On the Optimal Relaxation Parameters to the Improved SOR Method with Orderings

Abstract

To solve non-symmetric linear equations, we have proposed a generalized SOR method, named the improved SOR method with orderings, and for an $n\times n$ tridiagonal matrix, we have given $n$ selections of the multiple relaxation parameters which satisfy $\rho(\mathcal{L}_{\varPhi})=0$ and correspond to the reciprocal numbers of the pivots of Gaussian elimination, where $\mathcal{L}_{\varPhi}$ is the $n\times n$ iterative matrix of this method. In this paper, using the ``essential dimensions-reductions for error vectors'', we investigate the numbers of all conditions for the multiple relaxation parameters which satisfy $\rho(\mathcal{L}_{\varPhi})=0$. As a result, adding to $n$ known selections of the multiple relaxation parameters, we find another type of selections of the multiple relaxation parameters and we conclude that such numbers of conditions are totally $2^{n-1}$ cases for an $n\times n$ tridiagonal matrix. Examples of such selections of multiple relaxation parameters are also contained. For an $n\times n$ Hessenberg matrix, we also obtain the similar results.