{\prime}(x_{k+1}\big)A_{k}),\;k=0,1,…, \,A_{0}\in M_{n}({\Bbb R}), x_0 \in D,\end{align*} and we study its local convergence.

Ion Păvăloiu

nonlinear systems of equations in R^n; Schulz method.

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I. Păvăloiu, Local convergence of some Newton type methods for nonlinear systems, Rev. Anal. Numér. Théor. Approx., 33 (2004) 2, pp. 209-213.

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