Stability Analysis of Optimal Fourth-order Uniparametric Family Iterative Method

We mainly research the stability of a family of uniparametric optimal fourth-order iterative method by using complex dynamics tools. An affine transformation is used to simplify the analysis while the qualitative properties keep invariant. The fixed points of the iterative method are studied and the stability of these strange fixed points is analyzed. Subsequently, we show the parameter space related to the critical point, and select the parameter values in the parameter space to obtain an iterative family with stable and unstable behavior. Finally, we find that the most stable member of the iteration family is $k = - \frac{1}{2}$.