Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers

Abstract

In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented.

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

“…In these terms, Amat et al in [12] described the dynamical performance of some known families of iterative methods. More recently, in [9,[13][14][15][16][17], different authors analyze the qualitative behavior of several known methods or classes of iterative schemes. Most of these studies demonstrate some elements with very stable behavior, which is proven to be useful in practice, and also different pathological performances, such as attracting fixed points different from the solution of the problem, periodic orbits, etc.…”

Memory in the iterative processes for nonlinear problems

“…The convergence of an iterative method is not the only thing to analyze, but it is also important to study its stability in terms of the set of initial approximations that generate convergence or give rise to chaotic behavior (see, e.g., Chicharro et al, 15,16 and Sharma et al 17 ). This stability is analyzed using discrete real dynamics tools, which will allow us to differentiate family members with stable behavior from others with chaotic behavior.…”