🚀 go-pugleaf

In article <1993Aug12.192446.18556@ast.saic.com>, hilljr@jupiter.saic.com 
  (Jay R. Hill) writes:
|> In article <93222.174612GHGAPAC@cc1.kuleuven.ac.be>, 
|>   <GHGAPAC@cc1.kuleuven.ac.be> writes:
|> [...]
|> |> I ported part ... of fractINT to macintosh. ... I have some help 
|> |> ... They told me something about periodicity checking, which is
|> |> ...  Can anyone give me some help about this principle ?
|> |> 
|> |>    hope you can help....    jer
|> 
|> Last February and March, as part of the Mandelbrot area calculation thread,
|> Robert Munafo <mrob@world.std.com> suggested periodicity checking for the
|> Ocie Mitchell <omitchel@fenris.claremont.edu> code.  Ocie's final routine
[...] 
|> We used this idea in GIPPP#3.
[...]

Rollo  Silver <RSILVER%LANL.BITNET@uga.cc.uga.edu> wrote to the recipients 
of list FRAC-L <FRAC-L%GITVM1.BITNET@uga.cc.uga.edu>
>Jay Hill posted code to check an iteration for [eventual] periodicity.

>A more elegant test was given by Dewdney in Scientific American a few
>years ago: You have two (complex) series, f[i] and s[i]; they start out
>with the same c, but one series runs twice as fast as the other.
>The series are eventually periodic if and only if the CURRENT terms
>are ever equal.

>I say "elegant" because you don't have to remember the past behavior
>of the series, only the current terms!

>Here's a sketch of the code. I assume the language supports complex
>arithmetic:

>s <- f <- c;
>for (...) {
>  s <- s^2 + c;
>  f <- (f^2 + c)^2 + c;
>  if (s = f) return TRUE;
>}
>return FALSE;

>I left out (...) the usual details for breaking the loop: modulus
>greater than 4, or too many iterations -- a point might be in the
>set without its orbit becoming periodic.

>--Rollo

Thanks Rollo for the tip.  However, this method will go to the maxiter
when the sample is near chaotic or is chaotic.  I would add my chaos
test on the f point.  If f approaches zero, it is chaotic.

With apologies to the Twilight Zone:

Your following the fine line between the routine and the surprise,
between the periodic and the divergent. There's a sign post up ahead.
DeeDee DeeDee DeeDee DeeDee.  You are entering the CHAOS ZONE.  :-)

|> On about Jul 21 I posted two exact tests you can do before iterating which 
|> test for points in the period 1 cardioid or the period 2 bud.  I used these 
|> tests in GIPPP#3.  I also discussed how you can test for an orbit being 
|> chaotic without having to iterate to 'limit'.  Since the most interesting 
|> patterns are near component edges, these tests can save a lot of time.
|> 
Warmly,
Jay 'Not tonight honey, I've got a chaotic orbit' Hill
-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}