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Engine prototype #13

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glen merged 133 commits from engine-proto into main 2024-10-21 03:18:48 +00:00
Conversation 9 Commits 133 Files Changed 20 +18 -21
2 changed files with 18 additions and 21 deletions
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19
engine-proto/Engine.Numerical.jl
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@ -1,7 +1,8 @@
module Numerical module Numerical
using LinearAlgebra
using AbstractAlgebra using AbstractAlgebra
using HomotopyContinuation: Variable, Expression, System using HomotopyContinuation
using ..Algebraic using ..Algebraic
# --- polynomial conversion --- # --- polynomial conversion ---
@ -10,7 +11,7 @@ using ..Algebraic
# https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521 # https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521
function Base.convert(::Type{Expression}, f::MPolyRingElem) function Base.convert(::Type{Expression}, f::MPolyRingElem)
variables = Variable.(symbols(parent(f))) variables = Variable.(symbols(parent(f)))
f_data = zip(coefficients(f), exponent_vectors(f)) f_data = zip(AbstractAlgebra.coefficients(f), exponent_vectors(f))
sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data) sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data)
end end
@ -22,4 +23,18 @@ function System(I::Generic.Ideal)
System(eqns, variables = variables) System(eqns, variables = variables)
end end
# --- 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
end end

20
engine-proto/Engine.jl
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@ -82,25 +82,7 @@ n_planes = 36
for through_trivial in [false, true] for through_trivial in [false, true]
samples = [] samples = []
for _ in 1:n_planes for _ in 1:n_planes
cut_matrix = transpose(hcat( real_solns = solution.(Engine.Numerical.real_samples(system, freedom))
(normalize(randn(length(gens(coordring)))) for _ in 1:freedom)...
))
##cut_matrix = rand(binom, freedom, length(gens(coordring))) .- max_slope
##cut_matrix = [
## 1 1 1 1 0 1 1 0 1 1 0;
## 1 2 1 2 0 1 1 0 1 1 0;
## 1 1 0 1 0 1 2 0 2 0 0
##]
## display(cut_matrix) ## [verbose]
if through_trivial
cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)]
## display(cut_offset) ## [verbose]
cut_subspace = LinearSubspace(cut_matrix, cut_offset)
else
cut_subspace = LinearSubspace(cut_matrix, fill(0., freedom))
end
wtns = witness_set(system, cut_subspace)
real_solns = solution.(filter(isreal, results(wtns)))
nontrivial_solns = filter(is_nontrivial, real_solns) nontrivial_solns = filter(is_nontrivial, real_solns)
println("$(length(real_solns) - length(nontrivial_solns)) trivial solutions found") println("$(length(real_solns) - length(nontrivial_solns)) trivial solutions found")
for soln in nontrivial_solns for soln in nontrivial_solns