3.2.4 Least-Squares Adjustment 
230 
The functions (5) containing the unknown parameters as well as the 
observed quantities provide the non-linear model for the least-squares 
adjustment. The functional values are computed easily by using sub 
routines for the successive rotations. The derivatives with respect to 
the unknowns and observations however, cannot be expressed in an equally 
short and elegant form. For this reason the method of numerical diffe 
rentiation is applied. A great demand for computer time is the price to 
be paid for the benefits of a short and transparent formulation from the 
computer program. 
The linearized functions have the form of the least-squares model of the 
combined case: 
Ax + Bv + w = 0 
which yields the normal equation: 
A T (BP B T ) Ax- A T (BP -1 B T ) w 
N u 
and the solution vector: 
with its covariance matrix: 
N 1 u 
X = 
£x =a o^' 1 
The contribution of each pair of rays to the normal equations is com 
puted using matrix algebra. The normal equations are solved iteratively 
by the Cholesky decomposition method. The full covariance matrix of all 
unknowns and all residuals is computed in the last iterative step. 
3«3 Software 
The software is divided into two independent programs: 
- KALIB for the instrument calibration and 
- ORIENT for the orientation 
Both programs have a modular structure and are in FORTRAN. They are 
highly interactive. The operator is lead through the program by 
appropriate instructions and he can influence the program by choosing 
different options. Also, he can correct and edit the input data. 
The program KALIB requires the measurements of a number of plate carrier 
crosses in two Z-levels. The coordinates of these crosses are known. 
According to the principles of space-resection, the calibration of the 
instrument is determined and, if needed, an adjustment is made. After 
an initial adjustment, the instruments have remained stable up to now. 
The proper adjustment of the scales is an important prerequisite for the 
orientation. Normally the adjustment is checked once a month. The 
complete measuring and calculating procedure takes about 30 min. 
The program ORIENT consists of 12 subroutines and encompasses about 2500 
code lines. Around 40$ are needed for the actual calculating process 
and 60$ for the interactive organization and input-output. A represen 
tation of how the program runs is shown in Fig.4*