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#
# Tests of spherical Bessel functions.
#
import numpy as np
from numpy.testing import (assert_almost_equal, assert_allclose, dec,
assert_array_almost_equal)
from numpy import sin, cos, sinh, cosh, exp, inf, nan, r_, pi
from scipy.special import spherical_jn, spherical_yn, spherical_in, spherical_kn
from scipy.integrate import quad
class TestSphericalJn:
def test_spherical_jn_exact(self):
# http://dlmf.nist.gov/10.49.E3
# Note: exact expression is numerically stable only for small
# n or z >> n.
x = np.array([0.12, 1.23, 12.34, 123.45, 1234.5])
assert_allclose(spherical_jn(2, x),
(-1/x + 3/x**3)*sin(x) - 3/x**2*cos(x))
def test_spherical_jn_recurrence_complex(self):
# http://dlmf.nist.gov/10.51.E1
n = np.array([1, 2, 3, 7, 12])
x = 1.1 + 1.5j
assert_allclose(spherical_jn(n - 1, x) + spherical_jn(n + 1, x),
(2*n + 1)/x*spherical_jn(n, x))
def test_spherical_jn_recurrence_real(self):
# http://dlmf.nist.gov/10.51.E1
n = np.array([1, 2, 3, 7, 12])
x = 0.12
assert_allclose(spherical_jn(n - 1, x) + spherical_jn(n + 1,x),
(2*n + 1)/x*spherical_jn(n, x))
def test_spherical_jn_inf_real(self):
# http://dlmf.nist.gov/10.52.E3
n = 6
x = np.array([-inf, inf])
assert_allclose(spherical_jn(n, x), np.array([0, 0]))
def test_spherical_jn_inf_complex(self):
# http://dlmf.nist.gov/10.52.E3
n = 7
x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)])
assert_allclose(spherical_jn(n, x), np.array([0, 0, inf*(1+1j)]))
def test_spherical_jn_large_arg_1(self):
# https://github.com/scipy/scipy/issues/2165
# Reference value computed using mpmath, via
# besselj(n + mpf(1)/2, z)*sqrt(pi/(2*z))
assert_allclose(spherical_jn(2, 3350.507), -0.00029846226538040747)
def test_spherical_jn_large_arg_2(self):
# https://github.com/scipy/scipy/issues/1641
# Reference value computed using mpmath, via
# besselj(n + mpf(1)/2, z)*sqrt(pi/(2*z))
assert_allclose(spherical_jn(2, 10000), 3.0590002633029811e-05)
def test_spherical_jn_at_zero(self):
# http://dlmf.nist.gov/10.52.E1
# But note that n = 0 is a special case: j0 = sin(x)/x -> 1
n = np.array([0, 1, 2, 5, 10, 100])
x = 0
assert_allclose(spherical_jn(n, x), np.array([1, 0, 0, 0, 0, 0]))
class TestSphericalYn:
def test_spherical_yn_exact(self):
# http://dlmf.nist.gov/10.49.E5
# Note: exact expression is numerically stable only for small
# n or z >> n.
x = np.array([0.12, 1.23, 12.34, 123.45, 1234.5])
assert_allclose(spherical_yn(2, x),
(1/x - 3/x**3)*cos(x) - 3/x**2*sin(x))
def test_spherical_yn_recurrence_real(self):
# http://dlmf.nist.gov/10.51.E1
n = np.array([1, 2, 3, 7, 12])
x = 0.12
assert_allclose(spherical_yn(n - 1, x) + spherical_yn(n + 1,x),
(2*n + 1)/x*spherical_yn(n, x))
def test_spherical_yn_recurrence_complex(self):
# http://dlmf.nist.gov/10.51.E1
n = np.array([1, 2, 3, 7, 12])
x = 1.1 + 1.5j
assert_allclose(spherical_yn(n - 1, x) + spherical_yn(n + 1, x),
(2*n + 1)/x*spherical_yn(n, x))
def test_spherical_yn_inf_real(self):
# http://dlmf.nist.gov/10.52.E3
n = 6
x = np.array([-inf, inf])
assert_allclose(spherical_yn(n, x), np.array([0, 0]))
def test_spherical_yn_inf_complex(self):
# http://dlmf.nist.gov/10.52.E3
n = 7
x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)])
assert_allclose(spherical_yn(n, x), np.array([0, 0, inf*(1+1j)]))
def test_spherical_yn_at_zero(self):
# http://dlmf.nist.gov/10.52.E2
n = np.array([0, 1, 2, 5, 10, 100])
x = 0
assert_allclose(spherical_yn(n, x), -inf*np.ones(shape=n.shape))
def test_spherical_yn_at_zero_complex(self):
# Consistently with numpy:
# >>> -np.cos(0)/0
# -inf
# >>> -np.cos(0+0j)/(0+0j)
# (-inf + nan*j)
n = np.array([0, 1, 2, 5, 10, 100])
x = 0 + 0j
assert_allclose(spherical_yn(n, x), nan*np.ones(shape=n.shape))
class TestSphericalJnYnCrossProduct:
def test_spherical_jn_yn_cross_product_1(self):
# http://dlmf.nist.gov/10.50.E3
n = np.array([1, 5, 8])
x = np.array([0.1, 1, 10])
left = (spherical_jn(n + 1, x) * spherical_yn(n, x) -
spherical_jn(n, x) * spherical_yn(n + 1, x))
right = 1/x**2
assert_allclose(left, right)
def test_spherical_jn_yn_cross_product_2(self):
# http://dlmf.nist.gov/10.50.E3
n = np.array([1, 5, 8])
x = np.array([0.1, 1, 10])
left = (spherical_jn(n + 2, x) * spherical_yn(n, x) -
spherical_jn(n, x) * spherical_yn(n + 2, x))
right = (2*n + 3)/x**3
assert_allclose(left, right)
class TestSphericalIn:
def test_spherical_in_exact(self):
# http://dlmf.nist.gov/10.49.E9
x = np.array([0.12, 1.23, 12.34, 123.45])
assert_allclose(spherical_in(2, x),
(1/x + 3/x**3)*sinh(x) - 3/x**2*cosh(x))
def test_spherical_in_recurrence_real(self):
# http://dlmf.nist.gov/10.51.E4
n = np.array([1, 2, 3, 7, 12])
x = 0.12
assert_allclose(spherical_in(n - 1, x) - spherical_in(n + 1,x),
(2*n + 1)/x*spherical_in(n, x))
def test_spherical_in_recurrence_complex(self):
# http://dlmf.nist.gov/10.51.E1
n = np.array([1, 2, 3, 7, 12])
x = 1.1 + 1.5j
assert_allclose(spherical_in(n - 1, x) - spherical_in(n + 1,x),
(2*n + 1)/x*spherical_in(n, x))
def test_spherical_in_inf_real(self):
# http://dlmf.nist.gov/10.52.E3
n = 5
x = np.array([-inf, inf])
assert_allclose(spherical_in(n, x), np.array([-inf, inf]))
def test_spherical_in_inf_complex(self):
# http://dlmf.nist.gov/10.52.E5
# Ideally, i1n(n, 1j*inf) = 0 and i1n(n, (1+1j)*inf) = (1+1j)*inf, but
# this appears impossible to achieve because C99 regards any complex
# value with at least one infinite part as a complex infinity, so
# 1j*inf cannot be distinguished from (1+1j)*inf. Therefore, nan is
# the correct return value.
n = 7
x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)])
assert_allclose(spherical_in(n, x), np.array([-inf, inf, nan]))
def test_spherical_in_at_zero(self):
# http://dlmf.nist.gov/10.52.E1
# But note that n = 0 is a special case: i0 = sinh(x)/x -> 1
n = np.array([0, 1, 2, 5, 10, 100])
x = 0
assert_allclose(spherical_in(n, x), np.array([1, 0, 0, 0, 0, 0]))
class TestSphericalKn:
def test_spherical_kn_exact(self):
# http://dlmf.nist.gov/10.49.E13
x = np.array([0.12, 1.23, 12.34, 123.45])
assert_allclose(spherical_kn(2, x),
pi/2*exp(-x)*(1/x + 3/x**2 + 3/x**3))
def test_spherical_kn_recurrence_real(self):
# http://dlmf.nist.gov/10.51.E4
n = np.array([1, 2, 3, 7, 12])
x = 0.12
assert_allclose((-1)**(n - 1)*spherical_kn(n - 1, x) - (-1)**(n + 1)*spherical_kn(n + 1,x),
(-1)**n*(2*n + 1)/x*spherical_kn(n, x))
def test_spherical_kn_recurrence_complex(self):
# http://dlmf.nist.gov/10.51.E4
n = np.array([1, 2, 3, 7, 12])
x = 1.1 + 1.5j
assert_allclose((-1)**(n - 1)*spherical_kn(n - 1, x) - (-1)**(n + 1)*spherical_kn(n + 1,x),
(-1)**n*(2*n + 1)/x*spherical_kn(n, x))
def test_spherical_kn_inf_real(self):
# http://dlmf.nist.gov/10.52.E6
n = 5
x = np.array([-inf, inf])
assert_allclose(spherical_kn(n, x), np.array([-inf, 0]))
def test_spherical_kn_inf_complex(self):
# http://dlmf.nist.gov/10.52.E6
# The behavior at complex infinity depends on the sign of the real
# part: if Re(z) >= 0, then the limit is 0; if Re(z) < 0, then it's
# z*inf. This distinction cannot be captured, so we return nan.
n = 7
x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)])
assert_allclose(spherical_kn(n, x), np.array([-inf, 0, nan]))
def test_spherical_kn_at_zero(self):
# http://dlmf.nist.gov/10.52.E2
n = np.array([0, 1, 2, 5, 10, 100])
x = 0
assert_allclose(spherical_kn(n, x), inf*np.ones(shape=n.shape))
def test_spherical_kn_at_zero_complex(self):
# http://dlmf.nist.gov/10.52.E2
n = np.array([0, 1, 2, 5, 10, 100])
x = 0 + 0j
assert_allclose(spherical_kn(n, x), nan*np.ones(shape=n.shape))
class SphericalDerivativesTestCase:
def fundamental_theorem(self, n, a, b):
integral, tolerance = quad(lambda z: self.df(n, z), a, b)
assert_allclose(integral,
self.f(n, b) - self.f(n, a),
atol=tolerance)
@dec.slow
def test_fundamental_theorem_0(self):
self.fundamental_theorem(0, 3.0, 15.0)
@dec.slow
def test_fundamental_theorem_7(self):
self.fundamental_theorem(7, 0.5, 1.2)
class TestSphericalJnDerivatives(SphericalDerivativesTestCase):
def f(self, n, z):
return spherical_jn(n, z)
def df(self, n, z):
return spherical_jn(n, z, derivative=True)
def test_spherical_jn_d_zero(self):
n = np.array([1, 2, 3, 7, 15])
assert_allclose(spherical_jn(n, 0, derivative=True),
np.zeros(5))
class TestSphericalYnDerivatives(SphericalDerivativesTestCase):
def f(self, n, z):
return spherical_yn(n, z)
def df(self, n, z):
return spherical_yn(n, z, derivative=True)
class TestSphericalInDerivatives(SphericalDerivativesTestCase):
def f(self, n, z):
return spherical_in(n, z)
def df(self, n, z):
return spherical_in(n, z, derivative=True)
def test_spherical_in_d_zero(self):
n = np.array([1, 2, 3, 7, 15])
assert_allclose(spherical_in(n, 0, derivative=True),
np.zeros(5))
class TestSphericalKnDerivatives(SphericalDerivativesTestCase):
def f(self, n, z):
return spherical_kn(n, z)
def df(self, n, z):
return spherical_kn(n, z, derivative=True)
class TestSphericalOld:
# These are tests from the TestSpherical class of test_basic.py,
# rewritten to use spherical_* instead of sph_* but otherwise unchanged.
def test_sph_in(self):
# This test reproduces test_basic.TestSpherical.test_sph_in.
i1n = np.empty((2,2))
x = 0.2
i1n[0][0] = spherical_in(0, x)
i1n[0][1] = spherical_in(1, x)
i1n[1][0] = spherical_in(0, x, derivative=True)
i1n[1][1] = spherical_in(1, x, derivative=True)
inp0 = (i1n[0][1])
inp1 = (i1n[0][0] - 2.0/0.2 * i1n[0][1])
assert_array_almost_equal(i1n[0],np.array([1.0066800127054699381,
0.066933714568029540839]),12)
assert_array_almost_equal(i1n[1],[inp0,inp1],12)
def test_sph_in_kn_order0(self):
x = 1.
sph_i0 = np.empty((2,))
sph_i0[0] = spherical_in(0, x)
sph_i0[1] = spherical_in(0, x, derivative=True)
sph_i0_expected = np.array([np.sinh(x)/x,
np.cosh(x)/x-np.sinh(x)/x**2])
assert_array_almost_equal(r_[sph_i0], sph_i0_expected)
sph_k0 = np.empty((2,))
sph_k0[0] = spherical_kn(0, x)
sph_k0[1] = spherical_kn(0, x, derivative=True)
sph_k0_expected = np.array([0.5*pi*exp(-x)/x,
-0.5*pi*exp(-x)*(1/x+1/x**2)])
assert_array_almost_equal(r_[sph_k0], sph_k0_expected)
def test_sph_jn(self):
s1 = np.empty((2,3))
x = 0.2
s1[0][0] = spherical_jn(0, x)
s1[0][1] = spherical_jn(1, x)
s1[0][2] = spherical_jn(2, x)
s1[1][0] = spherical_jn(0, x, derivative=True)
s1[1][1] = spherical_jn(1, x, derivative=True)
s1[1][2] = spherical_jn(2, x, derivative=True)
s10 = -s1[0][1]
s11 = s1[0][0]-2.0/0.2*s1[0][1]
s12 = s1[0][1]-3.0/0.2*s1[0][2]
assert_array_almost_equal(s1[0],[0.99334665397530607731,
0.066400380670322230863,
0.0026590560795273856680],12)
assert_array_almost_equal(s1[1],[s10,s11,s12],12)
def test_sph_kn(self):
kn = np.empty((2,3))
x = 0.2
kn[0][0] = spherical_kn(0, x)
kn[0][1] = spherical_kn(1, x)
kn[0][2] = spherical_kn(2, x)
kn[1][0] = spherical_kn(0, x, derivative=True)
kn[1][1] = spherical_kn(1, x, derivative=True)
kn[1][2] = spherical_kn(2, x, derivative=True)
kn0 = -kn[0][1]
kn1 = -kn[0][0]-2.0/0.2*kn[0][1]
kn2 = -kn[0][1]-3.0/0.2*kn[0][2]
assert_array_almost_equal(kn[0],[6.4302962978445670140,
38.581777787067402086,
585.15696310385559829],12)
assert_array_almost_equal(kn[1],[kn0,kn1,kn2],9)
def test_sph_yn(self):
sy1 = spherical_yn(2, 0.2)
sy2 = spherical_yn(0, 0.2)
assert_almost_equal(sy1,-377.52483,5) # previous values in the system
assert_almost_equal(sy2,-4.9003329,5)
sphpy = (spherical_yn(0, 0.2) - 2*spherical_yn(2, 0.2))/3
sy3 = spherical_yn(1, 0.2, derivative=True)
assert_almost_equal(sy3,sphpy,4) # compare correct derivative val. (correct =-system val).
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