I have a set of two equations like this $$ \gamma_3=\left(\frac{1}{\sqrt{1+2c_3^2+6c_4^2}}\right)^3 \left( \alpha_1\,c_3^3 + \alpha_2\,c_3c_4^2 + \alpha_3\,c_3c_4 + \alpha_4\,c_4\right)\\ \gamma_4=\left(\frac{1}{\sqrt{1+2c_3^2+6c_4^2}}\right)^4\left( \beta_1\,c_3^4 + \beta_2\,c_4^4 + \beta_3\,c_3^2c_4^2 + \beta_4\,c_3^2 + \beta_5\,c_4^2 + \beta_6\,c_4^3 + \beta_7\,c_3^2c_4 + \beta_8\,c_4 + \beta_9\right) $$ where $\alpha_i$ and $\beta_j$ are constant $\in \mathbb{R}$, while $\gamma_3$, $\gamma_4$, $c_3$ and $c_4$ are variables $\in \mathbb{R}$.
I need to find the inverse equations to compute $c_3$ and $c_4$ starting from $\gamma_3$ and $\gamma_4$. In other words I need to find $f(\bullet)$ and $g(\bullet)$ in order to compute $$ c_3=f(\gamma_3,\gamma_4) \\ c_4=g(\gamma_3,\gamma_4) $$ I'm of course sure that for each couple $\gamma_3$ and $\gamma_4$ exists only one couple $c_3$ and $c_4$ that satisfy these equations.
I don't think that this system is invertible analytically, but could you suggest me at least a numerical strategy to achieve this?
Thank you very much.
PS: excuse me if my math terminology is not correct, but i'm not a mathematician nor english is my mother tongue.
Solved using an iterative process.
Being the domain of $c_3$ and $c_4$ limited, I created a grid of points in the $c_3-c_4$ plane. Then i computed $\gamma_3$ and $\gamma_4$ for each point and computed the error as $$E=(\gamma_3-\gamma_3^*)^2+(\gamma_4-\gamma_4^*)^2$$ then looked for $\min(E)$ and found $c_3^1$ and $c_4^1$ that returns $\gamma_3^1$ and $\gamma_4^1$.
At this point I define a grid of point in an interval $[c_3^1 \pm \delta ; c_4^1 \pm \delta]$ and repeated the process until $E$ is small enough.