`scipy.special._orthogonal:roots_sh_legendre`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:3515

Signature

def roots_sh_legendre(n, mu = False)

Summary

Gauss-Legendre (shifted) quadrature.

Extended Summary

Compute the sample points and weights for Gauss-Legendre quadrature. The sample points are the roots of the nth degree shifted Legendre polynomial $P_{n}^{*} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[0, 1]$ with weight function $w (x) = 1.0$ . See 22.2.11 in [AS] for details.

Parameters

n : int: quadrature order
mu : bool, optional: If True, return the sum of the weights, optional.

Returns

x : ndarray: Sample points
w : ndarray: Weights
mu : float: Sum of the weights

Examples

Compute nodes and weights for a 7th-order shifted Gauss-Legendre quadrature:

from scipy.special import roots_sh_legendre, eval_sh_legendre
x, w = roots_sh_legendre(7)

✓

x
w

✗

array([0.06474, 0.13985, 0.19092, 0.20898, 0.19092, 0.13985, 0.06474])

Verify that ``x`` are the roots of the degree-7 shifted Legendre polynomial:

eval_sh_legendre(7, x)

✗

array([ 0.,  0.,  0.,  0., -0., -0., -0.])

The sum of the weights for shifted Gauss-Legendre quadrature is always 1:

x, w, mu = roots_sh_legendre(10, mu=True)

✓

mu

✗

1.0

Aliases

scipy.special.ps_roots