Solving Linear Systems Ax=b Using MATLAB with Jacobi Iterative and Gauss-Seidel Iterative Methods

This article demonstrates how to solve linear systems Ax=b using Jacobi iterative and Gauss-Seidel iterative methods in MATLAB software. We begin by detailing the fundamental principles and computational procedures of both iterative algorithms, including their mathematical formulations and implementation approaches. The Jacobi method updates all variables simultaneously using values from the previous iteration, while the Gauss-Seidel method utilizes the most recent computed values for sequential updates, typically leading to faster convergence. Through concrete numerical examples, we implement both methods using MATLAB's matrix operations and looping structures, comparing their performance characteristics including solution accuracy, computational efficiency, and convergence behavior. The comparative analysis evaluates key aspects such as precision metrics, computation speed through timing functions like tic/toc, and convergence criteria implementation. By studying this article, readers will gain deeper understanding of both iterative methods' operational mechanisms and be equipped to apply them flexibly in practical linear system solving scenarios, with knowledge of appropriate method selection based on problem characteristics like matrix diagonal dominance.