Summary

DescriptionCenters8.png	English: Centers of 983 hyperbolic components of Mandelbrot set with respect to complex quadratic polynomial for period 1-10 Polski: Punkty centralne 983 składowych zbioru Mandelbrota dla okresów 1-10   This plot was created with Gnuplot.
Date	4.01.2009
Source	Own work
Author	Adam majewski

Long description

Program input

No input is needed

Program output

• png file : centers_9_new.png
• txt files with numerical values of centers in big float Maxima format ( one file for each period)

Parts of the program

• definition of functions and constants
• loading packages
• for periods 1-period_Max
  • computation of irreducible polynomials for each period
  • computation of centers for each period : centers[period]
  • saving centers to text files : centers_bf_p.txt
  • computes number of centers for each period ( l[period]) and for all periods 1-period_Max ( N_of_centers)
• drawing to centers_9_new.png file

Software needed

• Maxima CAS
  • cpoly package written in Lisp by Raymond Toy containing bfallroots function finding roots of complex polynomials by Jenkins-Traub algorithm. It is in file cpoly.lisp in src directory ( for example in Maxima-5.16.3\share\maxima\5.16.3\src )
  • draw package - Maxima-Gnuplot interface by Mario Rodriguez Riotorto archive copy at the Wayback Machine
• gnuplot for drawing ( creates png file )

Tested on versions:

• wxMaxima 0.7.6
• Maxima 5.16.3
• Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
• Gnuplot Version 4.2 patchlevel 3

Algorithm

See Wikibooks for detailes

Questions

• Can it be done for higher periods ? For me (GCL and wxMaxima) it fails for period 10 and precision^[1] fpprec:150 or 256, but for period 10 and fpprec:300 I can run it from console or XMaxima, not wxMaxima

• How to save it as svg file ?

Compare with :

• centers 1-12 archive copy at the Wayback Machine made with Maxima, Eigensolve and Gnuplot
• centers of period 10 made with MPSolve and Gnuplot
• http://math.stackexchange.com/questions/2205922/critical-polynomial-roots-bigger-than-2

Maxima CAS src code

 /*
 Maxima batch script 
 because :
 "this does works in the console and xMaxima, but not in wxMaxima " Julien B. O. - jul059
 http://sourceforge.net/tracker/index.php?func=detail&aid=1571099&group_id=4933&atid=104933
 handling of large factorials
 for periods >=10 run from console or XMaxima, not wxMaxima
 for example from console under windows run:
 cd C:\Program Files\Maxima-5.16.3\bin
 maxima
 batch("D:/doc/programming/maxima/batch/MandelbrotCenters/mset_centers_10_new_png.mac")$
 ----------------
 notation and idea is based on paper :
 V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost 
 ideal   circles ?
 */
 start:elapsed_run_time (); 
 load(cpoly); 
 period_Max:10;
 

/* basic funtion  = monic and centered complex quadratic polynomial 
 http://en.wikipedia.org/wiki/Complex_quadratic_polynomial
 */
 f(z,c):=z*z+c $
 
/* iterated function */
 fn(n, z, c) :=
  if n=1 	then f(z,c)
  else f(fn(n-1, z, c),c) $

 /* roots of Fn are periodic point of  fn function */
 Fn(n,z,c):=fn(n, z, c)-z $ 
 
/* gives irreducible divisors of polynomial Fn[p,z=0,c] */
 GiveG[p]:=
 block(
 [f:divisors(p),t:1],
 g,
 f:delete(p,f),
 if p=1 
 then return(Fn(p,0,c)),
 for i in f do t:t*GiveG[i],
 g: Fn(p,0,c)/t,  
 return(ratsimp(g))
  )$

 /* use :
 load(cpoly); 
 roots:GiveRoots_bf(GiveG[3]); 
 */
 GiveRoots_bf(g):=
 block(
 [cc:bfallroots(expand(%i*g)=0)],
 cc:map(rhs,cc),/* remove string "c=" */
 return(cc)
  )$ 
 
GiveCenters_bf(p):=
 block(
 [g,
 cc:[]],
 fpprintprec:10, /* number of digits to display */
 if p<7 then fpprec:16
 elseif p=7 then fpprec:32
 elseif p=8 then fpprec:64
 elseif p=9 then fpprec:128
 elseif p=10 then fpprec:300,
 g:GiveG[p],
 cc:GiveRoots_bf(g),
 return(cc)
 );
 N_of_centers:0;
 for period:1 thru period_Max step 1 do
 (
 centers[period]:GiveCenters_bf(period), /* compute centers */
 
/* save output to file as Maxima expressions */
 stringout(concat("centers_bf_",string(period),".txt"),centers[period]),
 l[period]: length(centers[period]),
 N_of_centers:N_of_centers+l[period]
 ); 
 
stop:elapsed_run_time ();
 time:fix(stop-start);  
 

load(draw);
 
draw2d(
 file_name = "centers_10_new",
 terminal  = 'png,
 pic_width=1000, 
 pic_height= 1000,
 yrange = [-1.5,1.5],
 xrange = [-2.5,0.5],
 title= concat("centers of ",string(N_of_centers)," hyperbolic components of Mandelbrot set for periods 1- ",string(period_Max)," made in ",string(time)," sec"),
 user_preamble="set size square;set key out;set key top left",
 xlabel     = "re ",
 ylabel     = "im",
 point_type    = filled_circle,
 points_joined = false,
 point_size    = 0.5,
 /* in reversed order of periods because number of centers is proportional to period */
 key = concat(string(l[10])," period 10 components"),
 color		  =purple,
 points(map(realpart, centers[10]),map(imagpart, centers[10])),
 key = concat(string(l[9])," period 9 components"),
 color		  =gray,
 points(map(realpart, centers[9]),map(imagpart, centers[9])),
 key = concat(string(l[8])," period 8 components"),
 color		  =black,
 points(map(realpart, centers[8]),map(imagpart, centers[8])),
 key = concat(string(l[7])," period 7 components"),
 color		  =navy,
 points(map(realpart, centers[7]),map(imagpart, centers[7])),
 key = concat(string(l[6])," period 6 components"),
 color		  =yellow,
 points(map(realpart, centers[6]),map(imagpart, centers[6])),
 key = concat(string(l[5])," period 5 components"),
 color		  =brown,
 points(map(realpart, centers[5]),map(imagpart, centers[5])),
 key = concat(string(l[4])," period 4 components"),
 color		  =magenta,
 points(map(realpart, centers[4]),map(imagpart, centers[4])),
 key = concat(string(l[3])," period 3 components"),
 color		  =blue,
 points(map(realpart, centers[3]),map(imagpart, centers[3])), 
 key = concat(string(l[2])," period 2 components"),
 color		  =green,
 points(map(realpart, centers[2]),map(imagpart, centers[2])),
 key = concat(string(l[1])," period 1 component "),
 color		  =red,
 points(map(realpart, centers[1]),map(imagpart, centers[1]))
 )$

References

1. ↑ wikibooks : Fractals and numerical precision

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

You may select the license of your choice.