On the Accelerated Overrelaxation Method

I. K. Youssef; M. M. Farid

doi:doi:10.11648/j.pamj.20150401.14

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On the Accelerated Overrelaxation Method

I. K. Youssef, M. M. Farid

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 1)

Received: 29 December 2014 Accepted: 25 January 2015 Published: 2 February 2015

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Abstract

A new variant of the accelerated over relaxation (AOR) method for solving systems of linear algebraic equations, the KAOR method is established. The treatment depends on the use of the extrapolation techniques introduced by Hadjidimos (1978) and the KSOR introduced by Youssef (2012) and its modified version published in (2013). The KAOR is an extrapolation version of the KSOR. This approach has enables us in establishing eigenvalue functional relations concerning the eigenvalues of the iteration matrices and their spectral radii. Moreover, our eigenvalue functional relation depends directly on the eigenvalues of Jacobi iteration matrix. The proposed method considers the advantages of the AOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the minimum value illustrates the smoothness in the choice of optimum parameters.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 1)
DOI	10.11648/j.pamj.20150401.14
Page(s)	26-31
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), 2015. Published by Science Publishing Group

Keywords

SOR, KSOR, AOR, QAOR, KAOR

References

[1]	A. Hadjidimos, Accelerated Overelaxation Method, Math. Comput. , 32, pp149-157, 1978.
[2]	D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
[3]	R.S. Varga, Matrix Iterative Analysis, prentice-Hall, Englewood cliffs, N. J., 1962.
[4]	A. Hadjidimos, A. Yeyios, The Principale of Extrapolation in Connection With the Accelerated Overrelaxation Method, Linear Algebra and its Applications 30, 115-128 , 1980.
[5]	N. M. Missirlis, D. J. Evans, The Extrapolated Successive Overrelaxation (ESOR) Method for Consistently Ordered Matrices, Internat. J. Math. & Math. sci. Vol. 7, No. 2, pp. 361-370, 1984.
[6]	I.K. Youssef, On the Successive Overrelaxation method, J. Math. Stat., 8 pp.176-184, 2012.
[7]	I.K. Youssef, A.A. Taha, On the Modified Successive Overrelaxation Method, Appl. Math.Comput., 219, pp. 4601-4613, 2013.
[8]	G. Avdelas and A. Hadjidimos, Optimum Accelerated Overrelaxation Method in a Special Case, Mathematics of computation Vol. 36, No. 153, 1981.
[9]	M. Madalena Martins, On an Accelerated Over relaxation Iterative Method for Linear Systems With Strictly Diagonally Dominant Matrix, Mathematics of Computation, Vol. 35, No. 152, pp. 1269-1273, 1980.
[10]	A.J. Hughes Hallet, The Convergence of Accelerated Overrelaxation Iterations, Mathematics of Computation, Vol. 47, No. 175, pp. 219-223, 1986.
[11]	Shi-liang Wu, Yu-Jun Liu, A new Version of the Accelerated Overrelaxation Iterative Method, Journal of Applied Mathematics, 2014.

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APA Style

I. K. Youssef, M. M. Farid. (2015). On the Accelerated Overrelaxation Method. Pure and Applied Mathematics Journal, 4(1), 26-31. https://doi.org/10.11648/j.pamj.20150401.14

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ACS Style

I. K. Youssef; M. M. Farid. On the Accelerated Overrelaxation Method. Pure Appl. Math. J. 2015, 4(1), 26-31. doi: 10.11648/j.pamj.20150401.14

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AMA Style

I. K. Youssef, M. M. Farid. On the Accelerated Overrelaxation Method. Pure Appl Math J. 2015;4(1):26-31. doi: 10.11648/j.pamj.20150401.14

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@article{10.11648/j.pamj.20150401.14,
  author = {I. K. Youssef and M. M. Farid},
  title = {On the Accelerated Overrelaxation Method},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {1},
  pages = {26-31},
  doi = {10.11648/j.pamj.20150401.14},
  url = {https://doi.org/10.11648/j.pamj.20150401.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150401.14},
  abstract = {A new variant of the accelerated over relaxation (AOR) method for solving systems of linear algebraic equations, the KAOR method is established. The treatment depends on the use of the extrapolation techniques introduced by Hadjidimos (1978) and the KSOR introduced by Youssef (2012) and its modified version published in (2013). The KAOR is an extrapolation version of the KSOR. This approach has enables us in establishing eigenvalue functional relations concerning the eigenvalues of the iteration matrices and their spectral radii. Moreover, our eigenvalue functional relation depends directly on the eigenvalues of Jacobi iteration matrix. The proposed method considers the advantages of the AOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the minimum value illustrates the smoothness in the choice of optimum parameters.},
 year = {2015}
}

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TY  - JOUR
T1  - On the Accelerated Overrelaxation Method
AU  - I. K. Youssef
AU  - M. M. Farid
Y1  - 2015/02/02
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150401.14
DO  - 10.11648/j.pamj.20150401.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 26
EP  - 31
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150401.14
AB  - A new variant of the accelerated over relaxation (AOR) method for solving systems of linear algebraic equations, the KAOR method is established. The treatment depends on the use of the extrapolation techniques introduced by Hadjidimos (1978) and the KSOR introduced by Youssef (2012) and its modified version published in (2013). The KAOR is an extrapolation version of the KSOR. This approach has enables us in establishing eigenvalue functional relations concerning the eigenvalues of the iteration matrices and their spectral radii. Moreover, our eigenvalue functional relation depends directly on the eigenvalues of Jacobi iteration matrix. The proposed method considers the advantages of the AOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the minimum value illustrates the smoothness in the choice of optimum parameters.
VL  - 4
IS  - 1
ER  -

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Author Information

I. K. Youssef

Math. Dept. Faculty of Science Ain Shams Uni., Cairo, Egypt
M. M. Farid

Basic Science Dept. British University in Egypt, Elshorouk City, Cairo, Egypt

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APA Style

I. K. Youssef, M. M. Farid. (2015). On the Accelerated Overrelaxation Method. Pure and Applied Mathematics Journal, 4(1), 26-31. https://doi.org/10.11648/j.pamj.20150401.14

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ACS Style

I. K. Youssef; M. M. Farid. On the Accelerated Overrelaxation Method. Pure Appl. Math. J. 2015, 4(1), 26-31. doi: 10.11648/j.pamj.20150401.14

Copy | Download

AMA Style

I. K. Youssef, M. M. Farid. On the Accelerated Overrelaxation Method. Pure Appl Math J. 2015;4(1):26-31. doi: 10.11648/j.pamj.20150401.14

Copy | Download

@article{10.11648/j.pamj.20150401.14,
  author = {I. K. Youssef and M. M. Farid},
  title = {On the Accelerated Overrelaxation Method},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {1},
  pages = {26-31},
  doi = {10.11648/j.pamj.20150401.14},
  url = {https://doi.org/10.11648/j.pamj.20150401.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150401.14},
  abstract = {A new variant of the accelerated over relaxation (AOR) method for solving systems of linear algebraic equations, the KAOR method is established. The treatment depends on the use of the extrapolation techniques introduced by Hadjidimos (1978) and the KSOR introduced by Youssef (2012) and its modified version published in (2013). The KAOR is an extrapolation version of the KSOR. This approach has enables us in establishing eigenvalue functional relations concerning the eigenvalues of the iteration matrices and their spectral radii. Moreover, our eigenvalue functional relation depends directly on the eigenvalues of Jacobi iteration matrix. The proposed method considers the advantages of the AOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the minimum value illustrates the smoothness in the choice of optimum parameters.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - On the Accelerated Overrelaxation Method
AU  - I. K. Youssef
AU  - M. M. Farid
Y1  - 2015/02/02
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150401.14
DO  - 10.11648/j.pamj.20150401.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 26
EP  - 31
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150401.14
AB  - A new variant of the accelerated over relaxation (AOR) method for solving systems of linear algebraic equations, the KAOR method is established. The treatment depends on the use of the extrapolation techniques introduced by Hadjidimos (1978) and the KSOR introduced by Youssef (2012) and its modified version published in (2013). The KAOR is an extrapolation version of the KSOR. This approach has enables us in establishing eigenvalue functional relations concerning the eigenvalues of the iteration matrices and their spectral radii. Moreover, our eigenvalue functional relation depends directly on the eigenvalues of Jacobi iteration matrix. The proposed method considers the advantages of the AOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the minimum value illustrates the smoothness in the choice of optimum parameters.
VL  - 4
IS  - 1
ER  -

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