def test_SWSH_conjugation(special_angles, ell_max):
# {s}Y{l,m}.conjugate() = (-1.)**(s+m) {-s}Y{l,-m}
indices1 = sf.LM_range(0, ell_max)
indices2 = np.array([[ell, -m] for ell, m in indices1])
neg1_to_m = np.array([(-1.)**m for ell, m in indices1])
for iota in special_angles:
for phi in special_angles:
R = quaternion.from_spherical_coords(iota, phi)
for s in range(1-ell_max, ell_max):
assert np.allclose(sf.SWSH(R, s, indices1),
(-1.)**s * neg1_to_m * np.conjugate(sf.SWSH(R, -s, indices2)),
atol=1e-15, rtol=1e-15)
def test_SWSH_signatures(Rs):
"""There are two ways to call the SWSH function: with an array of Rs, or with an array of (ell,m) values. This
test ensures that the results are the same in both cases."""
s_max = 5
ss = np.arange(-s_max, s_max + 1)
ell_max = 5
ell_ms = sf.LM_range(0, ell_max)
SWSHs1 = np.empty((Rs.size, ss.size, ell_ms.size // 2), dtype=np.complex)
SWSHs2 = np.empty_like(SWSHs1)
for i_s, s in enumerate(ss):
for i_ellm, (ell, m) in enumerate(ell_ms):
SWSHs1[:, i_s, i_ellm] = sf.SWSH(Rs, s, [ell, m])
for i_s, s in enumerate(ss):
for i_R, R in enumerate(Rs):
SWSHs2[i_R, i_s, :] = sf.SWSH(R, s, ell_ms)
assert np.array_equal(SWSHs1, SWSHs2)
def test_vector_as_ell_1_modes(special_angles):
indices = np.array([[1, -1], [1, 0], [1, 1]])
def nhat(theta, phi):
return np.array([
math.sin(theta) * math.cos(phi),
math.sin(theta) * math.sin(phi),
math.cos(theta)
])
np.random.seed(123)
for rep in range(1000):
vector = np.random.uniform(-1, 1, size=(3, ))
vec_ell_m = sf.vector_as_ell_1_modes(vector)
assert np.allclose(vector,
sf.vector_from_ell_1_modes(vec_ell_m),
atol=1.0e-16,
rtol=1.0e-15)
for theta in special_angles:
for phi in special_angles:
dot1 = np.dot(vector, nhat(theta, phi))
dot2 = np.dot(
vec_ell_m,
sf.SWSH(quaternion.from_spherical_coords(theta, phi), 0,
indices)).real
assert abs(dot1 - dot2) < 1e-15
def test_SWSH_spin_behavior(Rs, special_angles, ell_max):
# We expect that the SWSHs behave according to
# sYlm( R * exp(gamma*z/2) ) = sYlm(R) * exp(-1j*s*gamma)
# See http://moble.github.io/spherical_functions/SWSHs.html#fn:2
# for a more detailed explanation
# print("")
for i, R in enumerate(Rs):
# print("\t{0} of {1}: R = {2}".format(i, len(Rs), R))
for gamma in special_angles:
for ell in range(ell_max + 1):
for s in range(-ell, ell + 1):
LM = np.array([[ell, m] for m in range(-ell, ell + 1)])
Rgamma = R * np.quaternion(math.cos(gamma / 2.), 0, 0, math.sin(gamma / 2.))
sYlm1 = sf.SWSH(Rgamma, s, LM)
sYlm2 = sf.SWSH(R, s, LM) * cmath.exp(-1j * s * gamma)
# print(R, gamma, ell, s, np.max(np.abs(sYlm1-sYlm2)))
assert np.allclose(sYlm1, sYlm2, atol=ell ** 6 * precision_SWSH, rtol=ell ** 6 * precision_SWSH)
def test_SWSH_values(special_angles, ell_max):
for iota in special_angles:
for phi in special_angles:
for ell in range(ell_max + 1):
for s in range(-ell, ell + 1):
for m in range(-ell, ell + 1):
R = quaternion.from_euler_angles(phi, iota, 0)
assert abs(sf.SWSH(R, s, np.array([[ell, m]]))
- slow_sYlm(s, ell, m, iota, phi)) < ell_max ** 6 * precision_SWSH
def test_SWSH_WignerD_expression(special_angles, ell_max):
for iota in special_angles:
for phi in special_angles:
for ell in range(ell_max + 1):
for s in range(-ell, ell + 1):
R = quaternion.from_euler_angles(phi, iota, 0)
LM = np.array([[ell, m] for m in range(-ell, ell + 1)])
Y = sf.SWSH(R, s, LM)
LMS = np.array([[ell, m, -s] for m in range(-ell, ell + 1)])
D = (-1.) ** (s) * math.sqrt((2 * ell + 1) / (4 * np.pi)) * sf.Wigner_D_element(R.a, R.b, LMS)
assert np.allclose(Y, D, atol=ell ** 6 * precision_SWSH, rtol=ell ** 6 * precision_SWSH)
def test_SWSH_grid(special_angles, ell_max):
LM = sf.LM_range(0, ell_max)
# Test flat array arrangement
R_grid = np.array([quaternion.from_euler_angles(alpha, beta, gamma).normalized()
for alpha in special_angles
for beta in special_angles
for gamma in special_angles])
for s in range(-ell_max + 1, ell_max):
values_explicit = np.array([sf.SWSH(R, s, LM) for R in R_grid])
values_grid = sf.SWSH_grid(R_grid, s, ell_max)
assert np.array_equal(values_explicit, values_grid)
# Test nested array arrangement
R_grid = np.array([[[quaternion.from_euler_angles(alpha, beta, gamma)
for alpha in special_angles]
for beta in special_angles]
for gamma in special_angles])
for s in range(-ell_max + 1, ell_max):
values_explicit = np.array([[[sf.SWSH(R, s, LM) for R in R1] for R1 in R2] for R2 in R_grid])
values_grid = sf.SWSH_grid(R_grid, s, ell_max)
assert np.array_equal(values_explicit, values_grid)
def swsh(s, modes, theta, phi, psi=0):
"""
Return a value of a spin-weighted spherical harmonic of spin-weight s.
If passed a list of several modes, then a numpy array is returned with
SWSH values of each mode for the given point.
For one mode: swsh(s,[(l,m)],theta,phi,psi=0)
For several modes: swsh(s,[(l1,m1),(l2,m2),(l3,m3),...],theta,phi,psi=0)
"""
import spherical_functions as sf
import quaternion as qt
return sf.SWSH(qt.from_spherical_coords(theta, phi), s, modes) * np.exp(
1j * s * psi
)
def test_constant_as_ell_0_mode(special_angles):
indices = np.array([[0, 0]])
np.random.seed(123)
for imaginary_part in [0.0,
1.0j]: # Test both real and imaginary constants
for rep in range(1000):
constant = np.random.uniform(
-1, 1) + imaginary_part * np.random.uniform(-1, 1)
const_ell_m = sf.constant_as_ell_0_mode(constant)
assert abs(constant -
sf.constant_from_ell_0_mode(const_ell_m)) < 1e-15
for theta in special_angles:
for phi in special_angles:
dot = np.dot(
const_ell_m,
sf.SWSH(quaternion.from_spherical_coords(theta, phi),
0, indices))
assert abs(constant - dot) < 1e-15
