## 3 Theoretical Basis of the Method of Least Squares with Conditional Equations
 \right]_{\mathbf{X}=\mathbf{X}^{0}},\]  
\[\mathbf{\Delta}=\Psi\!\left(\mathbf{X}^{0}\right)=\left[\Psi_{1}\!\left( \mathbf{X}^{0}\right)\!,\;\Psi_{2}\!\left(\mathbf{X}^{0}\right)\!,\;\dots\;, \;\Psi_{w}\!\left(\mathbf{X}^{0}\right)\right]^{\!\mathsf{T}}.\]  
Taking into account the conditions binding the parameters being determined, and the application of the objective function of the least-squares method, it leads to the optimisation of the problem with the following form (e.g., [38]):  
\[\left. \begin{array}{l}\mathbf{A}d\mathbf{X}+\mathbf{1}=\mathbf{v}\\ \mathbf{B}d\mathbf{X}+\mathbf{\Delta}=0\\ \mathbf{C}_{\mathbf{v}}=\sigma_{0}^{2}\mathbf{P}^{-1}\\ \varphi(d\mathbf{X})=\mathbf{v}^{\mathsf{T}}\mathbf{P}\mathbf{v}=\min\end{array} \right\} \tag{6}\]  
In order to exactly solve this problem, it is necessary to replace the original objective function \(\varphi(d\mathbf{X})=\mathbf{v}^{\mathsf{T}}\mathbf{P}\mathbf{v}\) with the Lagrange function with the following form (e.g., [39])  
\[\varphi_{L}(d\mathbf{X})=\varphi(d\mathbf{X})-2\mathbf{x}^{\mathsf{T}}( \mathbf{B}d\mathbf{X}+\mathbf{\Delta})=\mathbf{v}^{\mathsf{T}}\mathbf{P} \mathbf{v}-2\mathbf{\kappa}^{\mathsf{T}}(\mathbf{B}d\mathbf{X}+\mathbf{\Delta}) \tag{7}\]  
where \(\mathbf{\kappa}\) is the Lagrange's unknown multiplier vector. The optimisation problem \(\min\!\varphi_{L}(d\mathbf{X})=\varphi_{L}(d\overset{\wedge}{\mathbf{X}})\) solves such quantities for which the following sufficient conditions are satisfied:  
\[\left. \begin{array}{l}\frac{\partial\varphi_{L}(d\mathbf{X})}{\partial d \mathbf{X}}\end{array} \right|_{\mathbf{\kappa},d\overset{\wedge}{\mathbf{X}}} =\left.\frac{\partial\varphi_{L}(d\mathbf{X})}{\partial\mathbf{v}}\frac{ \partial\mathbf{v}}{\partial  
\(\overset{\wedge}{\mathbf{v}_{*}}=0\) (within the calculation precision limits). Thus, task (6) is replaced with the optimisation problem, classical for the LS-method, with the following form:  
\[ \begin{array}{l}\mathbf{A}d\mathbf{X}+\mathbf{1}=\mathbf{v}\\ \mathbf{B}d\mathbf{X}+\mathbf{\Delta}=\mathbf{v}_{*}\\ \mathbf{C}_{\mathbf{v}}=\sigma_{0}^{2}\mathbf{P}^{-1}\\ \mathbf{C}_{\mathbf{v}_{*}}=\sigma_{0}^{2}\mathbf{P}^{-1}\\ \varphi(d\mathbf{X})=\mathbf{v}^{\mathsf{T}}\mathbf{P}\mathbf{v}+\mathbf{v}_{* }^{\mathsf{T}}\mathbf{P}_{*}\mathbf{v}_{*}=\min\end{array}  \tag{14}\]  
where:  
\[\widehat{\mathbf{A}}=\left[ \begin{array}{c}\mathbf{A}\\ \mathbf{B}\end{array} \right]\ \widehat{\mathbf{1}}=\left[ \begin{array}{c} \mathbf{1}\\ \mathbf{\Delta}\end{array} \right]\ \widehat{\mathbf{v}}=\left[ \begin{array}{c} \mathbf{v}\\ \mathbf{v}_{*}\end{array} \right]\text{. }\]  
Vector \(\widehat{\mathbf{v}}\) is the total vector of observation errors with the covariance matrix of:  
\[\widehat{\mathbf{C}}=\left[ \begin{array}{cc}\mathbf{C}_{\mathbf{v}}=\sigma_ {0}^{2}\mathbf{P}^{-1}&0\\ 0&\mathbf{C}_{\mathbf{v}_{*}}=\sigma_{0}^{2}\mathbf{P}^{-1}_{*}\end{array}  \right]=\sigma_{0}^{2}\left[ \begin{array}{cc}\mathbf{P}^{-1}&0\\ 0&\mathbf{P}^{-1}_{*}\end{array} \right]=\sigma_{0}^{2}\widehat{\mathbf{P}}^{-1} \tag{15}\]  
where \(\widehat{\mathbf{P}}=\text{diag}(\mathbf{P},\ \mathbf{P}_{*})\) is the total weight matrix (\(\sigma_{0}^{2}\)--variance coefficient common for both components of the model (15). The solution to task (14) is the increment estimator \(d\mathbf{X}\) with the following form:  
\[d\overset{\wedge}{\mathbf{X}}=-(\overset{\wedge}{\mathbf{A}}\overset{\wedge} {\mathbf{P}}\overset{\wedge}{\mathbf{A}})\overset{-1}{\mathbf{A}}\overset{ \wedge}{\mathbf{P}}\overset{\wedge}{\mathbf{1}} \tag{16}\]  
where: \(\overset{\wedge}{\mathbf{A}}\overset{\wedge}{\mathbf{P}}\overset{\wedge}{ \mathbf{A}}=\mathbf{A}^{\mathsf{T}}\mathbf{P}\mathbf{A}+\mathbf{B}^{\mathsf{T }}\mathbf{P}\mathbf{B},\overset{\wedge}{\mathbf{A}}\overset{\wedge}{\mathbf{P }}\overset{\wedge}{\mathbf{1}}=\mathbf{A}^{\mathsf{T}}\mathbf{P}\mathbf{I}+ \mathbf{B}^{\mathsf{T}}\mathbf{P}_{*}\mathbf{\Delta}\).