Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

In this work, two new iterative methods are proposed for finding simple roots of non-linear equations. The new methods are the modifications of the existing work proposed by Rafiullah and Jabeen (New eighth and sixteenth order iterative methods to solve nonlinear equations, International Journal of Applied and Computational Mathematics 3 (2017), 2467 – 2476). The first method obtained is of fifth-order two-step with memory method while the second scheme is three-point eight-order optimal without memory method. Firstly, the Hermite interpolation polynomial is employed to eliminate the first derivative. To maintain order, the conversion to a memory scheme was accomplished by introducing self-accelerated parameters, all without requiring any new function evaluations. Additionally, the Gauss quadrature approach was incorporated for the first derivative, aiming to attain optimal eighth-order convergence. In particular, the efficiency index is increased from 1. 4953 to 1. 7099 and 1. 5157 to 1. 6817 for fifth- and eighth-orders respectively. Some real-life application based problems, such as Kepler's equation, an ocean engineering problem, Planck's radiation law, a blood rheology model, and the charge between two parallel plates were presented to validate and demonstrate the superiority of the proposed scheme. Another benefit of the proposed scheme is on the restriction of the Newton's method that \ (f ' (v) 0\) can be eliminated close to the root.