The 3-Block KSOR Method for Full Rank Rectangular Systems

I. K. Youssef; Salwa M. Ali; M. A. Naser

doi:doi:10.11648/j.pamj.20160504.13

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The 3-Block KSOR Method for Full Rank Rectangular Systems

I. K. Youssef, Salwa M. Ali, M. A. Naser

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 4)

Received: 24 May 2016 Accepted: 2 June 2016 Published: 21 June 2016

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Abstract

A new version of the KSOR method is considered to introduce the least squares solution of a full rank over determinant system of linear algebraic equations. The treatment depends on introducing an augmented non-singular square system through splitting the coefficient matrix A into two matrices A₁, A₂ with non-singular part, A₁. Accordingly, a new version of the 3-block SOR method is introduced, the 3-block KSOR method. Selection of the relaxation parameter which guarantees the convergence in the sense of reducing the spectral radius of the iteration matrix is considered. Application of the theoretical results to a numerical example has confirmed the expected behaviour of the 3-block KSOR method.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 4)
DOI	10.11648/j.pamj.20160504.13
Page(s)	103-107
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), 2024. Published by Science Publishing Group

Keywords

SOR, KSOR, 3 Block SOR Method, Rectangular Systems

References

[1]	D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
[2]	I. K. Youssef, On the Successive Overrelaxation Method, Journal of Mathematics and Statistics. 8 (2): 176-184, 2012.
[3]	I. K. Youssef, A. A. Taha, On the Modified Successive Overrelaxation method, Applied Mathematics and Computation. 219, 4601-4613, 2013.
[4]	T. L. Markham, M. Neumann, R. J. Plemmons, Convergence of a Direct-Iterative Method for Large-Scale Least-Squares Problems, Linear Algebra and Its Applications. 69: 155-167, 1985.
[5]	V. A. Miller, M. Neumann, Successive Over Relaxation Methods for Solving the Rank Deficient Least Squares Problem, Linear Algebra Appl. 88-89: 533-557, 1987.
[6]	W. Niethammer, J. de pillis, R. S. Varga, Convergence of Block Iterative Methods Applied to Sparse Least-Squares Problems, Linear Algebra and Its Applications. 58: 327-341, 1984.
[7]	A. Bjorck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
[8]	A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
[9]	C. H. Santos, B. P. B. Silva, J. Y. Yuan, Block SOR Methods for Rank-Deficient Least-Squares Problems, Journal of Computational and Applied Mathematics. 100: 1-9, 1998.
[10]	M. T. Darvishi, F. Khani, S. Hamedi-Nezhad, B. Zheng, Symmetric Block-SOR Methods for Rank-Deficient Least Squares Problems, Journal of Computational and Applied Mathematics. 215: 14-27, 2007.
[11]	R. J. Plemmons, Adjustment by Least Squares in Geodesy Using Block Iterative Methods for Sparse Matrices, in Proceeding of the U. S. Army Conference on Numerical Analysis and Computers, EL Paso, Texas, 1979.
[12]	A. Berman, M. Neumann, Proper Splittings of Rectangular Matrices, SIAM J. Apll. math. 31: 307-312, 1976.
[13]	C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, SIAM, Philadelphia, 1995.

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APA Style

I. K. Youssef, Salwa M. Ali, M. A. Naser. (2016). The 3-Block KSOR Method for Full Rank Rectangular Systems. Pure and Applied Mathematics Journal, 5(4), 103-107. https://doi.org/10.11648/j.pamj.20160504.13

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ACS Style

I. K. Youssef; Salwa M. Ali; M. A. Naser. The 3-Block KSOR Method for Full Rank Rectangular Systems. Pure Appl. Math. J. 2016, 5(4), 103-107. doi: 10.11648/j.pamj.20160504.13

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AMA Style

I. K. Youssef, Salwa M. Ali, M. A. Naser. The 3-Block KSOR Method for Full Rank Rectangular Systems. Pure Appl Math J. 2016;5(4):103-107. doi: 10.11648/j.pamj.20160504.13

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@article{10.11648/j.pamj.20160504.13,
  author = {I. K. Youssef and Salwa M. Ali and M. A. Naser},
  title = {The 3-Block KSOR Method for Full Rank Rectangular Systems},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {103-107},
  doi = {10.11648/j.pamj.20160504.13},
  url = {https://doi.org/10.11648/j.pamj.20160504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.13},
  abstract = {A new version of the KSOR method is considered to introduce the least squares solution of a full rank over determinant system of linear algebraic equations. The treatment depends on introducing an augmented non-singular square system through splitting the coefficient matrix A into two matrices A1, A2 with non-singular part, A1. Accordingly, a new version of the 3-block SOR method is introduced, the 3-block KSOR method. Selection of the relaxation parameter which guarantees the convergence in the sense of reducing the spectral radius of the iteration matrix is considered. Application of the theoretical results to a numerical example has confirmed the expected behaviour of the 3-block KSOR method.},
 year = {2016}
}

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TY  - JOUR
T1  - The 3-Block KSOR Method for Full Rank Rectangular Systems
AU  - I. K. Youssef
AU  - Salwa M. Ali
AU  - M. A. Naser
Y1  - 2016/06/21
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.13
DO  - 10.11648/j.pamj.20160504.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 103
EP  - 107
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160504.13
AB  - A new version of the KSOR method is considered to introduce the least squares solution of a full rank over determinant system of linear algebraic equations. The treatment depends on introducing an augmented non-singular square system through splitting the coefficient matrix A into two matrices A1, A2 with non-singular part, A1. Accordingly, a new version of the 3-block SOR method is introduced, the 3-block KSOR method. Selection of the relaxation parameter which guarantees the convergence in the sense of reducing the spectral radius of the iteration matrix is considered. Application of the theoretical results to a numerical example has confirmed the expected behaviour of the 3-block KSOR method.
VL  - 5
IS  - 4
ER  -

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Author Information

I. K. Youssef

Math Department, Faculty of Science, Ain Shams University, Cairo, Abbassia, Egypt
Salwa M. Ali

Math Department, Faculty of Science, Ain Shams University, Cairo, Abbassia, Egypt
M. A. Naser

Math Department, Faculty of Science, Ain Shams University, Cairo, Abbassia, Egypt

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APA Style

I. K. Youssef, Salwa M. Ali, M. A. Naser. (2016). The 3-Block KSOR Method for Full Rank Rectangular Systems. Pure and Applied Mathematics Journal, 5(4), 103-107. https://doi.org/10.11648/j.pamj.20160504.13

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ACS Style

I. K. Youssef; Salwa M. Ali; M. A. Naser. The 3-Block KSOR Method for Full Rank Rectangular Systems. Pure Appl. Math. J. 2016, 5(4), 103-107. doi: 10.11648/j.pamj.20160504.13

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AMA Style

I. K. Youssef, Salwa M. Ali, M. A. Naser. The 3-Block KSOR Method for Full Rank Rectangular Systems. Pure Appl Math J. 2016;5(4):103-107. doi: 10.11648/j.pamj.20160504.13

Copy | Download

@article{10.11648/j.pamj.20160504.13,
  author = {I. K. Youssef and Salwa M. Ali and M. A. Naser},
  title = {The 3-Block KSOR Method for Full Rank Rectangular Systems},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {103-107},
  doi = {10.11648/j.pamj.20160504.13},
  url = {https://doi.org/10.11648/j.pamj.20160504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.13},
  abstract = {A new version of the KSOR method is considered to introduce the least squares solution of a full rank over determinant system of linear algebraic equations. The treatment depends on introducing an augmented non-singular square system through splitting the coefficient matrix A into two matrices A1, A2 with non-singular part, A1. Accordingly, a new version of the 3-block SOR method is introduced, the 3-block KSOR method. Selection of the relaxation parameter which guarantees the convergence in the sense of reducing the spectral radius of the iteration matrix is considered. Application of the theoretical results to a numerical example has confirmed the expected behaviour of the 3-block KSOR method.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The 3-Block KSOR Method for Full Rank Rectangular Systems
AU  - I. K. Youssef
AU  - Salwa M. Ali
AU  - M. A. Naser
Y1  - 2016/06/21
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.13
DO  - 10.11648/j.pamj.20160504.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 103
EP  - 107
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160504.13
AB  - A new version of the KSOR method is considered to introduce the least squares solution of a full rank over determinant system of linear algebraic equations. The treatment depends on introducing an augmented non-singular square system through splitting the coefficient matrix A into two matrices A1, A2 with non-singular part, A1. Accordingly, a new version of the 3-block SOR method is introduced, the 3-block KSOR method. Selection of the relaxation parameter which guarantees the convergence in the sense of reducing the spectral radius of the iteration matrix is considered. Application of the theoretical results to a numerical example has confirmed the expected behaviour of the 3-block KSOR method.
VL  - 5
IS  - 4
ER  -

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