# # Tests for the Ellipsoidal Harmonic Function, # Distributed under the same license as SciPy itself. # import numpy as np from numpy.testing import (assert_equal, assert_almost_equal, assert_allclose, assert_, suppress_warnings) from scipy.special._testutils import assert_func_equal from scipy.special import ellip_harm, ellip_harm_2, ellip_normal from scipy.integrate import IntegrationWarning from numpy import sqrt, pi def test_ellip_potential(): def change_coefficient(lambda1, mu, nu, h2, k2): x = sqrt(lambda1**2*mu**2*nu**2/(h2*k2)) y = sqrt((lambda1**2 - h2)*(mu**2 - h2)*(h2 - nu**2)/(h2*(k2 - h2))) z = sqrt((lambda1**2 - k2)*(k2 - mu**2)*(k2 - nu**2)/(k2*(k2 - h2))) return x, y, z def solid_int_ellip(lambda1, mu, nu, n, p, h2, k2): return (ellip_harm(h2, k2, n, p, lambda1)*ellip_harm(h2, k2, n, p, mu) * ellip_harm(h2, k2, n, p, nu)) def solid_int_ellip2(lambda1, mu, nu, n, p, h2, k2): return (ellip_harm_2(h2, k2, n, p, lambda1) * ellip_harm(h2, k2, n, p, mu)*ellip_harm(h2, k2, n, p, nu)) def summation(lambda1, mu1, nu1, lambda2, mu2, nu2, h2, k2): tol = 1e-8 sum1 = 0 for n in range(20): xsum = 0 for p in range(1, 2*n+2): xsum += (4*pi*(solid_int_ellip(lambda2, mu2, nu2, n, p, h2, k2) * solid_int_ellip2(lambda1, mu1, nu1, n, p, h2, k2)) / (ellip_normal(h2, k2, n, p)*(2*n + 1))) if abs(xsum) < 0.1*tol*abs(sum1): break sum1 += xsum return sum1, xsum def potential(lambda1, mu1, nu1, lambda2, mu2, nu2, h2, k2): x1, y1, z1 = change_coefficient(lambda1, mu1, nu1, h2, k2) x2, y2, z2 = change_coefficient(lambda2, mu2, nu2, h2, k2) res = sqrt((x2 - x1)**2 + (y2 - y1)**2 + (z2 - z1)**2) return 1/res pts = [ (120, sqrt(19), 2, 41, sqrt(17), 2, 15, 25), (120, sqrt(16), 3.2, 21, sqrt(11), 2.9, 11, 20), ] with suppress_warnings() as sup: sup.filter(IntegrationWarning, "The occurrence of roundoff error") sup.filter(IntegrationWarning, "The maximum number of subdivisions") for p in pts: err_msg = repr(p) exact = potential(*p) result, last_term = summation(*p) assert_allclose(exact, result, atol=0, rtol=1e-8, err_msg=err_msg) assert_(abs(result - exact) < 10*abs(last_term), err_msg) def test_ellip_norm(): def G01(h2, k2): return 4*pi def G11(h2, k2): return 4*pi*h2*k2/3 def G12(h2, k2): return 4*pi*h2*(k2 - h2)/3 def G13(h2, k2): return 4*pi*k2*(k2 - h2)/3 def G22(h2, k2): res = (2*(h2**4 + k2**4) - 4*h2*k2*(h2**2 + k2**2) + 6*h2**2*k2**2 + sqrt(h2**2 + k2**2 - h2*k2)*(-2*(h2**3 + k2**3) + 3*h2*k2*(h2 + k2))) return 16*pi/405*res def G21(h2, k2): res = (2*(h2**4 + k2**4) - 4*h2*k2*(h2**2 + k2**2) + 6*h2**2*k2**2 + sqrt(h2**2 + k2**2 - h2*k2)*(2*(h2**3 + k2**3) - 3*h2*k2*(h2 + k2))) return 16*pi/405*res def G23(h2, k2): return 4*pi*h2**2*k2*(k2 - h2)/15 def G24(h2, k2): return 4*pi*h2*k2**2*(k2 - h2)/15 def G25(h2, k2): return 4*pi*h2*k2*(k2 - h2)**2/15 def G32(h2, k2): res = (16*(h2**4 + k2**4) - 36*h2*k2*(h2**2 + k2**2) + 46*h2**2*k2**2 + sqrt(4*(h2**2 + k2**2) - 7*h2*k2)*(-8*(h2**3 + k2**3) + 11*h2*k2*(h2 + k2))) return 16*pi/13125*k2*h2*res def G31(h2, k2): res = (16*(h2**4 + k2**4) - 36*h2*k2*(h2**2 + k2**2) + 46*h2**2*k2**2 + sqrt(4*(h2**2 + k2**2) - 7*h2*k2)*(8*(h2**3 + k2**3) - 11*h2*k2*(h2 + k2))) return 16*pi/13125*h2*k2*res def G34(h2, k2): res = (6*h2**4 + 16*k2**4 - 12*h2**3*k2 - 28*h2*k2**3 + 34*h2**2*k2**2 + sqrt(h2**2 + 4*k2**2 - h2*k2)*(-6*h2**3 - 8*k2**3 + 9*h2**2*k2 + 13*h2*k2**2)) return 16*pi/13125*h2*(k2 - h2)*res def G33(h2, k2): res = (6*h2**4 + 16*k2**4 - 12*h2**3*k2 - 28*h2*k2**3 + 34*h2**2*k2**2 + sqrt(h2**2 + 4*k2**2 - h2*k2)*(6*h2**3 + 8*k2**3 - 9*h2**2*k2 - 13*h2*k2**2)) return 16*pi/13125*h2*(k2 - h2)*res def G36(h2, k2): res = (16*h2**4 + 6*k2**4 - 28*h2**3*k2 - 12*h2*k2**3 + 34*h2**2*k2**2 + sqrt(4*h2**2 + k2**2 - h2*k2)*(-8*h2**3 - 6*k2**3 + 13*h2**2*k2 + 9*h2*k2**2)) return 16*pi/13125*k2*(k2 - h2)*res def G35(h2, k2): res = (16*h2**4 + 6*k2**4 - 28*h2**3*k2 - 12*h2*k2**3 + 34*h2**2*k2**2 + sqrt(4*h2**2 + k2**2 - h2*k2)*(8*h2**3 + 6*k2**3 - 13*h2**2*k2 - 9*h2*k2**2)) return 16*pi/13125*k2*(k2 - h2)*res def G37(h2, k2): return 4*pi*h2**2*k2**2*(k2 - h2)**2/105 known_funcs = {(0, 1): G01, (1, 1): G11, (1, 2): G12, (1, 3): G13, (2, 1): G21, (2, 2): G22, (2, 3): G23, (2, 4): G24, (2, 5): G25, (3, 1): G31, (3, 2): G32, (3, 3): G33, (3, 4): G34, (3, 5): G35, (3, 6): G36, (3, 7): G37} def _ellip_norm(n, p, h2, k2): func = known_funcs[n, p] return func(h2, k2) _ellip_norm = np.vectorize(_ellip_norm) def ellip_normal_known(h2, k2, n, p): return _ellip_norm(n, p, h2, k2) # generate both large and small h2 < k2 pairs np.random.seed(1234) h2 = np.random.pareto(0.5, size=1) k2 = h2 * (1 + np.random.pareto(0.5, size=h2.size)) points = [] for n in range(4): for p in range(1, 2*n+2): points.append((h2, k2, np.full(h2.size, n), np.full(h2.size, p))) points = np.array(points) with suppress_warnings() as sup: sup.filter(IntegrationWarning, "The occurrence of roundoff error") assert_func_equal(ellip_normal, ellip_normal_known, points, rtol=1e-12) def test_ellip_harm_2(): def I1(h2, k2, s): res = (ellip_harm_2(h2, k2, 1, 1, s)/(3 * ellip_harm(h2, k2, 1, 1, s)) + ellip_harm_2(h2, k2, 1, 2, s)/(3 * ellip_harm(h2, k2, 1, 2, s)) + ellip_harm_2(h2, k2, 1, 3, s)/(3 * ellip_harm(h2, k2, 1, 3, s))) return res with suppress_warnings() as sup: sup.filter(IntegrationWarning, "The occurrence of roundoff error") assert_almost_equal(I1(5, 8, 10), 1/(10*sqrt((100-5)*(100-8)))) # Values produced by code from arXiv:1204.0267 assert_almost_equal(ellip_harm_2(5, 8, 2, 1, 10), 0.00108056853382) assert_almost_equal(ellip_harm_2(5, 8, 2, 2, 10), 0.00105820513809) assert_almost_equal(ellip_harm_2(5, 8, 2, 3, 10), 0.00106058384743) assert_almost_equal(ellip_harm_2(5, 8, 2, 4, 10), 0.00106774492306) assert_almost_equal(ellip_harm_2(5, 8, 2, 5, 10), 0.00107976356454) def test_ellip_harm(): def E01(h2, k2, s): return 1 def E11(h2, k2, s): return s def E12(h2, k2, s): return sqrt(abs(s*s - h2)) def E13(h2, k2, s): return sqrt(abs(s*s - k2)) def E21(h2, k2, s): return s*s - 1/3*((h2 + k2) + sqrt(abs((h2 + k2)*(h2 + k2)-3*h2*k2))) def E22(h2, k2, s): return s*s - 1/3*((h2 + k2) - sqrt(abs((h2 + k2)*(h2 + k2)-3*h2*k2))) def E23(h2, k2, s): return s * sqrt(abs(s*s - h2)) def E24(h2, k2, s): return s * sqrt(abs(s*s - k2)) def E25(h2, k2, s): return sqrt(abs((s*s - h2)*(s*s - k2))) def E31(h2, k2, s): return s*s*s - (s/5)*(2*(h2 + k2) + sqrt(4*(h2 + k2)*(h2 + k2) - 15*h2*k2)) def E32(h2, k2, s): return s*s*s - (s/5)*(2*(h2 + k2) - sqrt(4*(h2 + k2)*(h2 + k2) - 15*h2*k2)) def E33(h2, k2, s): return sqrt(abs(s*s - h2))*(s*s - 1/5*((h2 + 2*k2) + sqrt(abs((h2 + 2*k2)*(h2 + 2*k2) - 5*h2*k2)))) def E34(h2, k2, s): return sqrt(abs(s*s - h2))*(s*s - 1/5*((h2 + 2*k2) - sqrt(abs((h2 + 2*k2)*(h2 + 2*k2) - 5*h2*k2)))) def E35(h2, k2, s): return sqrt(abs(s*s - k2))*(s*s - 1/5*((2*h2 + k2) + sqrt(abs((2*h2 + k2)*(2*h2 + k2) - 5*h2*k2)))) def E36(h2, k2, s): return sqrt(abs(s*s - k2))*(s*s - 1/5*((2*h2 + k2) - sqrt(abs((2*h2 + k2)*(2*h2 + k2) - 5*h2*k2)))) def E37(h2, k2, s): return s * sqrt(abs((s*s - h2)*(s*s - k2))) assert_equal(ellip_harm(5, 8, 1, 2, 2.5, 1, 1), ellip_harm(5, 8, 1, 2, 2.5)) known_funcs = {(0, 1): E01, (1, 1): E11, (1, 2): E12, (1, 3): E13, (2, 1): E21, (2, 2): E22, (2, 3): E23, (2, 4): E24, (2, 5): E25, (3, 1): E31, (3, 2): E32, (3, 3): E33, (3, 4): E34, (3, 5): E35, (3, 6): E36, (3, 7): E37} point_ref = [] def ellip_harm_known(h2, k2, n, p, s): for i in range(h2.size): func = known_funcs[(int(n[i]), int(p[i]))] point_ref.append(func(h2[i], k2[i], s[i])) return point_ref np.random.seed(1234) h2 = np.random.pareto(0.5, size=30) k2 = h2*(1 + np.random.pareto(0.5, size=h2.size)) s = np.random.pareto(0.5, size=h2.size) points = [] for i in range(h2.size): for n in range(4): for p in range(1, 2*n+2): points.append((h2[i], k2[i], n, p, s[i])) points = np.array(points) assert_func_equal(ellip_harm, ellip_harm_known, points, rtol=1e-12) def test_ellip_harm_invalid_p(): # Regression test. This should return nan. n = 4 # Make p > 2*n + 1. p = 2*n + 2 result = ellip_harm(0.5, 2.0, n, p, 0.2) assert np.isnan(result)