`scipy.special._orthogonal:roots_sh_chebyu`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:3156

Signature

def roots_sh_chebyu(n, mu = False)

Summary

Gauss-Chebyshev (second kind, shifted) quadrature.

Extended Summary

Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree shifted 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 $[0, 1]$ with weight function $w (x) = x - x^{2}$ . See 22.2.9 in [AS] for more 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_sh_chebyu
roots, weights = roots_sh_chebyu(5)

✓

roots
weights

✗

array([0.03272, 0.09817, 0.1309 , 0.09817, 0.03272])

Verify that the values in `roots` are roots of the shifted Chebyshev polynomial of the second kind :math:`U^*_5(x)=U_5(2x-1)`.

from scipy.special import eval_chebyu, eval_sh_chebyu

✓

eval_chebyu(5, 2*roots-1)
eval_sh_chebyu(5, roots)

✗

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

The values of :math:`U_5(2x-1)` and :math:`U^*_5(x)` evaluated at the roots are indeed zero or very close to it. Verify that the sum of the weights equals the integral from 0 to 1 of :math:`\sqrt{x-x^2}` which evaluates to :math:`\pi/8`. There are two ways to obtain the sum of weights, both resulting in :math:`\pi/8` within numerical precision.

sum(weights)
roots, weights, sum_of_weights = roots_sh_chebyu(5, mu=True)
sum_of_weights
from math import pi
pi/8

✓

0.39269908169872414

Roots and weights obtained from the shifted Chebyshev polynomial of the second kind :math:`U^*_n(x)` are used in Gauss-Chebyshev quadrature where the integral from 0 to 1 of :math:`f(x)\sqrt{x-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)

✓

np.float64(0.06442719309119692)

The exact result is :math:`21\pi/1024`.

21*pi/1024

✓

0.06442719309119693

In general, Gauss-Chebyshev quadrature will only yield an approximate value of the integral. Consider the integral from 0 to 1 of :math:`\cos(x)\sqrt{x-x^2}`.

import numpy as np
weights @ np.cos(roots)
from scipy.integrate import quad
quad(lambda x: np.cos(x) * np.sqrt(x-x*x), 0, 1)

✓

(0.3339679082866854, 1.4951873072988064e-11)

The two results agree better than the estimated absolute error, i.e. the second value in the last output.

Aliases

scipy.special.roots_sh_chebyu