An Efficient Three Step Method For finding the Root Of Non-linear Equation with Accelerated convergence.

Abstract

We have made an effort to design an accurate numerical strategy to be applied in the vast computing domain of numerical analysis. The purpose of this research is to develop a novel hybrid numerical method for solving a nonlinear equation, That is both quick and computationally cheap, given the demands of today's technological landscape. Sixth-order convergence is demonstrated by combining the classical Newton method, on which this method is largely based, with another two-step third-order iterative process. The effectiveness index for this novel approach is close to 1.4309, and it requires only five evaluations of the functions without a second derivative. The findings are compared to standard practice. The provided technique demonstrates higher performance in terms of computational efficiency, productivity, error estimation, and CPU times. Moreover, its accuracy and performance are tested using a variety of examples from the existing literature.

Keywords: efficient scheme, nonlinear application, nonlinear functions, error estimation, computational cost.