Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

Abstract

The methods that use memory using accelerating parameters for computing multiple roots are almost non-existent in the literature. Furthermore, the only paper available in this direction showed an increase in the order of convergence of 0.5 from the without memory to the with memory extension. In this paper, we introduce a new fifth-order without memory method, which we subsequently extend to two higher-order with memory methods using a self-accelerating parameter. The proposed with memory methods extension demonstrate a significant improvement in the order of convergence from 5 to 7, making this the first paper to achieve at least a 2-order improvement. In addition to this improvement, our paper is also the first to use Hermite interpolating polynomials to approximate the accelerating parameter in the proposed with memory methods for multiple roots. We also provide rigorous theoretical proofs of convergence theorems to establish the order of the proposed methods. Finally, we demonstrate the potential impact of the proposed methods through numerical experimentation on a diverse range of problems. Overall, we believe that our proposed methods have significant potential for various applications in science and engineering.