Xivanki Share !!: Simple Newton's Method Fractal code in MATLAB

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Due to popular request I've sharing some very simple Newton's Method Fractal code in MATLAB. It produces the following 800x800 image (in about 2.5 seconds on my 2.4Ghz Macbook Pro):


>> [niters,solutions] = matlab_fractal;
>> imagesc(niters)

function [niters,solutions] = matlab_fractal
%Create Newton's Method Fractal Image
%Tomasz Malisiewcz (tomasz@cmu.edu)
%http://quantombone.blogspot.com/
NITER = 40;
threshold = .001;

[xs,ys] = meshgrid(linspace(-1,1,800), linspace(-1,1,800));
solutions = xs(:) + i*ys(:);
select = 1:numel(xs);
niters = NITER*ones(numel(xs), 1);

for iteration = 1:NITER
oldi = solutions(select);

%in newton's method we have z_{i+1} = z_i - f(z_i) / f'(z_i)
solutions(select) = oldi - f(oldi) ./ fprime(oldi);

%check for convergence or NaN (division by zero)
differ = (oldi - solutions(select));
converged = abs(differ) < threshold;
problematic = isnan(differ);

niters(select(converged)) = iteration;
niters(select(problematic)) = NITER+1;
select(converged | problematic) = [];
end

niters = reshape(niters,size(xs));
solutions = reshape(solutions,size(xs));

function res = f(x)
res = (x.^2).*x - 1;

function res = fprime(x)
res = 3*x.^2;