`scipy.special._orthogonal:roots_chebyc`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:2596

Signature

def roots_chebyc(n, mu = False)

Summary

Gauss-Chebyshev (first 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 first kind, $C_{n} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[- 2, 2]$ with weight function $w (x) = 1/ 1 - (x /2)^{2}$ . See 22.2.6 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_chebyc
roots, weights = roots_chebyc(5)

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roots
weights

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array([1.257, 1.257, 1.257, 1.257, 1.257])

Verify that the values in `roots` are roots of the Chebyshev polynomial of first kind :math:`C_5(x)`.

from scipy.special import eval_chebyc

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eval_chebyc(5, roots)

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

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

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

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6.283185307179586

Roots and weights obtained from the Chebyshev polynomial of the first kind :math:`C_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

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weights @ f(roots)

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np.float64(37.69911184307751)

The exact result is :math:`12\pi`.

from math import pi
12*pi

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37.69911184307752

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}` which evaluates to :math:`2\pi J_0(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
2*pi*jv(0, 2)

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np.float64(1.4067472539132013)

Aliases

scipy.special.c_roots