Add README

README.md

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+This repository contains code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680).
+
+# Installing
+To run the code You need `julia-v0.5` (should work on `v0.6`, but with warnings).
+You also need to install julia packages: `Nemo-v0.6.3`, `ArgParse`. To do so in `julia`'s REPL run:
+```julia
+Pkg.update()
+Pkg.add("Nemo")
+Pkg.add("ArgParse")
+```
+Then clone the main repository of `Groups.jl`, `GroupRings.jl` and `PropertyT.jl`:
+```julia
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git")
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git")
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git")
+Pkg.resolve()
+```
+This should resolve all dependencies (e.g. install `JuMP`, `SCS`, `IntervalArithmetic`, `JLD`, `Memento`). Exit julia and finally clone this repository:
+```shell
+git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git
+cd GroupswithPropertyT
+```
+
+# Running
+
+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`.
+
+Compare this to
+```shell
+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.
+```
+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 $\|B_d(e))\|*tol$)
+
+# Help
+
+```shell
+julia SL.jl --help
+usage: SL.jl [--tol TOL] [--iterations ITERATIONS]
+             [--upper-bound UPPER-BOUND] [--cpus CPUS] [-N N] [-p P]
+             [--radius RADIUS] [-h]
+
+optional arguments:
+  --tol TOL             set numerical tolerance for the SDP solver
+                        (type: Float64, default: 1.0e-6)
+  --iterations ITERATIONS
+                        set maximal number of iterations for the SDP
+                        solver (default: 20000) (type: Int64, default:
+                        50000)
+  --upper-bound UPPER-BOUND
+                        Set an upper bound for the spectral gap (type:
+                        Float64, default: Inf)
+  --cpus CPUS           Set number of cpus used by solver (type:
+                        Int64)
+  -N N                  Consider elementary matrices EL(N) (type:
+                        Int64, default: 2)
+  -p P                  Matrices over field of p-elements (p=0 => over
+                        ZZ) (type: Int64, default: 0)
+  --radius RADIUS       Radius of ball B_r(e,S) to find solution over
+                        (type: Int64, default: 2)
+  -h, --help            show this help message and exit
+```
+
+# Specific version of the article
+
+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`
+```julia
+Pkg.checkout("Groups", "1703.09680v1")
+Pkg.checkout("GroupRings", "1703.09680v1")
+Pkg.checkout("PropertyT", "1703.09680v1")
+Pkg.resolve()
+```