Enhanced Ninth-Order Memory-Based Iterative Technique for Efficiently Solving Nonlinear Equations

In this article, we present a novel three-step with-memory iterative method for solving nonlinear equations. We have improved the convergence order of a well-known optimal eighth-order iterative method by converting it into a with-memory version. The Hermite interpolating polynomial is utilized to compute a self-accelerating parameter that improves the convergence order. The proposed uni-parametric with-memory iterative method improves its R-order of convergence from 8 to 8.8989. Additionally, no more function evaluations are required to achieve this improvement in convergence order. Furthermore, the efficiency index has increased from 1.6818 to 1.7272. The proposed method is shown to be more effective than some well-known existing methods, as shown by extensive numerical testing on a variety of problems.