86 lines
3.7 KiB
Markdown
86 lines
3.7 KiB
Markdown
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This repository contains code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680).
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# Installing
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To run the code You need `julia-v0.5` (should work on `v0.6`, but with warnings).
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You also need to install julia packages: `Nemo-v0.6.3`, `ArgParse`. To do so in `julia`'s REPL run:
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```julia
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Pkg.update()
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Pkg.add("Nemo")
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Pkg.add("ArgParse")
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```
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Then clone the main repository of `Groups.jl`, `GroupRings.jl` and `PropertyT.jl`:
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```julia
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Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git")
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Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git")
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Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git")
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Pkg.resolve()
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```
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This should resolve all dependencies (e.g. install `JuMP`, `SCS`, `IntervalArithmetic`, `JLD`, `Memento`). Exit julia and finally clone this repository:
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```shell
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git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git
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cd GroupswithPropertyT
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```
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# Running
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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
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```shell
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julia SL.jl -N 2 -p 7 --radius 2 --iterations 100000
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```
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(~30 seconds, depending on hardware). The monotonous decreasing $\lambda$ during the optimisation is in column `pri obj` (or `dua obj`) of `solver.log`.
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Compare this to
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```shell
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julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000
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```
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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).
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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.
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```
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julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000 --tol 1e-9
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```
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to achieve a better estimate (the residuals $\ell_1$-norm should be around $\|B_d(e))\|*tol$)
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# Help
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```shell
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julia SL.jl --help
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usage: SL.jl [--tol TOL] [--iterations ITERATIONS]
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[--upper-bound UPPER-BOUND] [--cpus CPUS] [-N N] [-p P]
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[--radius RADIUS] [-h]
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optional arguments:
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--tol TOL set numerical tolerance for the SDP solver
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(type: Float64, default: 1.0e-6)
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--iterations ITERATIONS
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set maximal number of iterations for the SDP
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solver (default: 20000) (type: Int64, default:
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50000)
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--upper-bound UPPER-BOUND
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Set an upper bound for the spectral gap (type:
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Float64, default: Inf)
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--cpus CPUS Set number of cpus used by solver (type:
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Int64)
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-N N Consider elementary matrices EL(N) (type:
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Int64, default: 2)
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-p P Matrices over field of p-elements (p=0 => over
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ZZ) (type: Int64, default: 0)
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--radius RADIUS Radius of ball B_r(e,S) to find solution over
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(type: Int64, default: 2)
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-h, --help show this help message and exit
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```
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# Specific version of the article
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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`)
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```shell
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git checkout 1703.09680v1
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```
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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`
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```julia
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Pkg.checkout("GroupRings", "1703.09680v1")
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Pkg.checkout("PropertyT", "1703.09680v1")
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Pkg.resolve()
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```
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