dyna3/engine-proto/gram-test/gram-test.jl

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using LinearAlgebra
using AbstractAlgebra
function printgood(msg)
printstyled("", color = :green)
println(" ", msg)
end
function printbad(msg)
printstyled("", color = :red)
println(" ", msg)
end
F, gens = rational_function_field(AbstractAlgebra.Rationals{BigInt}(), ["a₁", "a₂", "b₁", "b₂", "c₁", "c₂"])
a = gens[1:2]
b = gens[3:4]
c = gens[5:6]
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# three mutually tangent spheres which are all perpendicular to the x, y plane
gram = [
-1 1 1;
1 -1 1;
1 1 -1
]
eig = eigen(gram)
n_pos = count(eig.values .> 0.5)
n_neg = count(eig.values .< -0.5)
if n_pos + n_neg == size(gram, 1)
printgood("Non-degenerate subspace")
else
printbad("Degenerate subspace")
end
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sig_rem = Int64[ones(1-n_pos); -ones(4-n_neg)]
unk = hcat(a, b, c)
M = matrix_space(F, 5, 5)
big_gram = M(F.([
diagm(sig_rem) unk;
transpose(unk) gram
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]))
r, p, L, U = lu(big_gram)
if isone(p)
printgood("Found a solution")
else
printbad("Didn't find a solution")
end
solution = transpose(L)
mform = U * inv(solution)
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vals = [0, 0, 0, 1, 0, -3//4]
solution_ex = [evaluate(entry, vals) for entry in solution]
mform_ex = [evaluate(entry, vals) for entry in mform]
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std_basis = [
0 0 0 1 1;
0 0 0 1 -1;
1 0 0 0 0;
0 1 0 0 0;
0 0 1 0 0
]
std_solution = M(F.(std_basis)) * solution
std_solution_ex = std_basis * solution_ex
println("Minkowski form:")
display(mform_ex)
big_gram_recovered = transpose(solution_ex) * mform_ex * solution_ex
valid = all(iszero.(
[evaluate(entry, vals) for entry in big_gram] - big_gram_recovered
))
if valid
printgood("Recovered Gram matrix:")
else
printbad("Didn't recover Gram matrix. Instead, got:")
end
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display(big_gram_recovered)
# this should be a solution
hand_solution = [0 0 1 0 0; 0 0 -1 2 2; 0 0 0 1 -1; 1 0 0 0 0; 0 1 0 0 0]
unmix = Rational{Int64}[[1//2 1//2; 1//2 -1//2] zeros(Int64, 2, 3); zeros(Int64, 3, 2) Matrix{Int64}(I, 3, 3)]
hand_solution_diag = unmix * hand_solution
big_gram_hand_recovered = transpose(hand_solution_diag) * diagm([1; -ones(Int64, 4)]) * hand_solution_diag
println("Gram matrix from hand-written solution:")
display(big_gram_hand_recovered)