dyna3/engine-proto/Engine.Numerical.jl

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module Numerical
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using LinearAlgebra
using AbstractAlgebra
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using HomotopyContinuation:
Variable, Expression, AbstractSystem, System, LinearSubspace,
nvariables, isreal, witness_set, results
import GLMakie
using ..Algebraic
# --- polynomial conversion ---
# hat tip Sascha Timme
# https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521
function Base.convert(::Type{Expression}, f::MPolyRingElem)
variables = Variable.(symbols(parent(f)))
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f_data = zip(coefficients(f), exponent_vectors(f))
sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data)
end
# create a ModelKit.System from an ideal in a multivariate polynomial ring. the
# variable ordering is taken from the polynomial ring
function System(I::Generic.Ideal)
eqns = Expression.(gens(I))
variables = Variable.(symbols(base_ring(I)))
System(eqns, variables = variables)
end
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# --- sampling ---
function real_samples(F::AbstractSystem, dim)
# choose a random real hyperplane of codimension `dim` by intersecting
# hyperplanes whose normal vectors are uniformly distributed over the unit
# sphere
# [to do] guard against the unlikely event that one of the normals is zero
normals = transpose(hcat(
(normalize(randn(nvariables(F))) for _ in 1:dim)...
))
cut = LinearSubspace(normals, fill(0., dim))
filter(isreal, results(witness_set(F, cut)))
end
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AbstractAlgebra.evaluate(pt::Point, vals::Vector{<:RingElement}) =
GLMakie.Point3f([evaluate(u, vals) for u in pt.coords])
function AbstractAlgebra.evaluate(sph::Sphere, vals::Vector{<:RingElement})
radius = 1 / evaluate(sph.coords[1], vals)
center = radius * [evaluate(u, vals) for u in sph.coords[3:end]]
GLMakie.Sphere(GLMakie.Point3f(center), radius)
end
end