Minimization of ℓ2-Norm of the KSOR Operator
I. K. Youssef and A. I. Alzaki
DOI : 10.3844/jmssp.2012.461.470
Journal of Mathematics and Statistics
Volume 8, Issue 4
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".
© 2012 I. K. Youssef and A. I. Alzaki. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.