On the Successive Overrelaxation Method

Abstract

Problem statement:A new variant of the Successive Overrelaxation (SOR) method for solving linear algebraic systems, the KSOR method was introduced. The treatment depends on the assumption that the current component can be used simultaneously in the evaluation in addition to the use the most recent calculated components as in the SOR method. Approach: Using the hidden explicit characterization of linear functions to introduce a new version of the SOR, the KSOR method. Prove the convergence and the consistency analysis of the proposed method. Test the method through application to well-known examples. Results: The proposed method had the advantage of updating the first component in the first equation from the first step which affected all the subsequent calculations. It was proved that the KSOR can converge for all possible values of the relaxation parameter, ω*∈R-[-2, 0] not only for (ω∈(0, 2) as in the SOR method. A new eigenvalue functional relation similar to that of the SOR method between the eigenvalues of the iteration matrices of the Jacobi and the KSOR methods was proved. Numerical examples illustrating this treatment, comparison with the SOR with optimal values of the relaxation parameter were considered. Conclusion: The relaxation parameter ω* in the proposed method, can take values, ω*∈R-[-2, 0] not only for (ω∈(0, 2) as in the SOR. The enlargement of the domain has the affect of relaxing the sensivity near the optimum value of the relaxation parameter. Moreover, all the advantages of the SOR method are conserved and the proposed method can be applied to any system. This approach is promising and will help in the numerical treatment of boundary value problems. Other extensions and applications for further work are mentioned.

MINIMIZATION OF ℓ₂-NORM OF THE KSOR OPERATOR

“…As it was in the SOR the rate of convergence of the KSOR method depends on the choice of the relaxation parameter ω*. For certain classes of matrices (2-cyclic consistently ordered with property A), Youssef (2012) established the eigenvalue functional relation Equation 11:…”

“…For simplicity we adapted the right hand side b as it was in Young (2003) and Youssef (2012) so that the exact solution is x 1 = 1, x 2 = 1, x 3 = 1, x 4 = 1.…”

“…Milleo et al (2006), considered systems with block skew symmetric Jacobi iteration matrix and used the SVD in studying the behavior of the SOR operator from the 2 l -norm point of view they determined theoretically the minimizing relaxation parameter of the 2 l -norm. Youssef (2012), defined the KSOR operator, we used the SVD in re-prove the functional eigenvalue relation for the KSOR operator and the corresponding unitary block 2×2 matrix ∆(ω * ). we employed the same argument as in Yin and Yuan (2002), also in Milleo et al (2006) for systems with block skew symmetric Jacobi iteration matrix and used the SVD in studying the behavior of the 2 l -norm of the KSOR operator.…”