Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Metaheuristic Method for Solving Systems of Equations

Samson Odan

TL;DR

This work addresses solving systems of equations in science and engineering by applying Genetic Algorithms (GAs) as a robust, flexible alternative to classical solvers. By formulating the problem as a multi-objective residual minimization, the GA evolves candidate solutions and can uncover multiple valid solution sets, unlike traditional methods that converge to a single solution. Across linear and nonlinear benchmarks, the GA demonstrated strong robustness and competitive performance in complex nonlinear landscapes, albeit with higher computational cost. The study highlights the GA’s value for exploring solution landscapes and suggests hybrid and parameter-tuning avenues to improve efficiency for large-scale problems.

Abstract

This study investigates the effectiveness of Genetic Algorithms (GAs) in solving both linear and nonlinear systems of equations, comparing their performance to traditional methods such as Gaussian Elimination, Newton's Method, and Levenberg-Marquardt. The GA consistently delivered accurate solutions across various test cases, demonstrating its robustness and flexibility. A key advantage of the GA is its ability to explore the solution space broadly, uncovering multiple sets of solutions -- a feat that traditional methods, which typically converge to a single solution, cannot achieve. This feature proved especially beneficial in complex nonlinear systems, where multiple valid solutions exist, highlighting the GA's superiority in navigating intricate solution landscapes.

Metaheuristic Method for Solving Systems of Equations

TL;DR

This work addresses solving systems of equations in science and engineering by applying Genetic Algorithms (GAs) as a robust, flexible alternative to classical solvers. By formulating the problem as a multi-objective residual minimization, the GA evolves candidate solutions and can uncover multiple valid solution sets, unlike traditional methods that converge to a single solution. Across linear and nonlinear benchmarks, the GA demonstrated strong robustness and competitive performance in complex nonlinear landscapes, albeit with higher computational cost. The study highlights the GA’s value for exploring solution landscapes and suggests hybrid and parameter-tuning avenues to improve efficiency for large-scale problems.

Abstract

This study investigates the effectiveness of Genetic Algorithms (GAs) in solving both linear and nonlinear systems of equations, comparing their performance to traditional methods such as Gaussian Elimination, Newton's Method, and Levenberg-Marquardt. The GA consistently delivered accurate solutions across various test cases, demonstrating its robustness and flexibility. A key advantage of the GA is its ability to explore the solution space broadly, uncovering multiple sets of solutions -- a feat that traditional methods, which typically converge to a single solution, cannot achieve. This feature proved especially beneficial in complex nonlinear systems, where multiple valid solutions exist, highlighting the GA's superiority in navigating intricate solution landscapes.
Paper Structure (21 sections, 26 equations, 2 figures, 2 tables)

This paper contains 21 sections, 26 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Systems of Equations
  3. Nonlinear Systems
  4. Linear Systems
  5. Methodology
  6. Genetic Algorithm (GA) Implementation
  7. Population Initialization
  8. Fitness Function
  9. Selection Mechanism
  10. Crossover and Mutation
  11. Elitism and Replacement
  12. Termination Criteria
  13. Pseudocode for the Genetic Algorithm
  14. Newton's Method
  15. Levenberg-Marquardt Algorithm
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Convergence behavior of the Genetic Algorithm (GA) for solving linear and nonlinear systems of equations. a) Linear system: The GA rapidly converges to a solution, minimising the fitness value within a few generations. b) Nonlinear system: The GA shows a more gradual convergence, reflecting the increased complexity of the nonlinear system.
  • Figure 2: Time Comparison of Different Methods for Solving Linear and Nonlinear Systems of Equations. The Genetic Algorithm (GA) shows significantly higher computational time, especially for the nonlinear system, compared to traditional methods like Gaussian Elimination, Newton’s Method, and Levenberg-Marquardt.