On the modified successive overrelaxation method

Keywords: SOR KSOR MSOR MKSOR and iterative methods a b s t r a c t New forms of the modified successive overrelaxation (MSOR); MKSOR, MKSOR1 and MKSOR2 are introduced. These forms depend on the updated successive overrelaxation method (KSOR) and the combination of the SOR and the KSOR methods. We present analysis of the convergence of the MKSOR methods with fixed parameters for solving linear system of equations AX ¼ b, where A belongs to a subclass of consistently ordered matrices. These matrices appear in the numerical treatment of elliptic partial differential equations corresponding to the red black ordering. Also, functional relations for the eigenvalues of the iteration matrices of the Jacobi, the MKSOR methods and the relaxation parameters are established. A general representative numerical example illustrating this treatment is considered. Different values of the relaxation parameters are used on each set of equations. We have implemented our treatment using computer algebra software ''Mathematica 8.0''. New results on this subject are presented and compared with those of Young. This approach is promising and will help in the numerical treatment of boundary value problems. Other extensions and applications for further work are mentioned. Solving sparse systems of linear equations is at the heart of scientific computing. Large sparse systems arise in using dis-cretization techniques in science and engineering problems [1–3]. The purpose of this paper is to introduce new versions of the MSOR method, MKSOR, MKSOR1 and MKSOR2. Establish theoretical relations analogues to those of Young [1] for the MSOR method and give simple proves that depend on a theorem in the partitioned determinants. Also, combinations of the two approaches (MSOR and MKSOR) and confirmation of the overall properties with a well-known numerical example are considered. Given the linear system AX ¼ b, where A 2 R m;m , b 2 R m and Det ðAÞ – 0 with non-vanishing diagonal elements, this system can be written in component wise form as: