Is there a simpler way to construct Mandelbrot set in Matlab?

Question

The code shown below for drawing a Mandelbrot set, I think my code a bit redundancy to construction the Matrix M. In Python I know there is a clean way do this,

M = [[mandel(complex(r, i)) for r in np.arange(-2, 0.5,0.005) ] for i in np.range(-1,1,0.005)]

Is there a similar way do this in Matlab?

function M=mandelPerf()
rr=-2:0.005:0.5;
ii=-1:0.005:1;
M = zeros(length(ii), length(rr));
id1 = 1;
for i =ii
    id2 = 1;
    for r = rr
        M(id1, id2) = mandel(complex(r,i));
        id2 = id2 + 1;
    end
    id1 = id1 + 1;
end
end

function n = mandel(z)
n = 0;
c = z;
for n=0:100
    if abs(z)>2
        break
    end
    z = z^2+c;
end
end

Note that you at least double computation as madelbrot set is linearly symmetric with respect to `y=0` — alko, Dec 07 '13 at 13:23

Luis Mendo · Answer 1 · 2013-12-07T18:40:07.690

8

You can avoid the loop altogether. You can do the iteration z = z.^2 + c in a vectorized manner. To avoid unnecessary operations, at each iteration keep track of which points c have already surpassed your threshold, and continue iterating only with the remaining points (that's the purpose of indices ind and ind2 in the code below):

rr =-2:0.005:0.5;
ii =-1:0.005:1;
max_n = 100;
threshold = 2;
c = bsxfun(@plus, rr(:).', 1i*ii(:)); %'// generate complex grid
M = max_n*ones(size(c)); %// preallocation.
ind = 1:numel(c); %// keeps track of which points need to be iterated on
z = zeros(size(c)); %// initialization
for n = 0:max_n;
    z(ind) = z(ind).^2 + c(ind);
    ind2 = abs(z(ind)) > threshold;
    M(ind(ind2)) = n; %// store result for these points...
    ind = ind(~ind2); %// ...and remove them from further consideration
end

imagesc(rr,ii,M)
axis equal

enter image description here

edited Dec 07 '13 at 18:40

answered Dec 07 '13 at 15:32

Luis Mendo

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Change the colormap :P – Ander Biguri Oct 28 '15 at 13:23
@AnderBiguri I know, I know... hg2 didn't exist back then :-) – Luis Mendo Oct 28 '15 at 13:38
@AnderBiguri Actually I've just tried `parula` and `viridis`, and I find the old rainbow colormap more suited to this problem, in which it's all about highlighting the edges – Luis Mendo Oct 28 '15 at 13:42
1

You are probably right. in the end , this image is 2 colors and then edges – Ander Biguri Oct 28 '15 at 13:55

Robert Seifert · Accepted Answer · 2013-12-07T13:38:38.027

1

You could at least avoid the for-loop:

function M=mandelPerf()

rr = -2:0.005:0.5;
ii = -1:0.005:1;

[R,I] = meshgrid(rr,ii);
M = arrayfun(@(x) mandel(x), R+1i*I);

end

function n = mandel(z)
n = 0;
c = z;
for n=0:100
    if abs(z)>2
        break
    end
    z = z^2+c;
end
end

edited Dec 07 '13 at 13:38

answered Dec 07 '13 at 13:33

Robert Seifert

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Note that `arrayfun` is usually just a `for` loop in disguise. It won't necessarily save time compared to the `for` loop. See [here](http://stackoverflow.com/questions/12522888/arrayfun-can-be-significantly-slower-than-an-explicit-loop-in-matlab-why) for example – Luis Mendo Dec 07 '13 at 15:33
@LuisMendo I'm aware of that, but not familiar enough with the theory behind Mandelbrot to skip unnecessary steps. The python example the OP gave, is also just looping over it, right? – Robert Seifert Dec 07 '13 at 15:55
About Python, I couldn't say anything... The `for` loop here can be vectorized easily; see my answer – Luis Mendo Dec 07 '13 at 18:19

Is there a simpler way to construct Mandelbrot set in Matlab?

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