scipy.special.

roots_chebyc#

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

Gauss-Chebyshev (first kind) quadrature.

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 \(2n - 1\) or less over the interval \([-2, 2]\) with weight function \(w(x) = 1 / \sqrt{1 - (x/2)^2}\). See 22.2.6 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

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_chebyc
>>> roots, weights = roots_chebyc(5)
>>> roots
array([-1.90211303, -1.1755705 ,  0.        ,  1.1755705 ,  1.90211303])
>>> weights
array([1.25663706, 1.25663706, 1.25663706, 1.25663706, 1.25663706])

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

>>> from scipy.special import eval_chebyc
>>> eval_chebyc(5, roots)
array([ 1.77635684e-15,  0.00000000e+00,  0.00000000e+00,  0.00000000e+00,
       -1.77635684e-15])

The values of \(C_5(x)\) 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 \(1/\sqrt{1-(x/2)^2}\) which evaluates to \(2\pi\). There are two ways to obtain the sum of weights, both resulting in \(2\pi\) within numerical precision.

>>> sum(weights)
np.float64(6.283185307179586)
>>> roots, weights, sum_of_weights = roots_chebyc(5, mu=True)
>>> sum_of_weights
6.283185307179586

Roots and weights obtained from the Chebyshev polynomial of the first kind \(C_n(x)\) are used in Gauss-Chebyshev quadrature where the integral from -2 to 2 of \(f(x)/\sqrt{1-(x/2)^2}\) 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(37.69911184307752)

The exact result is \(12\pi\).

>>> from math import pi
>>> 12*pi
37.69911184307752

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

>>> import numpy as np
>>> weights @ np.cos(roots)
np.float64(1.4067504148408285)
>>> from scipy.special import jv
>>> 2*pi*jv(0, 2)
np.float64(1.4067472539132013)