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

06/17/2022, 04:57 PM
Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.
Daniel
(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 tetrationwiki were really good. Unfortunately I did never understand the methods from Kneser on upwards: no cauchyintegral (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
06/17/2022, 10:33 PM
(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
06/17/2022, 11:10 PM
(06/17/2022, 08:47 PM)Gottfried Wrote:Why do not you have ten fingers? What happend?(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.
Please remember to stay hydrated.
ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
06/18/2022, 12:09 AM
(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. 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 ShellThron 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 45 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. (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. 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
06/18/2022, 07:00 AM
(06/17/2022, 11:10 PM)Catullus Wrote:(06/17/2022, 08:47 PM)Gottfried Wrote:Why do not you have ten fingers? What happend?(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. :) Well, one handful five fingers. Two handful  ten fingers
Gottfried Helms, Kassel
(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 Carlemanmatrix is simply a factorially similarity scaling of the Bell matrix, and transposed. The idea is to work on the coefficients of the taylorseries, and so the work is practically identical between Carleman and Bellnotation. But note: it provides its concise arithmetic/algebra on *taylorseries* 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 Carlemanansatz 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 Carlemanansatz 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 selfreflection of my knowledge.
Gottfried Helms, Kassel

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