MATLAB Implementation of Newton's Iteration Method for Nonlinear Equations

Newton's iteration method is an iterative algorithm specifically designed for solving nonlinear equations. This method employs a step-by-step approximation approach to determine the solution of equations. The core algorithm utilizes an initial guess value to calculate the derivative of the function and uses these derivatives to iteratively estimate the equation's root. The implementation typically involves calculating the function value (f(x)) and its derivative (f'(x)) at each iteration point, then updating the guess using the formula x_new = x_old - f(x_old)/f'(x_old). This process repeats until the solution converges within a specified tolerance threshold. In practical applications, Newton's method demonstrates extensive utility across various domains. It can effectively solve nonlinear equations, optimization problems, and least-squares fitting challenges. The algorithm is particularly valuable for calculus-related computations including finding function roots, determining curve tangents, and locating function extrema. From an implementation perspective, key considerations include proper handling of derivative calculations (either analytical or numerical differentiation), convergence criteria definition (absolute/relative error thresholds), and robustness against divergence cases. Due to its quadratic convergence properties and computational efficiency, Newton's iteration method remains widely adopted and actively researched in mathematics, physics, engineering, and computer science disciplines. The MATLAB implementation typically involves creating function handles for both the main equation and its derivative, followed by a while-loop structure that iteratively refines the solution until convergence criteria are met.