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("GroupRings", "1703.09680v1") Pkg.checkout("PropertyT", "1703.09680v1") Pkg.resolve() ```