Fitting

  Experimental data analysis frequently leads to the following set of m simultaneous equations for the n (< m) unknowns c_j (an overdetermined system):

Here the c_j are the unknowns and the f_j(u_i), b are known. If we introduce

the (m,n) matrix A = (f_j(u_i)) the (n,1) matrix the (m,1) matrix

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 f_i(u_j) = u_j^i-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 Q^Tb.

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 can be found by solving Ax = z with the least squares condition, where

Using the pseudoinverse, one gets x = A⁺b.