Stability Analysis of Numerical Methods for Solving Nonlinear Equations

Maha Abdulwahhab Albkoush; Amena Ramadan Alkoni Raash

المؤلفون

Maha Abdulwahhab Albkoush Department of Mathematics, Faculty of Education, Sabratha University, Sabratha, Libya
Amena Ramadan Alkoni Raash Department of General Subjects, Higher Institute of Marine Science Technologies Sabratha, Sabratha, Libya

الكلمات المفتاحية:

Numerical stability، nonlinear equations، Newton-Raphson method، fixed-point iteration، bisection method، solution convergence

الملخص

This research investigates the numerical stability of three widespread numerical methods used to solve nonlinear equations including Newton-Raphson method and fixed-point iteration method and bisection method. The research employed both theoretical foundations with numerical experiments using selected nonlinear equations and various initial conditions under a unified convergence criterion.

The Newton-Raphson method achieves the fastest convergence rate among the tested methods yet remains vulnerable to initial guess accuracy and derivative behavior since it can potentially diverge under certain conditions. The simplicity of the fixed-point method depends on both the form of g(x) and the derivative values around the root. The bisection method achieved reliable and stable convergence across all tested cases because it required neither derivative information nor exact root knowledge.

The research concludes that selecting a proper numerical method requires both speed performance and stability evaluation under different circumstances. The bisection method should serve as a starting point for safe calculations until efficiency methods can be applied after confirming stability conditions.