def check_S3_quadint_equals_numint(l=1, m=1, n=1, b=10):
# Create grids on the sphere
x = S3.meshgrid(b=b, grid_type='SOFT')
x = np.c_[x[0][..., None], x[1][..., None], x[2][..., None]]
# Compute quadrature weights
w = S3.quadrature_weights(b=b, grid_type='SOFT')
# Define a polynomial function, to be evaluated at one point or at an array of points
def f1(alpha, beta, gamma):
df = wigner_D_function(l=l, m=m, n=n, alpha=alpha, beta=beta, gamma=gamma)
return df * df.conj()
def f1a(xs):
d = np.zeros(x.shape[:-1])
for i in range(d.shape[0]):
for j in range(d.shape[1]):
for k in range(d.shape[2]):
d[i, j, k] = f1(xs[i, j, k, 0], xs[i, j, k, 1], xs[i, j, k, 2])
return d
# Obtain the "true" value of the integral of the function over the sphere, using scipy's numerical integration
# routines
i1 = S3.integrate(f1, normalize=True)
# Compute the integral using the quadrature formulae
i1_w = S3.integrate_quad(f1a(x), grid_type='SOFT', normalize=True, w=w)
# Check error
print(b, l, m, n, 'results:', i1_w, i1, 'diff:', np.abs(i1_w - i1))
assert np.isclose(np.abs(i1_w - i1), 0.0)
def naive_conv(l1=1, m1=1, l2=1, m2=1, g_parameterization='EA313'):
f1 = lambda t, p: sh(l=l1,
m=m1,
theta=t,
phi=p,
field='real',
normalization='quantum',
condon_shortley=True)
f2 = lambda t, p: sh(l=l2,
m=m2,
theta=t,
phi=p,
field='real',
normalization='quantum',
condon_shortley=True)
theta, phi = S2.meshgrid(b=3, grid_type='Gauss-Legendre')
f1_grid = f1(theta, phi)
f2_grid = f2(theta, phi)
alpha, beta, gamma = S3.meshgrid(b=3,
grid_type='SOFT') # TODO check convention
f12_grid = np.zeros_like(alpha)
for i in range(alpha.shape[0]):
for j in range(alpha.shape[1]):
for k in range(alpha.shape[2]):
f12_grid[i, j, k] = naive_S2_conv_v2(f1, f2, alpha[i, j, k],
beta[i, j, k], gamma[i, j,
k],
g_parameterization)
print(i, j, k, f12_grid[i, j, k])
return f1_grid, f2_grid, f12_grid
def compare_naive_and_spectral_conv():
f1 = lambda t, p: sh(l=2,
m=1,
theta=t,
phi=p,
field='real',
normalization='quantum',
condon_shortley=True)
f2 = lambda t, p: sh(l=2,
m=1,
theta=t,
phi=p,
field='real',
normalization='quantum',
condon_shortley=True)
theta, phi = S2.meshgrid(b=4, grid_type='Gauss-Legendre')
f1_grid = f1(theta, phi)
f2_grid = f2(theta, phi)
alpha, beta, gamma = S3.meshgrid(b=4,
grid_type='SOFT') # TODO check convention
f12_grid_spectral = spectral_S2_conv(f1_grid,
f2_grid,
s2_fft=None,
so3_fft=None)
f12_grid = np.zeros_like(alpha)
for i in range(alpha.shape[0]):
for j in range(alpha.shape[1]):
for k in range(alpha.shape[2]):
f12_grid[i, j, k] = naive_S2_conv(f1, f2, alpha[i, j, k],
beta[i, j, k], gamma[i, j,
k])
print(i, j, k, f12_grid[i, j, k])
return f1_grid, f2_grid, f12_grid, f12_grid_spectral