Old Research Paper

[andydude]

I could not find my own paper in this forum, so I'm posting it again. It's mostly general continuous iteration stuff.

[Gottfried]

I could not find my own paper in this forum, so I'm posting it again. It's mostly general continuous iteration stuff.

Hi Andrew - 

I took the opportunity to look again at the method that you proposed in your paper. I notice that I do not know exactly, whether it was shown whether the occuring series for the Abel-function are convergent, divergent, or convergent in a certain interval of bases. However I remember that Jay D. Fox has analyzed your ansatz with very large matrices and thus high accuracy - but again I do not remember any mention of range of convergences and/or maximal precision.        

Is there something explicite about this - here in our forum or elsewhere? (I think P. Walker in his own application had left this question open) .           

Before finding something I've just done some series expansions for bases b=e, up to b=e^2 and have found a tendency towards divergence of the coefficients when the matrix-sizes were increased up to (in some cases n=1024x1024). Examples for b=e^2 down to b=5 and also a bit smaller showed a significant pattern which suggest a general tendency to unbounded, diverging coefficients but where the divergence begin to occur at high indexes/at high powers of x in the associated power series.                   

If this is indeed an open problem and if it is also of interest for you I could a) simply sendout my Excelfile with datasets and pictures for a couple of bases b or b) could invest some time to comment on the pictures and insert them here (if this shall not be too complicated for the Forum's software, and I'm not too lazy "between years" ... )

Besides of that -

 I'd like to wish everyone a happy  new year

Gottfried

[andydude]

Unfortunately I have never heard or discovered a proof or disproof of convergence of Peter Walker's method or my variation of it. I was very close to some kind of closed form for the super-logarithm approximants, which I was hoping to use to find convergence properties, but I never finished that research. I can dig up some notes on it if you want to work together on it.

Regards,
Andrew Robbins

[andydude]

Also, I know there are several pari/gp implementations on here, and I have a Mathematica implementation, and a partial Sage implementation. I know pari/gp is free, but so is Sage, I think to make these subjects more accessible to all, I should revisit my Sage implementation at some point. SageMath also includes pari/gp as a way to implement functions, so maybe I might end up using the pari/gp code in the Sage implementation.