Hello, I'm trying to solve a non linear equation using a computer program. It's a discrete equation and I'm not really sure how to get the Jacobian. This is the equation and how I've approached it


$\displaystyle \sum_{i}\Bigg\{\sum_{j}\sum_{k}\frac{W_{ij}\exp^{\ alpha(\Delta X_i - G^k_{ij})}}{\sum_{i}W_{ij}\exp^{\alpha(\Delta X_i - G^k_{ij})}} \Bigg\}=\sum_{i} C_i $

I want to solve this equation for $\displaystyle \Delta X_i $

So, for example:

$\displaystyle \sum_{j}\sum_{k}\frac{W_{1j}\exp^{\alpha(\Delta X_1 - G^k_{1j})}}{\sum_{i}W_{ij}\exp^{\alpha(\Delta X_i - G^k_{ij})}}= C_1 $

I let $\displaystyle X_i =\exp^{\alpha(\Delta X_i)}$ , $\displaystyle G^k_{ij} = W_{ij}\exp^{\alpha(- G^k_{ij})}$ and $\displaystyle L = \sum_{i}W_{ij}\exp^{\alpha(\Delta X_i - G^k_{ij})} = \sum_{i}X_iG^k_{ij}
$ (Denominator)
This leaves me with:

$\displaystyle \sum_{j}\sum_{k}\frac{X_iG^k_{ij} }{L} = C_i $

At this stage my equation is a function $\displaystyle F(X_i, L)$

I differentiate the function next

Differentation with respect to $\displaystyle L$ gives me:

$\displaystyle \frac{\delta F(x_i,L)}{\delta L}= - \sum_{j}\sum_{k}\frac{X_iG^k_{ij} }{L^2} $ (1)

Differentation with respect to $\displaystyle X_i$ gives me:

$\displaystyle \frac{\delta F(x_i,L)}{\delta x_i}=\sum_{j}\sum_{k}\frac{G^k_{ij}\sum_{i}X_iG^k_ {ij} -X_iG^k_{ij}\sum_{i}G^k_ij}{L^2} = C_i (2)$

The program I'm using to solve this equation needs the Jacobian. I know the Jacobian is

$\displaystyle J =
\frac{\delta y_1}{\delta x_1}&.&.&.&\frac{\delta y_1}{\delta x_n}\\
\frac{\delta y_n}{\delta x_1}&.&.&.&\frac{\delta y_n}{\delta x_n}\\

The problem is I have no idea how to apply this to the two equations I obtained.

If anyone can help me with this I would be very grateful.