def test_nu_varying_gamma_mod(s, nu, N, h):
ht = HankelTransform(nu=nu, N=N, h=h)
ans = ht.integrate(lambda x: x ** (nu - 2 * s + 1) * gammainc_(s, x ** 2), False, False)
anl = 0.5 ** (2 * s - nu - 1) * gammaincc_(1 - s + nu, 0.25)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu_varying_powerlaw(s, nu, N, h):
ht = HankelTransform(nu=nu, N=N, h=h)
ans = ht.integrate(lambda x: x**(s + 1), False, False)
anl = 2**(s + 1) * gamma(0.5 * (2 + nu + s)) / gamma(0.5 * (nu - s))
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu0_x4():
"""
Result on wikipedia
"""
ht = HankelTransform(nu=0, N=150, h=10 ** -1.5)
ans = ht.integrate(lambda x: x ** 4, False, False)
print("Numerical Result: ", ans, " (required 9)")
assert np.isclose(ans, 9, rtol=1e-3)
def test_nu0_x4():
"""
Result on wikipedia
"""
ht = HankelTransform(nu=0, N=150, h=10**-1.5)
ans = ht.integrate(lambda x: x**4, False, False)
print("Numerical Result: ", ans, " (required 9)")
assert np.isclose(ans, 9, rtol=1e-3)
def test_nu_varying_powerlaw(s, nu, N, h):
ht = HankelTransform(nu=nu, N=N, h=h)
ans = ht.integrate(lambda x: x ** (s + 1), False, False)
anl = 2 ** (s + 1) * gamma(0.5 * (2 + nu + s)) / gamma(0.5 * (nu - s))
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu_varying_gamma_mod(s, nu, N, h):
ht = HankelTransform(nu=nu, N=N, h=h)
ans = ht.integrate(lambda x: x**(nu - 2 * s + 1) * gammainc_(s, x**2),
False, False)
anl = 0.5**(2 * s - nu - 1) * gammaincc_(1 - s + nu, 0.25)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu0_f_gauss():
"""
Result on wikipedia
"""
z = 2
ht = HankelTransform(nu=0, N=50, h=0.01)
ans = ht.integrate(lambda x: x * np.exp(-0.5 * z**2 * x**2), False, False)
anl = 1.0 / z**2 * np.exp(-0.5 / z**2)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu0_f_gauss():
"""
Result on wikipedia
"""
z = 2
ht = HankelTransform(nu=0, N=50, h=0.01)
ans = ht.integrate(lambda x: x * np.exp(-0.5 * z ** 2 * x ** 2), False, False)
anl = 1. / z ** 2 * np.exp(-0.5 / z ** 2)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu0_f_x2():
"""
Test f(x) = x^2, nu=0
Result on wikipedia
"""
ht = HankelTransform(nu=0, N=100, h=10 ** -1.5)
ans = ht.integrate(lambda x: x ** 2, False, False)
print("Numerical Result: ", ans, " (required -1)")
assert np.isclose(ans, -1, rtol=1e-3)
def test_nu0_f_x_on_x2():
"""
Test f(x) = x/(x**2 + 1), nu=0
This test is done in the Ogata (2005) paper, section 5"
"""
ht = HankelTransform(nu=0, N=50, h=10 ** -1.5)
ans = ht.integrate(lambda x: x / (x ** 2 + 1), False, False)
print("Numerical Result: ", ans, " (required %s)" % k0(1))
assert np.isclose(ans, k0(1), rtol=1e-3)
def test_nu0_f_unity():
"""
Test f(x) = 1, nu=0
This test is done in the Ogata (2005) paper, section 5"
"""
ht = HankelTransform(nu=0, N=50, h=0.1)
ans = ht.integrate(lambda x: 1, False, False)
print("Numerical Result: ", ans, " (required %s)" % 1)
assert np.isclose(ans, 1, rtol=1e-3)
def test_nu0_f_unity():
"""
Test f(x) = 1, nu=0
This test is done in the Ogata (2005) paper, section 5"
"""
ht = HankelTransform(nu=0, N=50, h=0.1)
ans = ht.integrate(lambda x: 1, False, False)
print("Numerical Result: ", ans, " (required %s)" % 1)
assert np.isclose(ans, 1, rtol=1e-3)
def test_nu0_f_x_on_x2():
"""
Test f(x) = x/(x**2 + 1), nu=0
This test is done in the Ogata (2005) paper, section 5"
"""
ht = HankelTransform(nu=0, N=50, h=10**-1.5)
ans = ht.integrate(lambda x: x / (x**2 + 1), False, False)
print("Numerical Result: ", ans, " (required %s)" % k0(1))
assert np.isclose(ans, k0(1), rtol=1e-3)
def test_nu0_f_x2():
"""
Test f(x) = x^2, nu=0
Result on wikipedia
"""
ht = HankelTransform(nu=0, N=100, h=10**-1.5)
ans = ht.integrate(lambda x: x**2, False, False)
print("Numerical Result: ", ans, " (required -1)")
assert np.isclose(ans, -1, rtol=1e-3)
def test_nu0_x_on_sqrt_x2_pz2():
"""
Result on wikipedia
"""
# Note that the number required is highly dependent on z .... smaller z is harder.
ht = HankelTransform(nu=0, N=50, h=10**-1.3)
z = 1
ans = ht.integrate(lambda x: x / np.sqrt(x**2 + z**2), False, False)
anl = np.exp(-z)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu0_1_on_sqrt_x():
"""
Result on wikipedia
"""
# NOTE: this is REALLY finnicky!! (check devel/)
ht = HankelTransform(nu=0, N=160, h=10**-3.5)
ans = ht.integrate(lambda x: 1.0 / np.sqrt(x), False, False)
m = -1.5
anl = 2**(m + 1) * gamma(m / 2 + 1) / gamma(-m / 2)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu0_1_on_sqrt_x():
"""
Result on wikipedia
"""
# NOTE: this is REALLY finnicky!! (check devel/)
ht = HankelTransform(nu=0, N=160, h=10 ** -3.5)
ans = ht.integrate(lambda x: 1. / np.sqrt(x), False, False)
m = -1.5
anl = 2 ** (m + 1) * gamma(m / 2 + 1) / gamma(-m / 2)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_nu0_x_on_sqrt_x2_pz2():
"""
Result on wikipedia
"""
# Note that the number required is highly dependent on z .... smaller z is harder.
ht = HankelTransform(nu=0, N=50, h=10 ** -1.3)
z = 1
ans = ht.integrate(lambda x: x / np.sqrt(x ** 2 + z ** 2), False, False)
anl = np.exp(-z)
print("Numerical Result: ", ans, " (required %s)" % anl)
assert np.isclose(ans, anl, rtol=1e-3)
def test_equivalence_of_integrate_and_transform():
ht = HankelTransform(N=50)
intg = ht.integrate(lambda x: 1, False)
tr = ht.transform(lambda x: 1.0 / x, ret_err=False)
assert intg == tr
def test_ret_err_and_cumsum():
ht = HankelTransform(N=50)
res, err, cumsum = ht.integrate(lambda x: 1.0 / x, True, True)
def test_ret_cumsum():
ht = HankelTransform(N=50)
res, cumsum = ht.integrate(lambda x: 1.0 / x, False, True)
def test_ret_err():
ht = HankelTransform(N=50)
res, err = ht.integrate(lambda x: 1.0 / x, True)
def test_equivalence_of_integrate_and_transform():
ht = HankelTransform(N=50)
int = ht.integrate(lambda x: 1, False)
tr = ht.transform(lambda x: 1. / x, ret_err=False)
assert int == tr
def test_ret_err_and_cumsum():
ht = HankelTransform(N=50)
res, err, cumsum = ht.integrate(lambda x: 1. / x, True, True)
def test_ret_cumsum():
ht = HankelTransform(N=50)
res, cumsum = ht.integrate(lambda x: 1. / x, False, True)
def test_ret_err():
ht = HankelTransform(N=50)
res, err = ht.integrate(lambda x: 1. / x, True)