`scipy.special._orthogonal:roots_gegenbauer`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:1938

Signature

def roots_gegenbauer(n, alpha, mu = False)

Summary

Gauss-Gegenbauer quadrature.

Extended Summary

Compute the sample points and weights for Gauss-Gegenbauer quadrature. The sample points are the roots of the nth degree Gegenbauer polynomial, $C_{n}^{α} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[- 1, 1]$ with weight function $w (x) = (1 - x^{2})^{α - 1/2}$ . See 22.2.3 in [AS] for more details.

Parameters

n : int: quadrature order.
alpha : float: alpha must be > -0.5.
mu : bool, optional: If True, return the sum of the weights, optional.

Returns

x : ndarray: Sample points
w : ndarray: Weights
mu : float: Sum of the weights

Examples

Special cases of Gauss-Gegenbauer quadrature are the Gauss-Chebyshev first kind quadrature (:math:`\alpha=0`) and Gauss-Chebyshev second kind quadrature (:math:`\alpha=1`). Therefore, roots and weights obtained from `roots_gegenbauer` should agree with those obtained from `roots_chebyt` and `roots_chebyu` for the appropriate values of :math:`\alpha`.

from scipy.special import roots_chebyt, roots_chebyu, roots_gegenbauer

✓

roots_gegenbauer(5, 0)  # doctest: +NORMALIZE_WHITESPACE
roots_chebyt(5)  # doctest: +NORMALIZE_WHITESPACE

✗

(array([-0.951, -0.588,  0.   ,  0.588,  0.951]), array([0.628, 0.628, 0.628, 0.628, 0.628]))

roots_gegenbauer(5, 1)  # doctest: +NORMALIZE_WHITESPACE
roots_chebyu(5)  # doctest: +NORMALIZE_WHITESPACE

✗

(array([-0.866, -0.5  ,  0.   ,  0.5  ,  0.866]), array([0.131, 0.393, 0.524, 0.393, 0.131]))

The sum of weights should equal the integral from -1 to 1 of :math:`(1-x^2)^{\alpha-1/2}` which evaluates to :math:`\sqrt{\pi}\Gamma(\alpha+1/2)/\Gamma(\alpha+1)`.

alpha = 0.7
roots, weights, sum_of_weights = roots_gegenbauer(5, alpha, mu=True)
sum(weights)
sum_of_weights
from math import gamma, pi, sqrt
sqrt(pi) * gamma(alpha+0.5) / gamma(alpha+1)

✓

1.7910437497388667

Roots and weights obtained from the Gegenbauer polynomial :math:`C^\alpha_n(x)` are used in Gauss-Gegenbauer quadrature where the integral from -1 to 1 of :math:`f(x)(1-x^2)^{\alpha-1/2}` is evaluated. Roots and weights for order :math:`n` are expected to yield the exact result for polynomials :math:`f(x)` of a maximal order of :math:`2n-1`.

f = lambda x: x**4
weights @ f(roots)

✓

np.float64(0.29265420747367116)

This result is indeed very close to the exact value of :math:`3\sqrt{\pi}\Gamma(\alpha+1/2)/4\Gamma(\alpha+3)`.

(3*sqrt(pi)/4) * gamma(alpha+0.5) / gamma(alpha+3)

✓

0.2926542074736711

In general, Gauss-Gegenbauer quadrature will only yield an approximate value of the integral. Consider the integral from -1 to 1 of :math:`\cos(x)(1-x^2)^{\alpha-1/2}` which evaluates to :math:`2^\alpha\sqrt{\pi}\Gamma(\alpha+1/2)J_\alpha(1)` where :math:`J_\alpha` is the Bessel function of first kind and order :math:`\alpha`.

import numpy as np
weights @ np.cos(roots)
from scipy.special import jv
2**alpha * sqrt(pi) * gamma(alpha+0.5) *jv(alpha, 1)

✓

np.float64(1.5395778706293377)

Aliases

scipy.special.cg_roots