def test_orthonormality_jacobi_1d(alpha, beta, ebound):
"""Verify that the Jacobi polymials are orthogonal in 1D."""
from modepy.modes import jacobi
from modepy.quadrature.jacobi_gauss import LegendreGaussQuadrature
max_n = 10
quad = LegendreGaussQuadrature(4 * max_n) # overkill...
class WeightFunction:
def __init__(self, alpha, beta):
self.alpha = alpha
self.beta = beta
def __call__(self, x):
return (1 - x)**self.alpha * (1 + x)**self.beta
from functools import partial
jac_f = [partial(jacobi, alpha, beta, n) for n in range(max_n)]
wf = WeightFunction(alpha, beta)
maxerr = 0
for i, fi in enumerate(jac_f):
for j, fj in enumerate(jac_f):
result = quad(lambda x: wf(x) * fi(x) * fj(x))
if i == j:
true_result = 1
else:
true_result = 0
err = abs(result - true_result)
maxerr = max(maxerr, err)
if abs(result - true_result) > ebound:
print(("bad", alpha, beta, i, j, abs(result - true_result)))
assert abs(result - true_result) < ebound
def test_gauss_quadrature(backend):
from modepy.quadrature.jacobi_gauss import LegendreGaussQuadrature
for s in range(9 + 1):
quad = LegendreGaussQuadrature(s, backend)
for deg in range(quad.exact_to + 1):
def f(x):
return x**deg
i_f = quad(f)
i_f_true = 1 / (deg+1) * (1 - (-1)**(deg + 1))
err = abs(i_f - i_f_true)
assert err < 2.0e-15, (s, deg, err, i_f, i_f_true)
def test_gauss_quadrature():
from modepy.quadrature.jacobi_gauss import LegendreGaussQuadrature
for s in range(9 + 1):
cub = LegendreGaussQuadrature(s)
for deg in range(2 * s + 1 + 1):
def f(x):
return x**deg
i_f = cub(f)
i_f_true = 1 / (deg + 1) * (1 - (-1)**(deg + 1))
err = abs(i_f - i_f_true)
assert err < 2e-15
def test_warp():
"""Check some assumptions on the node warp factor calculator"""
n = 17
from functools import partial
def wfc(x):
return nd.warp_factor(n, np.array([x]), scaled=False)[0]
assert abs(wfc(-1)) < 1e-12
assert abs(wfc(1)) < 1e-12
from modepy.quadrature.jacobi_gauss import LegendreGaussQuadrature
lgq = LegendreGaussQuadrature(n)
assert abs(lgq(partial(nd.warp_factor, n, scaled=False))) < 6e-14
def test_transformed_quadrature():
"""Test 1D quadrature on arbitrary intervals"""
def gaussian_density(x, mu, sigma):
return 1 / (sigma * np.sqrt(2 * np.pi)) * np.exp(-(x - mu)**2 /
(2 * sigma**2))
from modepy.quadrature import Transformed1DQuadrature
from modepy.quadrature.jacobi_gauss import LegendreGaussQuadrature
mu = 17
sigma = 12
tq = Transformed1DQuadrature(LegendreGaussQuadrature(20), mu - 6 * sigma,
mu + 6 * sigma)
result = tq(lambda x: gaussian_density(x, mu, sigma))
assert abs(result - 1) < 1.0e-9
def test_warp():
"""Check some assumptions on the node warp factor calculator"""
n = 17
from modepy.nodes import warp_factor
from functools import partial
from modepy.tools import accept_scalar_or_vector
wfc = accept_scalar_or_vector(1, 1)(partial(warp_factor, 17, scaled=False))
assert abs(wfc(-1)) < 1e-12
assert abs(wfc(1)) < 1e-12
from modepy.quadrature.jacobi_gauss import LegendreGaussQuadrature
lgq = LegendreGaussQuadrature(n)
assert abs(lgq(wfc)) < 6e-14
def __init__(self, N, dims, backend=None): # noqa: N803
from modepy.quadrature.jacobi_gauss import LegendreGaussQuadrature
super().__init__([LegendreGaussQuadrature(N, backend=backend)] * dims)