A numerical comparison of convergence among fixed point iteration processes

In this study, “ A numerical comparison of convergence among fixed point iteration processes”, considering one-self map iterated process i.e., T: X → has been proposed and analysed. As all schemes under consideration converge for Zamfirescu map which is a special case of quasi-contractive map, we applied this concept to find where these schemes converge to the fixed point. Seven iteration processes (Picard, Mann, Ishikawa, Halpern’s, Krasnoselskij, Noor, Abbas and Nazir) are compared at a time with respect to their rate of convergence, for both real and complex functions. Banach theory is the basic tool we applied in proving the convergence of these iterative schemes. The concept of fixed point and fixed point iteration is explained as well as how the convergence characteristics of these schemes occur. Using the Zamfirescu-operator (Z-oprator) we find all schemes converging to the fixed point even though it occurs at different rates. Finally we compared these iterative schemes due to their rate of convergence (i.e which are known to converge to the fixed point under Z-operators in Banach Space) is done and the methods which are stable are shown.