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.

Keywords

• nonlinear equation
• iterative algorithm
• local convergence
• <i>ω</i>-continuity condition
• parameter space
• dynamical plane