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Add a description of the symmetrised version

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kalmarek 2017-12-18 11:09:13 +01:00
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README.md
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@ -23,6 +23,8 @@ cd GroupswithPropertyT
# Running # 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 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 ```shell
julia SL.jl -N 2 -p 7 --radius 2 --iterations 100000 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 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A newer version of the software uses orbit and Wedderburn decomposition to effecitively find a (much) smaller optimisation problem to compute the spectral gap $\lambda$.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 # Help
@ -70,7 +89,7 @@ optional arguments:
-h, --help show this help message and exit -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`) 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 ```shell