OPTIMAL FOURTH-ORDER ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS: AN INNOVATIVE GENERAL CLASS WITH STABLE MEMBERS AND ENGINEERING APPLICATIONS

In this work, we construct a new class of two-step fourth-order iterative methods for solving nonlinear equations. Each iteration requires two function evaluations and one evaluation of the first derivative. Consequently, this family is optimal according to the Kung-Traub conjecture. The first step of the family coincides with the classical Newton's method, while the second step involves three parameters and a weight function, offering a wide range of options and including several well-known methods as special cases. Additionally, we identify three new particular cases that perform well compared to existing methods within the same family. The analysis of complex dynamics and basins of attraction shows that these methods have a wider range of initial points that ensure convergence. Furthermore, numerical examples using various test functions and real-life applications illustrate that, in general, the new methods produce good results in terms of accuracy.