Fitting
Experimental data analysis frequently leads to the following set of m simultaneous equations for the n (< m) unknowns cj (an overdetermined system):
Here the cj are the unknowns and the fj(ui), b File:Hepa img355.gif are known. If we introduce
| the (m,n) matrix | A = (fj(ui)) |
| the (n,1) matrix | File:Hepa img356.gif |
| the (m,1) matrix | File:Hepa img357.gif |
the problem to solve becomes
where the sign 
means that we want to find the vector x in the range of A which is closest to b according to some norm ( 
Branham90, Flowers95).
As an example we choose the fitting of a second-order polynomial. With fi(uj) = uji-1, the matrix A in the above equation becomes
and Ax = b can be solved e.g. by QR decomposition: QRx = b becomes x = R-1 QTb.
As a second example we look at the fitting of a second-order surface
through the 
neighbours of a point in an image. The coordinates u,v and the given values b are:
The coefficients of the second-order polynomial File:Hepa img365.gif can be found by solving Ax = z with the least squares condition, where
Using the pseudoinverse, one gets x = A+b.
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