An extension of Ostrowski’s method with improved convergence and complex geometry

Numerous higher-order iterative methods have been proposed in the literature for finding the roots of equations. Among these, methods with optimal order are particularly valued for their superior efficiency. However, the majority of such methods do not demonstrate consistent performance in every situation. Some yield low accuracy, others suffer from slow convergence, and some fail to maintain the desired convergence order in certain applications. This paper addresses these limitations by introducing a novel three-point iterative scheme, built upon the widely used two-point Ostrowski’s fourth-order method. The proposed scheme achieves eighth-order convergence with just four function evaluations per iteration. As a result, it is optimal according to the Kung–Traub conjecture, with an efficiency index of 1.682—exceeding those of Newton’s method (1.414) and Ostrowski’s method (1.587). To evaluate the performance and validate the theoretical properties of the method, we present several numerical examples. In addition, we assess its stability under various settings in which the input data are polluted with significant random noise. Furthermore, we provide graphical representations of the basins of attraction to illustrate and compare the stability and dynamic behavior of our proposed method against other well-established techniques. The computational results and convergence visualizations confirm that our scheme outperforms existing methods in the literature.