This should resolve all dependencies (e.g. install `JuMP`, `SCS`, `IntervalArithmetic`, `JLD`, `Memento`). Exit julia and finally clone this repository:
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
```
(~30 seconds, depending on hardware). The monotonous decreasing $\lambda$ during the optimisation is in column `pri obj` (or `dua obj`) of `solver.log`.
julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000
```
which finds $\lambda \geq 0.5857$ and decomposes $\Delta^2-\lambda\Delta$ into sum of $47$ hermitian squares in less than 20 seconds (including certification).
If You see in the output (or in `full.log`) that the upper end of the interval where $\lVert\Delta^2 - \lambda\Delta - \sum{\xi_i}^*\xi_i\rVert_1$ belongs to is too large (resulting in positive `Floating point distance`, but negative `The Augmentation-projected actual distance`), decrease the `--tol` parameter, e.g.
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
git checkout 1703.09680v1
```
Unfortunately: You need to link `~/.julia/v0.5/GroupRings` to `~/.julia/v0.5/GroupAlgebras` due to change in the name of the package. Then run in `julia`