Estimating Optimum Overrelaxation Parameters

finding the optimum overrelaxation parameter o¡¡, is an important and often a difficult part of the problem [1, p. 257]. When A is a 2-cyclic, consistently ordered, real symmetric, positive-definite matrix (this will be assumed in the rest of the paper), only the spectral radius p(£i) of the associated Gauss-Seidel iteration matrix £i is needed to determine the optimum parameter co¡, [2, p. 109]. When p(£i) is close to unity, small changes in the estimate for p(£i) can drastically affect the rate of convergence of the S.O.R. iterations and, thus, an accurate estimate of p(£i) is needed. In practice, two basic approaches are often used to estimate p(£i) or equivalently wbOne approach is to carry out a number of S.O.R. iterations with some o < 03b and then on the basis of numerical results obtain a new estimate for oib [1], [3], [4], [12]. A second approach is to obtain an estimate for p(£i), using the power method, prior to carrying out the S.O.R. iterations [1], [2], [5], [6]. The second approach is useful when the iterative process is slowly convergent, when the matrix problem (1.1) is to be solved a number of times for different vectors f, or when very accurate solutions are required. In this paper it is shown how to use Chebyshev polynomials to accelerate the convergence of the power method. The description of the procedure includes a method for terminating the iterations in an attempt to minimize the total computer time required to solve the problem. The method has worked well in practice in a number of difficult problems. More details are given in [13]. The numerical procedures given in this paper apply equally well when the cyclic Chebyshev semi-iterative method [2], [7] is used to solve the matrix problem (1.1).