Suppose I have the following non linear-model: $$ y=f(x ; \beta)+\varepsilon, $$ where $$ f(x ; \beta) =\sum_{j=0}^{p} w(\theta ; j)\cdot x_{j}. $$
With the exponential Almon structure: $$\omega_{j}\left(\theta_{1}, \theta_{2}\right)=\frac{\exp \left\{\theta_{1} j+\theta_{2} j^{2}\right\}}{\sum_{j=1}^{p} \exp \left\{\theta_{1} j+\theta_{2} j^{2}\right\}}.$$
I would like to obtain the normal equation
$$\boldsymbol{\Delta} \boldsymbol{\beta}=\left(\mathbf{J}^{\mathbf{T}} \mathbf{J}\right)^{-1}\mathbf{J}^{\mathbf{T}} \boldsymbol{\Delta} \mathbf{y}$$
In order to obtain the value of the parameters I need to construct J. Where $$\mathrm{J}=\left(\begin{array}{ccc} \frac{\partial f}{\partial \theta_{1}} & \frac{\partial f}{\partial \theta_{2}} \\ \end{array}\right)$$
Does anyone know a way to rewrite the sum of the given exponential Almon structure such that I can obtain J more easily without the summation sign? One option is to work out the sum manually but this procedure gets time-consuming when p becomes large.
Wiki for nls: https://en.wikipedia.org/wiki/Non-linear_least_squares