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So now we know, at least in theory, how to find the lines dividing the different convergence regions. At first sight one might assume that this is an artifact of having a finite iteration limit at all, but one would it turns out be wrong.

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The program will produce PNG, Windows. Replace power with the power you want; for example, replace it with 0. To raise the function to an overall power, use the -p option, for example -p 0.

But this is more than just a cycle between those two points: We find the roots of this using Newton-Raphson for its more conventional purpose and feed them into our fractal plotter, and get this picture: Oscillation Here's another curiosity.

Bryan Krofchok has a more varied — and more colourful — gallery.

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The interesting feature of this picture is the black areas, which are regions in which Newton-Raphson failed to converge to a root within the program's limit of iterations. So you wouldn't actually want to throw away your dedicated Julia-set plotting programs; but it's interesting, nonetheless, that something very like Julia vapid stanier online dating are a special case of Newton-Raphson fractals.


Hence, there's a sizeable region around each point of the cycle which all converges to the cycle itself, and hence never settles down to a root of the polynomial. Yes, as it turns out, there is. This gives rise to qualitatively more interesting fractal phenomena than the plot with seven roots in a line, in which all the large convergence regions are separated by boring curves which never meet, and the fractal blobs reflect this structure.

Here are some pre-generated larger versions of the above images: Even if you crank up the iteration limit to a much larger number, those black areas stay black, because they represent genuine non-convergence. This seemed like a good thing if we wanted to know the overall structure of the fractal; but merely clearer isn't good enough.

To zoom in on one of the boundary lines as I did above, replace "-x 2" with "-x 0. The program has a number of other options; just type newton on its own to list them.

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How often does this happen? That's a lot of maths to endure without a break for a pretty picture, and it's also a long and rather handwavey chain of reasoning to endure without some sort of reassurance that what I'm saying still makes sense.

Here are a few such pages; you can probably find more by googling for "Newton-Raphson fractal". It's also possible to imagine that we might encounter a point along this path at which there isn't a clear direction we should head in: BMP, or PPM files, and image processing software should be able to convert those to other formats if you prefer.

The roots of the polynomial have moved around, and the picture is distorted, but the region meeting points are still exactly where we asked for them. What's actually happening here is that for this particular polynomial the Newton-Raphson method gives rise to period-2 cyclic behaviour.

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I observed above that raising a polynomial to a real power greater than 1 has the effect of causing the iteration to take smaller steps towards the root, which in turn causes convergence to be slower but surer and reduces the incidence of overshoot leading to fractal phenomena.

So I wondered what would happen "in the limit", as the iteration speed slowed down more and more. MathWorld 's page on the Newton-Raphson method itself mentions its fractal property, and has some small examples and further references.

To raise an individual root to a power other than 1, put a slash after that root followed by the power value. It's not the only actual polynomial: I don't currently know the answers to these questions, but I'd be interested to hear them if anyone else does.

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This will slow down convergence, so you will probably also need to increase the number of shades of colour the -f option, as discussed above in order to make the result look nice. How do we find such points? So, here I present a sample polynomial Newton-Raphson fractal, with the roots of the polynomial's derivative marked as black blobs.

By default the program will use the cyclic shading behaviour with 16 shades of each colour. So I wondered, is there a way that we can predict how the convergence regions are going to be shaped, and thereby construct polynomials which have triple or quadruple meeting points exactly where we want them?

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You can specify -C no to turn off cyclic shading so the colours get uniformly darker as more iterations are needed-B yes to turn on blurring of the iteration boundaries, and -f 32 or -f 64 if you want to increase the number of shades used.

Observe that each dividing line between convergence regions has a blob somewhere on it, and that in particular the point where three regions and three such lines meet has a blob. This is interesting to me because it happens so rarely; I've plotted quite a lot of these fractals, including a lot of animated ones in which the points wander continuously around the plane, and this particular arrangement of roots is the only case I've found in which a polynomial gives rise to a non-converging area.

Alan Donovan was the person who introduced me to these fractals in the first place.