Enhanced Algorithms for Solving Nonlinear Systems: Beyond NEWTON-RAPHSON with Example of Advanced Nonlinear Solver

Abstract

Solving nonlinear systems of equations is a central challenge in scientific computing, impacting a wide range of fields such as engineering, physics, and applied mathematics.Although the Newton-Raphson method is popular for its quadratic convergence near solutions, it faces notable difficulties, including reliance on the initial guess, potential failure with ill-conditioned Jacobians, and complications when multiple or closely situated roots are present. In this study, we investigate the creation of new iterative algorithms aimed at overcoming these obstacles by promoting better global convergence and improving numerical stability. The proposed approaches utilize adaptive step-size management, quasi-Newton techniques, and hybrid strategies that integrate trust-region and homotopy concepts. Results from numerical tests on standard benchmark problems show that these algorithms provide enhanced robustness for a wide array of nonlinear systems. Compared to Newton-Raphson, the new methods expand the convergence domain and frequently deliver equal or better accuracy and computational speed. This work paves the way for developing more trustworthy solvers for complex nonlinear systems in contemporary computational practice.