有没有更简单的方法在 Matlab 中构造 Mandelbrot 集？

Question

下面显示的代码用于绘制Mandelbrot set，我认为我的代码对于构造矩阵 M 有点冗余。在 Python 我知道有一种干净的方法可以做到这一点，

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

在 Matlab 中是否有类似的方法？

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

Answer 1

您可以完全避免循环。您可以以矢量化方式进行迭代z = z.^2 + c。为避免不必要的操作，在每次迭代时跟踪哪些点 c 已经超过了您的阈值，并仅使用剩余点继续迭代（这就是下面代码中索引 ind 和 ind2 的目的）：

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

Answer 2

你至少可以避免for循环：

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