Abdallah Dawoud - Department of Electrical Engineering, College of Engineering, University of Prince Mugrin, P.O. Box ۴۲۲۴۱, Medina, Saudi Arabia.
MAlak Khashoqji - Department of Electrical Engineering, College of Engineering, University of Prince Mugrin, P.O. Box ۴۲۲۴۱, Medina, Saudi Arabia.
Tareq Al-hussain - Department of Civil Engineering, College of Engineering, University of Prince Mugrin, P.O. Box ۴۲۲۴۱, Medina, Saudi Arabia.
Ibrahim Al-Subaihi - Department of General Studies, University of Prince Mugrin, P.O. Box ۴۲۲۴۱, Medina, Saudi Arabia.

In this paper, a novel optimal class of eighth-order convergence methods for finding simple roots of nonlinear equations is derived based on the Predictor-Corrector of Halley method. By combining weight functions and derivative approximations,  an optimal class of iterative methods with eighth-order convergence is constructed. In terms of computational cost, the proposed methods require three function evaluations, and the first derivative is evaluated once per iteration. Moreover, the methods have efficiency indices equal to ۱.۶۸۱۷. The proposed methods have been tested with several numerical examples, as well as a comparison with existing methods for analyzing efficacy is presented.

Halley’s method, Non-linear equations, Iterative methods, Convergence analysis, Polynomiography