New optimal newton-householder methods for solving nonlinear equations and their dynamics

The classical iterative methods for finding roots of nonlinear equations, like the Newton method, Halley method, and Chebyshev method, have been modified previously to achieve optimal convergence order. However, the Householder method has so far not been modified to become optimal. In this study, we shall develop two new optimal Newton-Householder methods without memory. The key idea in the development of the new methods is the avoidance of the need to evaluate the second derivative. The methods fulfill the Kung-Traub conjecture by achieving optimal convergence order four with three functional evaluations and order eight with four functional evaluations. The efficiency indices of the methods show that methods perform better than the classical Householder's method. With the aid of convergence analysis and numerical analysis, the efficiency of the schemes formulated in this paper has been demonstrated. The dynamical analysis exhibits the stability of the schemes in solving nonlinear equations. Some comparisons with other optimal methods have been conducted to verify the effectiveness, convergence speed, and capability of the suggested methods.