Estimation of the successive over-relaxation factor

1. Introduction. Successive over-relaxation and its several variants are well-known methods for solving finite difference equations of elliptic type. To obtain the greatest rate of convergence one must know the spectral radius p of the basic simultaneous displacement iteration matrix. It is also well known (see Varga [5], for example) that using a value slightly larger than p is less serious than using an estimate too small by the same amount. In obtaining an estimate of p, one is willing to expend some small fraction of the total computing time required to solve a set of difference equations. If too little attention is given to the estimate of p, the convergence rate suffers; if too much effort is spent on estimating p, it will not be recovered in the improved convergence rate in the main calculation. It is the purpose of this paper to show how the Kohn-Kato formula for an upper bound of an eigenvalue (see Crandall [1]) is especially well suited for use in the estimation of p for successive over-relaxation.