40 lines
1.3 KiB
Julia
40 lines
1.3 KiB
Julia
module Numerical
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using LinearAlgebra
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using AbstractAlgebra
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using HomotopyContinuation
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using ..Algebraic
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# --- polynomial conversion ---
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# hat tip Sascha Timme
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# https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521
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function Base.convert(::Type{Expression}, f::MPolyRingElem)
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variables = Variable.(symbols(parent(f)))
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f_data = zip(AbstractAlgebra.coefficients(f), exponent_vectors(f))
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sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data)
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end
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# create a ModelKit.System from an ideal in a multivariate polynomial ring. the
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# variable ordering is taken from the polynomial ring
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function System(I::Generic.Ideal)
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eqns = Expression.(gens(I))
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variables = Variable.(symbols(base_ring(I)))
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System(eqns, variables = variables)
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end
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# --- sampling ---
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function real_samples(F::AbstractSystem, dim)
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# choose a random real hyperplane of codimension `dim` by intersecting
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# hyperplanes whose normal vectors are uniformly distributed over the unit
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# sphere
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# [to do] guard against the unlikely event that one of the normals is zero
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normals = transpose(hcat(
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(normalize(randn(nvariables(F))) for _ in 1:dim)...
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))
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cut = LinearSubspace(normals, fill(0., dim))
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filter(isreal, results(witness_set(F, cut)))
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end
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end |