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On the Line Successive Overrelaxation Method

Received: 17 May 2016     Accepted: 30 May 2016     Published: 13 June 2016
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Abstract

A line version of the KSOR method is introduced, LKSOR method. Comparison of the performance of some different iterative techniques with their line format (Jacobi – Gauss Seidel and SOR) are considered. Implementation of LKSOR method for several different formulas in different mesh geometries is discussed. The proposed method considers the advantages of the LSOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the optimum value illustrates the smoothness in the selection of relaxation parameters.

Published in Applied and Computational Mathematics (Volume 5, Issue 3)
DOI 10.11648/j.acm.20160503.12
Page(s) 103-106
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), 2016. Published by Science Publishing Group

Keywords

SOR, LSOR, KSOR Methods, Poisson Equation and Linear System

References
[1] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
[2] D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
[3] Z. I. Woznicki, H. A. Jedrzejec, A new class of modified line-SOR algorithms, (2001), 89-142.
[4] I. K. Youssef, “On the Successive Overrelaxation Method,” Journal of Mathematics and Statistics 8 (2): 176-184, 2012.
[5] R. J. Arms, L. D. Gates, and Zondek, B., A method of Block Iteration, J.Soc.Indust.App1.Math, (1956): Vol. 4, pp. 220-229.
[6] I. K. Youssef, A. A. Taha, On the Modified Successive Overrelaxation method, Applied Mathematics and Computation, 219, 4601-4613, 2013.
[7] S. V. Parter, Multi-line Iterative methods for Elliptic Difference Equations and Fundamental Frequencies, Numr. Math (1961):Vol. 3, pp.305-319.
[8] I. K. Youssef, A. I. Alzaki, Minimization of l2-Norm of the KSOR Operator, Journal of Mathematics and Statistics, 2012.
[9] S. V. Parter, Block Iterative methods, In Elliptic Problem Solvers, ed. Schultz, M. H., Academic Press, (1981): pp. 375-382.
[10] S. V. Parter, Steuerwalt, M., Block Iterative method for Elliptic Finite Element Equation, Society for Industrial and Applied Mathematics, (1985):Vol. 22, No. 1.
[11] D. J. Evans, M. J. Biggins, The Solution of Elliptic Partial Differential equation by a New Block Over-relaxation Technique, Intern. J. Computer. Math. (1982): 269–282.
[12] I. K. Youssef, M. M. Farid, On the Accelerated Overrelaxation Method, Pure and Applied Mathematics Journal, 2015; 4(1): 26-31.
[13] I. K. Youssef, A. M. Shukur, Precondition for discretized fractional boundary value problem, Pure and Applied Mathematics Journal, 2014; 3(1): 1-6.
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  • APA Style

    I. K. Youssef, Salwa M. Ali, M. Y. Hamada. (2016). On the Line Successive Overrelaxation Method. Applied and Computational Mathematics, 5(3), 103-106. https://doi.org/10.11648/j.acm.20160503.12

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    ACS Style

    I. K. Youssef; Salwa M. Ali; M. Y. Hamada. On the Line Successive Overrelaxation Method. Appl. Comput. Math. 2016, 5(3), 103-106. doi: 10.11648/j.acm.20160503.12

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    AMA Style

    I. K. Youssef, Salwa M. Ali, M. Y. Hamada. On the Line Successive Overrelaxation Method. Appl Comput Math. 2016;5(3):103-106. doi: 10.11648/j.acm.20160503.12

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  • @article{10.11648/j.acm.20160503.12,
      author = {I. K. Youssef and Salwa M. Ali and M. Y. Hamada},
      title = {On the Line Successive Overrelaxation Method},
      journal = {Applied and Computational Mathematics},
      volume = {5},
      number = {3},
      pages = {103-106},
      doi = {10.11648/j.acm.20160503.12},
      url = {https://doi.org/10.11648/j.acm.20160503.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160503.12},
      abstract = {A line version of the KSOR method is introduced, LKSOR method. Comparison of the performance of some different iterative techniques with their line format (Jacobi – Gauss Seidel and SOR) are considered. Implementation of LKSOR method for several different formulas in different mesh geometries is discussed. The proposed method considers the advantages of the LSOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the optimum value illustrates the smoothness in the selection of relaxation parameters.},
     year = {2016}
    }
    

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    AB  - A line version of the KSOR method is introduced, LKSOR method. Comparison of the performance of some different iterative techniques with their line format (Jacobi – Gauss Seidel and SOR) are considered. Implementation of LKSOR method for several different formulas in different mesh geometries is discussed. The proposed method considers the advantages of the LSOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the optimum value illustrates the smoothness in the selection of relaxation parameters.
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Author Information
  • Math. Dept., Faculty of Science, Ain Shams Uni., Cairo, Abbassia, Egypt

  • Math. Dept., Faculty of Science, Ain Shams Uni., Cairo, Abbassia, Egypt

  • Math. Dept., Faculty of Science, Ain Shams Uni., Cairo, Abbassia, Egypt

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