On the local convergence and complex geometry of eighth order iteration function

Abstract. The convergence order of numerous iterative methods is obtained using derivatives of higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high order derivatives, since their convergence is not guaranteed. In this paper, we study local convergence of an efficient eighth order method under hypotheses only on the first derivative. That is how we expand the applicability of some popular methods. Moreover, the study of local convergence of iteration functions is important because it provides the degree of difficulty for choosing initial points. We also verify the theoretical results on some numerical problems. Finally, stability of the method is checked through complex geometry shown by drawing basins of attraction of the solutions.