An efficient fourth order weighted-Newton method for systems of nonlinear equations
Given a system of \(n\) nonlinear equations in \(n\) unknowns, the quadratically convergent Newton's method is a basic procedure for finding its zero set. This method requires the inverse of the first Fréchet derivative, and hence \(n+n^2\) evaluations per iteration. Many efforts have been made to improve the convergence without increasing the number of evaluations by too much, thereby speeding up the overall calculation. In this paper, using Taylor's expansion on vector functions, the authors show that Theorem 1: Let \(\mathbf{F}:D \subseteq \mathbb R^n \rightarrow \mathbb R^n\) be four times Fréchet differentiable in a convex set \(D\) containing the root \(\mathbf{r}\) of \(\mathbf{F}(\mathbf{x}) = 0\). Then the sequence \([\mathbf{x}^{(k)}]_{k \geq 0}\) (\(\mathbf{x}^{(0)} \in D\)) defined by \[ \mathbf{y}^{(k)} = \mathbf{x}^{(k)}-\theta\mathbf{F}'(\mathbf{x}^{(k})^{-1} \mathbf{F}(\mathbf{x}^{(k)}) \] and \[ \mathbf{x}^{(k+1)} = \mathbf{x}^{(k)}-[a_1\mathbf{I}+a_2\mathbf{F}' (\mathbf{y}^{(k)})^{-1}\mathbf{F}'(\mathbf{x}^{(k)})+a_3\mathbf{F}' (\mathbf{x}^{(k)})^{-1}\mathbf{F}'(\mathbf{y}^{(k)})]\mathbf{F}' (\mathbf{x}^{(k)})^{-1}\mathbf{F}(\mathbf{x}^{(k)}) \] converges to \(\mathbf{r}\) with convergence order four, provided \(a_1 = -1/2, a_2 = 9/8, a_3 = 3/8\), and \(\theta = 2/3\). They note that their method requires only \(n+2n^2\) evaluations per iteration, and they call their method the ``weighted- Newton method''. To compare their method with other, higher order convergence methods, the authors use the efficiency index defined in [\textit{W. Gautschi}, Numerical Analysis. An Introduction. Boston: Birkhäuser (1997; Zbl 0877.65001)]. Additionally they carry out computational experiments in Mathematica to show that theory and practice generally agree. Besides Newton's method, the authors compare their method to the third order method by Homeier, the fourth order method by Cordero et. al., and the fourth order method by Darvishi et. al. They find that their method requires almost the same number of iterations as the other, fourth order mathods, but fewer total function evaluations than all of the other methods. Explicit computations of the efficiency index for each method, as well as the systems of equations for the computational tests are given.
scientific article; zbMATH DE number 6136620