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by wwalker3
2455 days ago
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The authors have shown a very nice and (to me) non-intuitive result. But they're playing a little fast and loose with their comparison to Mathematica. They're comparing their algorithm's accuracy (solution correctness vs. incorrectness), with Mathematica's ability to find the correct solution in less than 30 seconds. This is a very important distinction! Mathematica will never silently return an incorrect solution (barring software bugs, of course). And Mathematica can often take minutes to evaluate what appears to be a simple integral, so a 30-second timeout is far too short, unless you're simply trying to compare the computational efficiency of the two approaches. There may be other subtleties as well. Mathematica works in the complex domain by default, which makes many operations more difficult, but the authors discard expressions which contain complex-valued coefficients as "invalid", which makes me think they're implicitly working in the real domain. Do they restrict Mathematica to the real domain when they invoke it? Perhaps, but they don't say one way or the other. And do they try common tricks like invoking FullSimplify[] on an expression/equation before attempting to operate on it? I'd like to see more details of their methodology. |
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I had the same initial reaction as you, but then I realized that this is still extremely useful. In a ton of examples, only one direction of differentiation/integration is hard while the other direction is easy. You could build a system that attempted to solve it directly, and failing that attempted to guess-and-verify using this approach. My intuition is that such an overall system would be strictly superior to Mathematica's approach as it exists today.