Optimum Overrelaxation Parameter for the SOR Method for Solving the Eigenvalue Problem

Abstract

We study the eigenvalue problem $Ax = \lambda x$, where A is a consistently ordered positive definite matrix. The first eigenvalue of A is obtained with the eigenvector by the SOR method. We first introduce the Jacobi and SOR iteration matrices for the eigenvalue problem, and clarify that the spectral radii, that is the maximum eigenvalues, of both the matrices are unity, but the convergence rate, that is the ratio of the first two eigenvalues in radius, is smaller than unity. Next, we consider the optimum overrelaxation parameter of the SOR method. The optimum accelerating parameter minimizing the convergence rate is obtained from the first two eigenvalues (in radius) of the Jacobi iteration matrix. Since the eigenvalues are not known a priori, we propose a practical SOR method: in this method, the estimated overrelaxation parameters are used instead of the optimum value. Finally these results are confirmed by some numerical examples.