Commentary, By Anne Greenbaum

Abstract From the very early days, Gene Golub has been a driving force in the development and analysis of iterative methods for solving large sparse linear systems – problems for which Gaussian elimination is often prohibitive in terms of both storage and computation time. We review five of his seminal papers in this field. Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second-order Richardson iterative methods, Parts I and II, by Golub and Varga [10] This paper is probably less well-known today than it should be. In it the authors show the remarkable similarity between the Chebyshev semi-iterative method, the successive overrelaxation (SOR) method applied to an expanded matrix equation, and the second-order Richardson iterative method. They conclude that the Chebyshev semi-iterative method is to be preferred over the other two, since its iteration matrix has the smallest spectral norm, while the work per iteration is the same as that for the other methods. They present numerical results with the different methods used to solve elliptic difference equations.