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A parameterized multi-step Newton method for solving systems of nonlinear equations. (English) Zbl 1350.65046
The authors introduce a new multi-step method solving systems nonlinear equations \(\mathbf{F}(\mathbf{x})=0\), where \(\mathbf{F}: \Gamma \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^r\) is Fréchet differentiable at \(\mathbf{x}\in\mathrm{interior}(\Gamma)\) with \(\mathbf{F}(\mathbf{x^\ast})=0\) and \(\det(\mathbf{F}'(x^\ast))\neq 0\). They prove that the method needs \(m\) steps to obtain \(m+1\) convergence order. The method is a generalization of the multi-step Newton method based on a parameter \(\theta\). Applying the method for solving the nonlinear complex Zakharov system [A. H. Bhrawy, Appl. Math. Comput. 247, 30–46 (2014; Zbl 1339.65188)], the authors show that the appropriate choice of \(\theta\) leads to faster convergence and larger radius of convergence.

MSC:
65H10 Numerical computation of solutions to systems of equations
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
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