Timeline for How to estimate the accuracy of an integral?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 1, 2012 at 10:29 | vote | accept | MathematicalOrchid | ||
| Jun 27, 2012 at 20:08 | comment | added | Michael R. Chernick | I don't see how knowing the height of f at a single point x tells you anthing about the area. It does tell you that M must be > f(x). If you have f evaluated at several points you can approximate the integral by rectangles but who knows how good that approximation will be? | |
| Jun 27, 2012 at 19:59 | comment | added | MathematicalOrchid | Just wondering... When you evaluate $f(x)$, you then know exactly how much of that vertical slice is under the curve. Can you not use that knowledge somehow to improve the estimate a bit? | |
| Jun 26, 2012 at 20:43 | comment | added | MathematicalOrchid | What an ingenious idea... You're right, it doesn't work without bounds on $f$, but it looks like you can't do much of anything without that information. | |
| Jun 26, 2012 at 15:23 | comment | added | Michael R. Chernick | It is an assumption. I used the term mimimal to say that i am making as few assumptions as possible to reach a definitive answer. | |
| Jun 26, 2012 at 15:12 | comment | added | Macro | I certainly agree with your first sentence, which begins to address the OP's second question. But, the method you've described requires that you, a priori, know $M$, which is not a particularly minimal assumption. | |
| Jun 26, 2012 at 15:05 | comment | added | Michael R. Chernick | @Macro Without knowing anything about f I do not see how one could say anything about the statistical accuracy of an estimate of the integral based on evaluating it at a fixed finite set of points. My assumptions are rather minimal. If f is bounded on the interval [a,b] there should be some M large enough that it could be used as an upper bound on f. | |
| Jun 26, 2012 at 14:58 | comment | added | Macro | This would work under the assumptions you laid out in the first sentence but based on the problem description it seems unlikely that you can, a priori, bound the function values between $0$ and $M$. It appeared that all you're given is the ability to calculate $f$ and nothing else. | |
| Jun 26, 2012 at 14:51 | history | answered | Michael R. Chernick | CC BY-SA 3.0 |