Home – Home – Jornals – Numerical Analysis and Applications 2026 number 2

Numerous higher-order iterative methods have been proposed in the literature for locating roots of nonlinear equations. Among these, methods with optimal order are of particular interest due to their superior efficiency. However, not all of them exhibit consistent performance across all scenarios. Some offer low accuracy, while others suffer from slow convergence, yet there are methods that fail to maintain the desired convergence order in certain applications. This paper aims to address these shortcomings. Consequently, we introduce a novel three-point iterative scheme, whose formulation is based on the widely used two-point King's fourth-order method. This scheme attains eighth-order convergence at the cost of four function evaluations per step. As such, it is optimal according to the Kung-Traub conjecture and boasts an efficiency index of 1.682, which surpasses that of Newton's method and many other higher-order techniques. To assess the methods' performance and validate its theoretical properties, we present several numerical examples. Furthermore, we provide a detailed analysis of the complex dynamics through graphical representations of the basins of convergence, comparing our method with those of other established techniques. The computational results and convergence visualizations confirm that our scheme outperforms existing methods in the literature.