Implementasi Metode Jacobi dan Gauss-Seidel pada Penyelesaian Sistem Persamaan Linear: Analisis Konvergensi dan Efisiensi Komputasi menggunakan Python
DOI:
https://doi.org/10.55123/insologi.v5i3.8776Keywords:
Gauss-Seidel, Jacobi, Numerical Computation, Iterative Methods, Python, Systems of Linear EquationAbstract
Systems of linear equations are fundamental problems in numerical computation and are widely used in engineering, physics, economics, and computer science. Solving large systems of linear equations requires efficient numerical methods, particularly when the coefficient matrix is sparse or diagonally dominant. This study aims to compare the performance of the Jacobi and Gauss-Seidel methods in solving diagonally dominant systems of linear equations using Python. The experiments were conducted on three matrix sizes, namely 3×3, 5×5, and 10×10. The evaluation focused on the number of iterations, final error, Root Mean Square Error, computation time, and speedup. The implementation was carried out using Python 3.12 with the NumPy and Matplotlib libraries. The results show that both methods can produce numerical solutions close to the reference solution. However, the Gauss-Seidel method demonstrates better performance in all test scenarios. In the 10×10 matrix test, the Gauss-Seidel method reached convergence in 20 iterations, while the Jacobi method required 32 iterations. This indicates a 37.50% reduction in the number of iterations. The average speedup of Gauss-Seidel over Jacobi reached 1.85 times. These findings indicate that the Gauss-Seidel method is more efficient for solving diagonally dominant systems of linear equations in a Python-based computing environment.
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