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## 5.9 Test on linearization

Mathematical model of geodetic network adjustment in `gama-local` is defined as a set of known real-valued differentiable functions

 ```(11) L^* = f(X^*) ```

where L^* is a vector of theoretical correct observations and X^* is a vector of correct values of parameters. For the given sample set of observations `L` and the unknown vector of residuals `v` we can express the estimate of parameters `X` as a nonlinear set of equations

 ```(12) L + v = f(X). ```

With approximate values `X_0` of unknown parameters

 ```(12) X = X_0 + x ```

we can linearize the equations (12)

 ``` L + v = f(X_0) + f'(X_0)x. ```

yielding the linear set of equations (1)

Unknown parameters in `gama-local` mathematical model are points coordinates and orientation angles (transforming observed directions to bearings). The observables described by functions (12) belong into two classes

• linear observables: horizontal and slope distances, height differences, control coordinates and vectors (coordinate differences),
• angular observables: directions, horizontal and zenith angles.

Internally in `gama-local` unknown corrections to linear observables are computed in millimeters and corrections to angular observables in centigrade seconds. To reflect the internal units in used all partial derivatives of angular observables by coordinates are scaled by factor 2000/pi.

When computing coefficients of project equations (1) we expect that approximate coordinates of points are known with sufficient accuracy needed for linearization of generally nonlinear relations between observations and unknown paramters. Most often this is true but not always and generally we have to check how close our approximation is to adjusted parameters.

Generally we check linearization in adjustment by double calculation of residuals

 ``` v^I = Ax - b, v^II = ~l(~x) - l, ```

Program `gama-local` similarly computes and tests differences in values of adjusted observations once computed from residuals and once from adjusted coordinates. For measured directions and angles `gama-local` computes in addition transverse deviation corresponding to computed angle difference in the distance of target point (or the farther of two targets for angle). As a criterion of bad linearization is supposed positional deviation greater or equal to 0.0005 millimetres.

## Example

 ```Test of linearization error *************************** Diffs in adj. obs from residuals and from adjusted coordinates ************************************************************** i standpoint target observed r difference ================================= value = [mm|cc] = [cc] == [mm]= 2 3022184030 3022724008 dist. 28.39200 -7.070 -0.003 3 3022724002 dist. 72.30700 -18.815 -0.001 7 3000001063 dir. 286.305200 11.272 -0.002 -0.001 8 3022724008 dir. 357.800600 -23.947 0.037 0.002 ```

From the practical point of view it might seem that the tolerance 0.0005 mm for detecting poor linearization is too strict. Its exceeding in program `gama-local` results in repeated adjustment with substitute adjusted coordinates for approximate. Given tolerance was chosen so strict to guarantee that listed output results would never be influenced by linearization and could serve for verification and testing of numerical solutions produced by other programs.

Iterated adjustement with successive improvement of approximate unknown coordinates converges usually even for gross errors in initial estimates of unknown coordinates. If the influence of linearization is detected after adjustment, typically only one iteration is sufficient for recovering.

For any automatically controlled iteration we have to set up certain stopping criterion independent on the convergence and results obtained. Program `gama-local` computes iterated adjustment three times at maximum. If the bad linearization is detected even after three readjustments it signals that given network configuration is somehow suspicious.

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