Timeline for How to improve precision of NIntegrate while integrating a simple 2D Gaussian distribution?
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Nov 2, 2021 at 18:55 | comment | added | Michael E2 | @user64494 This is too short to completely document the workings of NIntegrate, but if you read it carefully, you will discover how NIntegrate handles infinite intervals. You may have to experiment to fill in missing details, but such is the nature of the documentation that we have. The option IntegrationMonitor, described on this site, is a useful tool for investigating the operation of NIntegrate. Good luck with your researches! | |
| Nov 1, 2021 at 18:13 | comment | added | user64494 | Thank you for your reply. You wrote " For the (centered) unit cube and infinite intervals, read the rest of the NIntegrate documentation". I repeat I don't find any example of the usage of "MultidimensionalRule" to improper integrals in the documentation to NIntegrate. Can you give a concrete reference? TIA. | |
| Nov 1, 2021 at 12:58 | comment | added | Michael E2 | @user64494 To follow the advice to "find its numeric support" when $t_{01}-t_{02}$ is too large for the standard recursive subdivision, usually one subdivides further at $\mu\pm a\sigma$, with $a$ around $5$ or $6$ for a precision goal of around 6. As you are a user who has complained at others' interest in ridiculously large or small numbers, I wonder why 160 is at all useful. f\[Tau]gaussian[1, 0, 160] is on the order of $10^{-5660}$. If you just want to stymie Mma, use 10^8. --- For the (centered) unit cube and infinite intervals, read the rest of the NIntegrate documentation. | |
| Nov 1, 2021 at 7:05 | comment | added | user64494 | Every method has its limitations: ListPlot[Thread[{Table[t02, {t02, 0, 160, 1}], Table[NIntegrate[(f\[Tau]gaussian[1, t1, 0]* f\[Tau]gaussian[1, t2, t02] + f\[Tau]gaussian[1, t2, 0]* f\[Tau]gaussian[1, t1, t02])^2, {t1, -Infinity, 0, t02, Infinity}, {t2, -Infinity, 0, t02, Infinity}, Method -> {"MultidimensionalRule", "Generators" -> 9}], {t02, 0, 160, 1}]}], PlotRange -> Full, Joined -> True, Frame -> True, FrameLabel -> {"t02-t01", "Overlap"}] ( 160 instead of 20) produces a bad plot | |
| Nov 1, 2021 at 5:58 | comment | added | user64494 | The documentation to "MultidimensionalRule" says only about the hypercube $ \left[-\frac{1}{2},\frac{1}{2}\right]^d \text{, } d\in \mathbb{N},d>1$. Is your usage to an improper integral documented? | |
| Nov 1, 2021 at 2:34 | vote | accept | Knife Lee | ||
| Nov 1, 2021 at 2:24 | history | answered | Michael E2 | CC BY-SA 4.0 |