(09/28/2021, 04:41 PM)Ember Edison Wrote:(09/25/2021, 03:00 AM)JmsNxn Wrote: See my thread here; I run a quick toy model for a \( 2 \pi i \) periodic tetration base \( b = 1/2 \). As far as I can tell this should work on the real positive line; and should work in the complex plane, but I'm not sure. I think the real trouble would be \( b <0 \).
I will not be as optimistic as you are. The real big trouble should be b=\( e^{-e} \)≈0.065988035845312537076790187596846424938577048252796
If you really want to give yourself some real meaningful trials, try deriving a numerical approximation of the tetration function for the following bases:
b=e^-e, b=10^-10(, b=-10^-10), b=1+10^-10, b=1-10^-10(, b=1+10^-10 * I)
Oh, Maybe the numerical approximation accuracy you already have is not up to 10^-10, so you can try from 10^-5.
Maybe a simpler sequence would be b=0.1, 0.07, 0.066, 0.06599
I'm not super optimistic yet; but I see no reason for it to fail at the moment. But you are absolutely right, I won't get ahead of myself. I still want to make sure everything is kosher with \( b=e \). I'll give \( b = e^{-e} \) a shot though. This should be easy to patch work code. I'll post something later tonight, on what it's shaping up to be.
I do think the code that I have is too patchwork at the moment to work for \( b = 0.001 \) or something like that. But mathematically, I can't see a difference between this b and b=1/2. But my code will surely crap out for this base. That's moreso a problem with my code than the math though. I'm not the greatest programmer.
EDIT:
I updated the other thread and handled the case where \( b = e^{-e} \)--no obvious errors, as expected. It runs slower than b = 1/2, but seems fine so far. I'm making a complex plane graph at the moment, and I'll see how it looks. I'm still working with the toy model case which is 2pi i periodic solutions.
I graphed some Taylor series for \( b= e^{-e} \) and there are no errors. The infinite composition method works fine here. Again, I'll say that for \( b > 0 \) we don't fall into the same traps we fall into when talking about Schroder functions. We're solving a schroder equation in the neighborhood of infinity; not a fixed point. So the neutral, attracting, repelling paradigm doesn't matter for us. We don't care about fixed points. All we care about is that \( b^z \) and \( \log_b(z) \) are well enough behaved.
We can always find an asymptotic solution, and we're just trying to solve for an error between the asymptotic and the actual tetration. Again, ember, I don't see anything glaringly wrong. This avoids all the problems that the theta mapping method has.
Anyway, that's enough for tonight. I have a zoom with sheldon tomorrow, to talk about \( b = e \) with \( 2 \pi i \)-period case. I'm focusing on \( b = e \) for now; if I can get this to work perfectly, I'll move on to \( b > 0 \); then if I dare \( b \in \mathbb{C} \).
Keep posting challenges though; let's try to break this method together. Find everything that could break it.