Adaptive improved block SOR method with orderings

Abstract

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 elimination applied to the system, then the spectral radius of the iterative matrix is reduced to zero. A proper choice of orderings and starting vectors for the iteration were also proposed.In this paper, we apply the above method to two-dimensional cases, and propose the "adaptive improved block SOR method with orderings" for block tridiagonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special 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, then this iterative method converges at most n2 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 example, 0(n) iterations) than n2 iterations. Results of several numerical examples show this efficiency.