Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System

Wondosen Lisanu Assefa; Ashenafi Woldeselassie Teklehaymanot

doi:doi:10.11648/j.pamj.20211001.13

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Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System

Wondosen Lisanu Assefa, Ashenafi Woldeselassie Teklehaymanot

Published in Pure and Applied Mathematics Journal (Volume 10, Issue 1)

Received: 6 October 2020 Accepted: 3 November 2020 Published: 4 March 2021

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Abstract

This paper describes a method for the numerical solution of linear system of equations. The main interest of refinement of accelerated over relaxation (RAOR) method is to minimize the spectral radius of the iteration matrix in order to increase the rate of convergence of the method comparing to the accelerated over relaxation (AOR) method. That is minimizing the spectral radius means increasing the rate of convergence of the method. This motivates us to refine the refinement of accelerated over relaxation method called second refinement of accelerated over relaxation method (SRAOR). In this paper, we proposed a second refinement of accelerated over relaxation method, which decreases the spectral radius of the iteration matrix significantly comparing to that of the refinement of accelerated over relaxation (RAOR) method. The method is a two-parameter generalization of the refinement of accelerated over relaxation methods and the optimal value of each parameter is derived. The third, fourth and in general the k^th refinement of accelerated methods are also derived. The spectral radius of the iteration matrix and convergence criteria of the second refinement of accelerated over relaxation (SRAOR) are discussed. Finally a numerical example is given in order to see the efficiency of the proposed method comparing with that of the existing methods.

Published in	Pure and Applied Mathematics Journal (Volume 10, Issue 1)
DOI	10.11648/j.pamj.20211001.13
Page(s)	32-37
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

AOR, Refinement AOR, Second Refinement AOR

References

[1]	V. B. Kumar Vatti, Ramadevi, M. S. Kumar Mylapalli, (2018). A Refinement of Accelerated Over Relaxation Method for the Solution of Linear System. International Journal of Pure and Applied mathematics. Vol. 118, No. 18, pp. 1571-1577.
[2]	Apostolos Hadjidimos, (January 1978). Accelerated over relaxation method. Mathematics of computations, Vol. 32, No. 141, pp. 149-157. doi: 10.2307/2006264.
[3]	A. K. YEYIOS, (1989). A necessary condition for the convergence of the accelerated over relaxation (AOR) method. J. Comput. Appl. Math, Vol. 26, No. 3, pp. 371-373.
[4]	Jinyun YUAN, Xiaoqing JIN, (1999). Convergence of the generalized AOR method. Appl. Math. Comput, Vol. 99, No. 1, pp. 35-46.
[5]	Reza Behzadi, (Jan 2019). A New Class AOR Preconditioner for L-Matrices. Journal of Mathematical Research with Applications, Vol. 39, No. 1, pp. 101-110. DOI: 10.3770/j.issn:2095-2651.2019.01.010.
[6]	Tesfaye Kebede Enyew, Gurju Awgichew, Eshetu Haile, Gashaye Dessalew Abie, (January 2020). Second refinement of Gauss-Seidel iterative method for solving a linear system of equation, Ethiop. J. Sci & Technol, Vol. 13, No. 1, pp. 1-15. DOI: 10.4314/ejstv.v13i1.1.
[7]	V. B. Kumar Vatti and Tesfaye Kebede Eneyew, (2011). A refinement of Gauss-Seidel method for solving a linear system of equations, Int. J. Contemp. Math. Sciences, Vol. 6, No. 3, pp. 117-121.
[8]	Richard L. Burden. J. Douglas Faires ”Numerical Analysis”, publisher: Richard Stratton, ninth edition.
[9]	G. Avdelas, G. and A. Hadjidimos, (1981). Optimum AOR Method in a Special Case. Mathematics of Computations, Vol. 36, No. 153.
[10]	K. Youssef and M. M. Farid, (January 31, 2015). On the Accelerated Over relaxation method. Pure and applied Mathematics Journal, vol. 4, No. 1, 2015, pp. 26-31. doi: 10.11648/j.pamj.20150401.14.
[11]	A. Hadjidimos, A. Yeyios, (1980). The Principle of Extrapolation in connection with Accelerated over relaxation method. Linear Algebra and its Applications, 30, pp. 115-128.
[12]	Shi-Liang Wu, Yu-Jun Liu, (26 August 2014). A new version of the accelerated over relaxation iterative method. Journal of applied Mathematics. volume 2014, Article ID725360, pp. 1-6.
[13]	http://dx.doi.org/10.1155/2014/725360
[14]	A. K. Yeyios, (1989). A. Necessary condition for the convergence of the accelerated over relaxation (AOR) method. Journal of Computational and Applied Mathematics. Vol. 26, No. 3, pp. 371-373.
[15]	Mykola Kryshchukb, Jurijs Lavendelsa, Iterative Method for Solving a System of Linear Equations, Procedia Computer Science 104 (2017) 133-137.
[16]	M. Madalena Martins, JULY 1981. Note on Irreducible Diagonally Dominant Matrices and the Convergence of the AOR Iterative Method. American Mathematical Society. VOLUME 37, NUMBER 155, pp. 101-103.

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Wondosen Lisanu Assefa, Ashenafi Woldeselassie Teklehaymanot. (2021). Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System. Pure and Applied Mathematics Journal, 10(1), 32-37. https://doi.org/10.11648/j.pamj.20211001.13

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ACS Style

Wondosen Lisanu Assefa; Ashenafi Woldeselassie Teklehaymanot. Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System. Pure Appl. Math. J. 2021, 10(1), 32-37. doi: 10.11648/j.pamj.20211001.13

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AMA Style

Wondosen Lisanu Assefa, Ashenafi Woldeselassie Teklehaymanot. Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System. Pure Appl Math J. 2021;10(1):32-37. doi: 10.11648/j.pamj.20211001.13

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@article{10.11648/j.pamj.20211001.13,
  author = {Wondosen Lisanu Assefa and Ashenafi Woldeselassie Teklehaymanot},
  title = {Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System},
  journal = {Pure and Applied Mathematics Journal},
  volume = {10},
  number = {1},
  pages = {32-37},
  doi = {10.11648/j.pamj.20211001.13},
  url = {https://doi.org/10.11648/j.pamj.20211001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211001.13},
  abstract = {This paper describes a method for the numerical solution of linear system of equations. The main interest of refinement of accelerated over relaxation (RAOR) method is to minimize the spectral radius of the iteration matrix in order to increase the rate of convergence of the method comparing to the accelerated over relaxation (AOR) method. That is minimizing the spectral radius means increasing the rate of convergence of the method. This motivates us to refine the refinement of accelerated over relaxation method called second refinement of accelerated over relaxation method (SRAOR). In this paper, we proposed a second refinement of accelerated over relaxation method, which decreases the spectral radius of the iteration matrix significantly comparing to that of the refinement of accelerated over relaxation (RAOR) method. The method is a two-parameter generalization of the refinement of accelerated over relaxation methods and the optimal value of each parameter is derived. The third, fourth and in general the kth refinement of accelerated methods are also derived. The spectral radius of the iteration matrix and convergence criteria of the second refinement of accelerated over relaxation (SRAOR) are discussed. Finally a numerical example is given in order to see the efficiency of the proposed method comparing with that of the existing methods.},
 year = {2021}
}

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TY  - JOUR
T1  - Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System
AU  - Wondosen Lisanu Assefa
AU  - Ashenafi Woldeselassie Teklehaymanot
Y1  - 2021/03/04
PY  - 2021
N1  - https://doi.org/10.11648/j.pamj.20211001.13
DO  - 10.11648/j.pamj.20211001.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 32
EP  - 37
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20211001.13
AB  - This paper describes a method for the numerical solution of linear system of equations. The main interest of refinement of accelerated over relaxation (RAOR) method is to minimize the spectral radius of the iteration matrix in order to increase the rate of convergence of the method comparing to the accelerated over relaxation (AOR) method. That is minimizing the spectral radius means increasing the rate of convergence of the method. This motivates us to refine the refinement of accelerated over relaxation method called second refinement of accelerated over relaxation method (SRAOR). In this paper, we proposed a second refinement of accelerated over relaxation method, which decreases the spectral radius of the iteration matrix significantly comparing to that of the refinement of accelerated over relaxation (RAOR) method. The method is a two-parameter generalization of the refinement of accelerated over relaxation methods and the optimal value of each parameter is derived. The third, fourth and in general the kth refinement of accelerated methods are also derived. The spectral radius of the iteration matrix and convergence criteria of the second refinement of accelerated over relaxation (SRAOR) are discussed. Finally a numerical example is given in order to see the efficiency of the proposed method comparing with that of the existing methods.
VL  - 10
IS  - 1
ER  -

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Author Information

Wondosen Lisanu Assefa

Mathematics Department, Faculty of Natural and Computational Science, Woldia University, Woldia, Ethiopia
Ashenafi Woldeselassie Teklehaymanot

Mathematics Department, Faculty of Natural and Computational Science, Woldia University, Woldia, Ethiopia

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APA Style

Wondosen Lisanu Assefa, Ashenafi Woldeselassie Teklehaymanot. (2021). Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System. Pure and Applied Mathematics Journal, 10(1), 32-37. https://doi.org/10.11648/j.pamj.20211001.13

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ACS Style

Wondosen Lisanu Assefa; Ashenafi Woldeselassie Teklehaymanot. Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System. Pure Appl. Math. J. 2021, 10(1), 32-37. doi: 10.11648/j.pamj.20211001.13

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AMA Style

Wondosen Lisanu Assefa, Ashenafi Woldeselassie Teklehaymanot. Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System. Pure Appl Math J. 2021;10(1):32-37. doi: 10.11648/j.pamj.20211001.13

Copy | Download

@article{10.11648/j.pamj.20211001.13,
  author = {Wondosen Lisanu Assefa and Ashenafi Woldeselassie Teklehaymanot},
  title = {Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System},
  journal = {Pure and Applied Mathematics Journal},
  volume = {10},
  number = {1},
  pages = {32-37},
  doi = {10.11648/j.pamj.20211001.13},
  url = {https://doi.org/10.11648/j.pamj.20211001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211001.13},
  abstract = {This paper describes a method for the numerical solution of linear system of equations. The main interest of refinement of accelerated over relaxation (RAOR) method is to minimize the spectral radius of the iteration matrix in order to increase the rate of convergence of the method comparing to the accelerated over relaxation (AOR) method. That is minimizing the spectral radius means increasing the rate of convergence of the method. This motivates us to refine the refinement of accelerated over relaxation method called second refinement of accelerated over relaxation method (SRAOR). In this paper, we proposed a second refinement of accelerated over relaxation method, which decreases the spectral radius of the iteration matrix significantly comparing to that of the refinement of accelerated over relaxation (RAOR) method. The method is a two-parameter generalization of the refinement of accelerated over relaxation methods and the optimal value of each parameter is derived. The third, fourth and in general the kth refinement of accelerated methods are also derived. The spectral radius of the iteration matrix and convergence criteria of the second refinement of accelerated over relaxation (SRAOR) are discussed. Finally a numerical example is given in order to see the efficiency of the proposed method comparing with that of the existing methods.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Second Refinement of Accelerated over Relaxation Method for the Solution of Linear System
AU  - Wondosen Lisanu Assefa
AU  - Ashenafi Woldeselassie Teklehaymanot
Y1  - 2021/03/04
PY  - 2021
N1  - https://doi.org/10.11648/j.pamj.20211001.13
DO  - 10.11648/j.pamj.20211001.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 32
EP  - 37
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20211001.13
AB  - This paper describes a method for the numerical solution of linear system of equations. The main interest of refinement of accelerated over relaxation (RAOR) method is to minimize the spectral radius of the iteration matrix in order to increase the rate of convergence of the method comparing to the accelerated over relaxation (AOR) method. That is minimizing the spectral radius means increasing the rate of convergence of the method. This motivates us to refine the refinement of accelerated over relaxation method called second refinement of accelerated over relaxation method (SRAOR). In this paper, we proposed a second refinement of accelerated over relaxation method, which decreases the spectral radius of the iteration matrix significantly comparing to that of the refinement of accelerated over relaxation (RAOR) method. The method is a two-parameter generalization of the refinement of accelerated over relaxation methods and the optimal value of each parameter is derived. The third, fourth and in general the kth refinement of accelerated methods are also derived. The spectral radius of the iteration matrix and convergence criteria of the second refinement of accelerated over relaxation (SRAOR) are discussed. Finally a numerical example is given in order to see the efficiency of the proposed method comparing with that of the existing methods.
VL  - 10
IS  - 1
ER  -

Copy | Download