Ball convergence for a family of eight-order iterative schemes under hypotheses only of the first-order derivative

Abstract

ABSTRACTIn this paper, we propose a local convergence study of an optimal family of eight-order iteration functions. Lotfi et al. [A new class of three-point methods with optimal convergence order eight and its dynamics, Numer. Algor. 68 (2015), pp. 261–288] used Taylor series expansions, and hypotheses up to the eight or higher-order derivative of the considered function in order to demonstrate the convergence order in their studies. However, the presented scheme did not involve second or higher-order derivative of the considered function. Such conditions restrict the usage of their scheme. Therefore, we extend the suitability of their iteration functions by considering suppositions solely on the first-order derivative. Furthermore, we present the convergence domain of the iteration functions and bounds on the error by using Lipschitz constants. Finally, we discuss a case where the earlier study is not applicable but our results are very useful for the same problem.