scipy.special.

roots_chebyt#

scipy.special.roots_chebyt(n, mu=False)[source]#

Gauss-Chebyshev (first kind) quadrature.

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 \(2n - 1\) or less over the interval \([-1, 1]\) with weight function \(w(x) = 1/\sqrt{1 - x^2}\). See 22.2.4 in [AS] for more details.

Parameters:

nint: quadrature order
mubool, optional: If True, return the sum of the weights, optional.

Returns:

xndarray: Sample points
wndarray: Weights
mufloat: Sum of the weights

See also

scipy.integrate.fixed_quad
numpy.polynomial.chebyshev.chebgauss

References

[AS]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples

>>> from scipy.special import roots_chebyt
>>> roots, weights = roots_chebyt(5)
>>> roots
array([-0.95105652, -0.58778525,  0.        ,  0.58778525,  0.95105652])
>>> weights
array([0.62831853, 0.62831853, 0.62831853, 0.62831853, 0.62831853])

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

>>> from scipy.special import eval_chebyt
>>> eval_chebyt(5, roots)
array([ 8.8817842e-16,  0.0000000e+00,  0.0000000e+00,  0.0000000e+00,
       -8.8817842e-16])

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

>>> sum(weights)
np.float64(3.141592653589793)
>>> roots, weights, sum_of_weights = roots_chebyt(5, mu=True)
>>> sum_of_weights
3.141592653589793

Roots and weights obtained from the Chebyshev polynomial of the first kind \(T_n(x)\) are used in Gauss-Chebyshev quadrature where the integral from -1 to 1 of \(f(x)/\sqrt{1-x^2}\) is evaluated. Roots and weights for order \(n\) are expected to yield the exact result for polynomials \(f(x)\) of a maximal order of \(2n-1\).

>>> f = lambda x: x**4
>>> weights @ f(roots)
np.float64(1.1780972450961724)

The exact result is \(3\pi/8\).

>>> from math import pi
>>> 3*pi/8
1.1780972450961724

In general, Gauss-Chebyshev quadrature will only yield an approximate value of the integral. Consider the integral from -1 to 1 of \(\cos(x)/\sqrt{1-x^2}\) which evaluates to \(\pi J_0(1)\), where \(J_0\) is the Bessel function of first kind and order 0.

>>> import numpy as np
>>> weights @ np.cos(roots)
np.float64(2.4039394322872774)
>>> from scipy.special import jv
>>> pi*jv(0, 1)
np.float64(2.4039394306344133)