Convergence and Efficiency Analysis of the Gauss-Seidel Method in Solving Linear Equation Systems: Implementation on Maple
DOI:
https://doi.org/10.31949/th.v11i2.18129Abstract
Iterative methods provide an efficient approach for solving large-scale systems of linear equations. This study aims to analyze the convergence and efficiency of the Gauss-Seidel method in solving linear systems through Maple-based implementation. Convergence analysis is conducted by examining the spectral radius of the iteration matrix and the diagonal dominance property of the coefficient matrix, while efficiency is evaluated based on the number of iterations and computational time. Numerical experiments are performed on various matrix sizes with an error tolerance of 10-5. The results indicate that the Gauss-Seidel method converges when the spectral radius of the iteration matrix is less than one and the coefficient matrix satisfies diagonal dominance conditions. The implementation in Maple enables systematic and accurate numerical and spectral analysis. This study contributes to computational-based numerical method analysis and provides a reference for selecting efficient iterative methods.
Keywords:
Gauss-Seidel, convergence analysis, computational efficiency, Linier systems, MapleDownloads
References
Ardhani, D. C., Solikah, N. H., & Wibowo, A. (2025). Pemanfaatan Python dalam Penyelesaian SPL dengan Metode Iterasi Jacobi dan Gauss-Seidel. J-PiMat: Jurnal Pendidikan Matematika, 7(1), 1685–1694.
Audu, K. J., & Essien, J. N. (2023). An Accelerated Iterative Technique : Third Refinement of Gauss Seidel Algorithm for Linear Systems †. Computer Sciences and Mathematics Forum, 1–6. https://doi.org/10.3390/IOCMA2023-14415
Bakari, A. I., & Dahiru, I. A. (2018). Comparison of Jacobi and Gauss-Seidel Iterative Methods for the Solution of Systems of Linear Equations. Asian Research Journal of Mathematics, 8(3), 1–7. https://doi.org/10.9734/ARJOM/2018/34769
Bhatti, N. (2023). Effectiveness Of Preconditioned M-Order Gauss-Seidel Method for Linear System. International Journal for Multidisciplinary Research (IJFMR), 5(2), 1–7. https://doi.org/10.36948/ijfmr.2023.v05i02.2600
Brahinets, O. B., & Vorobyova, A. I. (2022). Використання математичного пакету Maple до розв’язання СЛАР прямими чисельними методами. Computer-Integrated Technologies Education Science Production.
Denka, T., & Lhamo, D. (2018). Comparison of Iterative Methods for Sparse Linear Systems. Zorig Melong- A Technical Journal of Science Engineering and Technology, 4(1), 36–39. https://doi.org/https://doi.org/10.17102/zmv4.i1.007
Enyew, T. K., Awgichew, G., Haile, E., & Abie, G. D. (2020). Second-refinement of Gauss-Seidel iterative method for solving linear system of equations. Ethiopian Journal of Science and Technology, 13(January), 1–15. https://doi.org/10.4314/ejst.v13i1.1
Fan, W., & Xiong, J. (2023). A Model-Free Iteration Algorithm for Markov Jump Linear Systems Based on Gauss-Seidel Method. 2023 62nd IEEE Conference on Decision and Control (CDC), 1275–1280. https://doi.org/10.1109/CDC49753.2023.10383557
Grzegorski, S. M. (2019). On Optimal Parameter Not Only for the SOR Method. Applied and Computational Mathematics, 8(5), 82–87. https://doi.org/10.11648/j.acm.20190805.11
Haleem, B. A., Aghoury, I. M. El, Tork, B. S., & El-arabaty, H. A. (2020). A New Modified Version Of Gauss-Seidel Iterative Method Using Grouping Relaxation Approach. International Journal of Scientific and Technology Research, 9(10), 141–147.
Ibrahim, A., Aramide, T., Idowu, O., & Ayodele, A. (2025). Stationary Iterative Methods and Their Application to Some Problems Arising in Physics. Journal of Applied Sciences and Environmental Management, 29(8), 2654–2659. https://doi.org/https://doi.org/10.4314/jasem.v29i8.30
Ihsan, H., Wahyuni, M. S., & Waode, Y. S. (2024). Penerapan Metode Iterasi Jacobi dan Gauss-Seidel dalam Menyelesaikan Sistem Persamaan Linear Kompleks. Journal of Mathematics, Computations, and Statistics, 7(1), 34–54. https://doi.org/https://doi.org/10.35580/jmathcos.v7i1.1964
Jabnabillah, F., Sulistyono, E., Handayani, V. A., Batam, I. T., Mada, J. G., City, K. V., Ayu, T., Batam, K., & Riau, K. (2023). Pelatihan Penggunaan Aplikasi MAPLE Sebagai Media Pembelajaran Matematika. Jurnal SOLMA, 12(2), 415–424. https://doi.org/https://doi.org/10.22236/solma.v12i2.11236
Johar, D. A. (2020). Application of the Concept of Linear Equation Systems in Balancing Chemical Reaction Equations. International Journal of Global Operations Research, 1(4), 130–135. https://doi.org/10.47194/ijgor.v1i4.48
Krishnan, R. S. B. (2019). Probability Distributions, Random Processes and Numerical Methods.
Liu, X., Du, W., Yu, Y., & Qin, Y. (2018). Efficient Iterative Method for Solving the General Restricted Linear Equation. Journal of Applied Mathematics and Physics, 6(2), 418–428. https://doi.org/10.4236/jamp.2018.62039
Maharesi, R., & Widyatmini. (2025). Estimation of the Effective Reproduction Number (Rt) for the COVID-19 Pandemic in Jakarta: A Mathematical Modelling Approach. Jurnal Theorems (The Original Research of Mathematics), 10(1), 1–12. https://doi.org/10.31949/th.v10i1.13746
Meister, A. (2026). Iterative Methods. In: Numerical Methods for Linear Systems of Equations (A. Meister (ed.); pp. 81–248). Springer Fachmedien Wiesbaden. https://doi.org/10.1007/978-3-658-50260-7_4
Pujiono, I. P., Kamal, M. R., Prayogi, A., Sari, C. A., & Ikhsanuddin, R. M. (2025). Algoritma Counting Sort vs Algoritma Pengurutan Modern: Analisis Memori dan Waktu Komputasi. JITET (Jurnal Informatika Dan Teknik Elektro Terapan), 13(3), 135–142. https://doi.org/https://doi.org/10.23960/jitet.v13i3.6657
Purnama, D. A. (2024). Analisis Perbandingan Kompleksitas Waktu Algoritma Gauss, Gauss-Jordan, dan Kaidah Cramer dalam penyelesaian Sistem Persamaan Linier. https://informatika.stei.itb.ac.id/~rinaldi.munir/Matdis/2023-2024/Makalah2023/Makalah-Matdis-2023 (55).pdf
Ranga, P. (2019). A Study on Comparison of Jacobi and Gauss-Seidel Iterative Methods for the Solution of Systems of Linear Equations. Global Journal for Research Analysis, 8(4), 74–76. https://doi.org/https://www.doi.org/10.36106/gjra
Sagaut, P., Suman, V. K., Sundaram, P., Rajpoot, M. K., Bhumkar, Y. G., Sengupta, A., & Sengupta, T. K. (2023). Global spectral analysis : Review of numerical methods. Computers and Fluids, 261, pp.105915. https://doi.org/https://hal.science/hal-04546492v1
Sebro, A., Muleta, H., & Gebregiorgis, S. (2020). Extrapolated refinement of generalized gauss- seidel scheme for solving system of linear equations. Berhan International Research Journal of Science and Humanities (BIRJSH), 4(1), 12–23.
Smoktunowicz, A., Kierzkowski, J., & Wróbel, I. (2016). Forward Stability of Iterative Refinement with a Relaxation for Linear Systems. Journal of Advances in Applied & Computational Mathematics, 3(2), 68–73. https://doi.org/https://doi.org/10.15377/2409-5761.2016.03.02.1
Sobania, D., Schmitt, J., Köstler, H., & Rothlauf, F. (2022). Genetic Programming for Iterative Numerical Methods. Genetic Programming and Evolvable Machines, 23(2), 253–278. https://doi.org/10.1007/s10710-021-09425-5
Spielman, D. A. (2018). Iterative solvers for linear equations Why iterative methods ? First-Order Richardson Iteration. 1–8.
Sutrisno, A., Azis, D., Hasanah, A., Ruby, T., & Mega, N. (2025). Comparison of Gauss-Seidel Method , Newton-Raphson Method , and Broyden Method in Solving Nonlinear Equation Systems. International Journal of Advance Social Sciences and Education (IJASSE), 3(2), 47–58. https://doi.org/DOI: https://doi.org/10.59890/ijasse.v3i1.292
Taqwani, R. A., Prabawati, M. N., & Ratnaningsih, N. (2024). Hypothetical Learning Trajectory in Learning Three-Variable Linear Equation Systems. Jurnal Theorems (The Original Research of Mathematics), 9(2008), 134–147. https://doi.org/https://doi.org/10.31949/th.v9i1.9578
Vinsensia, D., Utami, Y., Siregar, F., & Arifin, M. (2024). Improve Refinement Approach Iterative Method for Solution Linear Equition of Sparse Matrices. Jurnal Teknik Informatika C.I.T Medicom, 15(6), 306–313. https://doi.org/https://doi.org/10.35335/cit.Vol15.2024.721.pp306-313
Zhang, J. (2025). Power Iteration on Non-Normal Matrices: Spectral Radius, Jordan Effects, and Krylov-Based Accelerations. Mathematical Modeling and Algorithm Application ISSN:, 6(2), 57–67. https://doi.org/https://doi.org/10.54097/81613487
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Nahdia Tuzzaro, Nadira Islam Madinah, Ari Wibowo
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
