Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$.

1. The only way to find the inverse I can think of is power series reversion at multiple points within the interval $(a,b)$, so that the radii of convergence of the inverse series overlap. One can try to improve the fit by using Newton's method on $f(y)-x = 0$.

2. Lagrange's theorem offers another method (see [Dominici]).

3. [Dominici] gives a third using nested derivatives.

4. Using Newton's method on the equation $f(y)-x = 0$ is probably the first choice. It gives the values of the inverse function at discrete points $x_i$. See StackExchange (thanks Antonio Vargas) and [Koepf]

Are there other methods to compute the inverse of a complicated function $f$?

• You could try to approximate $f$ by an analytically invertible function then use Newton's method to correct the approximation as in this question. – Antonio Vargas Jul 27 '14 at 10:11
• Is there a method to choose such a function from a certain class of functions? – jjack Jul 27 '14 at 12:34
• Not in general. – Antonio Vargas Jul 27 '14 at 12:37
• Is the function monotone? (Otherwise the inverse isn't defined). Also, if you are looking for a numerical approximation, then I don't understand what "choose from a certain class of functions" means. – user147263 Jul 27 '14 at 17:28
• Of course, the function is strictly monotone. – jjack Jul 27 '14 at 17:32