The comparison of standard ellipses with the artificial circles gives only a rough 
impression of the precision, because only a limited number of functions of the 
coordinates are considered. 
A better way to compare the real variance covariance matrix with the artificial 
one is possible by means of the general eigenvalue problem. Denote the artificial 
matrix by H, which is composed with the value c = l in the covariance function 
2 = c.l... One obtains the maximum eigenvalue uma, 
from the general eigenvalue problem: 
G -uH = 0 
| pH, 
If one now chooses c = U max: any function of the coordinates has a better 
precision than that of the function calculated from H. 
A complication in the judgement of Umax is caused by the dependancy of Umax on 
the size of the block. For homogeneous networks it has been shown, that In 
(u max) increases linear to the natural logarithm of the rank r of the matrices G 
and H (or the number of coordinates minus the number of degrees of freedom of 
the S-system). 
1n(u ax) 
  
  
  
1 r 1n(r) 
figure 4.2: Àn (u 4,4,) increases linear to In (1). 
The maximum eigenvalues resulting from the general eigenvalue problem applied 
to parts of G (parts of the block) show the same relationschip. This property can 
be used to test the homogenuity of the precision described by G (/11/). 
A value c, for which the artificial matrix H can reasonably replace the real 
variance covariance matrix G is found at the intersection of the line and the axis 
of In(u ), as indicated in figure 4.2. 
max 
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