Well for most of us winter break is coming to an end, or for the unlucky, might have already ended, which means its time to start thinking about teaching again. One of the topics commonly covered in a first or second semester of calculus is the use of Newton’s method to approximate roots of functions. Recall the basic approach of Newton’s method is that given a real valued differentiable function and an initial condition we let

$$x_n=x_{n-1}-\frac{f(x_{n-1})}{f’(x_{n-1})},$$

and then, under certain assumptions, should converge to a root of .

When I’ve taught Newton’s method, I tried to stress that this method is not guaranteed to always work and can be fairly sensitive to the initial condition . Going into the precise details of when the method will in fact work is well beyond my first semester calculus students. That said, I am not sure they’ve ever really appreciated these points. In fact, to be completely honest, I am not sure I completely appreciated some of the aspects of these things before I started teaching.

That said, one way to visualize some of these complexities is via a cool program called FractalStream. For example, using FractalStream, we can run the following script:

iterate z – (z^3 – 1)/(3*z^2) until z stops.

and then turn on autocoloring – under the color settings – and we get the following interesting picture of the complex plane colored red, blue, and green:

Newton’s method plot for z^3-1.

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