Comparison of the Accuracy of the Levenberg-Marquardt and Trust-Region Methods in Solving Multivariable Non-Linear Equation Systems

Abstract

Multivariable nonlinear equation systems are commonly found in various disciplines such as engineering, physics, economics, and artificial intelligence. Analytical solutions are often difficult to obtain, necessitating the use of numerical approaches. This study aims to evaluate and compare the performance of the Levenberg-Marquardt and Trust-Region methods in solving multivariable nonlinear equation systems. Computational simulations were performed using MATLAB software with an error tolerance of 0.001 and a maximum iteration limit of 100. The test system involved a combination of trigonometric, exponential, and polynomial functions to ensure computational complexity. The study's results show that both methods are capable of achieving solutions with high accuracy. The Levenberg-Marquardt method demonstrated higher efficiency, achieving convergence in only 2 iterations with a final error of 1.866 10. In contrast, the Trust-Region method required 27 iterations but yielded a smaller error of 4.768 10. Three-dimensional visualization revealed that the solution was obtained from the intersection point of the three function surfaces. These findings confirm that the selection of numerical methods should consider the priority between iteration efficiency and solution accuracy. The contribution of this research lies in presenting a comparison of the performance of two popular algorithms with controlled simulation parameters, which can serve as a basis for the development of numerical methods in larger dimensional systems