06/28/2011, 02:33 PM
(This post was last modified: 06/28/2011, 07:01 PM by sheldonison.)
(06/28/2011, 07:17 AM)Cherrina_Pixie Wrote: ....The problem is with the tetration/sexp code (which is used by pentation.gp). I haven't seen this bug before in kneser.gp, despite having tried a large number of different bases from 1.45 to 100,000 or so. But this particular base, exp(pi/2), seems to be a problem. I don't yet know why. Although in earlier versions of kneser.gp I saw it when pari-gp wasn't carrying out calculations with enough precision -- although I've fixed all of those bugs by manually updating the precision for the samples used to generate theta(z).
I ran through the loops for <sexp> of base ~4.81 and somehow, the indeterminacy of the 13th loop exceeded that of the 12th loop. Is there a way to solve this issue without changing the base? I'm not exactly certain about the possible consequences of this, other than the likely limit of precision... is a result like this 'normal' for larger bases?
I tried a few things, but this particular base seems to stop converging around 80 or 90 binary bits of precision (depending on which algorithm I use). I don't understand it yet, and I'm somewhat concerned that this means many other bases don't converge either. I know the error term is approximately a linear dot product from one iteration to the next -- I'm going to try some experiments and I'll report back results later.
Small update: The fixed point, L, for \( b=\exp(\pi/2) \), has L=i, with \( \Re(L)=0 \). I don't know what is special about the fixed point of L=i. I tried literally hundred of random bases between 1.5 and 10, (mostly using a faster version of kneser.gp, which uses a Taylor series for the Schroeder function and its inverse which is 3x faster than the version here). All of the random bases converged, so the problem is isolated. Then I tried bases nearby \( b=\exp(\pi/2)+\delta \). Using the default kneser.gp precision (\p 67), I get failures \( b=\exp(\pi/2)+10^{-25} \), but not for larger values of delta. Using a higher precision, \p 134, I get failures for \( \delta=10^{-28} \), but where the precision plateaus depends on how small delta is. I'll report some graphs later; hopefully I'll start to make some sense of what's going, and why this one particular base may not converge ....
- Sheldon