[HN Gopher] Derivative at a Discontinuity
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       Derivative at a Discontinuity
        
       Author : yuppiemephisto
       Score  : 11 points
       Date   : 2024-12-03 22:36 UTC (1 days ago)
        
 (HTM) web link (alok.github.io)
 (TXT) w3m dump (alok.github.io)
        
       | ogogmad wrote:
       | I 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?
       | 
       | -----
       | 
       | 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.
        
       | dhosek wrote:
       | One minor nit: A function can be differentiable at _a_ and
       | discontinuous at _a_ even with the standard definition of the
       | derivative. A trivial example would be the function _f_ ( _x_ ) =
       | ( _x_ 2-1)/( _x_ -1) which is undefined at _x_ =1, but _f_ '(1)=1
       | (in fact derivatives have exactly this sort of discontinuity in
       | them which is why they're defined via limits). In complex
       | analysis, this sort of "hole" in the function is called a
       | removable singularity1 which is one of three types of
       | singularities that show up in complex functions.
       | 
       | [?]
       | 
       | 1. Yes, this is mathematically the reason why black holes are
       | referred to as singularities.
        
         | dawnofdusk wrote:
         | I don't think it makes sense to allow derivatives of a function
         | f to have a larger domain than the domain of f.
         | 
         | >which is why they're defined via limits
         | 
         | They're defined via studying f(x+h) - f(x) with a limit h -> 0.
         | But, your example is taking two limits, h->0 and x->1,
         | simultaneously. This is not the same thing.
        
       | Animats wrote:
       | Hm. Back when I was working on game physics engines this might
       | have been useful.
       | 
       | In impulse/constraint mechanics, when two objects collide, their
       | momentum changes in zero time. An impulse is an infinite force
       | applied over zero time with finite energy transfer. You have to
       | integrate over that to get the new velocity. This is done as a
       | special case. It is messy for multi-body collisions, and is hard
       | to make work with a friction model. This is why large objects in
       | video games bounce like small ones, changing direction in zero
       | time.
       | 
       | I wonder if nonstandard analysis might help.
        
         | ogogmad wrote:
         | The following is just my opinion:
         | 
         | Integration can be done with its own special arithmetic:
         | Interval arithmetic. I base this suggestion on the fact that
         | this is apparently the only way of automatically getting error
         | bounds on integrals. It's cool that it works.
         | 
         | NSA does not work with a computable field so it's not directly
         | useful. But at the end of the article, there's a link to some
         | code that uses the Levi-Civita field, which is a "nice"
         | approximation to NSA because it's computable and still real-
         | closed. You might be able to do an "auto-limit" using it, in a
         | kind of generalisation of automatic differentiation. This might
         | for instance turn one numerical algorithm, like Householder QR,
         | into another one, like Gaussian elimination, by taking an
         | appropriate limit.
         | 
         | I don't know if these two things interact well in practice:
         | Levi-Civita for algebraic limits and interval arithmetic for
         | integrals. They might! This might suggest rather provocatively
         | that integration is only clumsily interpreted as a limit of
         | some function. Finally tbh, I'm not sure if this is the best
         | solution to the friction/collision detection problem you're
         | describing.
        
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