SORmenetelmä

The SOR method, or Successive Over-Relaxation method, is an iterative technique used for solving systems of linear equations. It is a modification of the Gauss-Seidel method and is particularly effective for solving large, sparse systems that arise from the discretization of partial differential equations. The core idea of SOR is to accelerate the convergence of the Gauss-Seidel method by introducing a relaxation parameter, denoted by omega (ω).

In the SOR method, each component of the solution vector is updated iteratively. For a system Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector, the update for the i-th component of x at iteration k+1, denoted by x_i^(k+1), is calculated based on the previously computed values and the current iteration's values. The formula incorporates the Gauss-Seidel update and then applies the relaxation factor. Specifically, the new value is a weighted average of the Gauss-Seidel estimate and the previous iteration's value for that component.

The choice of the relaxation parameter ω is crucial for the method's performance. If ω = 1, the SOR method is identical to the Gauss-Seidel method. If 0 < ω < 1, it is called under-relaxation, which can be used to achieve convergence for systems where Gauss-Seidel diverges. If 1 < ω < 2, it is called over-relaxation, which can significantly speed up convergence for systems where Gauss-Seidel converges. Values of ω outside the range (0, 2) generally lead to divergence. Determining the optimal value of ω often depends on the properties of the matrix A and can be a non-trivial task, sometimes requiring adaptive strategies. The SOR method is guaranteed to converge if the matrix A is symmetric and positive-definite, or if it is strictly diagonally dominant, provided that 0 < ω < 2.