How-tos#

Recipies to achieve common tasks using OpenQuad, targeted towards users familiar with the basic concepts.

For detailed description of the classes and methods contained in OpenQuad, see the API reference.

How to access quadrature points and weights#

Obtain Gauss-Legendre sample points and weights for degree 71 on the interval [-10, 5]:

>>> from openquad import Rn
>>> quad = Rn([
...     ('GaussLegendre', dict(degree=21, a=-10, b=5)),
... ])

Quadrature points and weights are stored in points and weights,

>>> quad.points.shape
(1, 11)
>>> quad.weights.shape
(11,)

How to integrate a python function#

Given you have a python function defined on the unit sphere in terms of spherical polar coordinates.

>>> def func(theta, phi):
...     """Spherical harmonic |Y_{2, 1}|^2."""
...     return ((np.sin(2*theta) * np.cos(phi))*np.sqrt(15/(16*np.pi)))**2

For example, initiate a \(\mathrm{S}^2\) spherical design of degree 7:

>>> from openquad import S2
>>> quad = S2([
...     ('S2-Design-Graef', dict(degree=7)),
... ])

To evaluate the integral of func over \(\mathrm{S}^2\) use the integrate() method:

>>> quad.integrate(func)
np.float64(0.9999999999975318)

How to export quadrature points and weights#

For example, create a quadrature method for an integral over the three Euler angles, using Lebedev-Laikov quadrature of degree 5 for the first two angles combined with the composite trapezoid rule with 6 sample points for the third angle.

>>> from openquad import SO3
>>> quad = SO3([
...     ('LebedevLaikov', dict(degree=5)),
...     ('Trapezoid', dict(size=6)),
... ])

Save sample points and weights to a text with the savetxt() method.

>>> quad.savetxt('points_and_weights.dat')
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