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test-matrix.jl
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test-matrix.jl
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using SymPy
using Test
using LinearAlgebra
@testset "Matrix" begin
## matrices
x, y = @vars x y
A = [x 1; 1 x]
B = [x 1; 0 2x]
v = [x, 2]
## These fail for older installations of SymPy
@test simplify(det(A)) == x^2 - 1
## we use inverse for A[:inv]()
# aliased to use inverse
@test simplify.(inv(A) * A) == [1 0; 0 1]
@test simplify.(A * inv(A)) == [1 0; 0 1]
##XXX @test simplify.(A[:inv]() - inv(A)) == zeros(2, 2)
@test SymPy.adjoint(B) == [adjoint(x) 0; 1 adjoint(2x)]
@test SymPy.adjoint(B) == B'
@test A.dual() == sympy.zeros(2, 2)
A1 = Sym[25 15 -5; 15 18 0; -5 0 11]
r = A1.cholesky()
@test r*r.transpose() == A1
# s = LUsolve(A, v)
s = A.LUsolve(B)
@test simplify.(A * s) == B
# norm
@test norm(A) == sqrt(2 * abs(x)^2 + 2)
# test norm for different subtypes of AbstractArray
## XXX @test norm(A) == norm(Symmetric(A)) LinearAlgebra.Symmetric no long works
@test norm(A) == norm(view(A, :, :))
# abs
@test all(convert.(Bool, abs.(A) .≧ 0))
@test abs.(A) == abs.(view(A, :, :))
# is_lower, is_square, is_symmetric much slower than julia only counterparts. May deprecate, but for now they are here
@test A.is_lower == istril(A)
@test A.is_square == true
@test A.is_symmetric() != issymmetric(A)
@vars x real=true
A = [x 1; 1 x]
@test A.is_symmetric() == issymmetric(A)
@test Set(eigvals(A)) == Set([x-1, x+1])
# issue with transpose being non-typestable
@test eltype(transpose(A)) == Sym
@test eltype(Symmetric(A)) == Sym
#numerical tests
M = Sym[1 0 0; 0 1 0; 0 0 x]
evecs = eigvecs(M)
@test evecs[:,1] == [1, 0, 0]
A = Sym[1 2 3; 3 6 2; 2 0 1]
q, r = A.QRdecomposition()
@test q * r == A
@test abs(det(q)) == 1
# for v0.4, the vector type is not correctly inferred
#L = Vector{Sym}[Sym[2,3,5], Sym[3,6,2], Sym[8,3,6]]
# L = collect(sympy.Matrix.(([[2,3,5]], [[3,6,2]], [[8,3,6]]]))
L = collect(sympy.Matrix.([[2,3,5], [3,6,2], [8,3,6]]))
out = sympy.GramSchmidt(L, true) # qualify, as L not SymbolicObject type
for i = 1:3, j = i:3
@test out[i].dot(out[j]) == (i == j ? 1 : 0)
end
A = Sym[4 3; 6 3]
L, U, _ = A.LUdecomposition()
@test L == Sym[1 0; 3//2 1]
A = Sym[1 0; 0 1] * 2
B = Sym[1 2; 3 4]
@test A.diagonal_solve(B) == B/2
M = Sym[1 2 0; 0 3 0; 2 -4 2]
P, D = M.diagonalize()
@test D == [1 0 0; 0 2 0; 0 0 3]
@test P == [-1 0 -1; 0 0 -1; 2 1 2]
@test D == inv(P) * M * P
# test SymPy's expm against Julia's expm
A = [1 2 0; 0 3 0; 2 -4 2]
M = Sym.(A)
## no exp(M)!
U = M.exp() - exp(A)
@test maximum(abs.(N.(U))) <= 1e-12
M = [x y; 1 0]
@test integrate.(M, x) == [x^2/2 x*y; x 0]
@test integrate.(M, Ref((x, 0, 2))) == [2 2y; 2 0]
M = Sym[1 3 0; -2 -6 0; 3 9 6]
@test M.nullspace()[1] == reshape(Sym[-3, 1, 0], 3, 1)
M = Sym[1 2 0; 0 3 0; 2 -4 2]
# M.cofactor uses 0-based indexing!
i, j = 1, 2
@test M.cofactor(i, j) == (-1)^(i+j) * det(M[setdiff(1:3, i+1), setdiff(1:3, j+1)])
@test M.adjugate() / M.det() == M.inv()
M = Sym[ 6 5 -2 -3;
-3 -1 3 3;
2 1 -2 -3;
-1 1 5 5]
P, J = M.jordan_form()
@test J == [2 1 0 0;
0 2 0 0;
0 0 2 1;
0 0 0 2]
@test J == inv(P) * M * P
ρ, ϕ = symbols("rho, phi")
X = [ρ*cos(ϕ), ρ*sin(ϕ), ρ^2]
Y = [ρ, ϕ]
@test X.jacobian(Y) == [cos(ϕ) -ρ*sin(ϕ);
sin(ϕ) ρ*cos(ϕ);
2ρ 0]
X = [ρ*cos(ϕ), ρ*sin(ϕ)]
@test convert(Matrix{Sym}, X.jacobian(Y)) == [cos(ϕ) -ρ*sin(ϕ);
sin(ϕ) ρ*cos(ϕ)]
@test X.jacobian(Y) == view(X, :, :).jacobian(view(Y, :, :))
## Issue #359
if VERSION >= v"1.2"
@vars a
@test eltype([a 1; 1 a] + I) == Sym
@test eltype([a 1; 1 a] + 2I) == Sym
end
end