On the higher-order method for the solution of a nonlinear scalar equation
The Newton method is the most popular method in numerical analysis for finding a simple root of a nonlinear scalar equation in one variable, i.e. for finding \(x^* \in \mathbb R\) such that \(f(x^*) = 0\). For a higher convergence rate, some algorithms were developed such as the Halley method [\textit{H. Bateman}, Am. Math. Mon. 45, 11--17 (1938; Zbl 0018.16103); \textit{M. A. Noor} et al., Appl. Math. Comput. 189, No. 2, 1268--1273 (2007; Zbl 1243.65053))] and the Chebyshev method [\textit{C. Chun}, Appl. Math. Comput. 189, No. 1, 597--601 (2007; Zbl 1122.65324)] and also generalized from time to time. A different method with higher order of convergence was developed by \textit{A. Germani} et al. [J. Optim. Theory Appl. 131, No. 3, 347--364 (2006; Zbl 1118.65034)] which is proved to be locally convergent with order \((m + 1)\), where \(m\) is the highest-order derivative used in the iterative formula. In this paper the authors generalise further by developing a new algorithm, which has a convergence rate with one order higher, i.e., \((m+2)\). Some numerical examples are used to demonstrate the new method.
scientific article; zbMATH DE number 5945533
scientific article