Stability analysis and dynamical behavior of optimal mean-based iterative methods

Abstract

In this work, we employed the techniques of complex dynamics to perform stability analysis of an optimal mean-based family of iterative methods of order four. Taking into consideration the stability aspect of the specified method, one can describe the method’s sensitivity to the initial guesses. A rational function corresponding to the iterative family is developed. The convergence and stability of a certain method can be analyzed upon finding the fixed points, critical points, periodic points, etc. of the rational function. Furthermore, the dynamical and parametric planes are drawn which help us to detect the stable as well as non-stable regions. It has been observed that stable iterative methods generally yield better performance on complex problems compared to unstable methods. This observation has been supported by numerical experiments that compare our proposed family with some existing methods for representing some chemistry problems, like conversion in a chemical reactor, equations of state, and continuous stirred tank reactor problem.