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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
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1 changed files with 51 additions and 29 deletions
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engine-proto/Engine.jl
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@ -227,6 +227,7 @@ end
using Random using Random
using Distributions using Distributions
using LinearAlgebra
using AbstractAlgebra using AbstractAlgebra
using HomotopyContinuation using HomotopyContinuation
@ -278,33 +279,53 @@ vbls = Variable.(symbols(coordring))
# test a random witness set # test a random witness set
system = CompiledSystem(System(eqns_ab_s, variables = vbls)) system = CompiledSystem(System(eqns_ab_s, variables = vbls))
max_slope = 2 sph_z_ind = indexin([sph.coords[5] for sph in ctx.spheres], gens(coordring))
println("sphere z variables: ", vbls[sph_z_ind])
trivial_soln = fill(0, length(gens(coordring)))
trivial_soln[sph_z_ind] .= 1
println("trivial solutions: $trivial_soln")
norm2 = vec -> real(dot(conj.(vec), vec))
is_nontrivial = soln -> norm2(abs.(real.(soln)) - trivial_soln) > 1e-4*length(gens(coordring))
max_slope = 5
binom = Binomial(2max_slope, 1/2) binom = Binomial(2max_slope, 1/2)
Random.seed!(6071) Random.seed!(6071)
n_planes = 36
for through_trivial in [false, true]
samples = [] samples = []
for _ in 1:3 for _ in 1:n_planes
cut_matrix = rand(binom, freedom, length(gens(coordring))) .- max_slope cut_matrix = rand(binom, freedom, length(gens(coordring))) .- max_slope
##cut_matrix = [ ##cut_matrix = [
## 1 1 1 1 0 1 1 0 1 1 0; ## 1 1 1 1 0 1 1 0 1 1 0;
## 1 2 1 2 0 1 1 0 1 1 0; ## 1 2 1 2 0 1 1 0 1 1 0;
## 1 1 0 1 0 1 2 0 2 0 0 ## 1 1 0 1 0 1 2 0 2 0 0
##] ##]
sph_z_ind = indexin([sph.coords[5] for sph in ctx.spheres], gens(coordring)) ## [verbose] display(cut_matrix)
if through_trivial
cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)] cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)]
println("sphere z variables: ", vbls[sph_z_ind]) ## [verbose] display(cut_offset)
display(cut_matrix)
display(cut_offset)
cut_subspace = LinearSubspace(cut_matrix, cut_offset) cut_subspace = LinearSubspace(cut_matrix, cut_offset)
wtns = witness_set(system, cut_subspace) else
append!(samples, solution.(filter(isreal, results(wtns)))) cut_subspace = LinearSubspace(cut_matrix, fill(0, 3))
end end
println("$(length(samples)) sample solutions:") wtns = witness_set(system, cut_subspace)
for soln in filter(is_nontrivial, solution.(filter(isreal, results(wtns))))
if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
push!(samples, soln)
end
end
end
if through_trivial
println("--- planes through trivial solution ---")
else
println("--- planes through origin ---")
end
println("$(length(samples)) sample solutions, not including the trivial one:")
for soln in samples for soln in samples
display([vbls round.(soln, digits = 6)]) ## [verbose] display([vbls round.(soln, digits = 6)])
k_sq = abs2(soln[1]) k_sq = abs2(soln[1])
if abs2(soln[end-2]) > 1e-12 if abs2(soln[end-2]) > 1e-12
if k_sq < 1e-12 if k_sq < 1e-12
println("center at infinity: z coordinates $(round(soln[end], digits = 6)) and $(round(soln[end-1], digits = 6))}") println(" center at infinity: z coordinates $(round(soln[end], digits = 6)) and $(round(soln[end-1], digits = 6))")
else else
sum_sq = soln[4]^2 + soln[7]^2 + soln[end-2]^2 / k_sq sum_sq = soln[4]^2 + soln[7]^2 + soln[end-2]^2 / k_sq
println(" center on z axis: r² = $(round(1/k_sq, digits = 6)), x² + y² + h² = $(round(sum_sq, digits = 6))") println(" center on z axis: r² = $(round(1/k_sq, digits = 6)), x² + y² + h² = $(round(sum_sq, digits = 6))")
@ -314,3 +335,4 @@ for soln in samples
println(" center at origin: r² = $(round(1/k_sq, digits = 6)); x² + y² + z² = $(round(sum_sq, digits = 6))") println(" center at origin: r² = $(round(1/k_sq, digits = 6)); x² + y² + z² = $(round(sum_sq, digits = 6))")
end end
end end
end