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2. Equation Solution using Matrices
Observation equations can be written in the form [C] x [U] = [K]. Using n for the number of unknowns and m for the number of observations, the matirces dimensions are:
| n[U]1 | m[K]1 | m[V]1 |
The unknowns in the U matrix are solved using the matrix algorithm Equation D-1:
| [U] = [CTC]-1 x [CTK] | |
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[Q] = [CTC]-1 |
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| [U] = [Q] x [CTK] | Equation D-1 |
[Q] is a symmetric m x m matrix; [CTK] is a m x 1 matrix. Equation D-1 is a simultaneous solution of m equations in m unknowns.
Because there are redundant measurements, a residual term is added to each observation equation so the matrix algorithm is [C] x [U] = [K] + [V]. Residuals are computed after the adjustment using Equation D-2.
| [V] = [CU] -[K] | Equation D-2 |