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fast accurate Kneser sexp algorithm
(10/15/2010, 04:03 PM)nuninho1980 Wrote: lastest your code for faster. Smile
I tried bases 1.5), 1.68, 1.7 and 100 and worked (not work on your older code). it's better. Smile but I get errors after try init(1.6) and init(200).

errors here:
base 1.6
base 1.5
Hi Nuninho,

I finally got around to debugging this (you reported it last time too). What's happening, is that the isuperf/superf equations starts misbehaving as the base approaches . Below, there is a a graph for base 1.7, which just barely works. You can see the isuperf "jumps" when the sexp(z) starts getting slightly less than zero; sexp(z)<0, for z<-1. For the riemup code to work, the superf/isuperf needs to be consistent, over at least the unit circle. The problem is occurring in the last logarithm of the isuperf equation, after subtracting the fixed point "L". There is a term trying to counteract this multi-valued logarithm problem, by multiplying by c^n, before the last log_c, and this term works for bases>=1.7, but not for smaller bases, approaching eta. I'm checking in a "patch", that adds a new "xterminc", to push the function back into the well behaved region for bases B>=1.49. It still fails for B=1.48, but for other reasons, since the algorithm to generate "L" failed. The number of log/exp iterations required for the superf/isuperf for bases near starts to grow very large. In my program, this is the variable "scnt", and the algorithm for generating "scnt" does not work for bases, B<1.49.
- Sheldon, small typo fixed in new code, uploaded a new version. Also, fixed so that it works down to B=1.47, but B<1.47 still fails. These bases close to eta run very slow.
xterminc = 0; /* xterminc used to stabilize isuperf for bases<1.7 */
if ((B<1.7),  xterminc=1);
if ((B<1.55), xterminc=2);

For the most recent code version: go to the Nov 21st, 2011 thread.

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Messages In This Thread
The pari-GP code - by sheldonison - 08/07/2010, 09:17 PM
updated code - by sheldonison - 08/19/2010, 02:35 AM
RE: updated code - by nuninho1980 - 08/19/2010, 12:08 PM
RE: updated code - by sheldonison - 08/20/2010, 01:05 AM
RE: fast accurate Kneser sexp algorithm - by sheldonison - 10/15/2010, 09:20 PM
update to support B<eta - by sheldonison - 11/15/2010, 02:53 PM
RE: update to support B<eta - by nuninho1980 - 11/15/2010, 03:26 PM
another new version - by sheldonison - 11/17/2010, 06:52 PM

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