Abstract
Problem statement: A new variant of the Successive Overrelaxation (SO R) method for solving linear algebraic systems, the KSOR method was introduced. The treatment depends on the assumption that the current component can be used s imultaneously 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 in troduce 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 firs t step which affected all the subsequent calculatio ns. It was proved that the KSOR can converge for all po ssible values of the relaxation parameter, ω*∈R-(- 2, 0) not only for ( ω∈(0, 2) as in the SOR method. A new eigenvalue funct ional relation similar to that of the SOR method between the eigenvalues of the it eration 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 con sidered. 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 relaxin g 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 .
Highlights
The problem of solving linear systems of algebraic equations appears as a final stage in solving many problems in different areas of science and engineering, it is the result of the discretization techniques of the mathematical models representing realistic problems (Saad and Vorst, 2000) and the references cited therein
In the second example we present the eigenvalues of the Jacobi μi,I = 1,2,3 and 4 and Gauss Seidel vi,i = 1,2,3 and 4 iteration matrices and obtained the eigenvalues λi,i = 1,2,3 and 4 of the Successive Over-Relaxation (SOR) iteration matrix as functions in ω and the eigenvalues βi,i = 1,2,3 and 4 of the KSOR iteration as functions in ω*
It remains to introduce an effective procedure for the estimation of the optimum value of the relaxation parameter ω*opt. which maximizes the rate of convergence of the proposed KSOR method and this will be the objective of a subsequent work
Summary
The problem of solving linear systems of algebraic equations appears as a final stage in solving many problems in different areas of science and engineering, it is the result of the discretization techniques of the mathematical models representing realistic problems (Saad and Vorst, 2000) and the references cited therein. Jacobi method: can be used in addition to the use of the most recent calculated ones (i.e., updating the residue simultaneously ( ) X[n +1] = TjX[n] + D-1b, Tj = D-1 L + U with the current new component). This (7) process leads to an implicit formula but it is explicit due to the linearity of the equations. Definition: The spectral radius of a matrix H, denoted ρ(H), is given by: of the proposed method with other well- known iterative methods especially with SOR with optimal values of the relaxation parameter ω has proved the efficiency and ρ(H) = max{ λi : λiis an eigen value of H}. The smaller the spectral radius of the iteration matrix is, the faster the rate of convergence
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