Abstract
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 suggested methods 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.