exploited. However, he does not go into details. 
Inkilä /9/ describes and compares methods for updating Cholesky 
factorization. The comparison is made theoretically by operation 
counts, and shows basicly no difference between Agee-Turner updating 
and orthogonal transformations, e.g. Givens transformations. 
An extensive review of the algorithmic aspects of on-line triangulation 
is given by Grün in /5/. This includes different methods for sequential 
adjustment. 
Encouraged by these and other publications this author decided to 
implement and test sequential adjustment based on Givens 
transformations (GT). One of the appealing features of GT was its true 
sequential nature, with row by row updating, while for instance TFU 
seemed to work best with groups of observations. 
GIVENS TRANSFORMATIONS IN SEQUENTIAL ADJUSTMENT 
The normal adjustment problem may be written 
v= AR =i (1) 
where A is the (m,n) design matrix, 1 is the observation vector, and v 
1s the vector of residuals. In the sequential case we want to add a 
new observation equation to the original problem 
Va Pi NS b (2) 
so that the complete problem is 
f deme] in 
The solutions of (1) and (3) respectively are given by: 
AIA woz A id (4) 
T T 
(ATA + ala) Y= A" 1 + 3 b (5) 
It may be shown (e.g. /9/) that, having reduced (4) to 
Rx = .t (6) 
by Cholesky factorization, the updated Cholesky reduction 
NX. .8:.¢t (7) 
of (5) (after adding (2)) may be obtained by an orthogonal 
transformation, expressed as 
0 b ; s È | (8) 
a b 0 e 
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