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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.

Aviram Serra
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  • you can invert this polynomial locally, but not globally. So obviously, any approximation is going to get worse the farther you are from the center of inversion. – Rushabh Mehta Mar 14 '21 at 01:48
  • @DonThousand that would be great. Could it be done in a range of lets say $0 < x < 80$? Is there only one range where it could be done or can it be done in different regions? Thanks for your response btw – Aviram Serra Mar 14 '21 at 01:50
  • There are certain points where it can't be done at all, but at all but finitely many points, there exists some neighborhood in which one can invert. $(0,80)$ doesn't work – Rushabh Mehta Mar 14 '21 at 02:07
  • You can invert the function on intervals where it doesn't turn around. That is, it is either increasing or decreasing over the entire interval. From the graph, it looks like $(0,80)$ might work. It depends on exactly where that maximum near $80$ is. It will be the upper limit to the interval. – Paul Sinclair Mar 14 '21 at 14:08

1 Answers1

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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 $x = F_X^{-1}(y)$.

Paul Sinclair
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