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MathGroup Archive 2006

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Functional decomposition (solving f[f[x]] = g[x] for given g)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71764] Functional decomposition (solving f[f[x]] = g[x] for given g)
  • From: "Kelly Jones" <kelly.terry.jones at gmail.com>
  • Date: Tue, 28 Nov 2006 06:04:18 -0500 (EST)

Given a function g[x], is there a standard methodology to find a
function f[x] such that:

f[f[x]] == g[x]

A "simple" example (that doesn't appear to have a closed form solution):

f[f[x]] == x^2+1

There are probably several (approximable? power-series-expressible?)
answers, but the most natural answer would be everywhere positive and
monotone increasing for x>0.

For x^2, there are at least two continuous solutions: x^Sqrt[2] and
x^-Sqrt[2], the former being more "natural". It's somewhat amazing
that x^2+1 is so much harder.

Of course, the next step would be to find f[x] such that:

f[f[f[x]]] == g[x]

for given g[x], and so on.

Thought: is there any operator that converts functional composition to
multiplication or something similar? That would reduce this problem to
find nth roots and applying the inverse operator?

Other thought: For some reason, taking the geometric average of the
identity function and g, and then iterating, seems like a good
idea. EG, Sqrt[x*g[x]], Sqrt[x*Sqrt[x*g[x]]], and so on.

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