You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
279 lines
9.4 KiB
Python
279 lines
9.4 KiB
Python
#
|
|
# 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)
|