A Mathematical Analysis of the Newton-Raphson Method for Solving Nonlinear Equations: Local Convergence Properties and the Impact of Initial Values

Abstract

One of the most basic iterative methods for analyzing nonlinear equations is the Newton-Raphson method. With an emphasis on local convergence qualities and the impact of the initial guess, this paper provides a simplified mathematical analysis of the method when used to solve a single nonlinear equation. There is discussion of both theoretical and practical elements, including difficulties arising from delayed convergence or bad initial values. pointing out Newton-Raphson's speed and precision benefits as well as its real-world drawbacks. The findings show that the approach is useful for resolving nonlinear equations in mathematics and engineering applications when the proper convergence conditions are satisfied.