[Date Index]
[Thread Index]
[Author Index]
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.
--
We're just a Bunch Of Regular Guys, a collective group that's trying
to understand and assimilate technology. We feel that resistance to
new ideas and technology is unwise and ultimately futile.
Prev by Date:
**Re: Mathematica "Interpolation" function**
Next by Date:
**Re: Using subscripts in function-parameter names**
Previous by thread:
**Re: sum of integrals over patial intervals != integral**
Next by thread:
**Re: Functional decomposition (solving f[f[x]] = g[x] for given g)**
| |