88 lines
2.5 KiB
Julia
88 lines
2.5 KiB
Julia
include("Engine.jl")
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using SparseArrays
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using AbstractAlgebra
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using PolynomialRoots
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using Random
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# initialize the partial gram matrix for an arrangement of seven spheres in
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# which spheres 1 through 5 are mutually tangent, and spheres 3 through 7 are
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# also mutually tangent
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J = Int64[]
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K = Int64[]
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values = BigFloat[]
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for j in 1:7
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for k in 1:7
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if (j <= 5 && k <= 5) || (j >= 3 && k >= 3)
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push!(J, j)
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push!(K, k)
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push!(values, j == k ? 1 : -1)
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end
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end
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end
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gram = sparse(J, K, values)
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# set the independent variable
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#
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# using gram[6, 2] or gram[7, 1] as the independent variable seems to stall
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# convergence, even if its value comes from a known solution, like
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#
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# gram[6, 2] = 0.9936131705272925
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#
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indep_val = -9//5
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gram[6, 1] = BigFloat(indep_val)
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gram[1, 6] = gram[6, 1]
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# in this initial guess, the mutual tangency condition is satisfied for spheres
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# 1 through 5
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Random.seed!(50793)
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guess = hcat(
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sqrt(1/BigFloat(2)) * BigFloat[
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1 1 -1 -1 0;
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1 -1 1 -1 0;
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1 -1 -1 1 0;
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0 0 0 0 -sqrt(BigFloat(6));
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1 1 1 1 2;
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] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
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Engine.rand_on_shell(fill(BigFloat(-1), 2))
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)
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# complete the gram matrix using gradient descent followed by Newton's method
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L, history = Engine.realize_gram_gradient(gram, guess, scaled_tol = 0.01)
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L_pol, history_pol = Engine.realize_gram_newton(gram, L)
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completed_gram = L'*Engine.Q*L
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println("Completed Gram matrix:\n")
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display(completed_gram)
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println("\nSteps: ", size(history.scaled_loss, 1), " + ", size(history_pol.scaled_loss, 1))
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println("Loss: ", history_pol.scaled_loss[end], "\n")
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# === algebraic check ===
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R, gens = polynomial_ring(AbstractAlgebra.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"])
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x = gens[1]
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t = gens[2:4]
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S, u = polynomial_ring(AbstractAlgebra.Rationals{BigInt}(), "u")
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M = matrix_space(R, 7, 7)
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gram_symb = M(R[
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1 -1 -1 -1 -1 t[1] t[2];
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-1 1 -1 -1 -1 x t[3]
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-1 -1 1 -1 -1 -1 -1;
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-1 -1 -1 1 -1 -1 -1;
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-1 -1 -1 -1 1 -1 -1;
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t[1] x -1 -1 -1 1 -1;
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t[2] t[3] -1 -1 -1 -1 1
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])
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rank_constraints = det.([
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gram_symb[1:6, 1:6],
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gram_symb[2:7, 2:7],
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gram_symb[[1, 3, 4, 5, 6, 7], [1, 3, 4, 5, 6, 7]]
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])
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# solve for x and t
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x_constraint = 25//16 * to_univariate(S, evaluate(rank_constraints[1], [2], [indep_val]))
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t₂_constraint = 25//16 * to_univariate(S, evaluate(rank_constraints[3], [2], [indep_val]))
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x_vals = PolynomialRoots.roots(x_constraint.coeffs)
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t₂_vals = PolynomialRoots.roots(t₂_constraint.coeffs)
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