def test_LMpM_range(ell_max):
for l_max in range(ell_max + 1):
assert np.array_equal(
sf.LMpM_range(l_max, l_max),
np.array([[l_max, mp, m] for mp in range(-l_max, l_max + 1)
for m in range(-l_max, l_max + 1)]))
for l_min in range(l_max + 1):
assert np.array_equal(
sf.LMpM_range(l_min, l_max),
np.array([[ell, mp, m] for ell in range(l_min, l_max + 1)
for mp in range(-ell, ell + 1)
for m in range(-ell, ell + 1)]))
def test_Wigner_D_matrices_representation_property(Rs, ell_max):
# Test the representation property for special and random angles
# Can't try half-integers because Wigner_D_matrices doesn't accept them
ell_max = min(8, ell_max)
LMpM = sf.LMpM_range(0, ell_max)
print("")
D1 = np.empty((LMpM.shape[0], ), dtype=complex)
D2 = np.empty((LMpM.shape[0], ), dtype=complex)
D12 = np.empty((LMpM.shape[0], ), dtype=complex)
for i, R1 in enumerate(Rs):
print("\t{0} of {1}: R1 = {2}".format(i + 1, len(Rs), R1))
for j, R2 in enumerate(Rs[i:]):
# print("\t\t{0} of {1}: R2 = {2}".format(j+1, len(Rs), R2))
R12 = R1 * R2
sf._Wigner_D_matrices(R1.a, R1.b, 0, ell_max, D1)
sf._Wigner_D_matrices(R2.a, R2.b, 0, ell_max, D2)
sf._Wigner_D_matrices(R12.a, R12.b, 0, ell_max, D12)
M12 = np.array([
np.sum([
D1[sf.LMpM_index(ell, mp, mpp, 0)] *
D2[sf.LMpM_index(ell, mpp, m, 0)]
for mpp in range(-ell, ell + 1)
]) for ell in range(ell_max + 1)
for mp in range(-ell, ell + 1) for m in range(-ell, ell + 1)
])
assert np.allclose(M12,
D12,
atol=ell_max * precision_Wigner_D_element)
def test_Wigner_D_element_symmetries(Rs, ell_max):
LMpM = sf.LMpM_range(0, ell_max)
# D_{mp,m}(R) = (-1)^{mp+m} \bar{D}_{-mp,-m}(R)
MpPM = np.array([
mp + m for ell in range(ell_max + 1) for mp in range(-ell, ell + 1)
for m in range(-ell, ell + 1)
])
LmMpmM = np.array([[ell, -mp, -m] for ell in range(ell_max + 1)
for mp in range(-ell, ell + 1)
for m in range(-ell, ell + 1)])
print()
for R in Rs:
print("\t", R)
assert np.allclose(sf.Wigner_D_element(R, LMpM), (-1.)**MpPM *
np.conjugate(sf.Wigner_D_element(R, LmMpmM)),
atol=ell_max**2 * precision_Wigner_D_element,
rtol=ell_max**2 * precision_Wigner_D_element)
# D is a unitary matrix, so its conjugate transpose is its
# inverse. D(R) should equal the matrix inverse of D(R^{-1}).
# So: D_{mp,m}(R) = \bar{D}_{m,mp}(\bar{R})
LMMp = np.array([[ell, m, mp] for ell in range(ell_max + 1)
for mp in range(-ell, ell + 1)
for m in range(-ell, ell + 1)])
for R in Rs:
print("\t", R)
assert np.allclose(sf.Wigner_D_element(R, LMpM),
np.conjugate(sf.Wigner_D_element(R.inverse(),
LMMp)),
atol=ell_max**4 * precision_Wigner_D_element,
rtol=ell_max**4 * precision_Wigner_D_element)
def test_Wigner_D_element_roundoff(Rs, ell_max):
LMpM = sf.LMpM_range(0, ell_max)
# Test rotations with |Ra|<1e-15
expected = [((-1.) ** ell if mp == -m else 0.0) for ell in range(ell_max + 1) for mp in range(-ell, ell + 1) for m in
range(-ell, ell + 1)]
assert np.allclose(sf.Wigner_D_element(quaternion.x, LMpM), expected,
atol=ell_max * precision_Wigner_D_element, rtol=ell_max * precision_Wigner_D_element)
expected = [((-1.) ** (ell + m) if mp == -m else 0.0) for ell in range(ell_max + 1) for mp in range(-ell, ell + 1)
for m in range(-ell, ell + 1)]
assert np.allclose(sf.Wigner_D_element(quaternion.y, LMpM), expected,
atol=ell_max * precision_Wigner_D_element, rtol=ell_max * precision_Wigner_D_element)
for theta in np.linspace(0, 2 * np.pi):
expected = [((-1.) ** (ell + m) * (np.cos(theta) + 1j * np.sin(theta)) ** (2 * m) if mp == -m else 0.0)
for ell in range(ell_max + 1) for mp in range(-ell, ell + 1) for m in range(-ell, ell + 1)]
assert np.allclose(sf.Wigner_D_element(np.cos(theta) * quaternion.y + np.sin(theta) * quaternion.x, LMpM),
expected,
atol=ell_max * precision_Wigner_D_element, rtol=ell_max * precision_Wigner_D_element)
# Test rotations with |Rb|<1e-15
expected = [(1.0 if mp == m else 0.0) for ell in range(ell_max + 1) for mp in range(-ell, ell + 1) for m in
range(-ell, ell + 1)]
assert np.allclose(sf.Wigner_D_element(quaternion.one, LMpM), expected,
atol=ell_max * precision_Wigner_D_element, rtol=ell_max * precision_Wigner_D_element)
expected = [((-1.) ** m if mp == m else 0.0) for ell in range(ell_max + 1) for mp in range(-ell, ell + 1) for m in
range(-ell, ell + 1)]
assert np.allclose(sf.Wigner_D_element(quaternion.z, LMpM), expected,
atol=ell_max * precision_Wigner_D_element, rtol=ell_max * precision_Wigner_D_element)
for theta in np.linspace(0, 2 * np.pi):
expected = [((np.cos(theta) + 1j * np.sin(theta)) ** (2 * m) if mp == m else 0.0)
for ell in range(ell_max + 1) for mp in range(-ell, ell + 1) for m in range(-ell, ell + 1)]
assert np.allclose(sf.Wigner_D_element(np.cos(theta) * quaternion.one + np.sin(theta) * quaternion.z, LMpM),
expected,
atol=ell_max * precision_Wigner_D_element, rtol=ell_max * precision_Wigner_D_element)
def test_Wigner_D_matrices_negative_argument(Rs, ell_max):
# For integer ell, D(R)=D(-R)
LMpM = sf.LMpM_range(0, ell_max)
a = np.empty((LMpM.shape[0], ), dtype=complex)
b = np.empty((LMpM.shape[0], ), dtype=complex)
for R in Rs:
sf._Wigner_D_matrices(R.a, R.b, 0, ell_max, a)
sf._Wigner_D_matrices(-R.a, -R.b, 0, ell_max, b)
assert np.allclose(a, b, rtol=ell_max * precision_Wigner_D_element)
def test_LMpM_index(ell_max):
for ell_min in range(ell_max + 1):
LMpM = sf.LMpM_range(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):
assert np.array_equal(
np.array([ell, mp, m]),
LMpM[sf.LMpM_index(ell, mp, m, ell_min)])
def test_Wigner_D_element_overflow(Rs, ell_max):
assert sf.ell_max >= 15 # Test can't work if this has been set lower
ell_max = max(sf.ell_max, ell_max)
LMpM = sf.LMpM_range(0, ell_max)
# Test |Ra|=1e-10
R = np.quaternion(1.e-10, 1, 0, 0).normalized()
assert np.all(np.isfinite(sf.Wigner_D_element(R, LMpM)))
# Test |Rb|=1e-10
R = np.quaternion(1, 1.e-10, 0, 0).normalized()
assert np.all(np.isfinite(sf.Wigner_D_element(R, LMpM)))
def test_Wigner_D_signatures(Rs):
"""There are two ways to call the WignerD function: with an array of Rs, or with an array of (ell,mp,m) values.
This test ensures that the results are the same in both cases."""
# from spherical_functions.WignerD import _Wigner_D_elements
ell_max = 6
ell_mp_m = sf.LMpM_range(0, ell_max)
Ds1 = np.zeros((Rs.size, ell_mp_m.shape[0]), dtype=np.complex)
Ds2 = np.zeros_like(Ds1)
for i, R in enumerate(Rs):
Ds1[i, :] = sf.Wigner_D_element(R, ell_mp_m)
for i, (ell, mp, m) in enumerate(ell_mp_m):
Ds2[:, i] = sf.Wigner_D_element(Rs, ell, mp, m)
assert np.allclose(Ds1, Ds2, rtol=3e-15, atol=3e-15)
def test_Wigner_D_matrix(Rs, ell_max):
for l_min in [0, 1, 2, ell_max // 2, ell_max - 1]:
print("")
for l_max in range(l_min + 1, ell_max + 1):
print("\tWorking on (l_min,l_max)=({0},{1})".format(l_min, l_max))
LMpM = sf.LMpM_range(l_min, l_max)
for R in Rs:
elements = sf.Wigner_D_element(R, LMpM)
matrix = np.empty(LMpM.shape[0], dtype=complex)
sf._Wigner_D_matrices(R.a, R.b, l_min, l_max, matrix)
assert np.allclose(elements, matrix,
atol=1e3 * l_max * ell_max * precision_Wigner_D_element,
rtol=1e3 * l_max * ell_max * precision_Wigner_D_element)
def test_Wigner_D_input_types(Rs, special_angles, ell_max):
LMpM = sf.LMpM_range(0, ell_max // 2)
# Compare with more explicit forms given in Euler angles
print("")
for alpha in special_angles:
print("\talpha={0}".format(alpha)) # Need to show some progress to Travis
for beta in special_angles:
for gamma in special_angles:
a = sf.Wigner_D_element(alpha, beta, gamma, LMpM)
b = sf.Wigner_D_element(quaternion.from_euler_angles(alpha, beta, gamma), LMpM)
assert np.allclose(a, b,
atol=ell_max ** 6 * precision_Wigner_D_element,
rtol=ell_max ** 6 * precision_Wigner_D_element)
for R in Rs:
a = sf.Wigner_D_element(R, LMpM)
b = sf.Wigner_D_element(R.a, R.b, LMpM)
assert np.allclose(a, b,
atol=ell_max ** 6 * precision_Wigner_D_element,
rtol=ell_max ** 6 * precision_Wigner_D_element)
def test_Wigner_D_linear_indices(ell_max):
for l_min in range(ell_max):
for l_max in range(l_min + 1, ell_max + 1):
LMpM = sf.LMpM_range(l_min, l_max)
assert len(LMpM) == sf._total_size_D_matrices(l_min, l_max)
for ell in range(l_min, l_max + 1):
assert list(LMpM[sf._linear_matrix_offset(ell, l_min)]) == [ell, -ell, -ell]
for ell in range(l_min, l_max + 1):
for mp in range(-ell, ell + 1):
i = sf._linear_matrix_diagonal_index(ell, mp)
o = sf._linear_matrix_offset(ell, l_min)
assert list(LMpM[i + o]) == [ell, mp, mp]
for m in range(-ell, ell + 1):
i = sf._linear_matrix_index(ell, mp, m)
o = sf._linear_matrix_offset(ell, l_min)
assert list(LMpM[i + o]) == [ell, mp, m]
