Error messages after inequality (141) or equation (A22) as well as error analyses after equation (129) or equation (A23): Calculating the covariances of the corrections

 

After closure of the iteration procedure, the corrected estimated values are available. The calculation of the covariance matrix of the corrections is then carried out according to VDI 2048, Sheet 1, equation (124). It has to be considered, however, that after eliminating the auxiliary variables, the transformed  functional matrix   instead of  is to be used in the scheme described here:

 (19)

This results by considering that in the formulation described here, equation (123) of VDI 2048 takes on the following form:

 (20)

Here according to our assumptions applies, wherein y(x) is a function that represents the auxiliary variables as a function of the measured variables. is defined by the equations . The theorem about implicit functions then says that

   (21)

Applying the chain rule to the terms in equation (20) thus results precisely in

 (22)

the expression already introduced in equation (10). Equation (124) according to VDI 2048 Sheet 2 thus turns into equation (19).

According to VDI2048, all further results that derive from the covariances of the corrections arise from this.