scipy.special.

roots_sh_chebyt#

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

Gauss-Chebyshev (first kind, shifted) quadrature.

Compute the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree shifted 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 \([0, 1]\) with weight function \(w(x) = 1/\sqrt{x - x^2}\). See 22.2.8 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_sh_chebyt
>>> roots, weights = roots_sh_chebyt(5)
>>> roots
array([0.02447174, 0.20610737, 0.5       , 0.79389263, 0.97552826])
>>> weights
array([0.62831853, 0.62831853, 0.62831853, 0.62831853, 0.62831853])

Verify that the values in roots are roots of the shifted Chebyshev polynomial of the first kind \(T^*_5(x)=T_5(2x-1)\).

>>> from scipy.special import eval_chebyt, eval_sh_chebyt
>>> eval_chebyt(5, 2*roots-1)
array([8.8817842e-16, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
       8.8817842e-16])
>>> eval_sh_chebyt(5, roots)
array([8.8817842e-16, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
       8.8817842e-16])

The values of \(T_5(2x-1)\) and 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 0 to 1 of \(1/\sqrt{x-x^2}\) 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_sh_chebyt(5, mu=True)
>>> sum_of_weights
3.141592653589793

Roots and weights obtained from the shifted Chebyshev polynomial of the first kind \(T^*_n(x)\) are used in Gauss-Chebyshev quadrature where the integral from 0 to 1 of \(f(x)/\sqrt{x-x^2}\) 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(0.8590292412159591)

The exact result is \(35\pi/128\).

>>> from math import cos, pi
>>> 35*pi/128
0.859029241215959

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

>>> import numpy as np
>>> weights @ np.cos(roots)
np.float64(2.5873677615532227)
>>> from scipy.special import jv
>>> pi * cos(0.5) * jv(0, 0.5)
np.float64(2.587367761551782)
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