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Weighted nonlinear least-squares fitting minimizes the function
where W = diag(w_1,w_2,...,w_n) is the weighting matrix,
and the weights w_i are commonly defined as w_i = 1/\sigma_i^2,
where \sigma_i is the error in the ith measurement.
A simple change of variables \tilde{f} = \sqrt{W} f yields
\Phi(x) = {1 \over 2} ||\tilde{f}||^2, which is in the
same form as the unweighted case. The user can either perform this
transform directly on their function residuals and Jacobian, or use
the gsl_multifit_fdfsolver_wset
interface which automatically
performs the correct scaling. To manually perform this transformation,
the residuals and Jacobian should be modified according to
where Y_i = Y(x,t_i).