The Square Root Method and its Use in Correlation and Regression

LTHOUGH many improvements in correlation calculational techniques have been introduced during recent years, we are desirous of discovering further improvements in techniques and more compact methods. Compact methods are those which enable us to write the entire solution in compact form with a minimum of transferring and recording of results. Such methods demand that the operations be designed in such a way that the machine is used to carry out many calculational steps as a single machine operation. It is the aim of this paper to present in some detail, and with illustrations to a correlation problem previously used by the author in discussing compact correlation techniques [3], the "square root" method [4] of solving equations. This solution enables one to replace the dual rows of a Doolittle solution by single rows. The resulting solution is much more compact and the calculational steps are easier than those of the Doolittle solution. The method does demand an additional operation (taking the square root of diagonal terms) as compared with the Doolittle solution, but this can be worked into the machine operation involving the computation of the diagonal term. The method as outlined here is applicable to symmetric equations. It can be extended to apply to non-symmetric cases, but then there is little advantage over conventional Doolittle methods. But if the matrix is symmetric, the compactness resulting makes possible (and this is not necessarily true with extremely compact methods) an easier identification of the terms of the operations following and, as a result, a tendency toward fewer errors, less computing time, and more pleasure for the computer. The method as outlined is especially valuable in case a modern calculating machine which uses the same keyboard for multiplicand and multiplier is available, but it is also applicable to other modern machines. It can be used to advantage, too, with manually operated machines. The presentation follows that used in the paper on "Recent Developments in Correlation Techniques" [3] published previously in this JOURNAL and indeed a first explanation of the method, for those familiar with the Doolittle Method, might be made on the basis that the dual