An efficient extension of three-step optimal iterative scheme into with memory and its stability

In this research paper, a novel approach is presented to solving nonlinear equations by modifying an existing optimal eight-order method. The proposed method introduces a three-step iterative process with memory, aiming to enhance the convergence rate without requiring additional function evaluations. To achieve this acceleration, the method incorporates a combination of a single parameter and double self-accelerating parameters. The values of these parameters are estimated using a Hermite interpolating polynomial technique. By leveraging this estimation approach, the convergence order of the original iterative method without memory is increased from eight to eleven. The paper provides a comprehensive analysis of the proposed method through numerous examples. These examples serve to support the theoretical findings and demonstrate its stability and the superior convergence rate achieved by the three-step with memory approach.