12/29/2017, 01:41 PM
(12/20/2017, 02:50 AM)andydude Wrote: 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
Gottfried Helms, Kassel