Adaptive improved block SOR method with orderings

Recently, a generalized SOR method with multiple relaxation parameters were considered for solving a linear system of equations and it was shown that if a pair of parameter values is computed from the pivots of the Gaussian e limination applied to the system, then the spectral radius of the iterative matrix is reduced to zero. A proper choice of orderings and starting vecto rs for the iteration were also proposed. In this paper, we apply the above method to two-dimensional cases, and prop ose the "adaptive improved block SOR method with orderings" for block tridi agonal matrices. The point of this method is to change the multiple relaxat ion parameters not only for each block but also for each iteration. If spec ial multiple relaxation parameters are selected with this method for an n x n block tridiagonal matrix whose block matrices are all n x n matrices, th en this iterative method converges at most n(2) iterations. Hence this is a direct method. In particular, if we select proper orderings and apply the admissible error bounds, then convergence occurs at fewer iterations (for e xample, O(n) iterations) than n(2) iterations. Results of several numerical examples show this efficiency.