Modified Square Root Method of Matrix Inversion
Ahmed Elsayed Sarhan, Bernard Greenberg, Eleanor Rorerts · Technometrics · 1962
When least squares procedures are applied to estimation problems in multiple regression, the main bulk of the calculation is concentrated in the process of inverting the matrix. When the design of the problem is such that the matrix has an identifiable pattern, such as in the analysis of variance or the fitting of response surfaces, specialized techniques for inversion have been suggested in [5], [6], and [8]. In cases where the matrix does not have a special pattern its inversion requires laborious computations by the calculator, either human or machine. Each writer on the subject of matrix inversion appears to have his own preferential method which is advocated because of some desired combination of speed, precision, and accessibility of checking devices throughout the procedure. The present authors, in continuation of the tradition that every writer has his own method, offer a procedure that has proved extremely useful to them. By experience with well over a thousand matrices that had to be inverted before high-speed machines were readily available, we concluded that the method proposed herein offered an optimal combination of the three factors mentioned above. In fact, unless an iterative method such as Hotelling's [7] is programmed for a high-speed computing machine, the present system offers excellent or better precision than most other non-iterative methods that might be used on an electronic computer where speed and checking procedures are less important. The methods proposed do not represent novel or radical departures from the currently published methods such as outlined in Dwyer [2]. The authors simply developed a short-cut and a computational check that proved beneficial. From time to time, many individuals have requested special instructions, however, from our computer personnel regarding these techniques. We decided several years ago, therefore, to record the steps to be followed in a report to the Office of Ordnance Research [9] since this agency was sponsoring the main effort for which the matrix inversion was a component part. In view of the limited distribution of the report cited above, requests for which are continually being received, a wider and more readily available reference source has been deemed desirable. The authors offer this method with no pretense that it is revolutionary but they do feel that computers faced with the problem of matrix inversion will find in its contents a practical and valuable document for this routine problem. This method is a slight modification of the square root method. In explaining it, formulas for calculating the elements of the triangular matrix, its inverse, and the inverse of the original matrix will be given. A more reliable check column is also furnished by working backward from the sum of the entries in the rows rather than in carrying a check column as in conventional methods.