An Optimal Fourth Order Newton–Householder Method in Finding Multiple Roots of Nonlinear Equations

Abstract

In this study, we develop an optimal Newton–Householder method without memory in solving nonlinear equations. The key idea in the development of the new method is the avoidance of the need to evaluate the second derivative. By using the function approach by Schroder, we modified the method which can find multiple roots of a nonlinear equation. The method fulfills the Kung–Traub conjecture by achieving optimal convergence order four with three functional evaluations. The efficiency index of the method shows that the method performs better than the classical Householder’s method. With the help of convergence analysis and numerical analysis, the efficiency of the scheme formulated in this paper can be demonstrated.  Some comparisons with other optimal methods have been conducted to verify the effectiveness, convergence speed, and capability of the suggested method