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Module « scipy.integrate »

Fonction lebedev_rule - module scipy.integrate

Signature de la fonction lebedev_rule

def lebedev_rule(n) 

Description

help(scipy.integrate.lebedev_rule)

Lebedev quadrature.

Compute the sample points and weights for Lebedev quadrature [1]_
for integration of a function over the surface of a unit sphere.

Parameters
----------
n : int
    Quadrature order. See Notes for supported values.

Returns
-------
x : ndarray of shape ``(3, m)``
    Sample points on the unit sphere in Cartesian coordinates.
    ``m`` is the "degree" corresponding with the specified order; see Notes.
w : ndarray of shape ``(m,)``
    Weights

Notes
-----
Implemented by translating the Matlab code of [2]_ to Python.

The available orders (argument `n`) are::

    3, 5, 7, 9, 11, 13, 15, 17,
    19, 21, 23, 25, 27, 29, 31, 35,
    41, 47, 53, 59, 65, 71, 77, 83,
    89, 95, 101, 107, 113, 119, 125, 131

The corresponding degrees ``m`` are::

    6, 14, 26, 38, 50, 74, 86, 110,
    146, 170, 194, 230, 266, 302, 350, 434,
    590, 770, 974, 1202, 1454, 1730, 2030, 2354,
    2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810

References
----------
.. [1] V.I. Lebedev, and D.N. Laikov. "A quadrature formula for the sphere of
       the 131st algebraic order of accuracy". Doklady Mathematics, Vol. 59,
       No. 3, 1999, pp. 477-481.
.. [2] R. Parrish. ``getLebedevSphere``. Matlab Central File Exchange.
       https://www.mathworks.com/matlabcentral/fileexchange/27097-getlebedevsphere.
.. [3] Bellet, Jean-Baptiste, Matthieu Brachet, and Jean-Pierre Croisille.
       "Quadrature and symmetry on the Cubed Sphere." Journal of Computational and
       Applied Mathematics 409 (2022): 114142. :doi:`10.1016/j.cam.2022.114142`.

Examples
--------
An example given in [3]_ is integration of :math:`f(x, y, z) = \exp(x)` over a
sphere of radius :math:`1`; the reference there is ``14.7680137457653``.
Show the convergence to the expected result as the order increases:

>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> from scipy.integrate import lebedev_rule
>>>
>>> def f(x):
...     return np.exp(x[0])
>>>
>>> res = []
>>> orders = np.arange(3, 20, 2)
>>> for n in orders:
...     x, w = lebedev_rule(n)
...     res.append(w @ f(x))
>>>
>>> ref = np.full_like(res, 14.7680137457653)
>>> err = abs(res - ref)/abs(ref)
>>> plt.semilogy(orders, err)
>>> plt.xlabel('order $n$')
>>> plt.ylabel('relative error')
>>> plt.title(r'Convergence for $f(x, y, z) = \exp(x)$')
>>> plt.show()

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