possible tetration extension part 2
#1
So, we saw in the previous part a formula which can be used to calculate half (or even complex) iteration of any function which converge. (it doesn't work if it doesn't)


Now it's easy to see how it could be used in tetration. For any x, \({^x}r\) converges to a finite value when r belongs to \([e^{-e}, e^{1/e}]\), and tetration can be defined as an iteration of the function \(r^x\), repeating it x times.
So, for example, with \(sqrt2\), infinite tetration converges to 2.
So the value \(\tau\) is equal to the value in which it converges, like in this example, and here \(\lambda\) equals the derivative of the function \(r^x\) at \(\tau\). So it equals \(\ln(\tau) \). Since we, with the formula, technically calculate half iteration of the function for big values (like 20.5),to get to lower values, we just need to apply the log a certain amount of times. 

So we can drive our final formula :

$${^r}x =(\lim_{n\rightarrow{+}\infty}(\log_{x} ^{n-k}((({^n}x-\tau)*\lambda^p) +\tau))$$

Where k+p=r, k being an integer and |p|<1, and \(\log_x^n\) means taking n times the log (or iterating n times the log base x).        

With this formula, we can define \({^r}x\) for values of r lower than -2, BUT NOT whole negatives below or equal to minus 2. For example, r=-1 is ok, same for r=-3.64, r=-15.9, but r=-2,r=-3,r=-4, ... is forbidden.    

The formula looks hard but is really easy to understand, for example, to calculate ${^{0.5}}(\sqrt2)$, we take a big value of ${^n}(\sqrt2)$ (like $n=23$), we substract $2$, multiply by $\ln(2)^{0.5}$, we add $2$, and take the $\log_{\sqrt2}$ $n$ times ($23$ times here). And we get a good approximation. 

With this formula, I have the following results :

\({^{-0.5}}({\sqrt2})\)≈0.6290566121
\({^{1.3}}({\sqrt2})\)≈1.49334127  
\({^{-{\sqrt2}}}({\sqrt2})\)≈-1   
\({^{-{2.5}}}({\sqrt2})\) ≈0.8390270267+9.064720284i
etc... 

Here is the graph of \({^{x}}({\sqrt2})\) from -2 to 4.
   
For example, here are some values of \({^x}0.5\) : 

\({^{0.5}}0.5\) ≈ 0.6297281585+0.2178921319i   
\({^{-1.9}}0.5\) ≈ 2.487456488+1.718168223i    
etc... 

Here is the graph of \({^x}0.5\) from -2 to 5. (red is the real part, blue is the imaginary part) 
   

Anyway, tell me what you think!
Reply
#2
(10/18/2022, 06:13 AM)Shanghai46 Wrote: So, we saw in the previous part a formula which can be used to calculate half (or even complex) iteration of any function which converge. (it doesn't work if it doesn't)
...
Anyway, tell me what you think!

At first reading your approach appears to be valid. I know because it was the first approach I developed. I suspect others here are likewise familiar. Currently there are about half a dozen good approaches to extending tetration.

A major problem with tetration is that without wide exposure, folks reinvent the same thing over and over. One of the major papers on tetration is named Exponentials Reiterated, due to this reinvention.
Daniel
Reply
#3
(10/18/2022, 06:37 AM)Daniel Wrote:
(10/18/2022, 06:13 AM)Shanghai46 Wrote: So, we saw in the previous part a formula which can be used to calculate half (or even complex) iteration of any function which converge. (it doesn't work if it doesn't)
...
Anyway, tell me what you think!

At first reading your approach appears to be valid. I know because it was the first approach I developed. I suspect others here are likewise familiar. Currently there are about half a dozen good approaches to extending tetration.

A major problem with tetration is that without wide exposure, folks reinvent the same thing over and over. One of the major papers on tetration is named Exponentials Reiterated, due to this reinvention.

I personally uploaded what I found. The problem is that it's probably the same case as for half dervivatives, where you have dozens of methods which are mathematically correct, but with different results. Also technically this method works with all numbers which infinite tetration converges. It's the red part in you diagram on your website
   
Reply
#4
(10/18/2022, 06:37 AM)Daniel Wrote:
(10/18/2022, 06:13 AM)Shanghai46 Wrote: So, we saw in the previous part a formula which can be used to calculate half (or even complex) iteration of any function which converge. (it doesn't work if it doesn't)
...
Anyway, tell me what you think!

At first reading your approach appears to be valid. I know because it was the first approach I developed. I suspect others here are likewise familiar. Currently there are about half a dozen good approaches to extending tetration.

A major problem with tetration is that without wide exposure, folks reinvent the same thing over and over. One of the major papers on tetration is named Exponentials Reiterated, due to this reinvention.

Also is there a blog here in which we compare several methods? Like we explain the methods and compare the results?
Reply
#5
(10/18/2022, 07:06 AM)Shanghai46 Wrote:
(10/18/2022, 06:37 AM)Daniel Wrote:
(10/18/2022, 06:13 AM)Shanghai46 Wrote: So, we saw in the previous part a formula which can be used to calculate half (or even complex) iteration of any function which converge. (it doesn't work if it doesn't)
...
Anyway, tell me what you think!

At first reading your approach appears to be valid. I know because it was the first approach I developed. I suspect others here are likewise familiar. Currently there are about half a dozen good approaches to extending tetration.

A major problem with tetration is that without wide exposure, folks reinvent the same thing over and over. One of the major papers on tetration is named Exponentials Reiterated, due to this reinvention.

I personally uploaded what I found. The problem is that it's probably the same case as for half dervivatives, where you have dozens of methods which are mathematically correct, but with different results. Also technically this method works with all numbers which infinite tetration converges. It's the red part in you diagram on your website

Well I'm not sure that techniques that give different results can both be correct, but you have identified an important step in tetration research, the categorization of the different methods and the reconciliation of their different results.
Daniel
Reply
#6
[quote pid="11382" dateline="1666073629"]
Well I'm not sure that techniques that give different results can both be correct, but you have identified an important step in tetration research, the categorization of the different methods and the reconciliation of their different results.
[/quote]

Technically, for half derivatives, there's at least 2 methods which are mathematically demonstrated and correct in a way, but give different results. I just hope it's not the case here, and if it is, we'll have to arbitrary chose one by some criterias. Mine has several advantages like consistency, easily demonstrable, easy to compute/understand,...
Reply
#7
[quote pid="11382" dateline="1666073629"]
Well I'm not sure that techniques that give different results can both be correct, but you have identified an important step in tetration research, the categorization of the different methods and the reconciliation of their different results.
[/quote]

Whats in your opinion the best extension method? And why?
Reply
#8
(10/18/2022, 07:49 AM)Shanghai46 Wrote: Well I'm not sure that techniques that give different results can both be correct, but you have identified an important step in tetration research, the categorization of the different methods and the reconciliation of their different results.

Whats in your opinion the best extension method? And why?

As for the best method, I will let others more knowledgeable weigh in. I do have my own approach which is one of the simpler attempts which is seen and being an extension of Schroeder's work. My work is also consistent with much if not all of the work Gottfried does with matrices. I suspect that things have been held up for the last fifteen years because of a lack of good convergence proof. Paulsen's work breaks the logjam with it's proof of convergence. It is also based on earlier work while providing better, simpler and cleaner proofs. Paulsen says his work is consistent with Kouznetsov and Trappmann's (Bo) work. There are links at the bottom of my Tetration.org page to Paulsen's and others work.

As far as the best technique I consider that to be mathematical religion. Some folks like me are comfortable with real values being mapped to complex values; others believe real values should map to real values. I'm agnostic - I believe at the different methods should be studied and reconciled.
Daniel
Reply
#9
(10/18/2022, 07:16 AM)Shanghai46 Wrote: [quote pid="11382" dateline="1666073629"]
Well I'm not sure that techniques that give different results can both be correct, but you have identified an important step in tetration research, the categorization of the different methods and the reconciliation of their different results.

Technically, for half derivatives, there's at least 2 methods which are mathematically demonstrated and correct in a way, but give different results. I just hope it's not the case here, and if it is, we'll have to arbitrary chose one by some criterias. Mine has several advantages like consistency, easily demonstrable, easy to compute/understand,...
[/quote]

I believe most researchers evolve from algorithmic, algebraic to multi-disciplined approaches. But it is typically the multi-disciplined approached that generate new mathematics. Ultimately we are looking for new unexpected mathematical principles.
Daniel
Reply


Possibly Related Threads…
Thread Author Replies Views Last Post
  possible tetration extension part 1 Shanghai46 6 607 10/31/2022, 09:45 AM
Last Post: Catullus
  possible tetration extension part 3 Shanghai46 11 1,022 10/28/2022, 07:11 PM
Last Post: bo198214
  Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 18 2,433 09/18/2022, 09:37 PM
Last Post: tommy1729
  On extension to "other" iteration roots Leo.W 34 5,184 08/30/2022, 03:29 AM
Last Post: JmsNxn
  Tetration extension for bases between 1 and eta dantheman163 23 37,929 07/05/2022, 04:10 PM
Last Post: Leo.W
  Non-trivial extension of max(n,1)-1 to the reals and its iteration. MphLee 9 9,153 06/15/2022, 10:59 PM
Last Post: MphLee
  Ueda - Extension of tetration to real and complex heights MphLee 4 1,729 05/08/2022, 11:48 PM
Last Post: JmsNxn
  Possible continuous extension of tetration to the reals Dasedes 0 3,364 10/10/2016, 04:57 AM
Last Post: Dasedes
  Andrew Robbins' Tetration Extension bo198214 32 81,372 08/22/2016, 04:19 PM
Last Post: Gottfried
  Tetration Extension to Real Heights chobe 3 11,110 05/15/2010, 01:39 AM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)