File:Jungreis.svg
Summary
| Description |
English: Boundary of Mandelbrot set as an image of unit circle under Jungreis function Polski: Brzeg Zbioru Mandelbrota jako obraz koła jednostkowego po zastosowaniu funkcji Jungreisa |
| Date | |
| Source | Own work : translation of maple code by G. A. Edgar; Help : G. A. Edgar and Richard J. Fateman, Robert P. Munafo and Richard J. Fateman.[1] See also : Bifurcation of Dynamic Rays in Complex Polynomials of Degree Two, Atela, P., Ergod Th & Dynam Sys (1991) 12, 401-423 |
| Author | Adam majewski |
![[1]](./File:Jungreis.svg#cite_note-1){kind=link}
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Maxima CAS src code
/* batch file for maxima
uses :
- symmetry around horizontal ( 0X ) axis
- Psi_M function to map conjugate plane to parameter plane
- jungreis algorithm to
time :
263 sec for jMax:100; and iMax:800;
4000 sec for jMax:400
*/
kill(all);
remvalue(all);
/* ============== functions ============================================================================*/
/* ------------ computes b coefficient of Jungreis function ----*/
betaF[n,m]:=block
(
[nnn:2^(n+1)-1],
if m=0
then 1.0
else if ((n>0) and (m < nnn))
then 0.0
else (betaF[n+1,m]- sum(betaF[n,k]*betaF[n,m-k],k,nnn,m-nnn)-betaF[0,m-nnn])/2.0
)$
b[m]:=betaF[0,m+1]$
/* -------------------------------*/
/* Power of w to j */
wn[w,j]:= if j=0 then 1 else w*wn[w,j-1]$
/* ---------Jungreis function ; c = Psi_M(w) ----------------------------- */
Psi_M(w):=w + sum(b[j]/wn[w,j],j,0,jMax)$
/* exponential for of complex number with angle in turns */
GiveCirclePoint(t):=R*%e^(%i*t*2*%pi)$ /* gives point of unit circle for angle t in turns */
compile(all);
start:elapsed_run_time ();
jMax:1000; /* precision = proportional to details and time of computations */
iMax:2000; /* number of points to draw */
/* --- w-plane -----------------------------------------*/
/*-------------- unit circle ------------*/
R:1;
circle_angles:makelist(i/iMax,i,0,iMax/2)$
CirclePoints:map(GiveCirclePoint,circle_angles)$
/* -------------------------parameter plane = c plane -----------------------------------*/
MPoints:map(Psi_M,CirclePoints)$ /* Mandelbrot set points */
/* add points below horizontal axis */
for w in CirclePoints do CirclePoints:cons(conjugate(w),CirclePoints)$
for c in MPoints do MPoints:cons(conjugate(c),MPoints)$
/* time */
stop:elapsed_run_time ();
time:fix(stop-start);
/* ---------------- draw *--------------------------------------------------------------------------*/
path : ""; /* pwd ; if you put here working directory name then graphic file will be saved in that dir */
FileName:concat(string(jMax),"jung", string(iMax))$ /* without extension which is the terminal name */
load(draw); /* Mario Rodríguez Riotorto http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
draw(file_name = concat(path,FileName),
dimensions=[2000, 1000],
terminal = 'svg,
columns = 2,
gr2d(title = " unit circle {w:abs(w)=1} ",
point_type = filled_circle,
points_joined =true,
point_size = 0.34,
color = red,
points(map(realpart, CirclePoints),map(imagpart, CirclePoints))
),
gr2d(title = " Image of circle under Psi_M(w) ",
points_joined =true,
point_type = filled_circle,
point_size =0.34,
color = blue,
points(map(realpart, MPoints),map(imagpart, MPoints))
)
);
