Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
fast accurate Kneser sexp algorithm
#31
(07/23/2011, 05:48 AM)deepblue Wrote: Is it continuous at 1/2? I got...
sexp(1/2)=1.6376...
sexp(1/2+10^-30)=1.6489...
You're using the initial seed approximation, which is a three term taylor series, along with iterative logs/exps. After initialization, you need to "loop" through iteratively calculating the Kneser theta mapping. After 13 iterations (which takes about 4 seconds), you'll get results accurate to 110 binary bits.
Code:
gp > init(exp(1));loop
   base          2.71828182845904523536029
   fixed point   0.318131505204764135312654 + 1.33723570143068940890116*I
   Pseudo Period 4.44695072006700782711228 + 1.05793999115693918376341*I
generating superf taylor series; inverse Schroder equation, scnt 235
generating isuperf taylor series; Schroder equation, scnt 235
sexp(-0.5)= 0.49855114028767537891166401244655
1=loopcnt  8.842977061 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856330616566898032298548529033
2=loopcnt  17.67523398 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328805247482985620720695870
3=loopcnt  26.43624250 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794127685577078081234199
4=loopcnt  35.02171218 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111357966546850464454
5=loopcnt  43.44600587 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443478679527238125
6=loopcnt  52.16036487 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443467429889486618
7=loopcnt  60.48849684 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443467961715088127
8=loopcnt  69.25315295 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443467961906029415
9=loopcnt  77.65865208 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443467961909246830
10=loopcnt  86.52036006 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443467961909249278
11=loopcnt  94.54807694 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443467961909249313
12=loopcnt  102.8988830 Riemann/sexp binary precision bits
sexp(-0.5)= 0.49856328794111443467961909249313
13=loopcnt  110.6414599 Riemann/sexp binary precision bits
%1 = 1
Code:
gp > sexp(0.5)
%2 = 1.6463542337511945809719240315921
gp > sexp(0.5+1E-30)
%3 = 1.6463542337511945809719240315937
gp >
The separate "init(b)" routine from the "loop" routine is a little confusing. Originally, the kneser.gp program was more of an experiment, although I kind of like to be able to initialize for a base, and see the fixed point, and the pseudo period, before doing the loop. Sometimes I want to manually control the looping....

Anyway, I could change the program in one of two ways that would make it simpler. Both changes I'm thinking of involve putting the "loop;" at the end of the init routine, so that anytime you initialize for a given base, via "init(b)", the program automatically loops through with the current precision, generating the sexp(z) series. Then the question becomes what to do when kneser.gp is first loaded into pari-gp. Since the loop only takes 4-5 seconds for base e, "\p 67", I could have the program fully initialize for base e when it loads, going through the loop of thirteen iterations. Or, I could completely remove the partial initialization for base e when the program loads. Any opinions, as to which makes more sense?
- Sheldon
For the most recent code version: go to the Nov 21st, 2011 thread.
Reply


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
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
RE: fast accurate Kneser sexp algorithm - by sheldonison - 07/23/2011, 12:54 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  "Kneser"/Riemann mapping method code for *complex* bases mike3 2 6,948 08/15/2011, 03:14 PM
Last Post: Gottfried
  Attempt to make own implementation of "Kneser" algorithm: trouble mike3 9 16,448 06/16/2011, 11:48 AM
Last Post: mike3
  Numerical algorithm for Fourier continuum sum tetration theory mike3 12 20,479 09/18/2010, 04:12 AM
Last Post: mike3
  regular sexp: curve near h=-2 (h=-2 + eps*I) Gottfried 2 6,189 03/10/2010, 07:52 AM
Last Post: Gottfried
  Attempting to compute the kslog numerically (i.e., Kneser's construction) jaydfox 11 19,182 10/26/2009, 05:56 PM
Last Post: bo198214
  regular sexp:different fixpoints Gottfried 6 12,254 08/11/2009, 06:47 PM
Last Post: jaydfox
  sexp(strip) is winding around the fixed points Kouznetsov 8 13,525 06/29/2009, 10:05 AM
Last Post: bo198214
  sexp and slog at a microcalculator Kouznetsov 0 3,283 01/08/2009, 08:51 AM
Last Post: Kouznetsov



Users browsing this thread: 1 Guest(s)