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

• To: mathgroup at smc.vnet.net
• Subject: [mg71885] Re: [mg71872] RE: [mg71839] RE: [mg71764] Functional decomposition (solving f[f[x]] = g[x] for given g)
• From: "Gregory Duncan" <gmduncan at gmail.com>
• Date: Sat, 2 Dec 2006 05:11:07 -0500 (EST)
• References: <200612011122.GAA25968@smc.vnet.net>

```The problem falls generally into the the area of functional equations.
See J. Aczel
Functional Equations: History, Applications and Theory or any of his
other books. One can also try to find a fixed point in a function
space.

On 12/1/06, Ingolf Dahl <ingolf.dahl at telia.com> wrote:
> Some further comments on the equation f(f(x)) = g(x):
> For the case g(x) = exp(x) some work have been done, see
> http://www.math.niu.edu/~rusin/known-math/index/39-XX.html  ,
> http://www.math.niu.edu/~rusin/known-math/99/sqrt_exp
> (found by a Google search for "f(f(x))")
>
> However, I have not penetrated deeply enough into all references to really
> see if there are good methods to find (precise) numerical values for at
> least one solution of the equation. Or a plot...
>
> But some thoughts further, somewhat fishing on deep water: If we have a
> function g(x) that is such that g(0)=0 (or g(x0)=x0 for some x0) and where
> g(x) also has a Maclaurin expansion with a positive first coefficient, it
> should be easy to find a Maclaurin series which composed to itself is
> identical to that of g(x). Examples of such functions g(x) are sin(x) and
> sinh(x). Then there is the question if this series converges in some range,
> I have not tested yet. (In the links above they refer to a theorem by Polya,
> stating that convergence range must be limited.) For some other functions,
> we could also think of expansions in x^(Sqrt(2)).
> Suppose that we find such a function f(x) for g(x) = sinh(x), name it
> fsinh(x), with convergence in some region near origo. Then it might be
> possible to extend that to the whole axis if we for any x apply ginv (the
> inverse of g) a number of times until we reach the convergence region. (we
> might have to use extended precision.) There we apply f, and then apply g
> the same number of times as we did with ginv. Then we should have obtained
> f(x).
>
> If we now look at the case g(x) = exp(x), we can note that if x > 20, exp
> and 2*sinh are essentially identical (to machine precision). Thus we could
> assume that fsinh(x)*Sqrt[2] is a good approximation of the solution f(x)
> for g(x)=exp(x) and x>20, and then we might use the same procedure as for
> g(x)=x^2+1 to extend this to zero. With the power of Mathematica available,
> just a few lines of code maybe are all that is needed. But maybe some devils
> are hiding in the details?
>
>
> Ingolf Dahl
>
>
> > -----Original Message-----
> > From: Ingolf Dahl [mailto:ingolf.dahl at telia.com]
> > Sent: den 30 november 2006 12:06
> > To: mathgroup at smc.vnet.net
> > Subject: [mg71839] RE: [mg71764] Functional decomposition
> > (solving f[f[x]] = g[x] for given g)
> >
> > Hi Kelly,
> > Here is a rapid amateur answer (but without scary rhymes, as
> > in my last answer). If you are satisfied with the numerical
> > values of the function, you can investigate
> >
> > g[x_]:=1+x^2;
> > ginv[x_]:=Sqrt[x-1];
> > f[x_Real]:=Nest[ginv,Power[Nest[g,x,9],Sqrt[2]],9]
> >
> > f[x_Real] appears to be a good solution to f[f[x]] == g[x] to
> > approximate machine precision. The solution is based on two
> > observations:
> > 1) Nest[g,x,9] is equivalent to Nest[f,x,18], and we reach
> > very large values when this is calculated. There g[x] is
> > approximately equal to x^2 (to machine precision), and f[x]
> > can be approximated by x^Sqrt[2]. Thus we might reach a good
> > approximation of Nest[f,x,19] by Power[Nest[g,x,9],Sqrt[2]].
> > 2) We can then apply the inverse function ginv of g 9 times
> > to reach f[x].
> > The relative error is reduced by the application of the
> > inverse function.
> >
> > We can note that f[0.] is 0.6420945043908285` which do not
> > appear to be a known mathematical constant. But now it is.
> > (If someone can check... I cannot reach the "The Inverse
> > Symbolic Calculator" at CESM. Is it down?)
> >
> > Generalize and write  f[x] as  DecompositionPower[x,1/2] and
> > the solution to f[f[f[x]]]\[Equal]g[x] as
> > DecompositionPower[x,1/3] For the given function g then we can try
> >
> > DecompositionPower[x_,p_]:=Nest[ginv,Power[Nest[g,x,9],Power[2,p]],9]
> >
> > and we might for instance plot
> >
> > Plot[DecompositionPower[0.,p],{p,0,1}]
> >
> > The function DecompositionPower[x_,p_] should be quite easy
> > to generalize to a large set of polynomial functions g. Maybe
> > it then should have the name FractionalNest?
> >
> > Best regards
> >
> > Ingolf Dahl
> >
> > -----Original Message-----
> > From: Kelly Jones [mailto:kelly.terry.jones at gmail.com]
> > Sent: 28 November 2006 12:04
> > To: mathgroup at smc.vnet.net
> > Subject: [mg71839] [mg71764] Functional decomposition
> > (solving f[f[x]] = g[x] for given g)
> >
> > 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.
> >
> >
> >
>
>
>

```

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