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total rewrite with the aim of modularisation
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SL(3,Z).jl
135
SL(3,Z).jl
@ -1,79 +1,104 @@
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using JuMP
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import SCS: SCSSolver
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import Mosek: MosekSolver
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function SL₃ℤ_generatingset()
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workers_processes = addprocs()
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@everywhere push!(LOAD_PATH, "./")
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using GroupAlgebras
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@everywhere include("property(T).jl")
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function E(i::Int, j::Int, N::Int=3)
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function E(i::Int, j::Int, N::Int=3)
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@assert i≠j
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k = eye(N)
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k[i,j] = 1
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return k
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end
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end
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function SL_3ZZ_generating_set()
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S = [E(1,2), E(1,3), E(2,3)];
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S = vcat(S, [x' for x in S]);
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S = vcat(S, [inv(x) for x in S]);
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return S
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end
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const ID = eye(3)
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function generate_B₂_and_B₄(B₁)
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B₂ = products(B₁, B₁);
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B₃ = products(B₁, B₂);
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B₄ = products(B₁, B₃);
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const S₁ = SL_3ZZ_generating_set()
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@assert B₄[1:length(B₂)] == B₂
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return B₂, B₄;
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end
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function prepare_Laplacian_and_constraints(S, identity)
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B₂, B₄ = generate_B₂_and_B₄(vcat([identity], S))
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product_matrix = create_product_matrix(B₄,length(B₂));
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sdp_constraints = constraints_from_pm(product_matrix, length(B₄))
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L_coeff = splaplacian_coeff(S, B₄);
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return GroupAlgebraElement(L_coeff, product_matrix), sdp_constraints
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end
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function prepare_Δ_sdp_constraints(name::String;cached=true)
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f₁ = isfile("$name.product_matrix")
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f₂ = isfile("$name.delta.coefficients")
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if cached && f₁ && f₂
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println("Loading precomputed pm, Δ, sdp_constraints...")
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product_matrix = readdlm("$name.product_matrix", Int)
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L = readdlm("$name.delta.coefficients")[:, 1]
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Δ = GroupAlgebraElement(L, product_matrix)
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sdp_constraints = constraints_from_pm(product_matrix)
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else
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println("Computing pm, Δ, sdp_constraints...")
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ID = eye(Int, 3)
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S₁ = SL₃ℤ_generatingset()
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Δ, sdp_constraints = prepare_Laplacian_and_constraints(S₁, ID)
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writedlm("$name.delta.coefficients", Δ.coefficients)
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writedlm("$name.product_matrix", Δ.product_matrix)
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end
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return Δ, sdp_constraints
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end
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const TOL=10.0^-7
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# const VERBOSE=true
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#solver = SCSSolver(eps=TOL, max_iters=ITERATIONS, verbose=VERBOSE);
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# solver = MosekSolver(MSK_DPAR_INTPNT_CO_TOL_REL_GAP=TOL,
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# # MSK_DPAR_INTPNT_CO_TOL_PFEAS=1e-15,
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# # MSK_DPAR_INTPNT_CO_TOL_DFEAS=1e-15,
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# # MSK_IPAR_PRESOLVE_USE=0,
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# QUIET=!VERBOSE)
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function compute_κ_A(name::String, Δ, sdp_constraints;
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cached = true,
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tol = TOL,
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verbose = VERBOSE,
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solver = MosekSolver(INTPNT_CO_TOL_REL_GAP=tol, QUIET=!verbose))
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# solver = SCSSolver(eps=TOL, max_iters=ITERATIONS, verbose=VERBOSE))
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# κ, A = solve_for_property_T(S₁, solver, verbose=VERBOSE)
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f₁ = isfile("$name.kappa")
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f₂ = isfile("$name.SDPmatrixA")
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if cached && f₁ && f₂
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println("Loading precomputed κ, A...")
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A = readdlm("$name.SDPmatrixA")
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κ = readdlm("$name.kappa")[1]
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else
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println("Solving SDP problem maximizing κ...")
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κ, A = solve_SDP(sdp_constraints, Δ, solver, verbose=verbose)
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writedlm("$name.kappa", kappa)
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writedlm("$name.SDPmatrixA", A)
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end
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return κ, A
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end
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const product_matrix = readdlm("SL3Z.product_matrix", Int)
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const L = readdlm("SL3Z.delta.coefficients")[:, 1]
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const Δ = GroupAlgebraElement(L, product_matrix)
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workers_processes = addprocs()
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@everywhere push!(LOAD_PATH, "./")
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using GroupAlgebras
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@everywhere include("property(T).jl")
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const A = readdlm("SL3Z.SDPmatrixA.Mosek")
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const κ = readdlm("SL3Z.kappa.Mosek")[1]
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const NAME = "SL3Z"
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const VERBOSE = true
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const TOL=1e-7
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const Δ, sdp_constraints = prepare_Δ_sdp_constraints(NAME)
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const κ, A = compute_κ_A(NAME, Δ, sdp_constraints)
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@assert isapprox(eigvals(A), abs(eigvals(A)), atol=TOL)
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@assert A == Symmetric(A)
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if κ > 0
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@time T = ℚ_distance_to_positive_cone(Δ, κ, A, tol=TOL, verbose=VERBOSE)
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const A_sqrt = real(sqrtm(A))
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if T < 0
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println("$NAME HAS property (T)!")
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else
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println("$NAME may NOT HAVE property (T)!")
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end
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const SOS_fp_diff, SOS_fp_L₁_distance = check_solution(κ, A_sqrt, Δ)
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@show SOS_fp_L₁_distance
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@show GroupAlgebras.ɛ(SOS_fp_diff)
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const κ_rational = rationalize(BigInt, κ, tol=TOL)
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const A_sqrt_rational = rationalize(BigInt, A_sqrt, tol=TOL)
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const Δ_rational = rationalize(BigInt, Δ, tol=TOL)
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const SOS_rational_diff, SOS_rat_L₁_distance = check_solution(κ_rational, A_sqrt_rational, Δ_rational)
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@assert isa(SOS_rat_L₁_distance, Rational{BigInt})
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@show float(SOS_rat_L₁_distance)
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@show float(GroupAlgebras.ɛ(SOS_rational_diff))
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const A_sqrt_augmented = correct_to_augmentation_ideal(A_sqrt_rational)
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const SOS_rational_aug_diff, SOS_aug_rat_L₁_distance = check_solution(κ_rational, A_sqrt_augmented, Δ_rational)
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@assert isa(SOS_aug_rat_L₁_distance, Rational{BigInt})
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@assert GroupAlgebras.ɛ(SOS_rational_aug_diff) == 0//1
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@show float(SOS_aug_rat_L₁_distance)
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@show float(κ_rational - 2^3*SOS_aug_rat_L₁_distance)
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else
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println("$κ < 0: $NAME may NOT HAVE property (T)!")
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end
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rmprocs(workers_processes)
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