`scipy.special._orthogonal:roots_chebys`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:2787

Signature

def roots_chebys(n, mu = False)

Summary

Gauss-Chebyshev (second kind) quadrature.

Extended Summary

Compute 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, $S_{n} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[- 2, 2]$ with weight function $w (x) = 1 - (x /2)^{2}$ . See 22.2.7 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_chebys
roots, weights = roots_chebys(5)

✓

roots
weights

✗

array([0.2618, 0.7854, 1.0472, 0.7854, 0.2618])

Verify that the values in `roots` are roots of the Chebyshev polynomial of first kind :math:`S_5(x)`.

from scipy.special import eval_chebys

✓

eval_chebys(5, roots)

✗

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

The values of :math:`S_5(x)` evaluated at the roots are indeed very close to zero. Verify that the sum of the weights equals the integral from -2 to 2 of :math:`\sqrt{1-(x/2)^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_chebys(5, mu=True)
sum_of_weights

✓

3.141592653589793

Roots and weights obtained from the Chebyshev polynomial of the second kind :math:`S_n(x)` are used in Gauss-Chebyshev quadrature where the integral from -2 to 2 of :math:`f(x)\sqrt{1-(x/2)^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(6.283185307179585)

The exact result is :math:`2\pi`.

from math import pi
2*pi

✓

6.283185307179586

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

import numpy as np
weights @ np.cos(roots)

✓

np.float64(1.8118352216010754)

Check against the result of `scipy.integrate.quad`.

from scipy.integrate import quad
result, abserror = quad(lambda x: np.cos(x) * np.sqrt(1-(x/2)**2), -2, 2)
result

✓

1.8118344191919165

The latter result has an estimated absolute error of

abserror

✓

1.5300099409643053e-08

Aliases

scipy.special.roots_chebys