Solving a system of non linear equations

Question

I have got a system of non-linear equations of the form

$$A x_1^B \exp \bigg(\frac{- C}{x_1} \bigg) = k_1$$ $$A x_2^B \exp \bigg(\frac{- C}{x_2} \bigg) = k_2$$ $$A x_3^B \exp \bigg(\frac{- C}{x_3} \bigg) = k_3$$

where $[x_1, x_2, x_3]$ and $[k_1, k_2, k_3]$ are known. The couple of constants $[A, B, C]$ is the unknown.

I'd like to know what is the best way to solve this kind of problem involving a non-linear system of equations.

Thanks a lot.

Answer 1

We have three equations of the form

$$\alpha \, x_i^{\beta} \exp \left(- \frac{\gamma}{x_i}\right) = k_i$$

Taking the logarithm of both sides,

$$\ln (\alpha) \, + \ln (x_i) \beta - \left(\frac{1}{x_i}\right) \gamma = \ln (k_i)$$

Let $\tilde{\alpha} := \ln (\alpha)$. We then obtain a linear system

$$\begin{bmatrix} 1 & \ln (x_1) & -\frac{1}{x_1}\\ 1 & \ln (x_2) & -\frac{1}{x_2}\\ 1 & \ln (x_3) & -\frac{1}{x_3}\end{bmatrix} \begin{bmatrix} \tilde{\alpha}\\ \beta\\ \gamma\end{bmatrix} = \begin{bmatrix} \ln (k_1)\\ \ln (k_2)\\ \ln (k_3)\end{bmatrix}$$