By further augmenting the matrix (8), the value s = Lm will also 
be maintained (see /3/): 
(3) 
© 
f c 7O 
JG uo rt 
u 
oo Oo 
C uU! ctf! 
Concerning weights, the row (a,b) must be multiplied by the square root 
of its weight. For correlated observations, the correlated observation 
equations should be multiplied by the Cholesky factor of the weight 
matrix. 
There are different methods for constructing the orthogonal matrices. 
In practical use, the orthogonal matrices themselves need not be 
constructed as such. Each Q is normally a product of several orthogonal 
matrices with rather simple structure, and the effect of the individual 
multiplications may be computed directly into the elements of the 
matrix operated upon. 
One of the methods is to apply Givens transformations to pairs of rows 
in (8). Q is then constructed as a product of n matrices Qi, i:1.:0ni 
each representing one Givens transformation. Each 03 will affect two 
rows of the matrix it is applied to, being of the form 
I a *0 
Qi z 0 gs (10) 
0 I 0 
0 - Ü c 
Calling the two rows 
z (ry, rp......T4, TO ) (11) 
= (34,85... 84, ^». * ) 
the elements are transformed by 
Pj? C T4 * S 8j (12) 
The desired operation (9) may be executed by sequentially applying 
Givens transformations to the last row (a,b) and the i'th row of R. For 
each pair of rows, c and s are constructed so that ài becomes zero. 
This is achieved when 
Cz Ti / d (13) 
za; VO 
where 
A (14) 
To remove observations, Blais /1/ recommends to apply negative weights 
and use the same formulae. After elaborating (12),(13), and (14) with 
the complex numbers caused by the square root of the negative weight, 
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