On the issue of calculating the weights of unknowns when solving normal equations using the Gauss method

Abstract

It is known from the theory of accuracy analysis of mediocre measurements that the solution of a system of normal equations by the Gauss method simultaneously yields the weight of the last unknown by order of elimination, which is equal to the coefficient of this unknown in the last equation. It is also known that the weight of the penultimate unknown is easily determined by the weight of the latter. To determine the weights of the remaining unknowns, it is necessary to make more or less significant additional calculations, the difficulty of which increases with the increase in the number of unknowns to be determined. Back in the early 30s of the last century, the astronomer Encke recommended, for example, to determine the weights of unknowns by performing repeated solutions of normal equations by rearranging not only the equations, but also the unknowns themselves, and the number of re-solutions of normal equations (according to Encke) is half the number of normal equations. It is easy to see that Encke's method is based only on theorems on the weights of the last and penultimate unknowns. A more advanced method for calculating the weights of unknowns is the method of undetermined factors (weight coefficients), which was proposed in the first half of the last century and has since become established in astronomical and geodetic practice.