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Exact numeric nth derivatives

jliszka.github.io

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Re: Exact numeric nth derivatives

#51

Post by Pitarou »

How does this technique compare to a computer implementation of the kinds of techniques we learnt in High School? Is it easier to implement? More efficient? Are there some situations where it isn't appropriate?

Re: Exact numeric nth derivatives

#52

Post by timmclean »

jordigh wrote: post #18

Am I missing something or is this begging the question? For any function that is not a combination of polynomials, you need to have its Taylor expansion up to the desired order of derivatives, so you can't just take an "arbitrary" function and use this method to compute its derivative in exact arithmetic. So for anything other than polynomials, you just reword the problem of finding exact derivatives to finding exact…

I am not a mathematician, but I don't think this is actually begging the question.

> you can't just take an arbitrary function and use this method

Actually, you can, AFAIK. The relation f(x + E) = f(x) + f'(x)E still holds.

If we try this method with a rational function:

    f(x) = (x - 1)/(x - 2)

    f(x + E)
     = (x + E - 1) / (x + E - 2)
     = (x + E - 1)(x - E - 2) / ((x + E - 2)(x - E - 2))
     = (x^2 - x E - 2 x + x E - E^2 - 2 E - x + E + 2) / (x^2 - x E - 2 x + x E - E^2 - 2 E - 2 x + 2 E + 4)
     = (x^2 - 3 x + 2 - E) / (x^2 - 4 x + 4)
     = (x^2 - 3 x + 2) / (x^2 - 4 x + 4) - (1 / (x^2 - 4 x + 4)) E
     = (x - 1) / (x - 2) - (1 / (x - 2)^2) E
so

    f'(x) = -1 / (x - 2)^2
since

    f(x + E) = f(x) + f'(x) E
Edit to add: the key idea here being that no knowledge of differentiation is needed, just tricks for manipulating expressions involving x and E until they are in normal form.

Re: Exact numeric nth derivatives

#53

Post by short_circut »

backprojection wrote: post #22

I think it's worth noting that the problem with numerical differentiation, fundamentally, is that differentiation is an unbounded operator. In finite-differences, (the more obvious approach), you assume that your data are samples of some, general, function. The problem then, is that that general functions have no (essential) bandlimit [1]. Remember that differentiation acts as a multiplication by a monomial, in the f…

Low pass filters are absolute miracles when working with numerical derivatives. I have had instances where that was the only reason I ever got anything meaningful out of them. It really comes down to the fact that the numerical representation of a function is in general not truly analytic. It is has kinks and bends all over the place.

Re: Exact numeric nth derivatives

#54

Post by fusiongyro »

Very neat article. This is essentially calculus with infinitesimals (also called "nonstandard analysis") implemented on the machine. If you like the approach, a more general and rigorous investigation can be had by reading H. Jerome Keisler's book Elementary Calculus, which is freely available online in the 2nd edition here:

  http://www.math.wisc.edu/~keisler/calc.html
The third edition is now in print. I've been studying calculus with it off-and-on for a while and I find the approach very intuitive, though Spivak's Calculus is probably a better book, the "standard analysis" is a little less intuitive (and now, evidently, harder to teach a machine).
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