How to find Parameters in nonlinear Regression Model?

Question

I have a nonlinear Regression Model with eleven observations of $x,y$. How do I find the parameters $a,b,c,d$ of the model: $ y=f(x)=a + b \sin cx e^{dx}$ by using the function: $$\Phi(a, b, c, d)=\sum_{i = 1}^{11}{(y_i - f(x_i))^2}$$ Subject to the following conditons: \begin{eqnarray} \frac{\partial\Phi}{\partial a} &=& 0 \\ \frac{\partial\Phi}{\partial b} &=& 0 \\ \frac{\partial\Phi}{\partial c} &=& 0 \\ \frac{\partial\Phi}{\partial d} &=& 0 \end{eqnarray} How can one implement it in Matlab and is there a possible Algorithm, how can I ensure measurement precision? Any help would be good, Thanks.

Answer 1

You should probably use fmincon or lsqnonlin and take your derivatives by hand. Levenberg-Marquardt is normally the method of choice for non-linear least-squares, you can do it with lsqnonlin. Mathematical precision is measured by your cost function $\Phi(a, b, c, d)$