numpy.polynomial.chebyshev.chebgauss — NumPy v1.13 Manual (original) (raw)

numpy.polynomial.chebyshev. chebgauss(deg)[source]¶

Gauss-Chebyshev quadrature.

Computes the sample points and weights for Gauss-Chebyshev quadrature. These sample points and weights will correctly integrate polynomials of degree $2*deg - 1$ or less over the interval $[-1, 1]$ with the weight function $f(x) = 1/\sqrt{1 - x^2}$ .

Parameters:	deg : int Number of sample points and weights. It must be >= 1.
Returns:	x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights.

Notes

New in version 1.7.0.

The results have only been tested up to degree 100, higher degrees may be problematic. For Gauss-Chebyshev there are closed form solutions for the sample points and weights. If n = deg, then

$x_i = \cos(\pi (2 i - 1) / (2 n))$

$w_i = \pi / n$