Test gradient descent for sphere in tetrahedron

engine-proto/gram-test/sphere-in-tetrahedron.jl

@@ -0,0 +1,43 @@
+include("Engine.jl")
+
+using SparseArrays
+using AbstractAlgebra
+using PolynomialRoots
+
+# initialize the partial gram matrix for a sphere inscribed in a regular
+# tetrahedron
+J = Int64[]
+K = Int64[]
+values = BigFloat[]
+for j in 1:5
+  for k in 1:5
+    push!(J, j)
+    push!(K, k)
+    if j == k
+      push!(values, 1)
+    elseif (j <= 4 && k <= 4)
+      push!(values, -1/BigFloat(3))
+    else
+      push!(values, -1)
+    end
+  end
+end
+gram = sparse(J, K, values)
+
+# set initial guess
+Random.seed!(99230)
+guess = sqrt(1/BigFloat(3)) * BigFloat[
+  1  1 -1 -1  0
+  1 -1  1 -1  0
+  1 -1 -1  1  0
+  1  1  1  1 -2
+  1  1  1  1  1
+] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5))
+
+# complete the gram matrix using gradient descent
+L, history = Engine.realize_gram(gram, guess)
+completed_gram = L'*Engine.Q*L
+println("Completed Gram matrix:\n")
+display(completed_gram)
+println("\nSteps: ", size(history.stepsize, 1))
+println("Loss: ", history.scaled_loss[end], "\n")