6 ms·

I see your edit - fair point, analyticity does make complex functions much more well-behaved than their real counterparts. Still, this isn't a death knell like

• ogogmad 4y ago
  I don't fully understand your bet (and I don't take kindly to being asked to silence myself). I don't have time at the moment to craft an example. I might play around a bit with the edges of floating point. Differentiation is indeed discontinuous, so with enough determination I should be able to win such a bet. My function would probably be a sigmoid function which would go from -a to +a (for small a) over a very small interval [-b,b], where b can equal a. > If I win, you agree to silence yourself for 1 year, regarding the split-complex and dual numbers, no longer touting them as a panacea. I never said these numbers were a panacea. I said the dual numbers were an exact method. In this comment [1], which is what you're referring to, I said that the connection with other planar algebras was cute, but you mischaracterised what I said, which was rude. The algebras are pertinent because the article mentioned the dual numbers. [1] - https://news.ycombinator.com/item?id=32305644 https://news.ycombinator.com/item?id=32305644
  • goldenkey 4y ago
    Fair enough. The bet I had in mind was the function you'd already stated as intractable: f(x) = a sin(x / a), the value of a is up to you, within reason (no IEEE fp shenanigans.) If you'd rather some other consequence than silencing, please suggest one. I am tired of hearing your panaceas. Symbolic differentiation is also exact but intractible. What makes the duals yield exactness in any way that isn't prone to the same finitude as other approximations? I see a lot of beasts but only one individual claiming to tame them with ease and exactness. If it comes across as rude, that's not my intent. Moreso to have some clarity and to have your claims be validated.
    • ogogmad 4y ago
      I'm not going to accept your bet because I don't understand its conditions (and what's the point?), but I accept your challenge. Also, I'm going to f*** around bigly with floating point. def f(x,a): return 1 + a * sin(x / a) Now let epsilon = 2.2204460492503133e-36. Consider f(x, epsilon). The function should always return 1 in floating point, so any finite differencing method will estimate its derivative as 0. But over the dual numbers, the function should have f(e, epsilon) = e, where e is the dual number imaginary. In other words, the dual numbers return the correct value of the derivative at x=0, which is 1. You should be able to use any Python implementation of the dual numbers; my own one uses Sympy. Actually, you might be able to use Scipy and represent dual numbers as matrices. Look up automatic differentiation because you're not aware of it, and it is the subject of the article. You seem to misunderstand symbol methods. The problem of finding an exact derivative at a single point is not as hard as finding the symbolic derivative everywhere. You're referring to the product rule and chain rule, but these are not needed to find derivatives at single points.
      • goldenkey 4y ago
        I am aware of AD. All AD does is use the chain rule on the Taylor (or otherwise) approximations of all arithmetic functions used in the calculation of the target function. It's symbolic but only at the single arithmetic operation level. The chain rule is concealed under rules on epsilon. The reason epsilon's square equals 0 is to eliminate any higher order derivatives as soon as they appear. If the derivative is abnormal in a way such that the normal truncated calculative formulas would conceal it, then AD would fail to give an accurate result as well.
        • ogogmad 4y ago
          You've just described the dual numbers, which provide a way of implementing forward-mode autodiff. > If the derivative is abnormal in a way such that the normal > truncated calculative formulas would conceal it, then AD > would fail to give an accurate result as well. What I'm seeing directly contradicts that. I've tested the example above (the one I called f(x,a) with a=1e-36, trying to find its derivative at x=0), and it gave the right answer with dual numbers, but not with finite differencing. So what you're saying isn't true. I'll post a minimal implementation here later. [edit] I've used Sympy out of laziness, which is inelegant. I don't know how clear this will be. First of all, I have to change the number 1 in the definition of f(x,a) into a matrix. We need this because we'll be representing dual numbers as matrices, and you can't add the scalar 1 to a matrix: one = eye(2) def f(x, a=1e-36): return one + a * sin(x / a) Sympy doesn't have its own implementation of `sin`, so we'll need to provide one: def sin(M): return im(exp(I*M).n()) At the dual number ε, we have f(ε) = f(0) + ε f'(0). The dual number ε will be represented as the matrix ⎡0 1⎤ ⎢ ⎥ ⎣0 0⎦ As code: epsilon = Matrix([[0,1],[0,0]]) Evaluating f on the above matrix gives: In: f(epsilon) Out: ⎡1 1.0⎤ ⎢ ⎥ ⎣0 1 ⎦ Which tells us that f(0) ≈ 1 and f'(0) ≈ 1. Both are correct. If we use finite differencing instead, we get: In: (f(h * eye(2)) - f(0 * eye(2)))/h Out: ⎡0 0⎤ ⎢ ⎥ ⎣0 0⎦ So it's claiming incorrectly that the derivative at 0 is 0. No finite differencing scheme can fix this because floating point causes the function f to become constant. Autodiff using the dual numbers has pulled off the seemingly impossible. [edit: An early typo caused the code to produce an incorrect result. Now fixed, and my claim holds.]