Matrix algebra method for deriving oxide factors and its relationship to the principle of least squares

Abstract

Multiple regression analysis of glass property-composition data involves the derivadon of 'normal equations', one for each oxide factor. The number of these is, in general, much smaller than the number of initial equations, which, for each composition, relate the property to the glass composition. The normal equations are readily derived by a matrix multiplication process. It is shown, by the use of a simple numerical example, that the method minimizes the sum of squares of the residuals in the initial set of equations. The method is used to derive oxide factors for calculating thermal expansion coefficients of glasses in a number of systems. Good agreement between calculated and measured expansion coefficients was obtained, the best being for a large group of alkaline earth aluminoborosilicate glasses, data for which is given in a recent patent. For oxides of both group I and group II elements, there is a linear dependence of the oxide factor on z/a, where z is the valency and a is the cation-oxygen distance. The method and results are compared with earlier work.