Timeline for Rigorous numerical integration
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Feb 12, 2020 at 14:38 | comment | added | Boris Bukh | Note from the future: a follow-up question was asked three years later at mathoverflow.net/questions/248486/…, with some useful answers and discussion. | |
| Mar 17, 2013 at 14:22 | vote | accept | CommunityBot | ||
| Mar 17, 2013 at 14:22 | history | bounty ended | H A Helfgott | ||
| Mar 17, 2013 at 9:58 | comment | added | vonjd | @H A Helfgott: Thank you. It would be helpful if you accepted one of our answers :-) | |
| Mar 13, 2013 at 17:58 | comment | added | H A Helfgott | Taking derivatives is easy. I can do (a) and (b) with my own code, and a friend just helped set up VNODE-LP to do (a) and (b). It is (c) that looks nasty right now; if you replace $\gamma$ by its definition, you get a double integral. By the way, that should really be $\int_{-\infty}^\infty |\gamma(it+1,-1)+\gamma(it+2,-1)| dt. | |
| Mar 13, 2013 at 16:38 | comment | added | Steve Huntsman | I'll add the trivial note that a change of variables for improper integrals will be helpful from the POV of implementation in silico. | |
| Mar 13, 2013 at 16:35 | comment | added | Steve Huntsman | Won't the following work for all of these? Use automatic differentiation (en.wikipedia.org/wiki/Automatic_differentiation, covered at a basic level in Tucker's book) to get expressions for any derivatives appearing in integrands, and then apply integration with IA (or Taylor forms) to the results. | |
| Mar 11, 2013 at 15:50 | comment | added | H A Helfgott | Vonjd, here are some fairly representative examples. (And yes, Henry, I've been coding things myself.) (a) $\int_{0+}^{1-} |h'''(x)| dx$, where $h(x) = x^2 (1-x)^2 e^x$. (easiest) (b) $\widehat{f}(t)$ at all points in $t\in (-655,655)\cap 0.001\mathbb{Z}$, where $f(x) = 4 x^{-2}$ if $x\in \lbrack 1/2,1\rbrack$, $f(x) = -4 x^{-2}$ if $x\in \lbrack 1/4,1/2\rbrack$ and $f(x)=0$ if $x<1/4$ or $x\geq 1$. (Already did this, though without interval arithmetic.) (c) $\int_{-\infty}^\infty |\gamma(it+1,-1) + \gamma(it+2,-2)| dt$, where $\gamma(s,x)=\int_0^x e^{-t} t^{s-1} dt$. | |
| Mar 10, 2013 at 16:37 | answer | added | vonjd | timeline score: 6 | |
| Mar 10, 2013 at 16:30 | comment | added | Henry Cohn | Gilead's suggestions look promising (more generally, searching for a validated ODE solver will lead to more results than rigorous numerical integration). However, if your examples aren't too complicated, it will probably be easier to code it yourself than to get a general-purpose system working. | |
| Mar 10, 2013 at 16:06 | comment | added | vonjd | Could you give us some of the integrals just to see what the problems might be? | |
| Mar 10, 2013 at 14:26 | comment | added | Gilead | Does VNODE do what you want? cas.mcmaster.ca/~nedialk/Software/VNODE/VNODE.shtml Or VSPODE? www3.nd.edu/~markst/lin-stadtherr-vspode-apnum.pdf | |
| Mar 10, 2013 at 14:21 | history | bounty started | H A Helfgott | ||
| Mar 10, 2013 at 14:20 | comment | added | H A Helfgott | Just to make myself clear: I would much appreciate a reference to open-source software that does this. | |
| Mar 5, 2013 at 23:25 | answer | added | Steve Huntsman | timeline score: 9 | |
| Mar 5, 2013 at 23:20 | comment | added | H A Helfgott | The ideal thing would be a program that could also come up with the derivative bounds on its own, given the symbolic expression for the function. (This should be possible in lots of cases where symbolic integration just isn't possible at all.) | |
| Mar 5, 2013 at 23:03 | history | asked | H A Helfgott | CC BY-SA 3.0 |