A family of higher order derivative free methods for nonlinear systems with local convergence analysis

Abstract

A wide general class of Ostrowski’s families without memory proposed by Behl et al. (Int J Comput Math 90(2):408–422, 2013) is being extended to solve systems of nonlinear equations. This extension uses multidimensional divided differences of first order. Many more new derivative free iterative families with higher order local convergence are presented. In addition, the proposed iterative family for $$\alpha _1=\mathbb {R}-\{0\}$$ and $$\alpha _2=0$$ are special cases of Grau et al. (J Comput Appl Math 237:363–372, 2013) for iterative schemes of fourth and sixth orders. The computational efficiency is compared with some known methods. It is proved that the proposed methods are equally competent with their existing counter parts. Moreover, we present the local convergence analysis of the proposed family of methods based on Lipschitz constants and hypotheses on the divided difference of order one in the more general settings of a Banach space. We expand this way the applicability of these methods, since we used higher derivatives to show convergence of the method in Sect. 3 although such derivatives do not appear in these methods. Numerical experiments are performed which support the theoretical results.