$$ \begin{aligned} J_{F}\left(x^{[i]}\right) \Delta x^{[i]} & =F\left(x^{[i]}\right) \\ x^{[i+1]} & =x^{[i]}-\Delta x^{[i]} \end{aligned} $$

$$ \left.\frac{\partial F_{i}(x)}{\partial x_{j}}\right|{x=x x^{[i]}}=\frac{F{i}\left(x^{[i]}+h e_{j}\right)-F_{i}\left(x^{[i]}\right)}{h} $$

$$ F\left(x_{1}, x_{2}, x_{3}\right)=\left(\begin{array}{c} \frac{x_{1}}{x_{2}}+\frac{x_{3}}{x_{1}} \\ \frac{1}{2} x_{2}^{3}-250 x_{2} x_{3}-75000 x_{3}^{2} \\ e^{-x_{3}}+x_{3} \cdot e^{1} \end{array}\right) $$

Straight forward just using numpy

Compile using the flags -static-libgcc and -static-libstdc++ for the std::vectors