In this paper, we provided a new fourth-order optimal method. This method demands three functional evaluations, and according to Kung-Traub it is considered as one of the optimal methods with efficiency indicator I that reaches 1.587. Furthermore, we can extend its convergence to obtain a new sixth-order method where its efficiency indicator is 1.565. In this paper, we also discuss the convergence analysis of our new methods as it was established that the new methods have convergence orders four and six. Moreover, we will illustrate our study of the stability criterion of the new methods, and we will present the stability theorems along with some examples which prove that our methods are stable. Finally, we have discussed attraction basins for those suggestedmethods and compared them with methods that have the same order, and we have applied them for numerical examples to clarify the performance and efficiency of the proposed methods.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.