06/29/2011, 10:36 PM
(This post was last modified: 06/29/2011, 10:42 PM by sheldonison.)

(06/29/2011, 05:05 AM)sheldonison Wrote: .... my superfunction and its inverse are only approximately inverses of each other. And that in turn messes up the theta calculation so it stops converging..... But I really don't understand it and I'm not sure how to fix it.

So, its the inverse Schroeder equation that is not working for . The Schroeder equation, which has the iterated logarithms, and corresponds to the inverse superfunction, is converging very nicely. But the inverse Schroeder equation which I use for the superfunction seems to conerge -- only up to a point -- and then stops converging. The defining characteristic of the inverse Schroeder equation, which I will call g(z), is

, for all z. The g(z) function should be entire.

The trouble I'm having, is that increasing the number of iterations and the precision doesn't seem to improve the results. I can sample g(z) around a unit circle with a reasonable radius so the function is well behaved, and yet there seems to be a lot of "noise" in the resulting Taylor series for the function. I was using \p 134, for 134 decimal digits of precision, and 336 iterations for n, and still only getting 25 digits of precision for g(z). And I can get the same or even better results, with 67 decimal digits precision and 165 iteration. The code for g(z) is in the previous post, where I graphed . I verified that the taylor series approximation for g(z) isn't following the defining equation definition too well, . I guess this could be some kind of pari-gp artifact for this particular base, but I still don't understand it, or it could be mathematical, but I still don't understand it.

I also verified that the Schroeder equation (inverse superfunction) is very well behaved for this base, and has nothing to do with the problem. Also, for all other bases I've tried, both the superfunction and the inverse superfunction are well behaved. Only bases very close to this base are affected, within a 10^-15 radius.

- Sheldon