Add a description of the symmetrised version

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+A newer version of the software uses orbit and Wedderburn decomposition to effecitively find a (much) smaller optimisation problem to compute the spectral gap $\lambda$.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