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#
# Tests for the Ellipsoidal Harmonic Function,
# Distributed under the same license as SciPy itself.
#
from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from numpy.testing import (assert_equal, assert_almost_equal, assert_allclose,
assert_, run_module_suite)
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 warnings.catch_warnings():
warnings.simplefilter("ignore", category=IntegrationWarning)
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, n*np.ones(h2.size), p*np.ones(h2.size)))
points = np.array(points)
with warnings.catch_warnings(record=True): # occurrence of roundoff ...
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 warnings.catch_warnings(record=True): # occurrence of roundoff ...
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)
if __name__ == "__main__":
run_module_suite()
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