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+--- Thread: The ultimate sanity check (/showthread.php?tid=1561)
The ultimate sanity check - Daniel - 07/03/2022
I no longer focus exclusively on tetration, I mostly work with smooth complex iterated functions \( f^n(z) \) to serve my attack on hyperoperations. Because I didn't have anyone to work with, I had to devise methods to independently check my work. My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \). For my check I was able to get to \( \mathcal{O}(z^{29}) \) where the Lyapunov multiplier \( \lambda \) is neither zero or a root of unity and the origin is set to a fixed point not infinity. I used no floating point in my calculations, only rational numbers, so I could obtain an exact answer.
My most general derivation only assumes there is a non-super attracting fixed point. It handles the entire complex plane - the Shell-Thron boundary, it's interior and external. My Mathematica software validated the sanity check out to \( \mathcal{O}(z^{8}) \). The only function with only a fixed point at infinity is the successor function which is trivial. My numerical computations indicate that my method can correctly compute the position of neighboring fixed points. Can any other tetration method correctly place the positions of the fixed points.
I have a admission to make. I don't trust any fractional iteration techniques, including those published, but my own and Carleman matrices, which are experimentally consistent with my work. I strongly suspect I am wrong in this attitude and I am trying to expand my math background with the material on this site. Unfortunately I only made it through sophomore year of college, so I am almost completely self-taught. For example, the Riemann mapping theorem was a pleasant surprise to me.
I would feel comfortable with these other techniques if they could pass my sanity test.
RE: The ultimate sanity check - tommy1729 - 07/04/2022
(07/03/2022, 01:46 PM)Daniel Wrote: I no longer focus exclusively on tetration, I mostly work with smooth complex iterated functions \( f^n(z) \) to serve my attack on hyperoperations. Because I didn't have anyone to work with, I had to devise methods to independently check my work. My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \). For my check I was able to get to \( \mathcal{O}(z^{29}) \) where the Lyapunov multiplier \( \lambda \) is neither zero or a root of unity and the origin is set to a fixed point not infinity. I used no floating point in my calculations, only rational numbers, so I could obtain an exact answer.
My most general derivation only assumes there is a non-super attracting fixed point. It handles the entire complex plane - the Shell-Thron boundary, it's interior and external. My Mathematica software validated the sanity check out to \( \mathcal{O}(z^{8}) \). The only function with only a fixed point at infinity is the successor function which is trivial. My numerical computations indicate that my method can correctly compute the position of neighboring fixed points. Can any other tetration method correctly place the positions of the fixed points.
I have a admission to make. I don't trust any fractional iteration techniques, including those published, but my own and Carleman matrices, which are experimentally consistent with my work. I strongly suspect I am wrong in this attitude and I am trying to expand my math background with the material on this site. Unfortunately I only made it through sophomore year of college, so I am almost completely self-taught. For example, the Riemann mapping theorem was a pleasant surprise to me.
I would feel comfortable with these other techniques if they could pass my sanity test.
My deep apologies but what was your own method again ?
I know you did alot of research , such as iterated integer interations and such.
Maybe you use those formula's and replace integer n with real x or complex z.
Im not sure what your analytic method is , what uniqueness it satisfies , what was proven or plotted etc.
Also I sometimes confuse you with Gadilakis and so. sorry.
I see you have a nice website with fractals and such, but i did not see it immediately.
regards
tommy1729
RE: The ultimate sanity check - Daniel - 07/04/2022
(07/04/2022, 12:52 PM)tommy1729 Wrote:Hey Tommy, thanks for replying. For my work on tetration, use the "Tetration" link in the menu. For iterated functions see "Combinatorics" and "Dynamics". The information has been up for almost twenty years and precedes the Tetration Forum by several years. In my post sanity check I explain that my method for determining \( f^n(z) \) perfectly satisfies \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}({z^8}) \) for almost any smooth iterated function and \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^{30}) \) for Schroeder's functional equation.(07/03/2022, 01:46 PM)Daniel Wrote: I no longer focus exclusively on tetration, I mostly work with smooth complex iterated functions \( f^n(z) \) to serve my attack on hyperoperations.
...
I would feel comfortable with these other techniques if they could pass my sanity test.
My deep apologies but what was your own method again ?
I know you did alot of research , such as iterated integer interations and such.
Maybe you use those formula's and replace integer n with real x or complex z.
Im not sure what your analytic method is , what uniqueness it satisfies , what was proven or plotted etc.
Also I sometimes confuse you with Gadilakis and so. sorry.
I see you have a nice website with fractals and such, but i did not see it immediately.
regards
tommy1729
What else does my derivation of \( f^n(z) \) do? Well, it is deeply consistent which Carleson and Gamelin book Complex Dynamics. It lead to the same theorems as the chapter on the classification of fixed points. Since I take the Taylor's series of \( f^n(z) \) it is guaranteed for integer tetrates of positive numbers. As far as what is plotted, that would be the fractals you mentioned.
Best Wishes,
Daniel
RE: The ultimate sanity check - JmsNxn - 07/04/2022
(07/04/2022, 12:52 PM)tommy1729 Wrote: My deep apologies but what was your own method again ?
....
Im not sure what your analytic method is , what uniqueness it satisfies , what was proven or plotted etc.
....
regards
tommy1729
Brute force, using Faa di Bruno's formula to construct what we call Schroder's iteration.
RE: The ultimate sanity check - Daniel - 07/05/2022
(07/04/2022, 11:33 PM)JmsNxn Wrote:(07/04/2022, 12:52 PM)tommy1729 Wrote: My deep apologies but what was your own method again ?
....
Im not sure what your analytic method is , what uniqueness it satisfies , what was proven or plotted etc.
....
regards
tommy1729
Brute force, using Faa di Bruno's formula to construct what we call Schroder's iteration.
The software I wrote to validate my work begins by enumerating the combinatorial structure Schroeder's Fourth Problem (also hierarchies and total partitions) and assigning each enumerated structure a value in a manner similar to how Feynman graphs are evaluated. I can evaluate any derivative of \( f^n(z) \) without evaluating the prior derivatives. For example, \( D^4f^n(z) \) is represented by 26 trees whose values added together give the fourth derivative without knowing \( D^2f^n(z) \) or \( D^3f^n(z) \).
As I have said in other postings, my methodology not only can use either Abel's or Schroeder's equations, it can be used to derive Abel's and Schroeder's functional equations AND THEIR PROPERTIES. My work explains why there is a Abel's and Schroeder's equation.
RE: The ultimate sanity check - JmsNxn - 07/05/2022
(07/05/2022, 12:43 AM)Daniel Wrote:(07/04/2022, 11:33 PM)JmsNxn Wrote:(07/04/2022, 12:52 PM)tommy1729 Wrote: My deep apologies but what was your own method again ?
....
Im not sure what your analytic method is , what uniqueness it satisfies , what was proven or plotted etc.
....
regards
tommy1729
Brute force, using Faa di Bruno's formula to construct what we call Schroder's iteration.
The software I wrote to validate my work begins by enumerating the combinatorial structure Schroeder's Fourth Problem (also hierarchies and total partitions) and assigning each enumerated structure a value in a manner similar to how Feynman graphs are evaluated. I can evaluate any derivative of \( f^n(z) \) without evaluating the prior derivatives. For example, \( D^4f^n(z) \) is represented by 26 trees whose values added together give the fourth derivative without knowing \( D^2f^n(z) \) or \( D^3f^n(z) \).
As I have said in other postings, my methodology not only can use either Abel's or Schroeder's equations, it can be used to derive Abel's and Schroeder's functional equations AND THEIR PROPERTIES. My work explains why there is a Abel's and Schroeder's equation.
Oh yes I apologize daniel, I'm not trying to minimize your work. But what I said Tommy would instantly be able to identify what your work is. I'm not discrediting your work, it still falls under the umbrella I wrote.
Have you ever considered writing a paper? I'm fascinated by your approach. And I apologize if I appear dismissive, plus I forgot that the method works for abel as well.
RE: The ultimate sanity check - Daniel - 07/12/2022
(07/03/2022, 01:46 PM)Daniel Wrote: ...
My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \). For my check I was able to get to \( \mathcal{O}(z^{29}) \) where the Lyapunov multiplier \( \lambda \) is neither zero or a root of unity and the origin is set to a fixed point not infinity. I used no floating point in my calculations, only rational numbers, so I could obtain an exact answer.
...
No one provided an answer to my question, so I will come at it from another direction. What software that members have written that is based on rational numbers and not floating point?
RE: The ultimate sanity check - JmsNxn - 07/12/2022
(07/12/2022, 03:05 AM)Daniel Wrote:(07/03/2022, 01:46 PM)Daniel Wrote: ...
My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \). For my check I was able to get to \( \mathcal{O}(z^{29}) \) where the Lyapunov multiplier \( \lambda \) is neither zero or a root of unity and the origin is set to a fixed point not infinity. I used no floating point in my calculations, only rational numbers, so I could obtain an exact answer.
...
No one provided an answer to my question, so I will come at it from another direction. What software that members have written that is based on rational numbers and not floating point?
Very difficult question to answer, Daniel. And I think I get the question more clearly now.
You want to approach the answer from \(\mathbb{Q}[z]\); the space of polynomials with rational coefficients \(p_m(z)\). This means, you are asking to look at iterations like:
\[
f^n(z) = \lim_{m\to\infty} p_m(z)
\]
In which, you are choosing the minimal rational polynomial near this solution.
I apologize if I'm skipping something, but does this sound what you're getting at?
If so, I have no record of anyone ever doing something like that. That is beyond fascinating. I've never seen anyone approach the question like that.
I don't see why you'd use that, but that's super cool!
It makes sense though, and I whole heartedly agree with you that the algorithm converges. Choosing rational polynomials, constructs the same limit. But choosing rational coefficients gets the number theorist part of me going, lol.
Regards
RE: The ultimate sanity check - Gottfried - 07/12/2022
(07/12/2022, 03:05 AM)Daniel Wrote:(07/03/2022, 01:46 PM)Daniel Wrote: ...
My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \). For my check I was able to get to \( \mathcal{O}(z^{29}) \) where the Lyapunov multiplier \( \lambda \) is neither zero or a root of unity and the origin is set to a fixed point not infinity. I used no floating point in my calculations, only rational numbers, so I could obtain an exact answer.
...
No one provided an answer to my question, so I will come at it from another direction. What software that members have written that is based on rational numbers and not floating point?
Hi Daniel -
I had (have?) difficulties to understand what you're really asking. Just to try one road, detecting the point "rational numbers": I've described the coefficients of the powerseries of the iteration of \( f_t(z) = t^z -1 \) using \(u = \log(t) \) in terms of polynomials of rational coefficients with keeping \( u \) symbolic/replacable by any numerical value ( real or complex).
By the usual fix-point shift, for \( b= t^{1/t} = \exp( u \exp (-u)) \) , I think this extends to the common tetration, which is then interpolated along the Schroeder-style.
The most detailed description of those polynomials (in \( u \) and \( u^h \) where \( h \) is the iteration height) is in coefficients on my webpages (btw. I've not seen that details anywhere else so far).
Gottfried
RE: The ultimate sanity check - bo198214 - 07/13/2022
Oh, Daniel, I really don't know where to start - there are so many implicit assumptions in your question.
Start with my first impression: Somehow it's asking like: "I want to sanity check an algorithm that I found (after long research and encountering a lot of interesting mathematical side topic) that can compute the length of a third side of a right triangle." People would answer "well there is something that is called theorem of Pythagoras, why not use that?!"
If you only would compare numerical results of this theorem with your own algorithm, you would miss a lot! First a numerical "equality " is no proof. (You could easily be fooled for example by the half-iterate developed at 2 and 4 of sqrt(2)^x, which coincide by many digits, but in the end are essentially different functions - this was already several times said on the forum).
Second an algorithm is not a proof. Theorem of Pythagoras contains a proof that an algorithm will always yield the right answer.
Similar issue with the Schroeder iteration: There we have a proof of convergence of different ways of computing. We have a proof of the resulting function being analytic, etc etc
You see how much richer that is than just having numerical coincidence? Of course as a sanity check on should do some numerical comparisons.
And then there is another category like the Kneser construction: which gives proof of existence, or the Leau-Fatou construction, which proofs existence and uniqueness. But both have no easily accessible means of numerical computation. (So the only sanity check is that many mathematicians looked at it and confirmed, typically by peer-reviewed publication or teaching it at university)
With Tetration we are unfortunately mostly (except regular iteration) in the situation of having some numerical methods with no proof of convergence, with no proof of holomorphy (in case of convergence) and with no clue about identity (which algorithms converge to the same function).
So what I want to say: it is essential to understand the well-known theory (e.g. Schröder-Iteration, Kneser construction, Leau-Fatou construction) if only to have a common base of understanding each other. Otherwise we praise ourselves in our snail house, never having seen the world in comparison.