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 )
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
/*
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]))
)$
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Captions
Add a one-line explanation of what this file represents
((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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