`scipy.special._orthogonal:roots_chebyt`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:2168

Signature

def roots_chebyt(n, mu = False)

Summary

Gauss-Chebyshev (first 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 first kind, $T_{n} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[- 1, 1]$ with weight function $w (x) = 1/ 1 - x^{2}$ . See 22.2.4 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_chebyt
roots, weights = roots_chebyt(5)

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roots
weights

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

Verify that the values in `roots` are roots of the Chebyshev polynomial of first kind :math:`T_5(x)`.

from scipy.special import eval_chebyt

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eval_chebyt(5, roots)

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

The values 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 -1 to 1 of :math:`1/\sqrt{1-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_chebyt(5, mu=True)
sum_of_weights

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3.141592653589793

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

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weights @ f(roots)

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np.float64(1.1780972450961722)

The exact result is :math:`3\pi/8`.

from math import pi
3*pi/8

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1.1780972450961724

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_0(1)`, 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*jv(0, 1)

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np.float64(2.4039394306344133)

Aliases

scipy.special.roots_chebyt