def test_fht_exact(n):
rng = np.random.RandomState(3491349965)
# for a(r) a power law r^\gamma, the fast Hankel transform produces the
# exact continuous Hankel transform if biased with q = \gamma
mu = rng.uniform(0, 3)
# convergence of HT: -1-mu < gamma < 1/2
gamma = rng.uniform(-1 - mu, 1 / 2)
r = np.logspace(-2, 2, n)
a = r**gamma
dln = np.log(r[1] / r[0])
offset = fhtoffset(dln, mu, initial=0.0, bias=gamma)
A = fht(a, dln, mu, offset=offset, bias=gamma)
k = np.exp(offset) / r[::-1]
# analytical result
At = (2 / k)**gamma * poch((mu + 1 - gamma) / 2, gamma)
assert_allclose(A, At)
def test_fht_special_cases():
rng = np.random.RandomState(3491349965)
a = rng.standard_normal(64)
dln = rng.uniform(-1, 1)
# let xp = (mu+1+q)/2, xm = (mu+1-q)/2, M = {0, -1, -2, ...}
# case 1: xp in M, xm in M => well-defined transform
mu, bias = -4.0, 1.0
with pytest.warns(None) as record:
fht(a, dln, mu, bias=bias)
assert not record, 'fht warned about a well-defined transform'
# case 2: xp not in M, xm in M => well-defined transform
mu, bias = -2.5, 0.5
with pytest.warns(None) as record:
fht(a, dln, mu, bias=bias)
assert not record, 'fht warned about a well-defined transform'
# case 3: xp in M, xm not in M => singular transform
mu, bias = -3.5, 0.5
with pytest.warns(Warning) as record:
fht(a, dln, mu, bias=bias)
assert record, 'fht did not warn about a singular transform'
# case 4: xp not in M, xm in M => singular inverse transform
mu, bias = -2.5, 0.5
with pytest.warns(Warning) as record:
ifht(a, dln, mu, bias=bias)
assert record, 'ifht did not warn about a singular transform'
def test_fht_identity(n, bias, offset, optimal):
rng = np.random.RandomState(3491349965)
a = rng.standard_normal(n)
dln = rng.uniform(-1, 1)
mu = rng.uniform(-2, 2)
if optimal:
offset = fhtoffset(dln, mu, initial=offset, bias=bias)
A = fht(a, dln, mu, offset=offset, bias=bias)
a_ = ifht(A, dln, mu, offset=offset, bias=bias)
assert_allclose(a, a_)
def test_fht_agrees_with_fftlog():
# check that fht numerically agrees with the output from Fortran FFTLog,
# the results were generated with the provided `fftlogtest` program,
# after fixing how the k array is generated (divide range by n-1, not n)
# test function, analytical Hankel transform is of the same form
def f(r, mu):
return r**(mu + 1) * np.exp(-r**2 / 2)
r = np.logspace(-4, 4, 16)
dln = np.log(r[1] / r[0])
mu = 0.3
offset = 0.0
bias = 0.0
a = f(r, mu)
# test 1: compute as given
ours = fht(a, dln, mu, offset=offset, bias=bias)
theirs = [
-0.1159922613593045E-02, 0.1625822618458832E-02,
-0.1949518286432330E-02, 0.3789220182554077E-02,
0.5093959119952945E-03, 0.2785387803618774E-01, 0.9944952700848897E-01,
0.4599202164586588, 0.3157462160881342, -0.8201236844404755E-03,
-0.7834031308271878E-03, 0.3931444945110708E-03,
-0.2697710625194777E-03, 0.3568398050238820E-03,
-0.5554454827797206E-03, 0.8286331026468585E-03
]
assert_allclose(ours, theirs)
# test 2: change to optimal offset
offset = fhtoffset(dln, mu, bias=bias)
ours = fht(a, dln, mu, offset=offset, bias=bias)
theirs = [
0.4353768523152057E-04, -0.9197045663594285E-05,
0.3150140927838524E-03, 0.9149121960963704E-03, 0.5808089753959363E-02,
0.2548065256377240E-01, 0.1339477692089897, 0.4821530509479356,
0.2659899781579785, -0.1116475278448113E-01, 0.1791441617592385E-02,
-0.4181810476548056E-03, 0.1314963536765343E-03,
-0.5422057743066297E-04, 0.3208681804170443E-04,
-0.2696849476008234E-04
]
assert_allclose(ours, theirs)
# test 3: positive bias
bias = 0.8
offset = fhtoffset(dln, mu, bias=bias)
ours = fht(a, dln, mu, offset=offset, bias=bias)
theirs = [
-7.343667355831685, 0.1710271207817100, 0.1065374386206564,
-0.5121739602708132E-01, 0.2636649319269470E-01,
0.1697209218849693E-01, 0.1250215614723183, 0.4739583261486729,
0.2841149874912028, -0.8312764741645729E-02, 0.1024233505508988E-02,
-0.1644902767389120E-03, 0.3305775476926270E-04,
-0.7786993194882709E-05, 0.1962258449520547E-05,
-0.8977895734909250E-06
]
assert_allclose(ours, theirs)
# test 4: negative bias
bias = -0.8
offset = fhtoffset(dln, mu, bias=bias)
ours = fht(a, dln, mu, offset=offset, bias=bias)
theirs = [
0.8985777068568745E-05, 0.4074898209936099E-04, 0.2123969254700955E-03,
0.1009558244834628E-02, 0.5131386375222176E-02, 0.2461678673516286E-01,
0.1235812845384476, 0.4719570096404403, 0.2893487490631317,
-0.1686570611318716E-01, 0.2231398155172505E-01,
-0.1480742256379873E-01, 0.1692387813500801, 0.3097490354365797,
2.759360718240186, 10.52510750700458
]
assert_allclose(ours, theirs)