Accelerated iterative method for Z-matrices

It has recently been reported that the convergence of the preconditioned Gauss-Seidel method which uses a matrix of the type (I + U) as a preconditioner is faster than the basic iterative method. In this paper, we generalize the preconditioner to the type (I + βU), where β is a positive real number. After discussing convergence of the method applied to Z-matrices, we propose an algorithm for estimating the optimum β. Numerical examples are also given, which show the effectiveness of our algorithm.