The general theory of relaxation methods applied to linear systems

Abstract During the last few years Southwell and his fellow-workers have developed a new method for the numerical solution of a very general type of problem in mathematical physics and engineering. The method was originally devised for the determination of stresses in frameworks, but it has proved to be directly applicable to any problem which is reducible to the solution of a system of non-homogeneous, linear, simultaneous algebraic equations in a finite number of unknown variables.* Southwell’s “ relaxation m ethod” is one of successive approximation and, in order to complete the previous investigations of this method, it is necessary to prove that the successive approximations do actually converge towards the exact solutions. This formal proof is given in § 4. Southwell’s “ relaxation” methods are not directly applicable to continuous systems, where the number of unknown variables is infinite, but it is shown here that simple extensions and modifications of the relaxation method render it suitable for application to either discrete or continuous systems (§ 5). The general theory of relaxation methods is then developed in terms of the theory of linear operators (§7) and sufficient conditions are prescribed for the convergence of the process of approximation (§ 8). These general methods are then applied to the solution of non-homogeneous, linear integral equations (§ 10) and to the solution of nonhomogeneous, linear differential equations (§§ 11, 12).