Research Article Open Access

Minimization of ℓ2-Norm of the KSOR Operator

I. K. Youssef1 and A. I. Alzaki1
  • 1 Ain Shams University, Egypt

Abstract

We consider the problem of minimizing the ℓ2-norm of the KSOR operator when solving a linear systems of the form AX = b where, A = I +B (TJ = -B, is the Jacobi iteration matrix), B is skew symmetric matrix. Based on the eigenvalue functional relations given for the KSOR method, we find optimal values of the relaxation parameter which minimize the ℓ2-norm of the KSOR operators. Use the Singular Value Decomposition (SVD) techniques to find an easy computable matrix unitary equivalent to the iteration matrix TKSOR. The optimum value of the relaxation parameter in the KSOR method is accurately approximated through the minimization of the ℓ2-norm of an associated matrix Δ(ω*) which has the same spectrum as the iteration matrix. Numerical example illustrating and confirming the theoretical relations are considered. Using SVD is an easy and effective approach in proving the eigenvalue functional relations and in determining the appropriate value of the relaxation parameter. All calculations are performed with the help of the computer algebra system "Mathematica 8.0".

Journal of Mathematics and Statistics
Volume 8 No. 4, 2012, 461-470

DOI: https://doi.org/10.3844/jmssp.2012.461.470

Submitted On: 17 August 2012 Published On: 8 January 2013

How to Cite: Youssef, I. K. & Alzaki, A. I. (2012). Minimization of ℓ2-Norm of the KSOR Operator. Journal of Mathematics and Statistics, 8(4), 461-470. https://doi.org/10.3844/jmssp.2012.461.470

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Keywords

  • KSOR Iterative Method
  • 2-Norm
  • Singular Value Decomposition (SVD)