def test_roots_chebyu():
weightf = orth.chebyu(5).weight_func
verify_gauss_quad(sc.roots_chebyu, orth.eval_chebyu, weightf, -1., 1., 5)
verify_gauss_quad(sc.roots_chebyu, orth.eval_chebyu, weightf, -1., 1., 25)
verify_gauss_quad(sc.roots_chebyu, orth.eval_chebyu, weightf, -1., 1., 100)
x, w = sc.roots_chebyu(5, False)
y, v, m = sc.roots_chebyu(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, -1, 1)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, sc.roots_chebyu, 0)
assert_raises(ValueError, sc.roots_chebyu, 3.3)
def test_roots_chebyu():
weightf = orth.chebyu(5).weight_func
verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 5)
verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 25)
verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 100)
x, w = sc.roots_chebyu(5, False)
y, v, m = sc.roots_chebyu(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, -1, 1)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, sc.roots_chebyu, 0)
assert_raises(ValueError, sc.roots_chebyu, 3.3)
def test_gausschebyshev2(self):
"""Test Gauss Chebyshev type 2 polynomial grid."""
points, weights = roots_chebyu(10)
grid = GaussChebyshevType2(10)
weights /= np.sqrt(1 - np.power(points, 2))
assert_allclose(grid.points, points)
assert_allclose(grid.weights, weights)
def __init__(self, npoints: int):
r"""Generate grid on :math:`[-1, 1]` interval based on Gauss-Chebyshev Type 2.
Parameters
----------
npoints : int
Number of grid points.
Returns
-------
OneDGrid
A 1-D grid instance containing points and weights.
"""
if npoints < 1:
raise ValueError(
f"Argument npoints must be an integer > 1, given {npoints}")
# compute points and weights for Gauss-Chebyshev quadrature (Type 2)
points, weights = roots_chebyu(npoints)
weights /= np.sqrt(1 - np.power(points, 2))
super().__init__(points, weights, (-1, 1))
def GaussChebyshevType2(npoints):
r"""Generate 1D grid on [-1, 1] interval based on Gauss-Chebyshev 2nd kind.
The fundamental definition of Gauss-Chebyshev quadrature is:
.. math::
\int_{-1}^{1} \sqrt{1-x^2} f(x) dx \approx \sum_{i=1}^n w_i f(x_i)
However, to integrate function :math:`g(x)` over [-1, 1], this is re-written as:
.. math::
\int_{-1}^{1}g(x) dx \approx \sum_{i=1}^n \frac{w_i}{\sqrt{1-x_i^2}} f(x_i)
= \sum_{i=1}^n w_i' f(x_i)
Where
.. math::
x_i = \cos\left( \frac{i}{n+1} \pi \right)
and the weights
.. math::
w_i = \frac{\pi}{n+1} \sin^2 \left( \frac{i}{n+1} \pi \right)
Parameters
----------
npoints : int
Number of grid points.
Returns
-------
OneDGrid
A 1D grid instance.
"""
if npoints < 1:
raise ValueError("npoints must be greater that one, given {npoints}")
points, weights = roots_chebyu(npoints)
weights /= np.sqrt(1 - np.power(points, 2))
return OneDGrid(points, weights, (-1, 1))
