Ok I had a look at wiki and it's clear to me how the dual numbers encode differentiation and the theory behind how autodiff works. How do the two fit together? I didn't follow that part

You define a dual number type in your programming language of choice, and overload the operators +, -, *, /, <, >, sin, cos, etc. Then you take a function which computes a numerical function f which is defined using the previous operations, and simply evaluate them on a dual number. It's "automatic" because, assuming the right programming language, a user-defined numerical function will often be able to take in many different kinds of numbers, like complex numbers, real numbers, arbitrary precision floats, and even user-defined classes like the dual numbers.

Ah thankyou, it was so neat and automatic that I missed the punchline. It looks like you'll always calculate f and f' together and it's not possible to calculate only f' though? Seems like this could waste time in some applications?

I can see how this is the solution to the problem of calculating the numerical derivative of a known function composed of finite combinations of rational functions and a set of specified functions, elementary or special. I don't see how it's relevant to solving ODEs numerically though, given that in that case you don't know the functional form of f or f'

> I can see how this is the solution to the problem of calculating the numerical derivative of a known function composed of finite combinations of rational functions and a set of specified functions, elementary or special.

Yes and no. It's worth taking into account that you can apply it to some functions defined using conditions and loops. For instance, if f is the sqrt function defined using Newton's method, then autodiff via the dual numbers will compute an estimate for f'(x) = 1/(2 * sqrt(x)). More explicitly:

  def f(a):
    x = 1
    oldx = 2
    threshold = 1e-10
    while abs(x - oldx) < threshold:
      oldx, x = x, (x + a/x) / 2
    return x

^ You can evaluate f on a dual number. This algorithm contains conditions and loops.

This was an interesting comment and I learnt something new, thank you. It does feel a little bit like you are proselytising the good news of the dual number autodiff method though, when you don't respond to questions on potential downsides and only add information about positives.

I'm more familiar with using differentiation in optimisation, where autodiff seems like the best method. I'm also aware of the fact that the differentiation operator is discontinuous, which makes me sceptical of finite-differencing methods. I'm not familiar with numerically solving differential equations, but the natural connection between Runge-Kutta and finite differencing seems to be an interesting one.