Matrix and other direct methods for the solution of systems of linear difference equations

Abstract The investigations described in this paper were initiated in an attempt to replace by direct methods the successive approximation methods such as those of Southwell and Thom for the solution of systems of difference equations arising in the approximate solutions of linear partial differential equations. Boundary problems of this type form the subject of part III, which is the kernel of the paper. As the work progressed it was found that the methods evolved were applicable, and capable of extension, to step-by-step solutions also, and to ordinary as well as partial differential equations. These topics are presented in parts I, II and IV. Matrix methods naturally predominate. The methods are illustrated by small-scale examples worked on desk machines, but the operations involved are, we believe, capable of being handled efficiently and simply by modern high-speed digital computers.