def get_orthpoly(n_deg,
f_weighting,
n_extra_point=10,
return_coef=True,
representation='chebyshev',
integral_method='legendre'):
"""
Get orthogonal polynomials with respect to the weighting (f_weighting).
The polynomials are represented by coefficients of Chebyshev polynomials.
Evaluate it as: np.dot(cheb.chebvander(set_of_x, n_deg), repr_coef)
See weighted_orthpoly1.py for validation of this algorithm.
"""
n_sampling = n_deg + 1 + n_extra_point
# Do the intergration by discrete sampling at good points.
if integral_method == 'chebyshev':
# Here we use (first kind) Chebyshev points.
x_sampling = chebyshev_roots(n_sampling)
# Put together the weighting and the weighting of the Gauss-Chebyshev quadrature.
diag_weight = np.array(
[f_weighting(x) / cheb.chebweight(x)
for x in x_sampling]) * (pi / n_sampling)
elif integral_method == 'legendre':
# Use Gauss-Legendre quadrature.
x_sampling, int_weight = lege.leggauss(n_sampling)
diag_weight = np.array(
[f_weighting(x) * w for x, w in zip(x_sampling, int_weight)])
if representation == 'chebyshev':
V = cheb.chebvander(x_sampling, n_deg)
else:
V = lege.legvander(x_sampling, n_deg)
# Get basis from chol of the Gramian matrix.
# inner_prod_matrix = dot(V.T, diag_weight[:, np.newaxis] * V)
# repr_coef = inv(cholesky(inner_prod_matrix).T)
# QR decomposition should give more accurate result.
repr_coef = inv(qr(sqrt(diag_weight[:, np.newaxis]) * V, mode='r'))
repr_coef = repr_coef * sign(sum(repr_coef, axis=0))
if return_coef:
return repr_coef
else:
if representation == 'chebyshev':
polys = [cheb.Chebyshev(repr_coef[:, i]) for i in range(n_deg + 1)]
else:
polys = [lege.Legendre(repr_coef[:, i]) for i in range(n_deg + 1)]
return polys
def test_weight(self):
x = np.linspace(-1, 1, 11)[1:-1]
tgt = 1./(np.sqrt(1 + x) * np.sqrt(1 - x))
res = cheb.chebweight(x)
assert_almost_equal(res, tgt)
def test_weight(self):
x = np.linspace(-1, 1, 11)[1:-1]
tgt = 1./(np.sqrt(1 + x) * np.sqrt(1 - x))
res = cheb.chebweight(x)
assert_almost_equal(res, tgt)
