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Question about tetration methods
#1
Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.
Daniel
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#2
(06/17/2022, 04:57 PM)Daniel Wrote: Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

Hi Daniel - there must have been a lightning in the air... frizzeling the electronic of our silicium cpus... I know of a handful of methods, but I think my hand has only 5 fingers ... ;-)) 


For the list of techniques a curated list in the tetration-wiki were really good. Unfortunately I did never understand the methods from Kneser on upwards: no cauchy-integral (Kouznetsov), no theta mapping, no fourier series, no beta method, no 2sin, no ... no no no ... ;-) otherwise I really would like to make it my hobby to undertake a catalogue of methods.

Gottfried
Gottfried Helms, Kassel
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#3
(06/17/2022, 04:57 PM)Daniel Wrote: Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

I think you are confused.

I recently said that.

Which ones are more important , better , faster , more practical , analytic , equal etc can be debated !

Here are some for bases larger than 2 that immediately cross my mind :

1) the base change method.  and their matrix analogue ideas see https://math.eretrandre.org/tetrationfor...php?tid=26

One of the simplest I guess but many believe it is not analytic.

2) the slog computed by an infinite linear system ; Andrew's slog method ... Peter Walker seems to be the original inventor.

see https://math.eretrandre.org/tetrationfor....php?tid=3

and https://math.eretrandre.org/tetrationfor...p?tid=1081

3) The carleman matrix method. Gottfriend in particular seems to prefer this matrix method.

4) Kouznetsov's method. Based on contour integrals.

5) Kneser's solution. Based on the fixpoint of ln(z). Uses Riemann mapping and fourier series. Apparently proven analytic. 

6) the gaussian method. And similar ofcourse. Uses the erf function from gauss. Posted by me as an improved or alternative version of James Nixons beta method. see https://math.eretrandre.org/tetrationfor...p?tid=1339

7) Using 2 sinh(x) as an asymptotic. see : https://math.eretrandre.org/tetrationfor...hp?tid=424

and related : https://math.eretrandre.org/tetrationfor...hp?tid=953

8 ) Using continuum sums and continuum products.

9) using interpolation

10) this taylor type solution NOT REQUIRING A FIXPOINT that I posted :  https://math.eretrandre.org/tetrationfor...hp?tid=791

11) An infinite sum method I made somewhere.

12) Iteration approximations of exp and in the limit getting tetration ( assuming that works ! ).  Such as iterating the taylor polynomials of exp(z) by growing degree.

13) Using fixpoints at real +/- infinity or trying at least.

14) moving fixpoints of approximations to infinity such as : https://math.eretrandre.org/tetrationfor...p?tid=1400

15) https://math.eretrandre.org/tetrationfor...p?tid=1356
 
another approximation method.

16) a special kind of interpolation method :  https://math.eretrandre.org/tetrationfor...p?tid=1257

17) not analytic but worth mentioning maybe : https://math.eretrandre.org/tetrationfor...p?tid=1188

18 ) nonconstructive methods that seem to exist though.

19)  secondary fixpoints and merged fixpoints ideas.

and many more.

and apparantly there are others not even discussed at this forum.



I also like this but it is not really tetration : https://math.eretrandre.org/tetrationfor...p?tid=1013

also nice is this concept : https://math.eretrandre.org/tetrationfor...hp?tid=166


yes yes im aware all of these need explaining and links ... but that would be half the forum :/ 

and maybe im biased sorry.


regards

tommy1729
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#4
Question 
(06/17/2022, 08:47 PM)Gottfried Wrote:
(06/17/2022, 04:57 PM)Daniel Wrote: Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

Hi Daniel - there must have been a lightning in the air... frizzeling the electronic of our silicium cpus... I know of a handful of methods, but I think my hand has only 5 fingers ... ;-)) 
Why do not you have ten fingers? What happend?
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ
Please remember to stay hydrated.
Sincerely: Catullus
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#5
Thank you much tommy1729.
Daniel
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#6
(06/17/2022, 11:18 PM)Daniel Wrote: Thank you much tommy1729.

your welcome.


regards

tommy1729
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#7
(06/17/2022, 08:47 PM)Gottfried Wrote:
(06/17/2022, 04:57 PM)Daniel Wrote: Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

Hi Daniel - there must have been a lightning in the air... frizzeling the electronic of our silicium cpus... I know of a handful of methods, but I think my hand has only 5 fingers ... ;-)) 


For the list of techniques a curated list in the tetration-wiki were really good. Unfortunately I did never understand the methods from Kneser on upwards: no cauchy-integral (Kouznetsov), no theta mapping, no fourier series, no beta method, no 2sin, no ... no no no ... ;-) otherwise I really would like to make it my hobby to undertake a catalogue of methods.

Gottfried

Tommy claimed your comment, Daniel. Tommy claimed about 10 of his formula's converge. Spoiler alert, they don't. But they are attempts.

As Gottfried spoke, he meant that he can think of less than 5 solutions to tetration which are intrinsically unique. This is true. There is Kneser. There is Sheldon. There is Carlemann (programmed by Gottfried). There is Kouznetsov. There is Paulsen and Cowgill.


The Only one that is really proven, which I referred to is, Kneser and Paulsen and Cowgill. This was pointed out by Gottfried, who said that even Sheldon's fatou.gp isn't mathematically proven. Even though we all believe it is true, it is not proven.

This whole thread is a dumpster fire of misunderstandings.

There are not 15 different tetrations.

There are maybe 4 different algorithms for Kneser. There are 2 or 3 offshoots of credible avenues. This includes Peter Walker/Andy's matrix avenue. The beta method (which includes Tommy's Gaussian method), which is an alternative method of constructing Kneser. Or the beta method itself which is used in the Shell-Thron region to create arbitrary periods. And the list goes on.

But Tommy, it's kind of dumb to say 15 different tetrations. There are not 15 different tetrations. Those are 15 different posts you made. They do not hold up in any sense that Daniel is asking. And if they do, show some evidence of that.

What Gottfried is saying, Daniel. Is that there are about 4-5 tetration formulas that hold up to scrutiny. As algorithms. And as mathematical ideas. And myself and Tommy are excluded from that list.

I apologize, but what Tommy is saying is wrong. There are not "15 tetrations." But, yes, there are 15 TOMMY TETRATIONS.
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#8
(06/18/2022, 03:25 AM)JmsNxn Wrote:
(06/17/2022, 08:47 PM)Gottfried Wrote:
(06/17/2022, 04:57 PM)Daniel Wrote: Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

Hi Daniel - there must have been a lightning in the air... frizzeling the electronic of our silicium cpus... I know of a handful of methods, but I think my hand has only 5 fingers ... ;-)) 
...
Gottfried

Tommy claimed your comment, Daniel. Tommy claimed about 10 of his formula's converge. Spoiler alert, they don't. But they are attempts.

As Gottfried spoke, he meant that he can think of less than 5 solutions to tetration which are intrinsically unique. This is true. There is Kneser. There is Sheldon. There is Carlemann (programmed by Gottfried). There is Kouznetsov. There is Paulsen and Cowgill.


The Only one that is really proven, which I referred to is, Kneser and Paulsen and Cowgill. This was pointed out by Gottfried, who said that even Sheldon's fatou.gp isn't mathematically proven. Even though we all believe it is true, it is not proven.
...
What Gottfried is saying, Daniel. Is that there are about 4-5 tetration formulas that hold up to scrutiny. As algorithms. And as mathematical ideas. And myself and Tommy are excluded from that list.

Thanks JmsNxn. I appreciate everybody's feedback as it gives me much to study and think about. 

My tetration research involves the range of complex numbers. Do all the techniques listed here deal with the range in the real numbers?

My work is experimentally consistent with Bell matrices. Are Bell matrices considered along with Carlemann matrices in people's research?
Daniel
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#9
(06/17/2022, 11:10 PM)Catullus Wrote:
(06/17/2022, 08:47 PM)Gottfried Wrote:
(06/17/2022, 04:57 PM)Daniel Wrote: Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

Hi Daniel - there must have been a lightning in the air... frizzeling the electronic of our silicium cpus... I know of a handful of methods, but I think my hand has only 5 fingers ... ;-)) 
Why do not you have ten fingers? What happend?

:-) Well, one handful -five fingers. Two handful - ten fingers
Gottfried Helms, Kassel
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#10
(06/18/2022, 04:29 AM)Daniel Wrote: My work is experimentally consistent with Bell matrices. Are Bell matrices considered along with Carlemann matrices in people's research?

The Carleman-matrix is simply a factorially similarity scaling of the Bell matrix, and transposed. The idea is to work on the coefficients of the taylor-series, and so the work is practically identical between Carleman- and Bell-notation.
But note: it provides its concise arithmetic/algebra on *taylor-series* only, not, for instance, laurent series - and for analysis and bulding a "toolbox" for many prominent problems, you need
the inclusion of $c_{-1} \cdot x^{-1}$ terms.

The only work with such an extension of the Carleman-ansatz I have ever seen is an article of Eri Jabotinsky, and when I read it it was over my head, and I had only spurious time for this matter then, so I did not step in. However I think it is a serious AND required extension of the range of applicability, it makes the Carleman-ansatz more useful and then surely wider known.

Gottfried

An exposition "from the ground" is in ContinuousfunctionalIteration (at my webspace for tetration) ; I think you'll find in it many things known to yourself (for instance the description in terms of derivatives) but I wrote this for the beginner as well as for a self-reflection of my knowledge.
Gottfried Helms, Kassel
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