uncomment certification

Manifest.toml

@@ -2,19 +2,13 @@
 
 [[AbstractAlgebra]]
 deps = ["InteractiveUtils", "LinearAlgebra", "Markdown", "Random", "SparseArrays", "Test"]
-git-tree-sha1 = "a45d75b3cad41fafe2642d0f64e17866873869c8"
+git-tree-sha1 = "56bdd9e9bb2cfdf0876d0846b337c8f395d9a4b7"
 uuid = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
-version = "0.4.4"
+version = "0.4.6"
 
 [[Base64]]
 uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"
 
-[[BenchmarkTools]]
-deps = ["JSON", "Printf", "Statistics", "Test"]
-git-tree-sha1 = "5d1dd8577643ba9014574cd40d9c028cd5e4b85a"
-uuid = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
-version = "0.4.2"
-
 [[BinDeps]]
 deps = ["Compat", "Libdl", "SHA", "URIParser"]
 git-tree-sha1 = "12093ca6cdd0ee547c39b1870e0c9c3f154d9ca9"
@@ -74,9 +68,9 @@ version = "0.2.0"
 
 [[Compat]]
 deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"]
-git-tree-sha1 = "195a3ffcb8b0762684b6821de18f83a16455c6ea"
+git-tree-sha1 = "84aa74986c5b9b898b0d1acaf3258741ee64754f"
 uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
-version = "2.0.0"
+version = "2.1.0"
 
 [[DataStructures]]
 deps = ["InteractiveUtils", "OrderedCollections", "Random", "Serialization", "Test"]
@@ -122,9 +116,9 @@ version = "0.2.0"
 
 [[FileIO]]
 deps = ["Pkg", "Random", "Test"]
-git-tree-sha1 = "c94b0787956629036fb2b20fccde9e52b89d079a"
+git-tree-sha1 = "da32159d4a2e526338506685e280e39ed2f18961"
 uuid = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549"
-version = "1.0.5"
+version = "1.0.6"
 
 [[ForwardDiff]]
 deps = ["CommonSubexpressions", "DiffResults", "DiffRules", "InteractiveUtils", "LinearAlgebra", "NaNMath", "Random", "SparseArrays", "SpecialFunctions", "StaticArrays", "Test"]
@@ -201,10 +195,10 @@ uuid = "1b4a561d-cfcb-5daf-8433-43fcf8b4bea3"
 version = "0.4.1"
 
 [[LibCURL]]
-deps = ["BinaryProvider", "Compat", "Libdl", "Printf"]
+deps = ["BinaryProvider", "Libdl", "Printf", "Test"]
-git-tree-sha1 = "6339c87cb76923a3cf947fcd213cbc364355c9c9"
+git-tree-sha1 = "d051c8057512ca38a273aaa514145a0b25f24d46"
 uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"
-version = "0.4.1"
+version = "0.5.0"
 
 [[LibExpat]]
 deps = ["Compat"]
@@ -257,10 +251,10 @@ uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
 version = "0.3.2"
 
 [[Nemo]]
-deps = ["AbstractAlgebra", "InteractiveUtils", "Libdl", "LinearAlgebra", "Markdown", "Test"]
+deps = ["AbstractAlgebra", "BinaryProvider", "InteractiveUtils", "Libdl", "LinearAlgebra", "Markdown", "Test"]
-git-tree-sha1 = "a41d652f0eefc6e08956bde976a10fed3535db57"
+git-tree-sha1 = "24b361effd69b4ca78014a7bab1b81d1de7f35c0"
 uuid = "2edaba10-b0f1-5616-af89-8c11ac63239a"
-version = "0.12.3"
+version = "0.13.2"
 
 [[OrderedCollections]]
 deps = ["Random", "Serialization", "Test"]
@@ -288,7 +282,7 @@ uuid = "9abbd945-dff8-562f-b5e8-e1ebf5ef1b79"
 
 [[PropertyT]]
 deps = ["AbstractAlgebra", "Dates", "GroupRings", "Groups", "IntervalArithmetic", "JLD", "JuMP", "LinearAlgebra", "Markdown", "MathProgBase", "Nemo", "Printf", "SparseArrays"]
-git-tree-sha1 = "4c4a5086b0c4dee30bba27ddf42e7ebb403765c2"
+git-tree-sha1 = "1307df01fc484b44fd24328f30304912ebd9bf2a"
 repo-rev = "master"
 repo-url = "https://github.com/kalmarek/PropertyT.jl"
 uuid = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d"

Project.toml

@@ -1,9 +1,23 @@
+name = "1712.07167"
+uuid = "c0ba6078-cfcc-57dc-8f34-6df289db1335"
+authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
+version = "0.1.0"
+
 [deps]
-BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
+AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
 GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
 Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
 IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
 JLD = "4138dd39-2aa7-5051-a626-17a0bb65d9c8"
+Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 PropertyT = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d"
 SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
+StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
+
+[extras]
+SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"
+Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+
+[targets]
+test = ["Test", "SCS"]

bin/check_SpNs.jl

@@ -0,0 +1,91 @@
+using Pkg
+Pkg.activate(".")
+using Test
+
+using PropertyT
+using Nemo
+
+using PropertyT.LinearAlgebra
+using PropertyT.SparseArrays
+using PropertyT.JuMP
+
+using PropertyT.AbstractAlgebra
+using PropertyT.Groups
+using PropertyT.GroupRings
+
+using PropertyT.JLD
+
+
+# include("../1712.07167/Roots.jl")
+include(joinpath("..", "src", "SpNs.jl"))
+
+using .SpNs
+using .SpNs.Roots
+
+BLAS.set_num_threads(Threads.nthreads())
+ENV["OMP_NUM_THREADS"] = Threads.nthreads()
+
+const N = 2
+const RADIUS = 3
+const UPPER_BOUND = 0.801
+
+root_system = SpNs.gens_roots(N)
+S = [g.matrix for g in root_system]
+G = parent(S[1])
+autS = WreathProduct(PermGroup(2), PermGroup(N))
+
+using SCS
+with_SCS = with_optimizer(SCS.Optimizer, linear_solver=SCS.Direct, max_iters=400000, eps=1e-11, alpha=1.95, acceleration_lookback=1, warm_start=true)
+
+sett = PropertyT.Settings("Sp($(2N),Z)_r$RADIUS", G, S, autS, with_SCS;
+    radius=RADIUS, upper_bound=UPPER_BOUND, warmstart=true)
+
+PropertyT.print_summary(sett)
+
+fp = PropertyT.fullpath(sett)
+isdir(fp) || mkpath(fp)
+
+Δ = PropertyT.Laplacian(S, RADIUS)
+RG = parent(Δ)
+Δs = SpNs.laplacians(RG, root_system)
+
+Sq  = sum( Δα^2 for (α, Δα) in Δs );
+# Op  = sum( Δα * sum(Δβ for (β, Δβ) in Δs if isorthogonal(α, β)) for (α, Δα) in Δs );
+Adj = sum( Δα * sum(Δβ for (β, Δβ) in Δs if !isproportional(α, β)) for (α, Δα) in Δs );
+
+const elt = Adj
+
+solver_logfile(sett) = joinpath(PropertyT.fullpath(sett), "Adj_solver_$(PropertyT.Dates.now()).log")
+
+λ, P = PropertyT.approximate_by_SOS(sett, elt, Δ, 
+    solverlog=solver_logfile(sett))
+
+save(PropertyT.filename(sett, :solution), "λ", λ, "P", P)
+λ < 0 && @warn "Solver did not produce a valid solution!"
+
+Q = real(sqrt(P));
+certified_λ = PropertyT.certify_SOS_decomposition(elt, Δ, λ, Q, R=RADIUS)
+
+@info "The obtained SOS can be use to certify λ ≥" certified_λ
+
+
+
+
+# 
+# od = PropertyT.OrbitData(RG, WreathProduct(PermGroup(2), PermGroup(N)))
+# orbit_data = PropertyT.decimate(od);
+# 
+# using SCS
+# using JuMP
+# warmstart = nothing
+# 
+# UB = 0.88 # for elt = Δ²;
+# SDP_problem, varP = PropertyT.SOS_problem(elt, Δ, orbit_data, upper_bound=UB);
+# # SDP_problem, varP = PropertyT.SOS_problem(elt, Δ, orbit_data);
+# 
+# sett = PropertyT.Settings("Sp($(2N),Z)", G, S, autS, solver(2000, accel=20);
+# upper_bound=1.3, warmstart=false)
+# 
+# with_SCS = with_optimizer(SCS.Optimizer, linear_solver=SCS.Direct, max_iters=100000, eps=1e-12, alpha=1.95, acceleration_lookback=1, warm_start=true)
+# 
+# status, warmstart = PropertyT.solve(SDP_problem, with_SCS, warmstart)

src/Roots.jl

@@ -0,0 +1,59 @@
+module Roots
+
+using StaticArrays
+using LinearAlgebra
+
+export Root, isproportional, isorthogonal, ~, ⟂
+
+abstract type AbstractRoot end
+
+struct Root{N} <: AbstractRoot
+    coord::SVector{N, Float64}
+    
+    Root(a::SVector{N}) where N = new{N}(a)
+end
+
+Root(a) = Root(SVector(a...))
+
+function Base.:(==)(r::Root{N}, s::Root{M}) where {M, N}
+    M == N || return false
+    r.coord == s.coord || return false
+    return true
+end
+
+Base.hash(r::Root, h::UInt) = hash(Root, hash(r.coord, h))
+
+Base.:+(r::Root{N}, s::Root{N}) where N = Root(r.coord+s.coord)
+Base.:-(r::Root{N}, s::Root{N}) where N = Root(r.coord-s.coord)
+Base.:-(r::Root{N}) where N = Root(-r.coord)
+
+Base.:*(a::Number, r::Root) = Root(a*r.coord)
+Base.:*(r::Root, a::Number) = a*r
+
+Base.length(r::Root) = norm(r, 2)
+
+LinearAlgebra.norm(r::Root, p::Real=2) = norm(r.coord, p)
+
+LinearAlgebra.dot(r::Root{N}, s::Root{N}) where N = dot(r.coord, s.coord)
+
+cos_angle(a,b) = dot(a, b) / (norm(a) * norm(b))
+
+function isproportional(α::Root{N}, β::Root{N}) where N 
+    return isapprox(abs(cos_angle(α, β)), 1.0, atol=eps(1.0))
+end
+
+Base.:~(α::Root{N}, β::Root{N}) where N = isproportional(α, β)
+
+function isorthogonal(α::Root{N}, β::Root{N}) where N
+    return isapprox(cos_angle(α, β), 0.0, atol=eps(1.0))
+end
+
+⟂(α::Root{N}, β::Root{N}) where N = isorthogonal(α, β)
+
+function Base.show(io::IO, r::Root{N}) where N
+    print(io, "$(r.coord): root of length $(norm(r))")
+end
+
+E(N, i::Integer) = Root(ntuple(k -> k == i ? 1 : 0, N))
+
+end # of module RootSystems

src/SpN_actions.jl

@@ -0,0 +1,39 @@
+function blkdiag(A, B)
+    new_size = (size(A,1) + size(B,1), size(A,2) + size(B,2))
+    res = zeros(eltype(A), new_size)
+    res[1:size(A,1), 1:size(A,2)] .= A
+    res[size(A,1)+1:end, size(A,2)+1:end] .= B
+    return res
+end
+
+function matrix_emb(n::DirectPowerGroupElem, p::perm)
+    Id = parent(n.elts[1])()
+    elt = Diagonal([(-1)^(el == Id ? 0 : 1) for el in n.elts])
+    elt = elt[:, p.d]
+    return blkdiag(elt, elt)
+end
+
+function (g::WreathProductElem)(A::MatElem)
+    g_inv = inv(g)
+    G = matrix_emb(g.n, g_inv.p)
+    G_inv = matrix_emb(g_inv.n, g.p)
+    M = parent(A)
+    return M(G)*A*M(G_inv)
+end
+
+function Base.:*(x::AbstractAlgebra.MatElem, P::Generic.perm)
+   z = similar(x)
+   m = nrows(x)
+   n = ncols(x)
+   for i = 1:m
+      for j = 1:n
+         z[i, j] = x[i,P[j]]
+      end
+   end
+   return z
+end
+
+function (p::perm)(A::MatElem)
+    length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
+    return p*A*inv(p)
+end

src/SpNs.jl

@@ -0,0 +1,119 @@
+module SpNs
+
+using LinearAlgebra
+using AbstractAlgebra
+using Nemo
+using Groups
+using GroupRings
+
+include("Roots.jl")
+using .Roots
+
+export Sp, isproportional, ~, isorthogonal, ⟂, positive
+
+# two families of generators
+
+function a(i::Int, j::Int, M::MatSpace, val=one(M.base_ring)) # short roots
+    @assert i ≠ j
+    m = one(M)
+    N = size(m,2)÷2
+    
+    m[i,j] = val
+    m[N+j, N+i] = -val
+    @assert issymplectic(m)
+    return m
+end
+
+function b(i::Int, j::Int, M::MatSpace, val=one(M.base_ring)) # long roots
+    m = one(M)
+    N = size(m,2)÷2
+    m[i, N+j] = val
+    m[j, N+i] = val
+    @assert issymplectic(m)
+    return m
+end
+
+# symplectic generator with associated root datum:
+
+struct Sp{N}
+    root::Root{N}
+    matrix::Nemo.fmpz_mat
+end
+
+function Sp{N}(typ::Val{:a}, i::Int, j::Int) where N
+    @assert i≠j
+    r = Roots.E(N, i) - Roots.E(N, j)
+    M = Nemo.MatrixSpace(Nemo.ZZ, 2N, 2N)
+    m = a(i,j, M)
+    return Sp{N}(r, m)
+end
+
+function Sp{N}(typ::Val{:b}, i::Int, j::Int=i) where N
+    r = Roots.E(N, i) + Roots.E(N, j)
+    M = Nemo.MatrixSpace(Nemo.ZZ, 2N, 2N)
+    m = b(i,j, M)
+    return Sp{N}(r, m)
+end
+
+Sp{N}(s::Symbol, i::Int, j::Int=i) where N = Sp{N}(Val(s), i, j)
+
+Base.inv(s::Sp) = Sp(s.root, inv(s.matrix))
+Base.:-(s::Sp) = Sp(-s.root, transpose(s.matrix))
+
+Roots.isproportional(x::Sp, y::Sp) = isproportional(x.root, y.root)
+Roots.:~(x::Sp,y::Sp) = x.root ~ y.root
+
+Roots.isorthogonal(x::Sp, y::Sp) = isorthogonal(x.root, y.root)
+Roots.:⟂(x::Sp,y::Sp) = x.root ⟂ y.root
+
+function issymplectic(M)
+    r = nrows(M)
+    c = ncols(M)
+    @assert r == c
+    @assert iseven(r)
+    n = r÷2
+    I = Diagonal(ones(Int, n))
+    
+    O = zeros(Int,n,n);
+    Ω = parent(M)([O  I; -I O])
+    return Ω == transpose(M)*Ω*M
+end
+
+indexing(n) = [(i,j) for i in 1:n for j in i+1:n]
+
+function gens_roots(N)
+    a_ijs = [Sp{N}(:a, i,j) for (i,j) in indexing(N)]
+    b_is = [Sp{N}(:b, i) for i in 1:N]
+    c_ijs = [Sp{N}(:b, i,j) for (i,j) in indexing(N)]
+    root_system = [a_ijs; b_is; c_ijs]
+    root_system = [root_system; [-r for r in root_system]];
+    root_system = [root_system; inv.(root_system)]
+    return root_system
+end
+
+function positive(root_system::Vector{SP}) where SP
+    roots = [r.root for r in root_system]
+    
+    pos_roots = eltype(roots)[]
+    for (idx,γ) in enumerate(roots)
+        if any(isproportional(s, γ) for s in roots[1:idx-1])
+            continue
+        end
+        push!(pos_roots, γ)
+        @info "" positive_root = γ
+    end
+    # TODO: what about their sums??
+    
+    return pos_roots
+end
+
+function laplacians(RG::GroupRing, root_system::Vector{SpNs.Sp{N}}) where N
+    positive_roots = positive(root_system)
+    Sαs = Dict(α => [w.matrix for w in root_system if isproportional(w.root, α)] for α in positive_roots)
+    Δs = Dict(α => RG(length(S)) - RG(S) for (α, S) in Sαs)
+    return Δs
+end
+
+include("SpN_actions.jl")
+
+end # of module SPNs

test/runtests.jl

@@ -0,0 +1,108 @@
+using Test
+
+using PropertyT
+include("../src/SpNs.jl")
+using .SpNs
+using .SpNs.Roots
+
+@testset "Roots" begin
+    @test Root((2,1)) isa Root
+    r = Root((2,1))
+    @test -r isa Root
+    @test 2r isa Root
+    @test length(2r) == 2length(r)
+
+    @test r+r isa Root
+    @test length(r+r) == 2length(r)
+    @test r-r isa Root;
+    @test length(r-r) == 0.0
+
+    r = Root((1,1));
+    s = Root((-1,1));
+    @test string(Root((1,1))) == "[1.0, 1.0]: root of length 1.4142135623730951"
+
+    @test isproportional(r, r)
+    @test isproportional(r, -r)
+    @test isproportional(r, 2r)
+    @test isproportional(-r, 2r)
+    @test isproportional(r+s, -s-r)
+
+    @test isorthogonal(r, s)
+    @test isorthogonal(r, -s)
+    @test isorthogonal(r, 2s)
+    @test isorthogonal(-r, 2s)
+
+    @test !isorthogonal(r, r+s)
+    @test !isorthogonal(r, -r)
+    @test !isorthogonal(r+s, 2s)
+    @test !isorthogonal(-r, 2s+r)
+
+    @test isorthogonal(r+s, -s+r)
+
+    N = 3
+    @test length(Roots.E(N, 1) - Roots.E(N,2)) == sqrt(2)
+end
+
+
+@testset "Symplectic Generators" begin
+
+    s = Sp{2}(:a, 1, 2) 
+    @test -s isa Sp 
+    @test inv(s) isa Sp
+    t = Sp{2}(:b, 1, 2)
+    @test -t isa Sp
+    @test inv(t) isa Sp
+
+    @test isproportional(s, -s)
+    @test isproportional(s, -s)
+    @test isproportional(s, inv(s))
+    
+    @test isproportional(t, -t)
+    @test isproportional(t, inv(t))
+    @test !isproportional(s, t)
+    @test !isproportional(inv(s), -t)
+    
+end
+
+@testset "Symplectic Group" begin
+
+    @testset "Symplectic Root System" begin
+        rs = SpNs.gens_roots(2)
+        S = [r.matrix for r in rs];
+        @test all(SpNs.issymplectic.(S))
+        @test length(S) == 16
+        
+        rs = SpNs.gens_roots(3)
+        S = [r.matrix for r in rs];
+        @test all(SpNs.issymplectic.(S))
+        @test length(S) == 36
+    end
+
+    @testset "Sp(4,Z)" begin
+
+        N = 2
+        root_system = SpNs.gens_roots(N)
+        S = [g.matrix for g in root_system]
+        
+        Δ = PropertyT.Laplacian(S, 2)
+        RG = parent(Δ)
+        
+        Δs = SpNs.laplacians(RG, root_system)
+        
+        @test sum(Δα for (α, Δα) in Δs) == Δ
+        
+        @testset "Orthogonality & commutation" begin
+            r = Root([2, 0])
+            s = Root([0, 2])
+
+            a = Δs[r]
+            b = Δs[s]
+            @test a*b == b*a
+        end
+        
+        Sq  = sum( Δα^2 for (α, Δα) in Δs );
+        Adj = sum( Δα * sum(Δβ for (β, Δβ) in Δs if !isproportional(α, β)) for (α, Δα) in Δs );
+        
+        @test Sq + Adj == Δ^2
+    end
+end