def test_WignerDindex(ell_max_slow):
k = 0
for ell_max in range(ell_max_slow + 1):
r = sf.WignerDrange(0, ell_max)
for ell in range(ell_max + 1):
for mp in range(-ell, ell + 1):
for m in range(-ell, ell + 1):
i = sf.WignerDindex(ell, mp, m)
assert np.array_equal(r[i], [ell, mp, m]), ((ell, mp, m),
i, ell_max, r)
k += 1
for ell_min in range(ell_max + 1):
r = sf.WignerDrange(ell_min, ell_max)
for ell in range(ell_min, ell_max + 1):
for mp in range(-ell, ell + 1):
for m in range(-ell, ell + 1):
i = sf.WignerDindex(ell, mp, m, ell_min)
assert np.array_equal(r[i],
[ell, mp, m]), ((ell, mp,
m), i, ell_min,
ell_max, r)
k += 1
for mp_max in range(ell_max + 1):
r = sf.WignerDrange(ell_min, mp_max, ell_max)
for ell in range(ell_min, ell_max + 1):
for mp in range(-min(ell, mp_max), min(ell, mp_max) + 1):
for m in range(-ell, ell + 1):
i = sf.WignerDindex(ell, mp, m, ell_min, mp_max)
assert np.array_equal(
r[i], [ell, mp, m]), ((ell, mp, m), i, ell_min,
mp_max, ell_max, r)
k += 1
def test_WignerDrange(ell_max_slow):
def r(ell_min, mp_max, ell_max):
return [(ℓ, mp, m) for ℓ in range(ell_min, ell_max + 1)
for mp in range(-min(ℓ, mp_max),
min(ℓ, mp_max) + 1) for m in range(-ℓ, ℓ + 1)]
for ell_max in range(ell_max_slow // 2 + 1):
for ell_min in range(ell_max + 1):
a = sf.WignerDrange(ell_min, ell_max) # Implicitly, mp_max=ell_max
b = r(ell_min, ell_max, ell_max)
assert np.array_equal(a, b), ((ell_min, mp_max, ell_max), a, b)
for mp_max in range(ell_max + 1):
a = sf.WignerDrange(ell_min, mp_max, ell_max)
b = r(ell_min, mp_max, ell_max)
assert np.array_equal(a, b), ((ell_min, mp_max, ell_max), a, b)
def test_Wigner_D_underflow(Rs, ell_max, eps):
# NOTE: This is a delicate test, which depends on the result underflowing exactly when expected.
# In particular, it should underflow to 0.0 when |mp+m|>32, but should never underflow to 0.0
# when |mp+m|<32. So it's not the end of the world if this test fails, but it does mean that
# the underflow properties have changed, so it might be worth a look.
epsilon = 1.e-10
ϵ = 5 * ell_max * eps
D = np.zeros(sf.WignerDsize(0, ell_max), dtype=complex)
wigner = sf.Wigner(ell_max)
ell_mp_m = sf.WignerDrange(0, ell_max)
# Test |Ra|=1e-10
R = quaternionic.array(epsilon, 1, 0, 0).normalized
wigner.D(R, out=D)
# print(R.to_euler_angles.tolist())
# print(D.tolist())
non_underflowing_indices = np.abs(ell_mp_m[:, 1] + ell_mp_m[:, 2]) < 32
assert np.all(D[non_underflowing_indices] != 0j)
underflowing_indices = np.abs(ell_mp_m[:, 1] + ell_mp_m[:, 2]) > 32
assert np.all(D[underflowing_indices] == 0j)
# Test |Rb|=1e-10
R = quaternionic.array(1, epsilon, 0, 0).normalized
wigner.D(R, out=D)
non_underflowing_indices = np.abs(ell_mp_m[:, 1] - ell_mp_m[:, 2]) < 32
assert np.all(D[non_underflowing_indices] != 0j)
underflowing_indices = np.abs(ell_mp_m[:, 1] - ell_mp_m[:, 2]) > 32
assert np.all(D[underflowing_indices] == 0j)
def test_WignerDsize_mpmax_ellmin(ell_max):
# k = 0
for ell_max in range(ell_max + 1):
for ell_min in range(ell_max + 1):
for mp_max in range(ell_max + 1):
a = sf.WignerDsize(ell_min, mp_max, ell_max)
b = len(sf.WignerDrange(ell_min, mp_max, ell_max))
assert a == b, ((ell_min, mp_max, ell_max), a, b) #, k)
def test_Wigner_D_symmetries(Rs, ell_max, eps):
# We have two obvious symmetries to test. First,
#
# D_{mp,m}(R) = (-1)^{mp+m} \bar{D}_{-mp,-m}(R)
#
# Second, since D is a unitary matrix, its conjugate transpose is its
# inverse; because of the representation property, D(R) should equal the
# matrix inverse of D(R⁻¹). Thus,
#
# D_{mp,m}(R) = \bar{D}_{m,mp}(\bar{R})
ϵ = 5 * ell_max * eps
D1 = np.zeros(sf.WignerDsize(0, ell_max), dtype=complex)
D2 = np.zeros(sf.WignerDsize(0, ell_max), dtype=complex)
wigner = sf.Wigner(ell_max)
ell_mp_m = sf.WignerDrange(0, ell_max)
flipped_indices = np.array(
[sf.WignerDindex(ell, -mp, -m) for ell, mp, m in ell_mp_m])
swapped_indices = np.array(
[sf.WignerDindex(ell, m, mp) for ell, mp, m in ell_mp_m])
signs = (-1)**np.abs(np.sum(ell_mp_m[:, 1:], axis=1))
for R in Rs:
wigner.D(R, out=D1)
wigner.D(R.inverse, out=D2)
# D_{mp,m}(R) = (-1)^{mp+m} \bar{D}_{-mp,-m}(R)
a = D1
b = signs * D1[flipped_indices].conjugate()
assert np.allclose(a, b, rtol=ϵ, atol=ϵ)
# D_{mp,m}(R) = \bar{D}_{m,mp}(\bar{R})
b = D2[swapped_indices].conjugate()
assert np.allclose(a, b, rtol=ϵ, atol=ϵ)
D1 = wigner.D(Rs)
D2 = wigner.D(Rs.inverse)
# D_{mp,m}(R) = (-1)^{mp+m} \bar{D}_{-mp,-m}(R)
a = D1
b = signs * D1[..., flipped_indices].conjugate()
assert np.allclose(a, b, rtol=ϵ, atol=ϵ)
# D_{mp,m}(R) = \bar{D}_{m,mp}(\bar{R})
b = D2[..., swapped_indices].conjugate()
assert np.allclose(a, b, rtol=ϵ, atol=ϵ)
def test_Wigner_D_vs_Wikipedia(special_angles, ell_max_slow, eps):
ell_max = ell_max_slow
ϵ = 5 * ell_max**6 * eps
D = np.zeros(sf.WignerDsize(0, ell_max), dtype=complex)
wigner = sf.Wigner(ell_max)
ell_mp_m = sf.WignerDrange(0, ell_max)
print("")
for alpha in special_angles:
print("\talpha={0}".format(alpha)) # Need to show some progress for CI
for beta in special_angles[len(special_angles) // 2:]: # Skip beta < 0
print("\t\tbeta={0}".format(beta))
for gamma in special_angles:
a = np.conjugate(
np.array([
Wigner_D_Wikipedia(alpha, beta, gamma, ell, mp, m)
for ell, mp, m in ell_mp_m
]))
b = wigner.D(quaternionic.array.from_euler_angles(
alpha, beta, gamma),
out=D)
assert np.allclose(a, b, rtol=ϵ, atol=ϵ)
