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.


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


Zbl 1339.65188
Full Text: DOI Link


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