Analisis Perbandingan Metode Gauss-Seidel dan Jacobi dalam Menyelesaikan Sistem Persamaan Linear Berdasarkan Jumlah Iterasi dan Waktu Komputasi

Abstract

Linear equation systems are one of the fundamental problems in computational mathematics with wide applications across various scientific fields. Numerical solution of linear equation systems using iterative methods is the primary choice when the system dimension increases. This study aims to compare the performance of the Gauss-Seidel and Jacobi methods in solving linear equation systems of order 3×3, 4×4, and 5×5 based on the number of iterations, convergence rate, computation time, and result accuracy. This is a comparative quantitative study using realistic simulation data. Results show that the Gauss-Seidel method consistently outperforms the Jacobi method in terms of iterations to convergence, with an average reduction of 44.7% for 3×3 systems, 42.1% for 4×4 systems, and 40.5% for 5×5 systems. In terms of computation time, Gauss-Seidel is also more efficient with an average of 38.6% faster. The accuracy of both methods is equivalent when both achieve convergence with error tolerance ε = 10⁻⁶. This study concludes that the Gauss-Seidel method is more recommended for iterative-based solutions of linear equation systems compared to the Jacobi method, particularly when computational efficiency is a priority.