Is there a way of approximating a polynomial's inverse? Suppose I have a fifth degree polynomial: $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5$ and that it does not factor nicely or have any nice roots. 
Is there a way to approximate the inverse function $f^{-1}$? Whether by hand or by computer aid.
I know that one can use Newton's Method and approximate roots, but I don't think that helps the problem at hand.
I was not able to find any information on this after a lot of searching of google and stackexchange.
 A: You can approximate the inverse function as long as you can properly define what "inverse" means. Suppose that, for any given $y$, you can establish some rule that allows you to choose one specific $x$ such that $y=f(x)$. This gives you a definition of the inverse function $x = g(y) = f^{-1}(y)$. I'm not sure that it's easy (or even possible) to make this inverse function continuous, even in the simple case of quintic polynomials. It's possible on intervals where $f$ is monotone, though, which might be enough for your purposes.
Now we need a method of approximating $g$. You can calculate a few specific values of $g(y)$, using some root-finding method (like Newton's method), and then interpolate these values with a polynomial. If $g$ is continuous on the interval of interest, this should work well both in theory and in practice. Use Chebyshev nodes in your interpolation to avoid nasty wiggliness.
To do the root-finding and construct the interpolating polynomial, I highly recommend the chebfun system.
A: There's no reason $f^{-1}$ should exist: $f$ is surjective but needn't be one-to-one. Newton's method is a very good way to compute a local inverse, though-applying it to $f-r$ gives you a point at which $f$ equals $r$, i.e. one possible choice for "$f^{-1}(r)$", though there might be up to four other possibilities. None of this is special to quintics or even to polynomials, of course.
A: This may be more of a computational science question/answer, but from a practical standpoint it's absolutely possible to approximate the inverse of a function on some domain where it is one-to-one.
Here's one way to do it: say you need your inverse to be accurate on $x_s < x < x_e$. Then, parameterize $f^{-1}$ however you see fit, and minimize squared errors on inversion efficacy:
\begin{equation}
   \int_{x_s}^{x_e} (f^{-1}(f(x)) - x)^2 dx
\end{equation}
In a practical/computational setting, $f^{-1}$ will be defined by some parameters. For example, you could have the inverse be another quintic, with different coefficients: $f^{-1}(x) = b_0 + b_1x + b_2x^2 + b_3x^3 + b_4x^4 + b_5x^5$. Then the integral would evaluate to something messy but exact, and you could solve for the inverse coefficients by setting the gradient of the integral to zero (linear problem).
