Given $\Delta t$, I want to solve in MATLAB the following system of equations with complex exponentials in unknowns $\alpha_k$ and $\nu_k$
$$\lambda_k = \exp \left((-\alpha_k + j2\pi\nu_k)\Delta t)\right)$$
where $k = 1, 2, \dots, K$ and $\lambda_k$ is a complex vector of size $K$. I am trying to find ways that is better than symbolic method. Could someone suggest me a simpler solution that can handle large $K$?
Hint.
Using de Moivre's identity and calling $\lambda_k = x_k + i y_k$ we have
$$ \cases{ 2\pi\nu_k\Delta t = \arctan\left(\frac{y_k}{x_k}\right)\\ e^{-\alpha_k\Delta t}=\sqrt{x_k^2+y_k^2} } $$