dyna3/engine-proto/gram-test/tetrahedron-radius-ratio.jl

include("Engine.jl")

using LinearAlgebra
using SparseArrays
using Random

# initialize the partial gram matrix for a sphere inscribed in a regular
# tetrahedron
J = Int64[]
K = Int64[]
values = BigFloat[]
for j in 1:11
  for k in 1:11
    filled = false
    if j == 11
      if k <= 4
        push!(values, 0)
        filled = true
      end
    elseif k == 11
      if j <= 4
        push!(values, 0)
        filled = true
      end
    elseif j == k
      push!(values, j <= 6 ? 1 : 0)
      filled = true
    elseif j <= 4
      if k <= 4
        push!(values, -1/BigFloat(3))
        filled = true
      elseif k == 5
        push!(values, -1)
        filled = true
      elseif 7 <= k <= 10 && k - j != 6
        push!(values, 0)
        filled = true
      end
    elseif k <= 4
      if j == 5
        push!(values, -1)
        filled = true
      elseif 7 <= j <= 10 && j - k != 6
        push!(values, 0)
        filled = true
      end
    elseif j == 6 && 7 <= k <= 10 || k == 6 && 7 <= j <= 10
      push!(values, 0)
      filled = true
    end
    if filled
      push!(J, j)
      push!(K, k)
    end
  end
end
gram = sparse(J, K, values)

# set initial guess
Random.seed!(99230)
guess = hcat(
  sqrt(1/BigFloat(3)) * BigFloat[
    1  1 -1 -1  0    0
    1 -1  1 -1  0    0
    1 -1 -1  1  0    0
    0  0  0  0  1.5  0.5
    1  1  1  1 -0.5 -1.5
  ] + 0.0*Engine.rand_on_shell(fill(BigFloat(-1), 6)),
  Engine.point([-0.5, -0.5, -0.5] + 0.3*randn(3)),
  Engine.point([-0.5,  0.5,  0.5] + 0.3*randn(3)),
  Engine.point([ 0.5, -0.5,  0.5] + 0.3*randn(3)),
  Engine.point([ 0.5,  0.5, -0.5] + 0.3*randn(3)),
  BigFloat[0, 0, 0, 0, 1]
)
frozen = vcat(
  [CartesianIndex(4, k) for k in 7:10],
  [CartesianIndex(j, 11) for j in 1:5]
)

# complete the gram matrix using Newton's method with backtracking
L, success, history = Engine.realize_gram(gram, guess, frozen)
completed_gram = L'*Engine.Q*L
println("Completed Gram matrix:\n")
display(completed_gram)
if success
  println("\nTarget accuracy achieved!")
else
  println("\nFailed to reach target accuracy")
end
println("Steps: ", size(history.scaled_loss, 1))
println("Loss: ", history.scaled_loss[end])
if success
  infty = BigFloat[0, 0, 0, 0, 1]
  radius_ratio = dot(infty, Engine.Q * L[:,5]) / dot(infty, Engine.Q * L[:,6])
  println("\nCircumradius / inradius: ", radius_ratio)
end