Abstract
We consider the problem of minimizing the e2-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 e2-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 e2-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".
Highlights
Where, TKSOR is the KSOR iteration matrix. As it was in the SOR the rate of convergence of the KSOR method depends on the choice of the relaxation parameter ω*
Our objective is to find the optimal value of the relaxation parameter ω* which minimizes the l2 -norm of the KSOR operator and illustrate the theoretical results through applications to a numerical example
We will prove the relation between the eigenvalue functional relation between the eigenvalues of TJ and TKSOR by using Singular Value Decomposition (SVD), theorem (2)
Summary
We consider linear systems of the form Equation 1:. The Jacobi Method in matrix form is Equation 4: X[n+1] = TJX[n] + D−1b Tj = D−1(L + U). Such linear systems arise in many different applications for example in the finite difference treatment of the Korteweg de Vries equation, Buckley (1977). For certain classes of matrices (2-cyclic consistently ordered) with property A, in the sense of Young (2003), for such systems there is a functional eigenvalue relation of the form Equation 7:. Most work on the choice of ω is to minimize ρ(TSOR) which is only an asymptotic criteria of the convergence rate of linear stationary iterative method, Hadjidimos and Neumann (1998). Norms of the SOR and MSOR operators for the skew symmetric case
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
