An efficient two-point iterative method with memory for solving non-linear equations and its dynamics

In this paper, we present a novel class of two-step iterative methods with memory for solving non-linear equations. By transforming an existing sixth-order scheme without memory into with memory method, we elevate both the order of convergence and computational efficiency. To attain an accelerated order of convergence, we explore several distinct approximations of self-accelerated parameters, calculated based on the current and previous iterations using Hermite interpolation polynomials. Additionally, we eliminate the need for the second order derivative in the existing without memory method by employing a third-degree Hermite interpolating polynomial. Specifically, the proposed two-step method with memory enhances the R-order of convergence from 6 to 6.7015, 7, and 7.2749 without the need for additional function evaluations. The efficiency index of our method increases from 1.37 to 1.64. Notably, our proposed approach remains effective even when the derivative approaches extremely small values near the desired root or when \(f'(u)\) equals 0. We validate and demonstrate the effectiveness of our proposed approach by conducting numerical comparisons with several existing methods across a range of application-based problems. Finally, we employ basin of attraction plots to visualize the fractal behavior and dynamic characteristics of our proposed method in comparison to some existing methods.