Timeline for Expected distance between two uniform points in distinct rectangles
Current License: CC BY-SA 4.0
7 events
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| Feb 22, 2021 at 14:46 | comment | added | Iosif Pinelis | @fedja : You are right. :-) | |
| Feb 22, 2021 at 6:09 | comment | added | fedja | in this case it is very easy to bound the derivatives of integrand of any order Erm... Derivatives of $|x|$? I'm not sure even the first ones exist everywhere :-). Of course, life is good if the rectangles are far apart but the overlapping case would require an adaptive net to get really high precision from generic cubature and the implementation of adaptive nets is quite a headache; I wouldn't rely on it being error-free in Mathematica, which was mainly designed for symbolic manipulations... | |
| Feb 21, 2021 at 22:42 | comment | added | Iosif Pinelis | @TomSolberg : I agree with fedja that PrecisionGoal is not a precision guarantee. Mathematica's internal code is unknown to me. However, in this case it is very easy to bound the derivatives of integrand of any order, and hopefully Mathematica uses such bounds to bound the error for sure. Here I used Mathematica just for an illustration. Of course, you can always write your own routine in any language of your choice and with a certain control of the error. | |
| Feb 21, 2021 at 20:59 | comment | added | fedja | @TomSolberg PrecisionGoal in Mathematica is, indeed, intended for that purpose, but from my experience it is just what it is called: a "goal", rather than a hard guarantee. Often it works but I would certainly try to compare it to other methods on a few "bad" examples before relying upon it. As to error bounds, IMHO, algebraic precision is quite a useless measure for $|x-y|$ even on the line. But by all means try all approaches people may suggest and compare for yourself :-) | |
| Feb 21, 2021 at 19:33 | comment | added | Iosif Pinelis | @TomSolberg : I am no expert in this area and just recalled hearing in the past about cubature formulas. However, Googling "cubature formulas error bounds" (without quotation marks) returns hundreds of thousands items. The second one of them is at ams.org/mcom/1970-24-111/S0025-5718-1970-0275673-6/… -- see especially Section 4. Extension to Higher Dimensions there; I guess references to that paper can be useful. | |
| Feb 21, 2021 at 18:57 | comment | added | Tom Solberg |
Pardon my ignorance, but do any of these cubature formulas give guaranteed bounds on the error, e.g. in terms of a Lipschitz constant for the function I'm integrating (i.e. Euclidean distance)? The thing that worried me about using cubature formulas was that I didn't know how to set things up to guarantee that my result was within a desired tolerance (I gather that that's the purpose of PrecisionGoal in your code, which I didn't know existed). Thank you!
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| Feb 21, 2021 at 16:59 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |