Notes on the Solution of Algebraic Linear Simultaneous Equations

In this paper four methods of solving simultaneous equations and inverting matrices are described from the viewpoint of the practical computer. The first three methods can be performed on ordinary desk calculating machines; the fourth uses Hollerith punched card equipment. The desk methods considered are the method of elimination or ‘pivotal condensation’, the method of ‘orthogonal vectors’, and the method of Choleski. The elimination method is, with slight variants, the simple method taught at school, used in such a way that solutions are obtained with the maximum possible speed and accuracy. In the method of orthogonal vectors, applicable only to symmetric matrices, a new set of variables is chosen in such a way that the matrix is transformed to a diagonal form, from which the solution of equations or the inverse of the matrix is immediately obtainable. The Choleski method, applied here again only to symmetric matrices, expresses the square matrix as the product of two triangular matrices, the reciprocation of which is a relatively simple operation. This method is quicker and more accurate than the others and can be used, with slight modifications, in the case of unsymmetric matrices. The procedure used with punched card equipment is essentially a mechanization of the elimination method, with a considerable advantage in speed over the corresponding desk method.