def test_spherical_quadrature(): """ Testing spherical quadrature rule versus numerical integration. """ b = 8 # 10 # Create grids on the sphere x_gl = S2.meshgrid(b=b, grid_type='Gauss-Legendre') x_cc = S2.meshgrid(b=b, grid_type='Clenshaw-Curtis') x_soft = S2.meshgrid(b=b, grid_type='SOFT') x_gl = np.c_[x_gl[0][..., None], x_gl[1][..., None]] x_cc = np.c_[x_cc[0][..., None], x_cc[1][..., None]] x_soft = np.c_[x_soft[0][..., None], x_soft[1][..., None]] # Compute quadrature weights w_gl = S2.quadrature_weights(b=b, grid_type='Gauss-Legendre') w_cc = S2.quadrature_weights(b=b, grid_type='Clenshaw-Curtis') w_soft = S2.quadrature_weights(b=b, grid_type='SOFT') # Define a polynomial function, to be evaluated at one point or at an array of points def f1a(xs): xc = S2.change_coordinates(coords=xs, p_from='S', p_to='C') return xc[..., 0]**2 * xc[..., 1] - 1.4 * xc[..., 2] * xc[ ..., 1]**3 + xc[..., 1] - xc[..., 2]**2 + 2. def f1(theta, phi): xs = np.array([theta, phi]) return f1a(xs) # Obtain the "true" value of the integral of the function over the sphere, using scipy's numerical integration # routines i1 = S2.integrate(f1, normalize=False) # Compute the integral using the quadrature formulae # i1_gl_w = (w_gl * f1a(x_gl)).sum() i1_gl_w = S2.integrate_quad(f1a(x_gl), grid_type='Gauss-Legendre', normalize=False, w=w_gl) print(i1_gl_w, i1, 'diff:', np.abs(i1_gl_w - i1)) assert np.isclose(np.abs(i1_gl_w - i1), 0.0) # i1_cc_w = (w_cc * f1a(x_cc)).sum() i1_cc_w = S2.integrate_quad(f1a(x_cc), grid_type='Clenshaw-Curtis', normalize=False, w=w_cc) print(i1_cc_w, i1, 'diff:', np.abs(i1_cc_w - i1)) assert np.isclose(np.abs(i1_cc_w - i1), 0.0) i1_soft_w = (w_soft * f1a(x_soft)).sum() print(i1_soft_w, i1, 'diff:', np.abs(i1_soft_w - i1)) print(i1_soft_w) print(i1)
def check_orthogonality(L_max=3, grid_type='Gauss-Legendre', field='real', normalization='quantum', condon_shortley=True): theta, phi = S2.meshgrid(b=L_max + 1, grid_type=grid_type) w = S2.quadrature_weights(b=L_max + 1, grid_type=grid_type) for l in range(L_max): for m in range(-l, l + 1): for l2 in range(L_max): for m2 in range(-l2, l2 + 1): Ylm = sh(l, m, theta, phi, field, normalization, condon_shortley) Ylm2 = sh(l2, m2, theta, phi, field, normalization, condon_shortley) dot_numerical = S2.integrate_quad(Ylm * Ylm2.conj(), grid_type=grid_type, normalize=False, w=w) dot_numerical2 = S2.integrate( lambda t, p: sh(l, m, t, p, field, normalization, condon_shortley) * \ sh(l2, m2, t, p, field, normalization, condon_shortley).conj(), normalize=False) sqnorm_analytical = sh_squared_norm(l, normalization, normalized_haar=False) dot_analytical = sqnorm_analytical * (l == l2 and m == m2) print(l, m, l2, m2, field, normalization, condon_shortley, dot_analytical, dot_numerical, dot_numerical2) assert np.isclose(dot_numerical, dot_analytical) assert np.isclose(dot_numerical2, dot_analytical)