Analysis and Enhancement of Newton Raphson Method

Abstract

Recently, Newton Raphson-based Algorithms 2.1 and 2.2 are developed to solve different nonlinear systems of nonlinear equations. These newly developed algorithms have demonstrated superior performance compared to the traditional Newton-Raphson Method (NRM) and the Fixed-Point Theorem. They took less time as compared to these conventional methods by reducing the number of iterations required to converge to an exact solution and thereby decreasing the computational work. It is also observed that these performance measures can be further improved. Accordingly, in this research, we present some efficient numerical algorithms for solving nonlinear and systems of nonlinear equations based on NRM. The Modified Adomian Decomposition Method (MADM) is applied to construct the numerical algorithm. A new Algorithm 2.3 is developed, an enhancement in previous relevant algorithms. Several nonlinear expressions are estimated via enhancements in NRM. The results of these tests are measured in terms of the number of iterations and presented in the form of tables to show the converging rate to an exact solution. The obtained results of Algorithm 2.3 are compared with other methods like NRM, Fixed Point Method (FPM), Algorithm 2.1, and Algorithm 2.2 to show the efficiency of the suggested enhancement. It is finally observed after solving several nonlinear equations that Algorithm 2.3 is converging more rapidly than other numerical methods. Consequently, Algorithm 2.3 significantly reduces the number of iterations needed to achieve an exact solution as compared to existing other iterative approaches in getting the exact solution. This efficiency makes it a valuable tool for solving nonlinear equations more effectively.