Simultaneous Equations

Abstract

MANY problems in engineering science resolve themselves into the solution of simultaneous linear equations. In general when more than three variables are present these equations tend to become troublesome to solve; their numerical solutions besides being tedious frequently involve differences between quantities very nearly equal, so that unless the calculations be carried to several significant figures considerable errors are likely to arise in the results. To overcome this difficulty recent research has been directed towards the development of approximate and successive approximation methods suitable for routine application. The most promising method appears to be that based on iteration. Such methods were originally given by Gauss and Seidel. In R. & M. 1711 the writer put forward an iteration process which has the advantage of being carried out in a simple tabular form and in which the iteration is carried out not with current values of the variables, but with the differences of consecutive current values. The iteration method is only applicable when the coefficients of the variables satisfy certain conditions which will be discussed in the sequel. For the present the convergency of the process will be assumed.