Richard S. Varga, Factorization and normalized iterative methods, Boundary Problems in Differential Equations (R. E. Langer, ed. ), pp. 121-142, University of Wisconsin Press, Madison, 1960. MR 22, 12704. Zbl. 100, p. 125. 

1. Introduction

In studying the literature on iterative methods for solving elliptic difference equations, one finds that these iterative methods can be phrased so that they all depend upon the ability to directly solve appropriate matrix equations in few unknowns. In some cases, such as the Young-Frankel successive overrelaxation iterative method [33,10] and the Richardson iterative method [24,39], the matrix equations to be directly solved involve only one linear equation in one unknown. In the other cases, such as the Peaceman-Rachford iterative method [21], and the Douglas-Rachford iterative method [8], tridiagonal matrix equations are directly solved. While the idea of group relaxation [19, 13, 4], the direct solution of matrix equations in few unknowns, has for some time been recognized to be advantageous, it appears that only the systematic direct solution of tridiagonal matrix equations has gained popularity in practical machine codes for solving elliptic difference equations.