`scipy.special._orthogonal:roots_chebyu`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:2388

Signature

def roots_chebyu(n, mu = False)

Summary

Gauss-Chebyshev (second kind) quadrature.

Extended Summary

Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree Chebyshev polynomial of the second kind, $U_{n} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[- 1, 1]$ with weight function $w (x) = 1 - x^{2}$ . See 22.2.5 in [AS] for details.

Parameters

n : int: quadrature order
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

from scipy.special import roots_chebyu
roots, weights = roots_chebyu(5)

✓

roots
weights

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array([0.1309, 0.3927, 0.5236, 0.3927, 0.1309])

Verify that the values in `roots` are roots of the Chebyshev polynomial of second kind :math:`U_5(x)`.

from scipy.special import eval_chebyu

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eval_chebyu(5, roots)

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array([-0., -0.,  0., -0.,  0.])

The values of :math:`U_5(x)` evaluated at the roots are indeed very close to zero. Verify that the sum of the weights equals the integral from -1 to 1 of :math:`\sqrt{1-x^2}` which evaluates to :math:`\pi/2`. There are two ways to obtain the sum of weights, both resulting in :math:`\pi/2` within numerical precision.

sum(weights)
roots, weights, sum_of_weights = roots_chebyu(5, mu=True)
sum_of_weights

✓

1.5707963267948966

Roots and weights obtained from the Chebyshev polynomial of the second kind :math:`U_n(x)` are used in Gauss-Chebyshev quadrature where the integral from -1 to 1 of :math:`f(x)\sqrt{1-x^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)

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np.float64(0.19634954084936204)

The exact result is :math:`\pi/16`.

from math import pi
pi/16

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0.19634954084936207

In general, Gauss-Chebyshev quadrature will only yield an approximate value of the integral. Consider the integral from -1 to 1 of :math:`\cos(x)\sqrt{1-x^2}` which evaluates to :math:`\pi J_1(1)`, where :math:`J_1` is the Bessel function of first kind and order 1.

import numpy as np

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weights @ np.cos(roots)

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np.float64(1.3824596877989563)

from scipy.special import jv
pi*jv(1, 1)

✓

np.float64(1.3824596873841686)

Aliases

scipy.special.roots_chebyu