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cosmetics
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@ -121,7 +121,7 @@ function correct_to_augmentation_ideal{T<:Rational}(sqrt_matrix::Array{T,2})
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return sqrt_corrected
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end
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function check_solution(κ, sqrt_matrix, Δ; verbose=true, augmented=false)
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function check_solution{T<:Number}(κ::T, sqrt_matrix::Array{T,2}, Δ::GroupAlgebraElement{T}; verbose=true, augmented=false)
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result = compute_SOS(sqrt_matrix, Δ)
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if augmented
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@assert GroupAlgebras.ɛ(result) == 0//1
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@ -155,33 +155,25 @@ end;
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ℚ(x, tol::Real) = rationalize(BigInt, x, tol=tol)
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function ℚ_distance_to_positive_cone(Δ::GroupAlgebraElement,
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κ::Float64,
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A::Array{Float64,2};
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tol=10.0^-7,
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verbose=true)
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function ℚ_distance_to_positive_cone(Δ::GroupAlgebraElement, κ, A;
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tol=10.0^-7, verbose=true)
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@show maximum(A)
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if maximum(A) < 1e-2
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warn("Solver might not solved the problem successfully and the positive solution is due to floating-point error, proceeding anyway...")
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end
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@assert isapprox(eigvals(A), abs(eigvals(A)), atol=TOL)
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isapprox(eigvals(A), abs(eigvals(A)), atol=tol) ||
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warn("The solution matrix doesn't seem to be positive definite!")
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@assert A == Symmetric(A)
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A_sqrt = real(sqrtm(A))
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println("")
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println("")
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println("Checking in floating-point arithmetic...")
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fp_distance = check_solution(κ, A_sqrt, Δ, verbose=VERBOSE)
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@time fp_distance = check_solution(κ, A_sqrt, Δ, verbose=verbose)
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println("Distance to positive cone ≈ $(Float64(trunc(fp_distance,8)))")
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println("-------------------------------------------------------------")
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println("")
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println("Checking in rational arithmetic...")
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κ_ℚ = ℚ(trunc(κ,Int(abs(log10(tol)))), TOL)
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@assert κ - κ_ℚ ≥ 0
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A_sqrt_ℚ, Δ_ℚ = ℚ(A_sqrt, TOL), ℚ(Δ, TOL)
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ℚ_distance = check_solution(κ_ℚ, A_sqrt_ℚ, Δ_ℚ, verbose=VERBOSE)
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κ_ℚ = ℚ(trunc(κ,Int(abs(log10(tol)))), tol)
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A_sqrt_ℚ, Δ_ℚ = ℚ(A_sqrt, tol), ℚ(Δ, tol)
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@time ℚ_distance = check_solution(κ_ℚ, A_sqrt_ℚ, Δ_ℚ, verbose=verbose)
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@assert isa(ℚ_distance, Rational)
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println("Distance to positive cone ≈ $(Float64(trunc(ℚ_distance,8)))")
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println("-------------------------------------------------------------")
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@ -189,7 +181,7 @@ function ℚ_distance_to_positive_cone(Δ::GroupAlgebraElement,
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println("Projecting columns of A_sqrt to the augmentation ideal...")
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A_sqrt_ℚ_aug = correct_to_augmentation_ideal(A_sqrt_ℚ)
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ℚ_dist_to_Σ² = check_solution(κ_ℚ, A_sqrt_ℚ_aug, Δ_ℚ, verbose=VERBOSE, augmented=true)
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@time ℚ_dist_to_Σ² = check_solution(κ_ℚ, A_sqrt_ℚ_aug, Δ_ℚ, verbose=verbose, augmented=true)
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@assert isa(ℚ_dist_to_Σ², Rational)
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s = (ℚ_dist_to_Σ² < 0? "≤": "≥")
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println("Distance to positive cone $s $(Float64(trunc(ℚ_dist_to_Σ²,8)))")
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