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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72009] Re: [mg71996] Re: [mg71972] RE: [mg71885] Re: [mg71872] RE: [mg71839] RE: [mg71764] Functional decomposition (solving f[f[x]] = g[x] for given g)
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 8 Dec 2006 06:17:57 -0500 (EST)
  • References: <200612061104.GAA02807@smc.vnet.net> <B2493AAB-0463-49C0-9E90-118BB276E47F@mimuw.edu.pl> <200612071126.GAA28291@smc.vnet.net>

A little after posting that I noticed that almost the same argument  
provides another counter-example: any smooth real function g with g(0) 
= 0 and g'(0)<0 cannot be represented as f(f(x)), with f a real  
function of class C1.

Andrzej Kozlowski

On 7 Dec 2006, at 20:26, Andrzej Kozlowski wrote:

> I meant "class C1", of course: differentiable, with continuous
> derivative.
>
> Andrzej Kozlowski
>
>
> On 7 Dec 2006, at 17:58, Andrzej Kozlowski wrote:
>
>> To my mind an equally interesting problem (though admittedly not if
>> one wishes to apply Mathematica) is to find the widest class of
>> functions g for which the equation admits no solutions. There is
>> one case which is immediately obvious: the functional equation f(f
>> (x))==g(x) cannot have any solutions f (of class C2) for smooth
>> functions g whose derivative is g'(x) <0 everywhere; g(x) = exp(-x)
>> is such a function. The proof is completely trivial, but as far as
>> I can tell no-one seems to have mentioned it explicitly so here it  
>> is:
>> if f(f(x)) = g(x) then f'(f(x)) f'(x)) = g'(x) is negative. But
>> that means that f' has opposite sign at x and f(x). That in turn
>> implies (if f is of class C2)  that f'(t) = 0 for some t between x
>> and f(x). But that would  mean that g'(t) = 0, contradicting the
>> fact that g'(x)<0 everywhere.
>> I have given very little thought to this but it seems to me that
>> the assumption g'(x)>0 for all x should suffice for the existence
>> of f (though may not suffice for the existence of a nice enough f,
>> e.g. analytic). I would expect also that examples can be
>> constructed where g'(x)>0 for some x and g'(y)< 0 for some why and
>> where no f of class C2 can exists, but I can't find any obvious ones.
>>
>> Andrzej Kozlowski
>>
>> On 6 Dec 2006, at 20:04, Ingolf Dahl wrote:
>>
>>> Hi again dear MathGroup,
>>> I have checked the suggested procedure below for solving the  
>>> equation
>>> f(f(x))=exp(x), and it works smoothly.
>>> Maybe this is already well-known. Maybe it is known, but not
>>> implemented in
>>> Mathematica code. Maybe it is included in the book by Aczel, I
>>> have not been
>>> able to check that yet, since I have not the book available. If
>>> you know
>>> where to find an already published solution, please inform me.
>>> But anyway, I have not spent too many hours on it, and it is a nice
>>> demonstration of the powers of Mathematica. Thus it is relevant to
>>> publish
>>> it here.
>>> Moreover, the solution I obtain is characterized by some  
>>> mathematical
>>> constants, which should be thumbprints for uniqueness. I have not
>>> been able
>>> to localize these constants anywhere else.
>>>
>>> But what I have found is that the solution is related to half-
>>> integer values
>>> of "tetration" in Wikipedia and PowerTower in MathWorld (naturally
>>> extending
>>> into other fractional values). Wikipedia states; "At this time
>>> there is no
>>> commonly accepted solution to the general problem of extending
>>> tetration to
>>> the real or complex numbers, although it is an active area of
>>> research."
>>>
>>> What I claim is that (in the Mathematica notation from
>>> FurtherExamples to
>>> "Power" in Help)
>>>
>>> PowerTower[E,-1/2] = "HalfwayFromZero" =
>>> 0.4987433645316708791828373750416860523968424`43.
>>>
>>> PowerTower[E,-3/2] = "HalfwayToZero" =
>>> -0.6956636150637165750917227913152630811483903`42.
>>>
>>> (PowerTower[E,-2] = -Infinity, PowerTower[E,-1] = 0, PowerTower[E,
>>> 0] = 1 by
>>> natural extension)
>>>
>>> Here is my solution:
>>> I started by doing a routine for decomposition of some functions g
>>> (x) into q
>>> equal functions. Thus f(f(f...(f(x))...))=g(x), with q f:s. First
>>> the series
>>> expansion to the power n is derived. Then the function is composed
>>> p times
>>> with itself. We thus get the series expansion of the function to the
>>> decompositional power p/q. The requirement on g is that g[0)=0, with
>>> positive derivative at origin and with a Maclaurin expansion
>>> around origin.
>>>
>>> Attributes[FractionalSeries]= {HoldFirst};
>>> FractionalSeries[ func_/; And[ func[0.]==0., SeriesCoefficient
>>> [ Series[
>>> func[var], { var,0,1}],1]>0],x_,
>>> { p_Integer/; p>0, q_Integer/; q>0}, n_Integer/; n>0]:=
>>> Module[ { a,f1,fn,f1s},
>>> f1[var_]= Sum[ var^i* a[i], { i,1,n}];
>>> fn[var_]:=
>>> Nest[ f1, var,q];
>>> a[1]= a[1] /. Solve[ Reduce[ { SeriesCoefficient[ Series[ fn[var], {
>>> var,0,1}],1]== SeriesCoefficient[ Series[ func[var], { var,0,1}],1],
>>> a[1]>0}, a[1]], a[1]][ [1]];
>>> Do[ a[i]= a[i] /. Solve[ { SeriesCoefficient[ Series[ fn[var], {
>>> var,0,i}],i]== SeriesCoefficient[ Series[ func[var], { var,
>>> 0,i}],i]}, a[i]][
>>> [1]], { i,2,n}];
>>> Series[ f1[x], { x,0,n}];
>>>  ComposeSeries@@ Table[ Series[ f1[x], { x,0,n}], {p}]
>>>
>>>
>>> Example of use: Sinh is such a function, and let us take it to the
>>> decompositional power 2/3. Define
>>>
>>> ftwothirds[y_] := Normal[FractionalSeries[Sinh[#1] & , var, {2,
>>> 3}, 29]] /.
>>> {var -> y};
>>>
>>> Then
>>>
>>> ftwothirds[ftwothirds[ftwothirds[0.3]]] - Sinh[Sinh[0.3]]
>>>
>>> evaluates to
>>>
>>> 5.551115123125783*^-17
>>>
>>> and
>>>
>>> ftwothirds[ftwothirds[ftwothirds[SetAccuracy[0.3, 60]]]] -
>>> Sinh[Sinh[SetAccuracy[0.3, 60]]]
>>>
>>> evaluates approx to
>>>
>>> 1.65*^-24
>>>
>>> The problem is that Exp is not such a function, since Exp[0} == 1.
>>> But Exp
>>> behaves asymptotically as 2*Sinh. Let us find the series expansion
>>> of 2*Sinh
>>> to the decompositional power 1/2. (There is a more complicated
>>> relation
>>> between the series expansions for 2*Sinh and Sinh than I thought
>>> from the
>>> beginning.)
>>>
>>> Simplify[FractionalSeries[2*Sinh[#1] & , y, {1, 2}, 29]]
>>>
>>> (returns long expression)
>>>
>>> If we use this expression, we might build a function f2sinh to
>>> extend the
>>> series expansion to whole real axis. f2sinh should now be THE smooth
>>> solution to the equation f2sinh[f2sinh[x]] == Sinh[x] .
>>>
>>> f2sinh[x_Real] := Module[{y, steps},
>>> y = If[MachineNumberQ[x], SetAccuracy[x, 24], x]; steps = 0;
>>> While[Abs[y] > 0.07, steps = steps + 1; y = ArcSinh[y/2]];
>>> y = fpoly[y];
>>> (If[MachineNumberQ[x], N[#1], #1] & )[Nest[2*Sinh[#1] & , y,  
>>> steps]]]
>>>
>>> where
>>>
>>> fpoly[y_] := Sqrt[2]*y + y^3/(9*Sqrt[2]) - (11*y^5)/(2700*Sqrt[2]) +
>>> (479*y^7)/(1020600*Sqrt[2]) - (426679*y^9)/(6246072000*Sqrt[2]) +
>>> (85589473*y^11)/(7557747120000*Sqrt[2]) -
>>> (1403953837837*y^13)/(689720002171200000*Sqrt[2]) +
>>> (1031743413846247*y^15)/(2669216408402544000000*Sqrt[2]) -
>>> (1496209724379058102331*y^17)/(19591834900362000756480000000*Sqrt
>>> [2]) +
>>> (17812978070670318115902941*y^19)/
>>> (1145769688642970528240463360000000*Sqrt[2
>>> ]) -
>>> (7533260191446785723514332836231*y^21)/
>>> (232533958310090868706402038912000000
>>> 0000*Sqrt[2]) +
>>> (136561976266545063524556174223350449*y^23)/
>>> (1985445516947137017537762080849
>>> 52576000000000*Sqrt[2]) -
>>> (719728529960035731908565151209614126425839277*y^25)/
>>> (4853778747825775229320
>>> 514450035010550004224000000000000*Sqrt[2]) +
>>> (100427683629367623630378641852188624798406912708777*y^27)/
>>> (3101826723913052
>>> 963398192041352673632022399364096000000000000*Sqrt[2]) -
>>> (230559003784630013844879523283194881704293155047113459273*y^29)/
>>> (3226456204
>>> 1869955579438812968736894041769959385166794752000000000000*Sqrt 
>>> [2]) +
>>> SetAccuracy[0., 44]*(y/SetAccuracy[0.07, 60])^29;
>>>
>>> The limit 0.07 in this definition is a bit arbitrarily chosen, and
>>> you might
>>> decrease it to increase accuracy, which anyway is good along the
>>> whole real
>>> axis. I have estimated the truncation error to be as large as the
>>> last term
>>> in the series.
>>> Now we can assume that Exp to a fractional decompositional power
>>> asymptotically should be identical to 2*Sinh to the same fractional
>>> decompositional power, with a relative error that is less than the
>>> relative
>>> error of Exp compared to 2*Sinh, if the decompositional power is
>>> less than
>>> one. Thus for any x, we might climb up to a large value of Exp...
>>> [Exp[x]]
>>> (say x>60), fetch a value using f2sinh[x], and then climb down
>>> again. Thus
>>> we might calculate the (smooth) solution to the equation fexp[fexp
>>> [x]]==
>>> Exp[x] as
>>>
>>> fexp[x_Real] :=
>>> Module[{y, steps},
>>> y = If[MachineNumberQ[x], SetAccuracy[x, 24], x]; steps = 0;
>>> While[y < 60, steps = steps + 1; y = Exp[y]];
>>> y = f2sinh[y]*(1 + SetAccuracy[0., y/1.15]);
>>> (If[MachineNumberQ[x], N[#1], #1] & )[Nest[Log[#1] & , y, steps]]]
>>>
>>> The limit of fexp[x] when x->-Infinity is the constant
>>> HalfwayToZero defined
>>> above.
>>> And fexp[HalfwayToZero] = 0,
>>> fexp[0]= HalfwayFromZero,
>>> fexp[HalfwayFromZero] = 1,
>>> fexp[1] = Exp[HalfwayFromZero],
>>> fexp[Exp[HalfwayFromZero] = E und so weiter.
>>>
>>> Any devils hiding anywhere?
>>> One little devil is that fexp[x] approaches HalfwayToZero fast
>>> when z ->
>>> -Infinity, so there easily is a loss of accuracy. For large
>>> negative values
>>> of x, fexp will approximately follow the formula
>>>
>>> fexp[x] ~ HalfwayToZero + Exp[x]/f´[HalfwayToZero]
>>>
>>> where
>>>
>>> f´[HalfwayToZero] = 0.5688037784984972069377219276490924736217`40.
>>>
>>> For calculations of derivatives etc. of f(x) in the region around
>>> x = 0, a
>>> slightly simpler formula might be used instead of f(x):
>>>
>>> fexpsimpl[x_Real] := Nest[Log, Nest[2Sinh[#] &,
>>>     fpoly[Nest[ArcSinh[#/2] &, Nest[Exp, x, 4], 8]], 8], 4]
>>>
>>> Thus we can obtain the zeroth, first second and third derivatives
>>> of fexp(x)
>>> at x=0:
>>>
>>> {0.498743364531670879182837375041686052396842401697695,
>>> 0.87682850111901739234161348908444926737095344685410,
>>> 0.489578037438247862041903165991088781934758253721,
>>> 0.1222616742183662064640316453590024071635142693}
>>>
>>> It should be possible to calculate Exp to other fractional
>>> decompositional
>>> powers, and to calculate PowerTower for other first arguments than
>>> E and for
>>> fractional second argument, following the general scheme here.
>>>
>>> That all for today!
>>>
>>> Ingolf Dahl
>>>
>>>> -----Original Message-----
>>>> From: Gregory Duncan [mailto:gmduncan at gmail.com]
>>>> Sent: den 2 december 2006 11:11
>>>> 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)
>>>>
>>>> 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
>>>>> and
>>>> http://www.newton.dep.anl.gov/newton/askasci/1993/math/ 
>>>> MATH023.HTM .
>>>>> (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?
>>>>>
>>>>> Is there anyone who knows more about this?
>>>>>
>>>>> 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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