I am currently working on a problem involving the minimization of the $\chi^2$ deviation between a model matrix ($C_{model}$) and a measured matrix ($C_{measured}$). by finding the best fit parameters values :$\Delta g$ $\Delta x$ $\Delta y$ and $\Delta p_l$ by performing iterations.
The minimization problem:
$$ \chi^2 =\sum_{i,j} C^2_{i,j} =\sum_{i,j} \frac{(C_{model,i,j}-\hat{C_{i,j}})^2}{\sigma^2_{i}} = \sum_{i,j} (C_{model,i,j}-\hat{C_{i,j}})^2 W_{i} $$
And:
$$ C_{model,i,j} = \hat{C_{i,j}} + \sum_{k}\frac{\partial{\hat{C_{i,j}}}}{\partial{g_k}} \Delta g_k+ \hat{C_{i,j}} \Delta x^i -\hat{C_{i,j}} \Delta y^j + \sum_{l}\frac{\partial{\hat{C_{i,j}}}}{\partial{p_l}} \Delta p_l $$
Where:
$C_{\text{model}}$ : $i \times j$ matrices
$C_{\text{measured}}$ : $i \times j$ matrices
$\sigma_i$ : vector of length $i$
$\Delta x^i$ : vector of length $i$
$\Delta y^i$ : vector of length $j$
$\Delta g_k$ : vector of length $k$
I have tried to solve this problem before with only one unknown $\Delta g_k$ as following:
$$(C_{model,i,j}-\hat{C_{i,j}}) = \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} \Delta g_k$$ $$ \chi^2 = |(C_{model,i,j}-\hat{C_{i,j}})-\sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} \Delta g_k|^2 $$ $$ \chi^{2}_w = |(C_{model,i,j}-\hat{C_{i,j}})-\sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} \Delta g_k|^{T}W|(C_{model,i,j}-\hat{C_{i,j}})-\sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} \Delta g_k|$$ $$ \frac{\partial{\chi^{2}_w}}{\partial{g_k}} = 2 \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}}^{T} W \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} \Delta g_k - 2 \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} W (C_{model,i,j}-\hat{C_{i,j}}) =0$$
$$ (\sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}}\partial{g_k})^{T}W \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} \partial{g_k} = \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} W(C_{model,i,j}-\hat{C_{i,j}})$$
$$ \Delta g_k =( \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} ^{T}W\frac{\partial{C_{i,j}}}{\partial{g_k}})^{-1} \sum_{k}\frac{\partial{C_{i,j}}}{\partial{g_k}} W(C_{model,i,j}-\hat{C_{i,j}}) $$
However in the problem i am working in the model and measured matrices are not liner in the fit parameters so it Saied the the solution must be iterated until it converges.
Starting from my solution in one variable can some one please explain to me how to extend the problem to include many unknowns, by performing iterations.