Minimization of ℓ₂-Norm of the KSOR Operator

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 (T_J = -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 T_KSOR. 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".