Author(s): I. K. Youssef | A. I. Alzaki
Journal: Journal of Mathematics and Statistics
ISSN 1549-3644
Volume: 8;
Issue: 4;
Start page: 461;
Date: 2012;
Original page
Keywords: KSOR Iterative Method | â2-Norm | Singular Value Decomposition (SVD)
ABSTRACT
We consider the problem of minimizing the â2-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 â2-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 â2-norm of an associated matrix Î(Ï*) 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 âMathematica 8.0â.
Journal: Journal of Mathematics and Statistics
ISSN 1549-3644
Volume: 8;
Issue: 4;
Start page: 461;
Date: 2012;
Original page
Keywords: KSOR Iterative Method | â2-Norm | Singular Value Decomposition (SVD)
ABSTRACT
We consider the problem of minimizing the â2-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 â2-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 â2-norm of an associated matrix Î(Ï*) 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 âMathematica 8.0â.
