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make augmented compute_sos fast
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@ -5,80 +5,50 @@ function augment_columns!(Q::AbstractMatrix)
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return Q
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end
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function _fma_SOS_thr!(
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result::AbstractVector{T},
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mstructure::AbstractMatrix{<:Integer},
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Q::AbstractMatrix{T},
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acc_matrix=zeros(T, size(mstructure)...),
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) where {T}
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function __sos_via_sqr!(
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res::StarAlgebras.AlgebraElement,
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P::AbstractMatrix;
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augmented::Bool
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)
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StarAlgebras.zero!(res)
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A = parent(res)
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b = basis(A)
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@assert size(A.mstructure) == size(P)
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e = b[one(b[1])]
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s1, s2 = size(mstructure)
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@inbounds for k = 1:s2
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let k = k, s1 = s1, s2 = s2, Q = Q, acc_matrix = acc_matrix
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Threads.@threads for j = 1:s2
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for i = 1:s1
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@inbounds acc_matrix[i, j] =
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muladd(Q[i, k], Q[j, k], acc_matrix[i, j])
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end
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for i in axes(A.mstructure, 1)
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x = StarAlgebras._istwisted(A.mstructure) ? StarAlgebras.star(b[i]) : b[i]
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for j in axes(A.mstructure, 2)
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p = P[i, j]
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xy = b[A.mstructure[i, j]]
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# either result += P[x,y]*(x*y)
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res[xy] += p
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if augmented
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# or result += P[x,y]*(1-x)*(1-y) == P[x,y]*(2-x-y+xy)
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y = b[j]
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res[e] += p
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res[x] -= p
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res[y] -= p
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end
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end
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end
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@inbounds for j = 1:s2
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for i = 1:s1
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result[mstructure[i, j]] += acc_matrix[i, j]
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end
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end
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return result
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return res
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end
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function _cnstr_sos!(res::StarAlgebras.AlgebraElement, Q::AbstractMatrix, cnstrs)
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function __sos_via_cnstr!(res::StarAlgebras.AlgebraElement, Q²::AbstractMatrix, cnstrs)
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StarAlgebras.zero!(res)
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Q² = Q' * Q
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for (g, A_g) in cnstrs
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res[g] = dot(A_g, Q²)
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end
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return res
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end
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function _augmented_sos!(res::StarAlgebras.AlgebraElement, Q::AbstractMatrix)
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A = parent(res)
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StarAlgebras.zero!(res)
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Q² = Q' * Q
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N = LinearAlgebra.checksquare(A.mstructure)
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augmented_basis = [A(1) - A(b) for b in @view basis(A)[1:N]]
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tmp = zero(res)
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for (j, y) in enumerate(augmented_basis)
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for (i, x) in enumerate(augmented_basis)
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# res += Q²[i, j] * x * y
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StarAlgebras.mul!(tmp, x, y)
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StarAlgebras.mul!(tmp, tmp, Q²[i, j])
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StarAlgebras.add!(res, res, tmp)
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end
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end
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return res
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end
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function compute_sos(A::StarAlgebras.StarAlgebra, Q::AbstractMatrix; augmented::Bool)
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if augmented
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z = zeros(eltype(Q), length(basis(A)))
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res = StarAlgebras.AlgebraElement(z, A)
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return _augmented_sos!(res, Q)
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cnstrs = constraints(basis(A), A.mstructure; augmented=true)
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return _cnstr_sos!(res, Q, cnstrs)
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else
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@assert size(A.mstructure) == size(Q)
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z = zeros(eltype(Q), length(basis(A)))
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_fma_SOS_thr!(z, A.mstructure, Q)
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return StarAlgebras.AlgebraElement(z, A)
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end
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Q² = Q' * Q
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res = StarAlgebras.AlgebraElement(zeros(eltype(Q²), length(basis(A))), A)
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res = __sos_via_sqr!(res, Q², augmented=augmented)
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return res
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end
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function sufficient_λ(residual::StarAlgebras.AlgebraElement, λ; halfradius)
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@ -159,7 +129,7 @@ function certify_solution(
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true, compute_sos(parent(elt), Q_int, augmented=augmented)
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end
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@info "Checking in $(eltype(sos_int)) arithmetic with" λ
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@info "Checking in $(eltype(sos_int)) arithmetic with" λ_int
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λ_certified =
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sufficient_λ(elt, orderunit, λ_int, sos_int, halfradius=halfradius)
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@ -6,37 +6,20 @@ function check_positivity(elt, unit, wd; upper_bound=Inf, halfradius=2, optimize
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@time status, _ = PropertyT.solve(sos_problem, optimizer)
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Q = let Ps = Ps
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flPs = [real.(sqrt(JuMP.value.(P))) for P in Ps]
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PropertyT.reconstruct(flPs, wd)
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Qs = [real.(sqrt(JuMP.value.(P))) for P in Ps]
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PropertyT.reconstruct(Qs, wd)
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end
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λ = JuMP.value(sos_problem[:λ])
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sos = let RG = parent(elt), Q = Q
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z = zeros(eltype(Q), length(basis(RG)))
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res = AlgebraElement(z, RG)
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cnstrs = PropertyT.constraints(basis(RG), RG.mstructure, augmented=true)
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PropertyT._cnstr_sos!(res, Q, cnstrs)
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end
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residual = elt - λ * unit - sos
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λ_fl = PropertyT.sufficient_λ(residual, λ, halfradius=2)
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λ_fl < 0 && return status, false, λ_fl
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sos = let RG = parent(elt), Q = [PropertyT.IntervalArithmetic.@interval(q) for q in Q]
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z = zeros(eltype(Q), length(basis(RG)))
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res = AlgebraElement(z, RG)
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cnstrs = PropertyT.constraints(basis(RG), RG.mstructure, augmented=true)
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PropertyT._cnstr_sos!(res, Q, cnstrs)
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end
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λ_int = PropertyT.IntervalArithmetic.@interval(λ)
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residual_int = elt - λ_int * unit - sos
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λ_int = PropertyT.sufficient_λ(residual_int, λ_int, halfradius=2)
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return status, λ_int > 0, PropertyT.IntervalArithmetic.inf(λ_int)
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certified, λ_cert = PropertyT.certify_solution(
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elt,
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unit,
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λ,
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Q,
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halfradius=2
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)
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return status, certified, λ_cert
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end
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@testset "1712.07167 Examples" begin
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