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fast accurate Kneser sexp algorithm
#12
(10/15/2010, 04:03 PM)nuninho1980 Wrote: ... but I get errors after try ... init(200).
base 200
Nuinho noticed that large bases don't work, so I modified the program (attached below), so that now kneser.gp uses "centerat=-0.5" for base>20. sexp(0) will still return "1", but if you evaluate the Taylor series, TaylorSeries(z=0.5)=1. This is for bases>20, which may also require more than the 13 or so iterations required for smaller bases to get optimal precision. I also changed the initial sexp estimate to use the linear estimate, sexp(z)=log(e), sexp(z-1)=log(log(e)), and I slightly improved the renormsub routine. So now, bases from 1.47 to 50,000 will converge by simply typing init(1.47) ...init(50000). Some bases higher then 50,000 converge too, but centerat needs to be manualy modified closer to z=-1. If anyone wants to examine bases larger than 50,000, here's the code to try. This is for the new kneser.gp code, attached below. As far as I know, nobody else has tried generating an analytic version of tetration for such large bases before.
Code:
init(50000);  
loop(25); /* larger bases need more than 13 iterations */
dumparray;
init(300000);centerat=-0.8;recenterup; /* manually adjust centerat */
loop(25);
dumparray;

The problem for computing a Taylor series for sexp(z), for larger bases is that iterating with the Taylor series centered around z=0 doesn't work too well. This is a parametric plot of sexp_150, around the unit circle. What is amazing to me is that the program worked as well as it did, somehow converging up to B=100.
ploth(t=0,1,z=exp(Pi*I*2*t);y=sexp(z);[real(y),imag(y)],1);
also, notice the little green dot representing sexp_e(y). The behavior around the unit circle gets increasingly poorly behaved as the base gets larger.
   
ploth(t=0,1,z=exp(Pi*I*2*t)-0.5;y=sexp(z);[real(y),imag(y)],1);
Now, we plot the unit circle, with the Taylor series for sexp_150, and with "centerat"=-0.5, so that taylorseries(0.5)=1. But the program still sets sexp(0)=1. Also, the green circle is Taylor series for sexp_e, also centered at -0.5
   
And finally, a plot for sexp_150(z) and sexp_e(z), for z from -1.99 to 0.5. Notice that the sexp_150(z) has a smaller slope. The fixed point for B=150 is L=-0.169+0.394*I, so that value will not be taken, so the slope for z<0 at the real axis needs to be small.
   
For the most recent code version: go to the Nov 21st, 2011 thread.


Attached Files
.gp   kneser.gp (Size: 16.22 KB / Downloads: 381)
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Messages In This Thread
The pari-GP code - by sheldonison - 08/07/2010, 09:17 PM
updated kneser.gp code - by sheldonison - 08/19/2010, 02:35 AM
RE: updated kneser.gp code - by nuninho1980 - 08/19/2010, 12:08 PM
RE: updated kneser.gp code - by sheldonison - 08/20/2010, 01:05 AM
kneser.gp modified for large bases - by sheldonison - 10/19/2010, 03:33 AM
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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