An Improved Numerical Analysis for Solving Nonlinear Equations Using a Hybrid Algorithm Between Newton’s Method and Metaheuristic Approaches

Abstract

The problem of finding all roots of a system of nonlinear equations is a common difficulty in scientific computing and various engineering disciplines. Classical numerical methods like Newton’s method and Levenberg-Marquardt usually have drawbacks, sensitive to initial guesses, local convergence, etc., and are not able to find multiple solutions. This paper explores the potential of metaheuristic algorithms, and more specifically Genetic Algorithms (GAs), for handling these challenges. The study shows that GAs represents a powerful and versatile solution to solve linear and nonlinear system, with the ability to explore the global solution space, avoiding local minima, and the capability of converging to the correct solution from poor initial guesses. A complementary strength of GA is that it can retrieve multiple solution sets, something deterministic methods cannot do as well (especially in systems where there are multiple physically meaningful equilibria). The behavior of the GA is examined on a series of benchmark problems and compared to traditional methods, i.e., Gaussian Elimination, and Newton's scheme. . It is established that GA will provide the exact solutions for the determination of the global optimum and is superior compared to other optimisation algorithms for the solution of complex and multimodal spaces of solutions. Also, the analysis reveals the possibility of the hybrid form of the metaheuristics with the local refinement for the target of minimising convergence and augmenting precision. The analysis highlights the need for the evolutionary analysis for numerical analysis and suggests the integration of the usage of the methodology of the metaheuristic within the process of the solutions for the complex mathematical problems. The results are especially significant for optimisation, control and computational physics problems where reliability and diversity can be crucial among solutions.