Stow algebraic engine prototype

We're using the Gram matrix engine for the next stage of development,
so the algebraic engine shouldn't be at the top level anymore.

engine-proto/alg-test/Engine.Numerical.jl

@@ -0,0 +1,53 @@
+module Numerical
+
+using Random: default_rng
+using LinearAlgebra
+using AbstractAlgebra
+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)))
+  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
+
+# --- sampling ---
+
+function real_samples(F::AbstractSystem, dim; rng = default_rng())
+  # 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(rng, nvariables(F))) for _ in 1:dim)...
+  ))
+  cut = LinearSubspace(normals, fill(0., dim))
+  filter(isreal, results(witness_set(F, cut, seed = 0x1974abba)))
+end
+
+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