We construct an optimal eighth-order scheme which will work for multiple zeros with multiplicity [Formula: see text], for the first time. Earlier, the maximum convergence order of multi-point iterative schemes was six for multiple zeros in the available literature. So, the main contribution of this study is to present a new higher-order and as well as optimal scheme for multiple zeros for the first time. In addition, we present an extensive convergence analysis with the main theorem which confirms theoretically eighth-order convergence of the proposed scheme. Moreover, we consider several real life problems which contain simple as well as multiple zeros in order to compare with the existing robust iterative schemes. Finally, we conclude on the basis of obtained numerical results that our iterative methods perform far better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

In this paper, we present a new and interesting optimal scheme of order eight in a general way for solving nonlinear equations, numerically. The beauty of our scheme is that it is capable of producing further new and interesting optimal schemes of order eight from every existing optimal fourth-order scheme whose first substep employs Newton’s method. The construction of this scheme is based on rational functional approach. The theoretical and computational properties of the proposed scheme are fully investigated along with a main theorem which establishes the order of convergence and asymptotic error constant. Several numerical examples are given and analyzed in detail to demonstrate faster convergence and higher computational efficiency of our methods.

In this paper, we present a local convergence analysis of some iterative methods to approximate a locally unique solution of nonlinear equations in a Banach space setting. In the earlier study [Babajee et al. (2015) “On some improved harmonic mean Newton-like methods for solving systems of nonlinear equations,” Algorithms 8(4), 895–909], demonstrate convergence of their methods under hypotheses on the fourth-order derivative or even higher. However, only first-order derivative of the function appears in their proposed scheme. In this study, we have shown that the local convergence of these methods depends on hypotheses only on the first-order derivative and the Lipschitz condition. In this way, we not only expand the applicability of these methods but also proposed the theoretical radius of convergence of these methods. Finally, a variety of concrete numerical examples demonstrate that our results even apply to solve those nonlinear equations where earlier studies cannot apply.