`scipy.special._orthogonal:roots_sh_chebyt`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:2986

Signature

def roots_sh_chebyt(n, mu = False)

Summary

Gauss-Chebyshev (first kind, shifted) quadrature.

Extended Summary

Compute the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree shifted Chebyshev polynomial of the first kind, $T_{n} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[0, 1]$ with weight function $w (x) = 1/ x - x^{2}$ . See 22.2.8 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_chebyt
roots, weights = roots_sh_chebyt(5)

✓

roots
weights

✗

array([0.62832, 0.62832, 0.62832, 0.62832, 0.62832])

Verify that the values in `roots` are roots of the shifted Chebyshev polynomial of the first kind :math:`T^*_5(x)=T_5(2x-1)`.

from scipy.special import eval_chebyt, eval_sh_chebyt

✓

eval_chebyt(5, 2*roots-1)
eval_sh_chebyt(5, roots)

✗

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

The values of :math:`T_5(2x-1)` and of :math:`T^*_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:`1/\sqrt{x-x^2}` which evaluates to :math:`\pi`. There are two ways to obtain the sum of weights, both resulting in :math:`\pi` within numerical precision.

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

✓

3.141592653589793

Roots and weights obtained from the shifted Chebyshev polynomial of the first kind :math:`T^*_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(0.8590292412159591)

The exact result is :math:`35\pi/128`.

from math import cos, pi
35*pi/128

✓

0.859029241215959

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}` which evaluates to :math:`\pi\cos(1/2) J_0(1/2)`, where :math:`J_0` is the Bessel function of first kind and order 0.

import numpy as np
weights @ np.cos(roots)
from scipy.special import jv
pi * cos(0.5) * jv(0, 0.5)

✓

np.float64(2.587367761551782)

Aliases

scipy.special.roots_sh_chebyt