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 ordinary measurements that the solution of a system of normal equations by the Gauss method simultaneously yields the weight of the last unknown in the 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 from the weight of the last unknown. To determine the weights of the remaining unknowns, it is necessary to perform more or less significant additional calculations, the difficulty of which increases with the number of unknowns to be determined. As early as the beginning of the 1930s, the astronomer Encke recommended, for example, to determine the weights of unknowns by repeated solutions of the normal equations, rearranging not only the equations but also the unknowns themselves, with the number of re-solutions (according to Encke) being half the number of normal equations. It is easy to see that Encke’s method is based solely on the theorems concerning the weights of the last and penultimate unknowns. A more advanced method for calculating the weights of unknowns is the method of undetermined multipliers (weight coefficients), which was proposed in the first half of the last century and has since become established in astronomical and geodetic practice. It should be noted that in practice, weight coefficients are computed in various ways (by Hansen’s method or using certain multipliers derived from the coefficients of the normal equations). Finally, Prof. I. M. Bakhrin recommends (in mine surveying practice) using the general formula for the error of any function of unknowns to calculate the weights of unknowns.

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