By S. A. Gaal
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Extra info for Linear Analysis & Representation Theory (Grundlehren Der Mathematischen Wissenschaften Series, Vol 198)
Example text
These factorizations can be computed in a numerically stable way using a simple algorithm. For example, consider the factorization Equating terms we have tg2 = 2122, t 2 2 t 1 2 = v 1 2 and T11Tz = V11 - t 1 2 t T 2 . The problem is now reduced to finding the factorization of the modified 46 submatrix VII - t12tT2. The complete factorization can be achieved by repeating this step: I Set T ( i , j )= V ( i , j ) for , all i 2 j. I1 For k = n : -1 : 1, set T ( k ,k ) = T ( k ,k)1/2 and T(1:k-1,k) =T(l:k-l,k)/T(k,k).
For example, we have generated exact q for yj = 1, j = 1,.. 0],and then subtracted 10 from y2, simulating a discrepant value. Fig. 3 shows the perturbations y - q and corresponding i yi - qi f 2Uii associated with the input (first intervals yi - t ~ if 2 ~ i and + + 49 -. 04 I Figure 2. Residuals r = y - Ca along with error bars r izt2uii associated with the least squares fit t o the input data and f i f 20ii associated with the fit for the three adjusted uncertainties corresponding t o algorithms B, C and D.
Simple adjustment model To investigate adjustment procedures for determining estimates of the physical constants we consider a much simpler model involving five (fictitious) 48 6 . 64' 0 2 4 6 8 10 2 Figure 1. Input and adjusted uncertainties corresponding to procedures B, C, and D and for the 10 measurements of G. physical constants a = ( a ~ a2, , C Z ~ and ) ~ p = (PI, P 2 ) T and nine measurements with associated observation matrix -100 100 5 0 -5 100 -100 0 5 -5 100 0 100 -5 5 100 0 -100 -5 5 C= 0 100 100 5 -5 0 100 -100 5 5 5 0 0 100 100 0 0 -5 100 100 0 5 0 100 -100 The first six (last three) observations contain strong information about a (p).