In line iterative methods, one considers the linear systems obtained from the discretization of PDE over grids as described in Figure (1) or Figure (2). Also, block iterative techniques is faster than the corresponding point wise form. The treatment of partitioned systems is easier than the treatment of the overall system, since the subsystems are much smaller than the original system. Partitioning large linear system into block forms is a problem in itself, in many cases the matrix of the smaller system has properties which make its solution particularly convenient. Īccordingly, equation (1) can be approximated at the point P by the equation, Let, P denote a typical grid point of the grid defined in Figure (1) or Figure (2), R denote to the point to the right of the point P and L denote to the point to the left of the point P along the horizontal grid line similarly, T denote to the point above the point P and B denote to the point below the point P along the vertical grid line as shown in figure (3). We also analyze the convergence behavior.Īs usual, the discussion of the treatment is introduced through a well-known model problem, equation as

Implementation of LKSOR algorithms for several different grid labelling (natural and red black ordering) formulas in different mesh geometries are discussed. Our main objective in this work is to introduce the line version of the KSOR method introduced in 2012, as well as the comparison of its performance. Moreover, the LSOR method can be adapted to give the same improvement done in the SOR, it is well known that the LSOR is asymptotically faster by a factor of over the point SOR. In addition, the line Gauss Seidel performs faster than the point Gauss Seidel method. It is well- known that, the line Jacobi method performs like the point Gauss Seidel method for some classes of linear algebraic systems. The LKSOR method is adapted from the KSOR method by the same philosophy as done in the line successive overrelaxation method, LSOR. The paper is devoted to describe a new version of efficient iterative algorithm for solving systems of linear equations, the line KSOR, LKSOR.

The popularity of SOR algorithms is a great measure due to their simplicity from the programming and computational points of view. Large linear systems appears in many areas of science and engineering, especially from the discretization of elliptic partial differential equations equation (3). The successive overrelaxation (SOR) method and its line variants are among the most popular and efficient iterative methods used for solving large and sparse linear systems.

Gauss seidel sor python