An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics
This is an incremental improvement for researchers in numerical analysis seeking more efficient iterative methods for nonlinear equations.
The paper presents a three-point iterative method for solving nonlinear equations that achieves eighth-order convergence with four function evaluations per iteration, supporting Kung and Traub's conjecture. Numerical tests show it outperforms existing eighth-order methods in accuracy and basin of attraction.
We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung and Traub's conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basin of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basin of attraction.