02/14/2016, 03:25 AM
(This post was last modified: 02/14/2016, 03:30 AM by sheldonison.)
(02/13/2016, 06:07 AM)marraco Wrote: ...
I have no idea on how to improve precision. Increasing the decimals in Pari/GP does nothing.
Maybe using a more accurate numerical difference equation? I used finite difference of order 40.
See http://math.eretrandre.org/tetrationforu...729&page=2 post#15,
In that post, there is a Taylor series for the 1st derivative of sexp_b(z) developed around b=2, and a Taylor series for sexp_b(-0.5), also developed around b=2. At the time I posted that, I also had developed Taylor series for the first 50 or so derivatives in the neighborhood of b=2. The key is to treat sexp_b(z) as analytic for the base b in the complex plane, in a circle around b=2. Then sample a set of points around a unit circle, and use Cauchy/Fourier which will give really good approximations to the derivatives. There is a mild singularity at b=eta which barely effects results at all, and a much much much more severe singularity at b=1. There is also reduced precision on the Shell Thron boundary. As I recall, the Taylor series posted were accurate to >25 decimal digits; using a relatively small number of sample points. I can post some pari-gp code that would work with fatou.gp to recreate that effort, if Maracco is interested. Of course, these results will be numeric, not a series.
- Sheldon
- Sheldon
