A Comparative Analysis of the Jacobian and Gauss-Seidel Iterative Methods for Solving a System of Linear Equations

This paper presents a comprehensive comparative analysis of the Jacobi and Gauss-Seidel iterative methods for solving systems of linear equations. Both methods are fundamental iterative techniques widely used in numerical linear algebra for finding approximate solutions to large-scale linear systems, particularly when direct methods become computationally expensive or impractical. The analysis begins with a detailed theoretical overview of each method, elucidating their mathematical foundations, assumptions, and operational mechanisms. We then apply both methods to a specific example system of linear equations to demonstrate their step-by-step iterative processes. Through these practical applications, we illustrate the convergence behaviors, highlighting how the initial guesses and the structure of the coefficient matrix influence the rate and reliability of convergence. Our findings reveal that while both methods are capable of effectively solving linear systems, the Gauss-Seidel method generally exhibits faster convergence compared to the Jacobi method. This is primarily due to the Gauss-Seidel method’s utilization of the most recent updates within each iteration, leading to accelerated convergence rates. Furthermore, the computational efficiency and resource requirements of both methods are examined, providing insights into their respective advantages and limitations in various practical scenarios.