Tommy's Gaussian method.

[JmsNxn]

An interesting idea is this :

Are all these " beta methods " equivalent ? as James asks.

And 

Is there a way to accelerate the convergeance of the iterations ? 

Series acceleration is well known but iteration acceleration not so much.

Those are 2 nice questions , but what is the interesting idea you might ask ??

Well that those 2 questions are related !!

let 

f_1(s) = exp( t(1 s) * f(s-1))

f_2(s) = exp( t(2 s) * f(s-1))

and the resp analytic tetrations from them : F1(s) and F2(s).

Remember that tetration(s + theta(s)) is also tetration where theta(s) is a suitable analytic real 1-periodic function. 

so F2(s) = F1(s + theta1(s))  , F1(s) = F2(s + theta2(s)).

BUT THIS ALSO IMPLIES THAT 

f*_2(s) = exp( t( 2*(s + theta2(s)) ) * f(s-1))    .. RESULTING IN F2_*(s) is actually equal to 

F2_*(s) = F2(s + theta(s)) = F1(s)

HENCE USING t(1 s) = t(s) is the same as using t( 2 s + 2 theta(s)) !!

So this relates to the main questions posed :

when are 2 solutions equal ?

How to accelerate convergeance ?

As for the acceleration , ofcourse the complexity and difficulty of theta and computing theta are key.

But numerically it is expected using t( 2 s + 2 theta(s) ) converges faster. ( because using t(2s) does converge faster than using t(s) ) .

---

Tom(s,v) = exp( t(v * s)   *   exp(Tom(s-1,v)) )

resulting in 

tet(s+1,v) = exp( tet(s,v) ).

I Like that notation.

---

regards

tommy1729

Oh you must've posted this right as I posted mine, I missed it.

Yes, I agree with you entirely here. I think it's similar to what Kouznetsov did when he constructed his general form of the superfunction equation. Where, Kouznetsov chose an asymptotic function,

\(
f_M(z) = L +\sum_{n=1}^M a_n exp{zL n}\\
\text{tet}_{K}(z) = \lim_{n\to\infty} \exp^{\circ n} f_M(z-n)
\)

Where \( M \) was just a degree of "how well we are approximating." But, it had no effect on the final tetration--it still created Kneser.

I think we are in a similar situation here. Where all these asymptotic tetrations are all going to be \( \text{tet}_\beta \). And they are characterized by the fact \( \lim_{\Im(s) \to \infty} \text{tet}(s) = \infty \). I can't think of an obvious uniqueness condition though. Kneser has the benefit of being normal at infinity; non-normality tends to mean there's lots of room for errors and slight adjustments. Plus; we don't have the added benefit of a unique Fourier theta mapping--where we can just call on the uniqueness of Fourier coefficients (like what Paulsen and Cogwill did).


My only thoughts how we might do this was hit with a dead-end as I tried to write it up. It doesn't feel natural. But if we talk about,

\(
F_\lambda(s) = \lim_{n\to\infty} \log^{\circ n} \beta_\lambda(s+n)\\
\)

Which is the unique tetration with period \( 2\pi i / \lambda \) and holomorphy on an almost cylinder \( \mathbb{T} \) (which just means \( \overline{\mathbb{T}} \simeq \mathbb{C}/2\pi\mathbb{Z} \)). And then using a different kind of mapping we can transform between tetrations by creating a \( 1 \)-periodic function \( \lambda(s+1) = \lambda(s) \); then,

\(
\text{tet}_{WEIRD}(s) = F_{\lambda(s)}(s)\\
\)

Is a tetration function; and we'd be able to find a \( \lambda \) for Kneser; or any tetration really. But I can't think of a uniqueness condition that would guarantee that, there exists a unique \( \lambda \) such that,

\(
\text{tet}(s) = F_{\lambda(s)}(s)\\
\lim_{\Im(s) \to \infty} \text{tet}(s) = \infty\\
\)

I just made it to the point where there exists \( \lambda^+, \lambda^- \) which are holomorphic in the upper/lower half planes (resp.) in which,

\(
\text{tet}_\beta(s) = F_{\lambda^+(s)}(s)\,\,\text{for}\,\,\Im(s) > 0\\
\text{tet}_\beta(s) = F_{\lambda^-(s)}(s)\,\,\text{for}\,\,\Im(s) < 0\\
\lim_{|\Im(s)| \to \infty} \lambda^{\pm}(s) = 0\\
\)

So that we have a fourier series,

\(
\lambda^+(s) = \sum_{k=1}^\infty c_k e^{2\pi i ks}\\
\lambda^-(s) = \sum_{k=1}^\infty \overline{c_k} e^{-2\pi i ks}\\
\)

I couldn't think of any obvious arguments that the sequence \( c_k \) is unique though... So I gave up on that paper and focused on better investigating the programming. And no matter how you change the initial asymptotic tetration function--they all seem to give \( \text{tet}_\beta \). So I'm at least reinforcing the numerical evidence, lol.

Regards, James.