y 
LI" 
le 
8 
y 
d 
8 
le 
[S 
d 
[t 
n 
peu) =o vive ll o 2 Ape Y g (10) 
which shall be minimized. By differentiation with respect to x 
we get the normal equation system 
ax - a = 0 -CHBDIMEAS 
The solution of this symmetric definite equation system 
corresponds to the- search for the minimum of the guadratlc 
funotion-cF(x). The main principle of the relaxation 
computation is^ the following. Starting from a trial veotor 
x(0) one chooses a direction vector h. The length of this 
vector h is determined in such a way, that the quadratic 
function F(x) decreases. in this way one gets a better 
approximation x(1) for the veotor of the unknowns. These steps 
are repeated until the minimum of the function F(x) is reached. 
This: isu the. case, if the System (11) is valid: without 
contradictions. The various methods distinguish by the choice 
of the direction h (relaxation directions) and the choice of 
the length of these vectors. 
A possible strategy is to choose the directions in such a 
way, that in the local area of the approximate solutions they 
point to the direction of the largest descent of the quadratic 
function F(x). 
grad(FG0) 2 Ax D - aT = pf) = field (12) 
In the procedure of conjugate gradients the relaxation 
direction h form a system of conjugate directions and the 
vectors of residuals r form an orthogonal system. From a 
theoretical point of view, this procedure leads to an exact 
solution after m steps (m = number of unknown parameters). 
3.4 Mathematical models for error detection: 
  
Conventional least squares adjustment minimizes the sum of 
the squares of the residuals. It minimizes the 2-norm: 
: 1 
vl, = I v - Min cs) 
subject to certain constraints. 
It is generally known that in the presence of blunders the 
result “of a least squares adjustment is usually distorted and 
falsified. Therefore alternatives to least squares adjustment 
have recently received new attention. One other alternative is 
obtained by minimizing the norm of residuals in a different 
way, namely minimizing the 1-norm: 
n 
irs Y, dts Min (12) 
1*1 
Minimizing this norm is equivalent to minimizing the sum of