Filed under: Uncategorized

I’ve made a minor change to my comments policy for this blog in the light of the growing number of spam comments making it past the spam filters which I have to delete by hand.

Shame about the title though. I think the words “Yes we can” should probably be banned. Anyway, here’s Robin Hahnel on “Change how the world works? Yes we can“:

Until capitalism is replaced, we want the tail to stop wagging the dog. Finance should serve the real economy instead of the other way around. If the financial sector improves the efficiency of the real economy, it is helpful. But if it misdirects investment resources to where they are less productive, it reduces production in the real economy by obstructing the flow of credit altogether. Then it is failing to accomplish its only social purpose. Jobs producing useful goods and services, and investments which help us to produce what we need with less human toil and less strain on the environment, are what count. Increases in the profit rates and stock prices of financial corporations count for nothing when they fail to correspond to real increases in productivity, as has too often been the case.

We have offered several positive alternatives to capital liberalization and to the governing structures and policies of the International Monetary Fund (IMF) and the World Bank, such as capital controls and a Tobin tax to protect smaller economies from volatile speculative flows. We have made suggestions on how national governments can restore competent regulation of their traditional financial sectors, and stressed the urgency of extending regulation to cover new financial institutions which were allowed to grow outside existing regulatory structures.

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:

Uncommented version:

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

Commented version:

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.

The Mathematics Genealogy Project has a huge database of mathematicians, showing who was supervised by whom, and what students everyone had. If you’re a mathematician, you can use this to trace back who your mathematical ancestors were and it can be quite fun. Below is a chart I made of my own mathematical genealogy. It’s nice to see exciting names from the history of mathematics and science there, such as Poisson, Laplace, Lagrange, d’Alembert, Euler, the Bernoullis, Leibniz, and Huygens (I stopped at that point). The dates are when they finished their doctorate, or if they didn’t do one, when they lived.

Reg article reporting on Nigel Inkster, former Assistant Chief of MI6:

There are limits to what we can sensibly aspire to…

Efforts to establish a global repository of counterterrorist information are unlikely ever to succeed. We need to be wary of rebuilding our world to deal with just one problem, one which might not be by any means the most serious we face.

…

We need to keep terrorism in some kind of context, for example, every year in the UK, more people die in road accidents than have been killed by terrorists in all of recorded history.

…

We should keep our nerve and our faith in our own values. Our own behaviour – especially with respect to the rule of law – is very important.