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Tienstra's method was developed to solve parameter adjustment with linear equality constraints, which has been otherwise often carried out by directly applying the conventional (or standard) method of Lagrange multipliers to quadratic optimization problems with a positive definite matrix. We analyze the computational complexity of the celebrated Tienstra's method and compare it with that of the method of Lagrange multipliers. We show that Tienstra's method is not only statistically elegant but also interestingly of significant computational advantage over the method of Lagrange multipliers to solve weighted least squares adjustment with linear equality constraints, with the saving of computational cost by a minimum of about 38% to a maximum of 100%. Tienstra's method can be very important for large scale problems of adjustment and inversion with linear equality constraints. [ABSTRACT FROM AUTHOR]

The effects of incorrect weights on the weighted least squares (WLS) estimate, its accuracy, relative goodness and reliability have been thoroughly investigated in statistical and geodetic literature. Although the variance of unit weight has been one of the most important quantities in statistics and geodesy, little has been done to understand the effect of incorrect weights of observations and incorrect prior information on this quantity, except for an earlier study of Koch who showed that incorrect weights of observations would create a bias in the estimation of the variance of unit weight, though they do not affect the unbiasedness of the WLS estimates of parameters and the corrections to the observations. Since Koch did not directly give the formula to compute the bias, we will derive the bias of the estimated variance of unit weight due to incorrect weights of observations in this paper. In the case of incorrect prior information, both incorrect prior mean and incorrect prior weights can independently contribute a bias to the estimate of the variance of unit weight. Two simulated examples clearly show that the bias of the estimated variance of unit weight due to incorrect weights can be much larger than the variance of unit weight itself. [ABSTRACT FROM AUTHOR]