Filed under: Mathematics, Programming, python | Tags: fast mandelbrot, mandelbrot, mandelbrot set, numpy, vectorisation, vectorization
This will be of little interest to people who regularly read my blog, but might be of some interest to people who find their way here by the power of Google.
The standard way to compute fractals like the Mandelbrot set using Python and numpy is to use vectorisation and do the operations on a whole set of points. The problem is that this is slower than it needs to be because you keep doing computations on points that have already escaped. This can be avoided though, and the version below is about 3x faster than the standard way of doing it with numpy.
The trick is to create a new array at each iteration that stores only the points which haven’t yet escaped. The slight complication is that if you do this you need to keep track of the x, y coordinates of each of the points as well as the values of the iterate z. The same trick can be applied to many types of fractals and makes Python and numpy almost as good as C++ for mathematical exploration of fractals.
I’ve included the code below, both with and without explanatory comments. This 400×400 image below using 100 iterations took 1.1s to compute on my 1.8GHz laptop:
def mandel(n, m, itermax, xmin, xmax, ymin, ymax): ix, iy = mgrid[0:n, 0:m] x = linspace(xmin, xmax, n)[ix] y = linspace(ymin, ymax, m)[iy] c = x+complex(0,1)*y del x, y img = zeros(c.shape, dtype=int) ix.shape = n*m iy.shape = n*m c.shape = n*m z = copy(c) for i in xrange(itermax): if not len(z): break multiply(z, z, z) add(z, c, z) rem = abs(z)>2.0 img[ix[rem], iy[rem]] = i+1 rem = -rem z = z[rem] ix, iy = ix[rem], iy[rem] c = c[rem] return img
from numpy import * def mandel(n, m, itermax, xmin, xmax, ymin, ymax): ''' Fast mandelbrot computation using numpy. (n, m) are the output image dimensions itermax is the maximum number of iterations to do xmin, xmax, ymin, ymax specify the region of the set to compute. ''' # The point of ix and iy is that they are 2D arrays # giving the x-coord and y-coord at each point in # the array. The reason for doing this will become # clear below... ix, iy = mgrid[0:n, 0:m] # Now x and y are the x-values and y-values at each # point in the array, linspace(start, end, n) # is an array of n linearly spaced points between # start and end, and we then index this array using # numpy fancy indexing. If A is an array and I is # an array of indices, then A[I] has the same shape # as I and at each place i in I has the value A[i]. x = linspace(xmin, xmax, n)[ix] y = linspace(ymin, ymax, m)[iy] # c is the complex number with the given x, y coords c = x+complex(0,1)*y del x, y # save a bit of memory, we only need z # the output image coloured according to the number # of iterations it takes to get to the boundary # abs(z)>2 img = zeros(c.shape, dtype=int) # Here is where the improvement over the standard # algorithm for drawing fractals in numpy comes in. # We flatten all the arrays ix, iy and c. This # flattening doesn't use any more memory because # we are just changing the shape of the array, the # data in memory stays the same. It also affects # each array in the same way, so that index i in # array c has x, y coords ix[i], iy[i]. The way the # algorithm works is that whenever abs(z)>2 we # remove the corresponding index from each of the # arrays ix, iy and c. Since we do the same thing # to each array, the correspondence between c and # the x, y coords stored in ix and iy is kept. ix.shape = n*m iy.shape = n*m c.shape = n*m # we iterate z->z^2+c with z starting at 0, but the # first iteration makes z=c so we just start there. # We need to copy c because otherwise the operation # z->z^2 will send c->c^2. z = copy(c) for i in xrange(itermax): if not len(z): break # all points have escaped # equivalent to z = z*z+c but quicker and uses # less memory multiply(z, z, z) add(z, c, z) # these are the points that have escaped rem = abs(z)>2.0 # colour them with the iteration number, we # add one so that points which haven't # escaped have 0 as their iteration number, # this is why we keep the arrays ix and iy # because we need to know which point in img # to colour img[ix[rem], iy[rem]] = i+1 # -rem is the array of points which haven't # escaped, in numpy -A for a boolean array A # is the NOT operation. rem = -rem # So we select out the points in # z, ix, iy and c which are still to be # iterated on in the next step z = z[rem] ix, iy = ix[rem], iy[rem] c = c[rem] return img if __name__=='__main__': from pylab import * import time start = time.time() I = mandel(400, 400, 100, -2, .5, -1.25, 1.25) print 'Time taken:', time.time()-start I[I==0] = 101 img = imshow(I.T, origin='lower left') img.write_png('mandel.png', noscale=True) show()
George Monbiot has an interesting article linking capitalism and privatisation with growing prison populations:
This revolting trade in human lives creates a permanent incentive to lock people up; not because prison works; not because it makes us safer, but because it makes money. Privatisation appears to have locked this country into mass imprisonment.
It’s not clear to me that this is enough to explain the whole problem, but it’s worth considering.
Alderson has an interesting piece on religion over at Directionless Bones:
[Alderson’s view] also implies a certain set of priorities, that changing people’s lives is more important than changing their minds (though obviously not unrelated), and that often religion will persist regardless of rational arguments if the conditions that produce it persist.