Start tetrahedron radius ratio example

Add the vertices of the tetrahedron to the `sphere-in-tetrahedron`
example.

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

@@ -0,0 +1,87 @@
+include("Engine.jl")
+
+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 1:10
+  for k in 1:10
+    filled = false
+    if j == 10
+      if k <= 4
+        push!(values, 0)
+        filled = true
+      end
+    elseif k == 10
+      if j <= 4
+        push!(values, 0)
+        filled = true
+      end
+    elseif j == k
+      push!(values, j <= 5 ? 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 k <= 9 && k - j != 5
+        push!(values, 0)
+        filled = true
+      end
+    elseif k <= 4
+      if j == 5
+        push!(values, 1)
+        filled = true
+      elseif j <= 9 && j - k != 5
+        push!(values, 0)
+        filled = true
+      end
+    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
+    1 -1  1 -1  0
+    1 -1 -1  1  0
+    0  0  0  0  1.5
+    1  1  1  1  0.5
+  ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
+  Engine.point([-1, -1, -1]),
+  Engine.point([ 1, -1, -1]),
+  Engine.point([-1,  1, -1]),
+  Engine.point([ 1, -1,  1]),
+  BigFloat[0, 0, 0, 0, 1]
+)
+frozen = vcat(
+  [CartesianIndex(4, k) for k in 6:9],
+  [CartesianIndex(j, 10) for j in 1:5]
+)
+
+# complete the gram matrix
+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], "\n")