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import warnings
import numpy as np
import pytest
from scipy.fft._fftlog import fht, ifht, fhtoffset
from scipy.special import poch
from scipy.conftest import array_api_compatible
from scipy._lib._array_api import xp_assert_close
pytestmark = array_api_compatible
def test_fht_agrees_with_fftlog(xp):
# 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 = xp.asarray(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, +0.4599202164586588E+00,
+0.3157462160881342E+00, -0.8201236844404755E-03,
-0.7834031308271878E-03, +0.3931444945110708E-03,
-0.2697710625194777E-03, +0.3568398050238820E-03,
-0.5554454827797206E-03, +0.8286331026468585E-03]
theirs = xp.asarray(theirs, dtype=xp.float64)
xp_assert_close(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,
+0.1339477692089897E+00, +0.4821530509479356E+00,
+0.2659899781579785E+00, -0.1116475278448113E-01,
+0.1791441617592385E-02, -0.4181810476548056E-03,
+0.1314963536765343E-03, -0.5422057743066297E-04,
+0.3208681804170443E-04, -0.2696849476008234E-04]
theirs = xp.asarray(theirs, dtype=xp.float64)
xp_assert_close(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 = [-7.3436673558316850E+00, +0.1710271207817100E+00,
+0.1065374386206564E+00, -0.5121739602708132E-01,
+0.2636649319269470E-01, +0.1697209218849693E-01,
+0.1250215614723183E+00, +0.4739583261486729E+00,
+0.2841149874912028E+00, -0.8312764741645729E-02,
+0.1024233505508988E-02, -0.1644902767389120E-03,
+0.3305775476926270E-04, -0.7786993194882709E-05,
+0.1962258449520547E-05, -0.8977895734909250E-06]
theirs = xp.asarray(theirs, dtype=xp.float64)
xp_assert_close(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,
+0.1235812845384476E+00, +0.4719570096404403E+00,
+0.2893487490631317E+00, -0.1686570611318716E-01,
+0.2231398155172505E-01, -0.1480742256379873E-01,
+0.1692387813500801E+00, +0.3097490354365797E+00,
+2.7593607182401860E+00, 10.5251075070045800E+00]
theirs = xp.asarray(theirs, dtype=xp.float64)
xp_assert_close(ours, theirs)
@pytest.mark.parametrize('optimal', [True, False])
@pytest.mark.parametrize('offset', [0.0, 1.0, -1.0])
@pytest.mark.parametrize('bias', [0, 0.1, -0.1])
@pytest.mark.parametrize('n', [64, 63])
def test_fht_identity(n, bias, offset, optimal, xp):
rng = np.random.RandomState(3491349965)
a = xp.asarray(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)
xp_assert_close(a_, a)
def test_fht_special_cases(xp):
rng = np.random.RandomState(3491349965)
a = xp.asarray(rng.standard_normal(64))
dln = rng.uniform(-1, 1)
# let x = (mu+1+q)/2, y = (mu+1-q)/2, M = {0, -1, -2, ...}
# case 1: x in M, y in M => well-defined transform
mu, bias = -4.0, 1.0
with warnings.catch_warnings(record=True) as record:
fht(a, dln, mu, bias=bias)
assert not record, 'fht warned about a well-defined transform'
# case 2: x not in M, y in M => well-defined transform
mu, bias = -2.5, 0.5
with warnings.catch_warnings(record=True) as record:
fht(a, dln, mu, bias=bias)
assert not record, 'fht warned about a well-defined transform'
# case 3: x in M, y 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: x not in M, y 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'
@pytest.mark.parametrize('n', [64, 63])
def test_fht_exact(n, xp):
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 = xp.asarray(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 = xp.asarray((2/k)**gamma * poch((mu+1-gamma)/2, gamma))
xp_assert_close(A, At)
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