alt Hacker NewsI think you can get a generalisation of autodiff using this idea of "nonstandard real numbers": You just need a computable field with infinitesimals in it. The Levi-Civita field looks especially convenient because it's real-closed. You might be able to get an auto-limit algorithm from it by evaluating a program infinitely close to a limit. I'm not sure if there's a problem with numerical stability when something like division by infinitesimals gets done. Does this have something to do with how Mathematica and other CASes take limits of algebraic expressions?
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Concerning the Dirac delta example: I think this is probably a pleasant way of using a sequence of better and better approximations to the Dirac delta. Terry Tao has some nice blog posts where he shows that a lot of NSA can be translated into sequences, either in a high-powered way using ultrafilters, or in an elementary way using passage to convergent subsequences where necessary.
An interesting question is: What does distribution theory really accomplish? Why is it useful? I have an idea myself but I think it's an interesting question.
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Distribution theory has lots of applications in physics. The charge density of a point particle is the delta function.
Also when Fourier transforming over the whole real line (not just an interval where the function is periodic), one has identities that involve delta functions. E.g. \int dx e^(i * k1 * x) e^(-i * k2 * x) = 2 * pi * delta (k1 - k2).
Thanks a bunch for pointing me towards Levi-Civita field. Where can I learn more ? Any pedagogic text ?
> I think this is probably a pleasant way of using a sequence of better and better approximations to the Dirac delta.
That can give wrong answers because derivative of the limit is not always the limit of the derivative.
When modeling phenomena with Dirac delta, I think the question becomes do I really need a discontinuity to have a useful model or can I get away with smoothening the discontinuity out.