Abstract
The main aim of this manuscript is to propose two new schemes having three and four substeps of order eight and sixteen, respectively. Both families are optimal in the sense to Kung-Traub conjecture. The derivation of them are based on the weight function approach. In addition, theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. Further, we also provide the local convergence of them in Banach space setting under weak conditions. From the numerical experiments, we find that they perform better than the existing ones when we checked the performance of them on a concrete variety of non-linear scalar equations. Finally, we analyze the complex dynamical behavior of them which also provide a great extent to this.
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