scipy.special.

roots_chebyu#

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

Gauss-Chebyshev (second 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 second kind, \(U_n(x)\). These sample points and weights correctly integrate polynomials of degree \(2n - 1\) or less over the interval \([-1, 1]\) with weight function \(w(x) = \sqrt{1 - x^2}\). See 22.2.5 in [AS] for 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_chebyu
>>> roots, weights = roots_chebyu(5)
>>> roots
array([-8.66025404e-01, -5.00000000e-01,  6.12323400e-17,  5.00000000e-01,
        8.66025404e-01])
>>> weights
array([0.13089969, 0.39269908, 0.52359878, 0.39269908, 0.13089969])

Verify that the values in roots are roots of the Chebyshev polynomial of second kind \(U_5(x)\).

>>> from scipy.special import eval_chebyu
>>> eval_chebyu(5, roots)
array([-1.33226763e-15, -1.77635684e-15,  3.67394040e-16, -8.88178420e-16,
        1.33226763e-15])

The values of \(U_5(x)\) evaluated at the roots are indeed very close to zero.

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

>>> sum(weights)
np.float64(1.5707963267948966)
>>> roots, weights, sum_of_weights = roots_chebyu(5, mu=True)
>>> sum_of_weights
1.5707963267948966

Roots and weights obtained from the Chebyshev polynomial of the second kind \(U_n(x)\) 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(0.19634954084936204)

The exact result is \(\pi/16\).

>>> from math import pi
>>> pi/16
0.19634954084936207

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 \(\pi J_1(1)\), where \(J_1\) is the Bessel function of first kind and order 1.

>>> import numpy as np
>>> weights @ np.cos(roots)
np.float64(1.3824596877989561)
>>> from scipy.special import jv
>>> pi*jv(1, 1)
np.float64(1.3824596873841686)