File:Jung200.png
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editDescriptionJung200.png |
English: Boundary of Mandelbrot set as an image of unit circle of Jungreis function
Polski: Brzeg Zbioru Mandelbrota jako obraz koła jednostkowego po zastosowaniu funkcji Jungreisa |
Date | 4 January 2009 (upload 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 |
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Maxima source code
edit/* 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 */ start:elapsed_run_time (); jMax:300; /* precision = proportional to details and time of computations */ iMax:800; /* number of points to draw */ /* ------------ 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 */ /* -------------------------------------- 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 *--------------------------------------------------------------------------*/ load(draw); /* Mario Rodríguez Riotorto http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */ draw(file_name = "jung300", dimensions=[1000, 500], terminal = 'png, 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)) ) );
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current | 13:28, 4 January 2009 | 1,000 × 500 (8 KB) | Soul windsurfer (talk | contribs) | {{Information |Description={{en|1=Boundary of Mandelbrot set as an image of unit circle}} {{pl|1=Brzeg Zbioru Mandelbrota jako obraz koła jednostkowego}} |Source=Own work |Author=Adam majewski |Date=4.01.2009 |Permission= |other_ve |
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