Bulletin of the
Korean Mathematical Society BKMS

For $Ax=b$, it has recently been reported that the convergence of the preconditioned Gauss-Seidel iterative method which uses a matrix of the type $P=I+S\left( \alpha \right) $ to perform certain elementary row operations on is faster than the basic Gauss-Seidel method. In this paper, we discuss the adaptive Gauss-Seidel iterative method which uses $P=I+S\left( \alpha \right) +\bar{K}\left( \beta \right) $ as a preconditioner. We present some comparison theorems, which show the rate of convergence of the new method is faster than the basic method and the method in [7] theoretically. Numerical examples show the effectiveness of our algorithm.

Keywords: Gauss-Seidel iterative method, preconditioned method, $Z$-matrix, diagonal dominant matrix

MSC numbers: 65F08