From 515d8e0e1473a9bcc00ba2139788d28d9d02865b Mon Sep 17 00:00:00 2001 From: kalmarek Date: Mon, 18 Dec 2017 11:09:13 +0100 Subject: [PATCH] Add a description of the symmetrised version --- README.md | 23 +++++++++++++++++++++-- 1 file changed, 21 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 906401f..85c4c83 100644 --- a/README.md +++ b/README.md @@ -23,6 +23,8 @@ cd GroupswithPropertyT # Running +## Naive implementation + To check that $\Delta^2-\lambda\Delta$ is not decomposable to a sum of hermitian squares of elements in the ball of radius $2$ in $SL(2,7)$ run ```shell julia SL.jl -N 2 -p 7 --radius 2 --iterations 100000 @@ -39,7 +41,24 @@ If You see in the output (or in `full.log`) that the upper end of the interval w ``` julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000 --tol 1e-9 ``` -to achieve a better estimate (the residuals $\ell_1$-norm should be around $\|B_d(e))\|*tol$) +to achieve a better estimate (the residuals $\ell_1$-norm should be around $\|B_d(e))\|\cdot tol$) + +## Symmetrization enhanced implementation + +A newer version of the software uses orbit and Wedderburn decomposition to effecitively find a (much) smaller optimisation problem to compute the spectral gap $\lambda$. In particular the solution to the original (naive) optimisation problem can be reconstructed from the solution of the symmetrised one. + +E.g. Run +```shell +julia SL_orbit.jl -N 4 --radius 2 --upper-bound 1.3 +``` +to find (and certify) the spectral gap for $SL(4, \mathbb{Z})$ is at least `1.2999...` in just under $2$ minutes time (for comparison this result requires over `5` hours in the old implementation on the same hardware). + +To replicate the results of _$\operatorname{Aut}(\textbf{F}_5)$ has property (T)_ article (currently not published) You neet to run (on a `4`-core CPU) +```shell +julia ../AutFN_orbit.jl -N 5 --upper-bound 1.2 --iterations 24000000 --cpus 4 +``` + +Note that this computation took more than `12` days and required at least `32`GB of ram (and possible more). # Help @@ -70,7 +89,7 @@ optional arguments: -h, --help show this help message and exit ``` -# Specific version of the article +# Specific version of [1703.09680](https://arxiv.org/abs/1703.09680) To checkout the specific versions of packages used for [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680) run (inside the cloned `GroupswithPropertyT`) ```shell