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Usually minimizing $$\chi^2 = \sum_i w_i (y_i - f(a_0, a_1, \dots, a_n, x_i))^2$$ where $a_k$ are parameters is done by taking $w_i = 1/\sigma_i^2$ where $\sigma_i$ is a observation error of $y_i$ while value of $x_i$ assumed to be exact. In this case errors of $a_k$ are calculated correctly in linear and non-linear cases (e.g. in GSL package).

However what should I take as $w_i$ in case when errors of both $x_i$ and $y_i$ are known? Something like $w_i = 1/(\sigma_{x,i}^2+\sigma_{y,i}^2)$? Is there any way to make e.g. GSL to calculate $\sigma_{a_i}$ at least approximately in this case?

All errors assumed to be uncorrelated and Gaussian.

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