Improved SOR method with orderings and direct methods

A generalized SOR method with multiple relaxation parameters is considered for solving a linear system of equations. Optimal choices of the parameters are examined under the assumption that the coefficient matrix is tridiagon al and regular, It is shown that the spectral radius of the iterative matri x is reduced to zero for a pair of parameter values which is computed from the pivots of the Gaussian elimination applied to the system. A proper choi ce of orderings and starting vectors for the iteration is also proposed. When the system is well-conditioned, it is solved stably Ly Gaussian elimin ation; there is little advantage in using an iterative method. However, whe n the system is ill-conditioned, the direct method is not necessarily stabl e and it is often required to improve the numerical solution by a certain i terative algorithm. Some numerical examples are presented which show that t he proposed method is more efficient than the standard iterative refinement method. It is also discussed how to apply our technique to a class of systems which includes Hessenberg systems.