Timeline for answer to Most harmful heuristic? by Spencer
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| when toggle format | what | by | license | comment | |
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| Dec 10, 2020 at 12:29 | comment | added | Jacques Carette | @BillyRubina There are no elementary examples, else this would be well-known. But recall that Weierstrass's example of a continuous but differentiable nowhere function is computable. So it would be a torture test to any purported differentiation algorithm. As far as I know, there is no literature that makes this point, because there's no one who studies both kinds of analysis simultaneously. | |
| Dec 10, 2020 at 3:03 | comment | added | Red Banana | @JacquesCarette Could you provide some elementary example on what you said or elementary literature on it? I got a bit curious about it. | |
| Oct 20, 2016 at 18:43 | comment | added | LSpice | For anyone who's been waiting 6 and a half years, @DanPiponi's link got mangled, but (I think) was meant to point to mathoverflow.net/questions/11540/… . | |
| Jun 26, 2016 at 18:44 | comment | added | Thomas Antony | @J.M. Unless you do complex-step differentiation which gives you near machine-tolerance accuracy by avoiding the round-off error completely. Of course, then you need a function that can work with complex numbers. | |
| Jun 9, 2016 at 4:12 | comment | added | J. M. isn't a mathematician | @user8823741, in general, numerical differentiation is an unstable process. Think about how the derivative is defined; you are in effect subtracting two nearly equal quantities to get a tiny result, and then dividing that tiny result by another tiny value to get a result that is almost often far from tiny. That's a lot of opportunities for a computer to slip up. | |
| Aug 21, 2013 at 1:26 | comment | added | Lenar Hoyt | I’m reading this as a second year undergraduate student and I didn’t go through this kind of reversal yet. I’d be glad if someone would give a short explanation in layman terms why it’s the other way around for computable functions! | |
| Apr 18, 2013 at 10:02 | comment | added | Andrew T. Barker | I had a similar reversal in a different area. In numerical analysis, numerical integration is (relatively) well understood, stable, and generally nice, while numerical differentiation is kind of a mess. But until graduate school I would have said the opposite. | |
| Apr 26, 2010 at 9:40 | comment | added | Neel Krishnaswami | @Jacques: that's really well-phrased! I had an "a-ha" moment reading your comment. | |
| Apr 26, 2010 at 0:51 | comment | added | James Weigandt | I'm upvoting this partially because I agree, but mostly because you used the term "pet hate" as opposed to "pet peeve". | |
| Mar 13, 2010 at 3:50 | comment | added | Jacques Carette | That's because on formulas differentiation is nice and integration is hard, but on computable functions differentiation is hard and integration is nice. In theory, we have a denotational semantics between formulas that functions that should transport these notions back-and-forth, but we really really don't. There are tons and tons of papers in computer algebra which basically boil down to this massive gulf between abstract analysis (the study of functions given by properties) and concrete analysis (study of functions given by formulas). | |
| Mar 1, 2010 at 5:03 | comment | added | Dan Piponi | I went through the same reversal as you recently. Slightly different but related reason. My explanation is <a href="mathoverflow.net/questions/11540/…>. | |
| Feb 28, 2010 at 17:10 | history | answered | Spencer | CC BY-SA 2.5 |