Araştırma Makalesi

One of the primary difficulties in linear algebra, considering its widespread application in many different domains, is solving linear system of equations. It is nevertheless apparent that there is a need for a quick, effective approach that can handle a variety of linear systems. In the realm of large and sparse systems, iterative methods play a crucial role in finding solutions. This research paper makes a significant contribution by introducing an enhancement to the current methodology Successive and Accelerated Over Relaxation methods, referred to as the "Third Refinement of Successive and Accelerated Over Relaxation Methods." This new iterative approach demonstrates its effectiveness when applied to coefficient matrices exhibiting properties such as 𝑀-matrix, irreducible diagonal dominance, positive definiteness and symmetry characteristics. Significantly, the proposed method substantially reduces the spectral radius, resulting in fewer iterations and notably enhancing the convergence rate. Numerical experiments were conducted to evaluate its performance compared to existing second refinement of Successive and Accelerated Over Relaxation methods. The outcomes underscore the "Third Refinement of Successive and Accelerated Over Relaxation" methods potentially to boost the efficiency of solving linear systems, thus rendering it a valuable asset within the arsenal of numerical methodologies utilized in

scientific and engineering realms.

SOR Method AOR Method Third Refinement Linear System Irreducible Matrix.

- T. K. Eneyew, G. Awgichew, E. Haile, and D. A. Gashaye, “Second refinement of Gauss-Seidel iteration method for solving linear system of equations,” Ethiopia Journal of Science and Technology, vol. 13, no. 1, pp. 1-15, 2020.
- H. Firew, G. G. Genanew, and M. C. Hailu, “Second degree generalized Successive over-relaxation method for solving system of linear equations.” Momona Ethiopia Journal of Sciences, vol. 12, no. 1, pp. 60-71, 2020.
- M. Saha, and J. Chakrabarty, “Convergence of generalized Jacobi, Gauss-Seidel and SOR methods for linear systems.” Internationl Journal of Applied Computational Mathematics, vol. 77, pp. 1-6, 2020.
- A. Fiseha, “Accelerated over relaxation method for solving a class of complex linear system of equations.” Master’s Thesis, Bahir Dar University, Ethiopia. Unpublished, 2020.
- K. J. Audu, Y. A. Yahaya, K. R. Adeboye, and U. Y. Abubakar, “Convergence of triple accelerated over relaxation (TAOR) for M-matrix linear systems.” Iranian Journal of Optimization, vol. 13, no. 2, 2021.
- V. B. Kumar Vatti, G. Chinna Rao, and Srinesh S. Pai, “Reaccelerated over-relaxation (ROR) method.” Bulettin of the International Mathematical Virtual Institute, vol. 10, no. 2, pp. 315-324, 2020.
- N. A. Tuah, N. H. Ngo, H. L. Nguyen, “Convergent result for linear conformable pseudo-parabolic equation,” Cankaya University Journal of Science and Engineering, vol. 20, no. 1, pp. 022-027, 2023.
- M., Bendehiba, and D. Rafik, “Pseudo-spectrum and the numerical range for Ricci tensor on the oscillator group of dimensions four.” Cankaya University Journal of Science and Engineering, vol. 20, no. 1, pp. 009-021, 2023.
- B. Khalef,and D. Rafik, “Abderrahmane, S. Pseudo-spectrum and numerical range of matrices Walker of Dimension Three.” Cankaya University Journal of Science and Engineering, vol. 20, no. 01, pp. 001-008, 2023.
- K. J. Audu, Y. A. Yahaya, K. R. Adeboye, and U. Y. Abubakar, “Extended Accelerated over relaxation (EAOR) method for solution of a large and sparse linear system.” Journal of Science,Technology, Mathematics and Education, vol. 17, no. 1, pp. 228-236, 2021.
- K. J. Audu, Y. A. Yahaya, K. R. Adeboye, and U. Y. Abubakar, “Refinement of extended accelerated over relaxation (EAOR) method for solution of linear systems.” Nigerian Annals of Pure and Applied Sciences, vol. 4, no. 1, pp. 51-61, 2021.
- M. S. R. Baloch, Z. A. Kalhoro, M. S. Khalil, A. W. Shaikh, “A New Improved Classical Iterative Algorithm for Solving System of Linear Equation.” Proceedings of the Pakistan Academy of Sciences: A Physical and Computational Sciences, vol. 58, no. 4, pp. 69-81, 2021.
- L. W. Assefa, and A. W. Teklehaymanot, “Second refinement of accelerated over relaxation method for the solution of linear system.” Pure and Applied Mathematics Journal, vol. 10, pp. 32-37, 2021.
- N. Ahmad, and F. Shaheen, “Study of Numerical Solution of Linear System of Equation by Using SOR Algorithm.” Communications in Mathematics and Applications, vol. 12, no. 4, pp. 853-867, 2021.
- K. J. Audu, J. N. Essien, A. B. Zhiri, and A. R. Taiwo, “A Third Refinement of Jacobi Method for Solutions to System of Linear Equations,” FUDMA Journal of Sciences, vol. 7, no. 5, pp. 234-239, 2023.
- K. J. Audu, J. N. Essien, “An Accelerated Iterative Technique: Third Refinement of Gauss Seidel Algorithm for Linear System.” Computer Science and Mathematics Forum, vol. 2, pp. 1-6, 2023.
- X. Zhang, Q. Wang, and T. Li, “The accelerated over-relaxation splitting method for solving symmetric tensor equations.” Computational and Applied Mathematics, vol. 39, no. 155, pp. 1-14, 2020.
- K. Vatti, R. Sri, and M. S. Mylapalli, “A refinement of accelerated over relaxation method for the solution of linear systems.” International Journal of Pure and Applied Mathematics, vol. 118, pp. 1571-1577, 2018.
- V.B.K. Vatti, Numerical Analysis Iterative Methods, I. K International Publishing House, Pvt, Limited, New Delhi, India, 2016.
- R. Abdullahi, and R. Muhammad, “Refinement of preconditioned overrelaxation algorithm for solution of the linear algebraic system 𝑨𝒙=𝒃.” Science World Journal, vol. 16, no. 3, pp. 26 – 31, 2021.

Yıl 2024,
Cilt: 21 Sayı: 1, 18 - 32, 01.05.2024
### Öz

### Kaynakça

- T. K. Eneyew, G. Awgichew, E. Haile, and D. A. Gashaye, “Second refinement of Gauss-Seidel iteration method for solving linear system of equations,” Ethiopia Journal of Science and Technology, vol. 13, no. 1, pp. 1-15, 2020.
- H. Firew, G. G. Genanew, and M. C. Hailu, “Second degree generalized Successive over-relaxation method for solving system of linear equations.” Momona Ethiopia Journal of Sciences, vol. 12, no. 1, pp. 60-71, 2020.
- M. Saha, and J. Chakrabarty, “Convergence of generalized Jacobi, Gauss-Seidel and SOR methods for linear systems.” Internationl Journal of Applied Computational Mathematics, vol. 77, pp. 1-6, 2020.
- A. Fiseha, “Accelerated over relaxation method for solving a class of complex linear system of equations.” Master’s Thesis, Bahir Dar University, Ethiopia. Unpublished, 2020.
- K. J. Audu, Y. A. Yahaya, K. R. Adeboye, and U. Y. Abubakar, “Convergence of triple accelerated over relaxation (TAOR) for M-matrix linear systems.” Iranian Journal of Optimization, vol. 13, no. 2, 2021.
- V. B. Kumar Vatti, G. Chinna Rao, and Srinesh S. Pai, “Reaccelerated over-relaxation (ROR) method.” Bulettin of the International Mathematical Virtual Institute, vol. 10, no. 2, pp. 315-324, 2020.
- N. A. Tuah, N. H. Ngo, H. L. Nguyen, “Convergent result for linear conformable pseudo-parabolic equation,” Cankaya University Journal of Science and Engineering, vol. 20, no. 1, pp. 022-027, 2023.
- M., Bendehiba, and D. Rafik, “Pseudo-spectrum and the numerical range for Ricci tensor on the oscillator group of dimensions four.” Cankaya University Journal of Science and Engineering, vol. 20, no. 1, pp. 009-021, 2023.
- B. Khalef,and D. Rafik, “Abderrahmane, S. Pseudo-spectrum and numerical range of matrices Walker of Dimension Three.” Cankaya University Journal of Science and Engineering, vol. 20, no. 01, pp. 001-008, 2023.
- K. J. Audu, Y. A. Yahaya, K. R. Adeboye, and U. Y. Abubakar, “Extended Accelerated over relaxation (EAOR) method for solution of a large and sparse linear system.” Journal of Science,Technology, Mathematics and Education, vol. 17, no. 1, pp. 228-236, 2021.
- K. J. Audu, Y. A. Yahaya, K. R. Adeboye, and U. Y. Abubakar, “Refinement of extended accelerated over relaxation (EAOR) method for solution of linear systems.” Nigerian Annals of Pure and Applied Sciences, vol. 4, no. 1, pp. 51-61, 2021.
- M. S. R. Baloch, Z. A. Kalhoro, M. S. Khalil, A. W. Shaikh, “A New Improved Classical Iterative Algorithm for Solving System of Linear Equation.” Proceedings of the Pakistan Academy of Sciences: A Physical and Computational Sciences, vol. 58, no. 4, pp. 69-81, 2021.
- L. W. Assefa, and A. W. Teklehaymanot, “Second refinement of accelerated over relaxation method for the solution of linear system.” Pure and Applied Mathematics Journal, vol. 10, pp. 32-37, 2021.
- N. Ahmad, and F. Shaheen, “Study of Numerical Solution of Linear System of Equation by Using SOR Algorithm.” Communications in Mathematics and Applications, vol. 12, no. 4, pp. 853-867, 2021.
- K. J. Audu, J. N. Essien, A. B. Zhiri, and A. R. Taiwo, “A Third Refinement of Jacobi Method for Solutions to System of Linear Equations,” FUDMA Journal of Sciences, vol. 7, no. 5, pp. 234-239, 2023.
- K. J. Audu, J. N. Essien, “An Accelerated Iterative Technique: Third Refinement of Gauss Seidel Algorithm for Linear System.” Computer Science and Mathematics Forum, vol. 2, pp. 1-6, 2023.
- X. Zhang, Q. Wang, and T. Li, “The accelerated over-relaxation splitting method for solving symmetric tensor equations.” Computational and Applied Mathematics, vol. 39, no. 155, pp. 1-14, 2020.
- K. Vatti, R. Sri, and M. S. Mylapalli, “A refinement of accelerated over relaxation method for the solution of linear systems.” International Journal of Pure and Applied Mathematics, vol. 118, pp. 1571-1577, 2018.
- V.B.K. Vatti, Numerical Analysis Iterative Methods, I. K International Publishing House, Pvt, Limited, New Delhi, India, 2016.
- R. Abdullahi, and R. Muhammad, “Refinement of preconditioned overrelaxation algorithm for solution of the linear algebraic system 𝑨𝒙=𝒃.” Science World Journal, vol. 16, no. 3, pp. 26 – 31, 2021.

Toplam 20 adet kaynakça vardır.

Birincil Dil | İngilizce |
---|---|

Konular | Sayısal Analiz |

Bölüm | Makaleler |

Yazarlar | |

Yayımlanma Tarihi | 1 Mayıs 2024 |

Gönderilme Tarihi | 6 Şubat 2024 |

Kabul Tarihi | 27 Mart 2024 |

Yayımlandığı Sayı | Yıl 2024 Cilt: 21 Sayı: 1 |