I have the following polynomial which has the following form:
$F_X \left ({x} \right)= Ax^2-Bx^4 + Cx^6$
Where $A$, $B$ and $C$ are real positive and non-zero. The roots are $\pm 161.634$, $\pm 125.993$ and $0$ . I am now in need of finding (or approximate) its inverse.
I have found this similar post from a couple of years ago where the Lagrange inversion method is recommended, but I do not think it can be applied to my situation. I found this nice paper where a method was outlined based on finding some coefficients of a pre-defined polynomial, but it yields poor results when far from zero.
This question is part of a problem that I am solving. Any help/guidance would be highly appreciated.
Note that $F_X(x) = P(g(x))$, where $P(x) = Cx^3 - Bx^2 + Ax$ and $g(x) = x^2$. Over domains where they are invertible, $$F_X^{-1}(y) = g^{-1}(P^{-1}(y))$$
Since you are interested in $x > 0$, $g$ is invertible with $g^{-1}(y) = \sqrt y$. And $P$ is a cubic, for which we have well-known means of inverting. For calculations, I recommend the trigonometric method (applied to the equation $P(t) - y = 0$ to solve for $t$). Once you find this $t$, set $x = \sqrt t$ to find $x = F_X^{-1}(y)$.