File:Centers8.png
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Summary
[edit]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 by n. |
Date | 4.01.2009 |
Source | Own work |
Author | Adam majewski |
Long description
[edit]Program input
[edit]No input is needed
Program output
[edit]- 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
[edit]- 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
[edit]- 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
[edit]See Wikibooks for detailes
Questions
[edit]- 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 :
[edit]- 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
[edit]/* 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
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[edit]I, the copyright holder of this work, hereby publish it under the following licenses:
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- to share – to copy, distribute and transmit the work
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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 |
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 10:02, 6 May 2017 | 1,000 × 797 (781 KB) | Shiftchange (talk | contribs) | Cropped white space above and below image. | |
13:24, 30 March 2009 | 1,000 × 1,000 (13 KB) | Soul windsurfer (talk | contribs) | centers for periods 1-10 | ||
13:53, 15 March 2009 | 1,000 × 1,000 (10 KB) | Soul windsurfer (talk | contribs) | periods 1-9 | ||
12:33, 4 January 2009 | 1,000 × 1,000 (9 KB) | Soul windsurfer (talk | contribs) | {{Information |Description={{en|1=Centers of hyperebolic components of Mandelbrot set with respect to complex quadratic polynomial for period 1-8 }} {{pl|1=Punkty centralne składowych zbioru Mandelbrota dla okresów 1-8 }} |Source=Own work |Author=[[User |
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