Timeline for How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 31, 2016 at 17:48 | comment | added | Igor Rivin | @Lembik I think that might be partly due to a desire on their part to get provable bounds... | |
| Mar 31, 2016 at 16:17 | comment | added | Simd | This is a shame as you can naively approximate the sum from below by only looking at vectors $x$ with small coefficients (say -$x_i \in \{-2,-1,0,1,2\}$). This seems to run in roughly the same amount of time and be accurate to within a few decimal places. In other words, it appears the naive approach is almost as good. | |
| Mar 31, 2016 at 16:12 | comment | added | Igor Rivin | @Lembik If you read the paper, you will see that the radius GROWS with $g$ (which is basically the dimension), and since the number of evaluations is something like $R^g,$ the running time will be superexponential. Presumably this cannot really be helped. | |
| Mar 31, 2016 at 14:58 | comment | added | Simd | It was essentially identical to the code at github.com/abelfunctions/abelfunctions/issues/121 . | |
| Mar 31, 2016 at 9:15 | comment | added | Igor Rivin | @Lembik what was the exact code you ran? | |
| Mar 31, 2016 at 6:19 | comment | added | Simd | For anyone interested, I tried the code as implemented at github.com/abelfunctions/abelfunctions/wiki/Getting%20Started and it is very slow. About 10 seconds for a 9 by 9 matrix and too slow to compute for a 12 by 12 matrix. I don't know if the version implemented by Mathematica is much faster as I don't have access to it. | |
| Mar 21, 2016 at 9:48 | vote | accept | Simd | ||
| Mar 20, 2016 at 22:53 | comment | added | Branimir Ćaćić | In your case, $S_M = \Theta\left(0 \vert \tfrac{1}{\pi} i M\right)$, where $\tfrac{1}{\pi} i M$ is indeed complex symmetric with strictly positive definite imaginary part. | |
| Mar 20, 2016 at 21:16 | comment | added | Simd |
Thank you for this. Please excuse my ignorance but would you be able explicitly to show an example? The Riemann theta function seems to require that $Im(\Omega)$ is strictly positive definite but my matrix $M$ is real. mathoverflow.net/questions/64261/… shows that there is a Maple function RiemannTheta. How would you use it in my case? Or equivalently, how would you use the Mathematica function SiegelTheta?
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| Mar 20, 2016 at 19:48 | history | answered | Igor Rivin | CC BY-SA 3.0 |