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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 $f$?

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  • $\begingroup$ You could try to approximate $f$ by an analytically invertible function then use Newton's method to correct the approximation as in this question. $\endgroup$ – Antonio Vargas Jul 27 '14 at 10:11
  • $\begingroup$ Is there a method to choose such a function from a certain class of functions? $\endgroup$ – jjack Jul 27 '14 at 12:34
  • $\begingroup$ Not in general. $\endgroup$ – Antonio Vargas Jul 27 '14 at 12:37
  • $\begingroup$ 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. $\endgroup$ – user147263 Jul 27 '14 at 17:28
  • $\begingroup$ Of course, the function is strictly monotone. $\endgroup$ – jjack Jul 27 '14 at 17:32

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