Newton Fractal, roots of equation z ^ 4 = 1

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Newton Fractal, roots of equation z ^ 4 = 1

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#1 by (2 votes)

5

I'm wanting to plot a graphic of an image, called Newton Fractal:

Followtemplate:

Theproblemistoplottherootsofthez^4=1equation,whereithas4rootsbeing(-1,1,iand-i),wheretofindtherootsIuseNewton'smethodtoapproximate.

Thedifferentcolors(red,green,yelloworblue)intheimagemeantowhichrootNewton'smethodisapproaching(andblackifitisnotany),butIamnotabletomakethecodethatgeneratestheimage.I'musingscilabwhichisverysimilartomatlab.

Here'swhatIwasabletodo:

function [z] = f(x,y)
    clc
    f(x,y) = z^4 -1 = 0;
    f1(x,y) = 4*z^3;
    niter = 100;
    x0 = 0.5;

    for i=1:niter
        for j=1:niter;
    end

   [X,Y] = meshgrid(x,y);

    Z(i,j) = f(x(i), y(j) );
  end
end

surf(X,Y,Z)
endfunction.

matlab

asked by anonymous 03.12.2015 / 01:15

1 answer

Identify what is in the USB port Get File Size in Bytes in Java Script

score 2 · Answer 1

Made in Matlab

%Cria matriz de valores iniciais
[X,Y]=meshgrid(-1:.002:1,-1:.002:1);
Z=X+i*Y;

%Cria funções
f=@(z) z.^4-1;
fp=@(z) 4*z.^3;

plot(Z,'.');

%NEWTON RAPHSON
for k=1:50
    Z=Z-f(Z)./fp(Z);
end

z=[1 i -1 -i]; %Raízes

%A matriz W contém o número da raíz que cada elemento Z converge 
W=5*ones(size(Z));%Cria matriz W

%Aproxima cada raiz
for k=1:4, 
    I=(abs(Z-z(k))<.3); 
    W(I)=k;
end 

%Cria o mapa de cores referentes as raizes
cmap=[0 0 1;1 0 0;0 1 0;1 1 0;0 0 0]; %[Azul;Vermelho;Verde;Amarelo;Preto]
%OS pontos que não convergem estarão em preto
%define o mapa de cores
colormap(cmap);
%Usando o mapa de cores cria imagem (W)
pcolor(X,Y,W),shading flat

And if you want to leave the image with shadows just follow the one described in the link

link

For more information:

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