Seek sample solutions by cutting with a hyperplane
The example hyperplane yields a single solution, with multiplicity six. You can find it analytically by hand, and homotopy continuation finds it numerically.
This commit is contained in:
Aaron Fenyes
parent
43cbf8a3a0
commit
45aaaafc8f
@ -2,12 +2,13 @@ include("HittingSet.jl")
|
||||
|
||||
module Engine
|
||||
|
||||
export Construction, mprod
|
||||
export Construction, mprod, codimension, dimension
|
||||
|
||||
import Subscripts
|
||||
using LinearAlgebra
|
||||
using AbstractAlgebra
|
||||
using Groebner
|
||||
using HomotopyContinuation: Variable, Expression, System
|
||||
using ..HittingSet
|
||||
|
||||
# --- commutative algebra ---
|
||||
@ -27,6 +28,34 @@ end
|
||||
dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
|
||||
length(gens(base_ring(I))) - codimension(I, maxdepth)
|
||||
|
||||
# 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
|
||||
|
||||
## [to do] not needed right now
|
||||
# create a ModelKit.System from a list of elements of a multivariate polynomial
|
||||
# ring. the variable ordering is taken from the polynomial ring
|
||||
##function System(eqns::AbstractVector{MPolyRingElem})
|
||||
## if isempty(eqns)
|
||||
## return System([])
|
||||
## else
|
||||
## variables = Variable.(symbols(parent(f)))
|
||||
## return System(Expression.(eqns), variables = variables)
|
||||
## end
|
||||
##end
|
||||
|
||||
# --- primitve elements ---
|
||||
|
||||
abstract type Element{T} end
|
||||
@ -189,39 +218,75 @@ function realize(ctx::Construction{T}) where T
|
||||
append!(eqns, [sum(sph.coords[k] for sph in ctx.spheres) for k in 3:4])
|
||||
end
|
||||
|
||||
Generic.Ideal(coordring, eqns)
|
||||
(Generic.Ideal(coordring, eqns), eqns)
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
# ~~~ sandbox setup ~~~
|
||||
|
||||
using AbstractAlgebra
|
||||
using HomotopyContinuation
|
||||
|
||||
CoeffType = Rational{Int64}
|
||||
|
||||
a = Engine.Point{CoeffType}()
|
||||
s = Engine.Sphere{CoeffType}()
|
||||
a_on_s = Engine.LiesOn{CoeffType}(a, s)
|
||||
ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
|
||||
ideal_a_s = Engine.realize(ctx)
|
||||
println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freeom")
|
||||
##ideal_a_s = Engine.realize(ctx)
|
||||
##println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freedom")
|
||||
|
||||
b = Engine.Point{CoeffType}()
|
||||
b_on_s = Engine.LiesOn{CoeffType}(b, s)
|
||||
Engine.push!(ctx, b)
|
||||
Engine.push!(ctx, s)
|
||||
Engine.push!(ctx, b_on_s)
|
||||
ideal_ab_s = Engine.realize(ctx)
|
||||
println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freeom")
|
||||
ideal_ab_s, eqns_ab_s = Engine.realize(ctx)
|
||||
println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freedom")
|
||||
|
||||
spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
|
||||
tangencies = [
|
||||
Engine.AlignsWithBy{CoeffType}(
|
||||
spheres[n],
|
||||
spheres[mod1(n+1, length(spheres))],
|
||||
CoeffType(-1//1)
|
||||
)
|
||||
for n in 1:3
|
||||
##spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
|
||||
##tangencies = [
|
||||
## Engine.AlignsWithBy{CoeffType}(
|
||||
## spheres[n],
|
||||
## spheres[mod1(n+1, length(spheres))],
|
||||
## CoeffType(-1//1)
|
||||
## )
|
||||
## for n in 1:3
|
||||
##]
|
||||
##ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
|
||||
##ideal_tan_sph = Engine.realize(ctx_tan_sph)
|
||||
##println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freedom")
|
||||
|
||||
# --- test rational cut ---
|
||||
|
||||
cut = [
|
||||
sum(vcat(a.coords, (s.coords - [0, 0, 0, 0, 1])))
|
||||
sum(vcat([2, 1, 1] .* a.coords, [1, 2, 1, 1, 1] .* s.coords - [0, 0, 0, 0, 1]))
|
||||
sum(vcat([1, 2, 0] .* a.coords, [1, 1, 0, 1, 2] .* s.coords - [0, 0, 0, 0, 1]))
|
||||
]
|
||||
ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
|
||||
ideal_tan_sph = Engine.realize(ctx_tan_sph)
|
||||
println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freeom")
|
||||
cut_ideal_ab_s = Generic.Ideal(base_ring(ideal_ab_s), [gens(ideal_ab_s); cut])
|
||||
cut_dim = Engine.dimension(cut_ideal_ab_s)
|
||||
println("Two points on a sphere, after cut: ", cut_dim, " degrees of freedom")
|
||||
if cut_dim == 0
|
||||
vbls = Variable.(symbols(base_ring(ideal_ab_s)))
|
||||
cut_system = System([eqns_ab_s; cut], variables = vbls)
|
||||
cut_result = HomotopyContinuation.solve(cut_system)
|
||||
println("non-singular solutions:")
|
||||
for soln in solutions(cut_result)
|
||||
display(soln)
|
||||
end
|
||||
println("singular solutions:")
|
||||
for sing in singular(cut_result)
|
||||
display(sing.solution)
|
||||
end
|
||||
|
||||
# test corresponding witness set
|
||||
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]
|
||||
cut_subspace = LinearSubspace(cut_matrix, [1, 1, 1])
|
||||
witness = witness_set(System(eqns_ab_s, variables = vbls), cut_subspace)
|
||||
println("witness solutions:")
|
||||
for wtns in solutions(witness)
|
||||
display(wtns)
|
||||
end
|
||||
end
|
||||
Loading…
Reference in New Issue
Block a user