"""Gauss-Gegenbauer quadrature rule."""
import numpy
import chaospy
from .hypercube import hypercube_quadrature
[docs]def gegenbauer(order, alpha, lower=-1, upper=1, physicist=False):
"""
Gauss-Gegenbauer quadrature rule.
Compute the sample points and weights for Gauss-Gegenbauer quadrature. The
sample points are the roots of the nth degree Gegenbauer polynomial. These
sample points and weights correctly integrate polynomials of degree
:math:`2N-1` or less.
Gaussian quadrature come in two variants: physicist and probabilist. For
Gauss-Gegenbauer physicist means a weight function
:math:`(1-x^2)^{\alpha-0.5}` and weights that sum to :math`2^{2\alpha-1}`,
and probabilist means a weight function is
:math:`B(\alpha+0.5, \alpha+0.5) (x-x^2)^{\alpha+1/2}` (where :math:`B` is
the beta normalizing constant) which sum to 1.
Args:
order (int):
The quadrature order.
alpha (float):
Gegenbauer shape parameter.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Returns:
abscissas (numpy.ndarray):
The ``order+1`` quadrature points for where to evaluate the model
function with.
weights (numpy.ndarray):
The quadrature weights associated with each abscissas.
Examples:
>>> abscissas, weights = chaospy.quadrature.gegenbauer(3, alpha=2)
>>> abscissas
array([[-0.72741239, -0.26621648, 0.26621648, 0.72741239]])
>>> weights
array([0.10452141, 0.39547859, 0.39547859, 0.10452141])
See also:
:func:`chaospy.quadrature.gaussian`
"""
order = int(order)
coefficients = chaospy.construct_recurrence_coefficients(
order=order, dist=chaospy.Beta(alpha + 0.5, alpha + 0.5, lower, upper)
)
[abscissas], [weights] = chaospy.coefficients_to_quadrature(coefficients)
weights *= 2 ** (2 * alpha - 1) if physicist else 1
return abscissas[numpy.newaxis], weights