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


\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 \Delta X_i

So, for example:

\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 X_i =\exp^{\alpha(\Delta X_i)} , G^k_{ij} = W_{ij}\exp^{\alpha(- G^k_{ij})} and L = \sum_{i}W_{ij}\exp^{\alpha(\Delta X_i - G^k_{ij})} = \sum_{i}X_iG^k_{ij}<br />
This leaves me with:

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

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

I differentiate the function next

Differentation with respect to L gives me:

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

Differentation with respect to X_i gives me:

\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

J = <br />
\left[<br />
\begin{array}{ccccc}<br />
\frac{\delta y_1}{\delta x_1}&.&.&.&\frac{\delta y_1}{\delta x_n}\\<br />
.&.&.&.&.\\<br />
\frac{\delta y_n}{\delta x_1}&.&.&.&\frac{\delta y_n}{\delta x_n}\\<br />
\end{array}<br />
\right]<br />

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.