(11/10/2013, 11:10 PM)sheldonison Wrote:(11/09/2013, 07:36 AM)mike3 Wrote: .... The case \( e^{-e} \) is really interesting since it lies on the Shell-Thron border and has pseudo-period 2 at the principal fixed point, and as far as I can tell, there's hasn't yet been a good construction of the merged/bipolar superfunction (i.e. the tetrational, \( \mathrm{tet} \)) at this base. sheldonison mentioned some work toward this, though. Nonetheless, with the HAM, it looks to be possible to construct what would be that superfunction, and perhaps this might point the way towards tetrating it with sheldonison's merge method, or just providing a new, independent method, though it seems choosing the right initial guess is one of the tricky aspects here. I have managed, however, to successfully tetrate base \( -1 \).I'm also interested in any results for \( \exp^{-e} \), with pseudo period 2, as per our earlier discussion on this forum. I haven't gotten any further with that base, though I believe it has a solution possible due to results I've gotten for a base with pseudo period=5, via a cumbersome indirect method.
- Sheldon
Hmm. Well I got it to work for base \( -1 \), which I was not able to do via the Kneser method. (I can post a graph to show you what \( \mathrm{tet}_{-1}(z) =\ ^{z} (-1) \) looks like, if you want.) I tried it for \( e^{-e} \), but the problem there is that the initial guesses I use for the method are not good enough in that they do not wrap the right way around 0, which matters because log is multivalued.
There are apparently methods by which the initial approximation for the HAM (and also, its other parameters) can be constructed, but I'll have to consult with the local university's library to get the paper on how to do it. The approximation I am using right now is apparently not good enough to do base \( e^{-e} \) as it wraps the wrong way around 0. So you may have to wait some, though I'm fiddling with it right now so maybe I might get something.
Added: I've tried forcing the initial guess to wrap the other way, but apparently it isn't close enough to the true solution so that, as iteration proceeds, they jump back across 0 and the method fails. I'm giving up on fiddling with it for now. I'll have to get that paper and see what information it contains.