A Comparative Analysis of Numerical Methods for Solving Nonlinear Equations: Accuracy, Convergence, and Computational Efficiency

Nonlinear equations arise frequently in scientific and engineering applications, yet closed-form analytical solutions are often unavailable, necessitating numerical methods. Selecting an appropriate method involves trade-offs among accuracy, convergence behavior, and computational cost. This study presents a comparative analysis of three widely used root-finding methods: the Bisection method, Newton Raphson method, and Secant method. The objective is to evaluate their performance in terms of solution accuracy, convergence rate, and computational efficiency across representative nonlinear problems. A set of polynomial and transcendental test functions is employed, and each method is implemented under consistent stopping criteria and initial conditions. Performance metrics include absolute error, number of iterations to convergence, and execution time. The comparative framework enables a systematic assessment of both robustness and efficiency. The results indicate that the Newton Raphson method achieves the fastest convergence and highest accuracy when a suitable initial guess is available, owing to its quadratic convergence. The Secant method offers a balance between efficiency and reduced computational overhead by avoiding derivative evaluations, though it converges slightly more slowly. The Bisection method, while robust and guaranteed to converge under appropriate conditions, demonstrates slower convergence and higher iteration counts. In short, no single method is universally optimal; the choice depends on problem characteristics and computational constraints. The findings provide practical guidance for selecting appropriate numerical methods in applied contexts.