Numerical Simulation and Error Analysis of the Newton–Raphson Method for Solving Nonlinear Equations

Abbas Mohsin Kazar (Teacher at Al-Shahimiya Middle School for Boys / Al-Suwaira Education Department, Wasit Governorate)

Academia Open
Vol. 11 No. 1 (2026): June

General Background: Nonlinear equations frequently arise in engineering and applied sciences and require reliable iterative numerical techniques for accurate root approximation. Specific Background: Among classical root-finding approaches, the Newton–Raphson method utilizes first-order Taylor series expansion and derivative information to generate successive approximations with theoretically quadratic convergence. Knowledge Gap: Despite its theoretical advantages, detailed numerical simulation integrating convergence verification, residual decay analysis, sensitivity to initial guesses, and comparison with alternative classical methods across representative nonlinear equations remains limited in a unified framework. Aims: This study analytically derives the Newton–Raphson iteration from Taylor expansion, verifies its quadratic convergence, implements a complete Python-based simulator with iteration logging, and evaluates performance using four representative nonlinear equations, including polynomial, transcendental, and exponential cases. Results: Numerical simulations confirm machine-precision solutions within four to five iterations, empirical convergence order p≈2.00±0.05p \approx 2.00 \pm 0.05p≈2.00±0.05, rapid residual decay, and substantially fewer iterations compared with bisection and secant methods, while also demonstrating sensitivity patterns related to initial guesses near inflection or flat regions. Novelty: The study integrates rigorous derivation, numerical simulation, graphical convergence interpretation, residual evaluation, and comparative benchmarking within a single structured analysis of four canonical test equations. Implications: These findings provide validated guidance for selecting initial approximations, interpreting convergence behavior, and applying the Newton–Raphson method efficiently in numerical analysis and engineering computation contexts. Highlights: • Machine-Precision Solutions Obtained Within Five Iterations Across Representative Nonlinear Test Cases• Iteration Counts Substantially Lower Than Classical Interval-Halving and Derivative-Free Alternatives• Convergence Behavior Varies With Starting Approximation Near Inflection or Flat Function Regions Keywords: Newton Raphson Method, Nonlinear Equations, Quadratic Convergence, Numerical Simulation, Error Analysis.