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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 $f$?

jjack
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  • You could try to approximate $f$ by an analytically invertible function then use Newton's method to correct the approximation as in [this question](http://math.stackexchange.com/q/871516/5531). – 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. –  Jul 27 '14 at 17:28
  • Of course, the function is strictly monotone. – jjack Jul 27 '14 at 17:32
  • What do you mean by "compute" $f^{-1}$? If you want to solve $f(x) = y$ for $x$, given $y$, there are numerical methods for that. – littleO Jul 27 '14 at 17:34
  • I would like to have a power series representation for taking derivatives and integrating. How would you go about numerically computing $f^{-1}$ for a known function $f(x)$? – jjack Jul 27 '14 at 17:57
  • [This answer of mine](http://math.stackexchange.com/questions/800476/equations-solved-with-newtons-method-by-finding-the-zeros-of-functions/803185#803185) may give you an idea of how one could numerically compute an inverse. – Antonio Vargas Jul 27 '14 at 18:02

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