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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