A Hollerith Technique for the Solution of Normal Equations

Abstract

IN THlE critical study of the results of large-scale sample surveys it is frequently necessary to consider data classified in several different ways, and to attempt to disentangle the effects of the various classifications which will usually not be orthogonal to one another. One way of doing this is to fit constants by least squares, assuming the effects implied by the classifications to be additive; for a discussion of the method, see Yates [6, p. 137 et seq.]. The process of fitting involves the solution of the normal equations, a set of simultaneous equations equal in number to the total number of categories in all the classifications. If this number is at all large, the computations become very lengthy, and it is desirable to use Hollerith machinery, more especially as the main computations of the survey will often be done on Hollerith machines, and the data will already be punched on cards. At least two methods of solving simultaneous equations with the aid of Hollerith machines have been published [3, 4]. Both employ a technique of pivotal condensation, and demand the use of a range of machines outside the scope of a small installation. In the present coiltext the large number of equations may lead to a serious accumulation of rounding errors, and there are advantages in the alternative technique of successive approximation, as described, for example, by Stevens [5]. The present paper gives a method of mechanising this technique, using only the basic Hollerith machines, the sorter and tabulator. For producing the working pack, a reproducer is desirable though not essential.