openquad.S2#

class openquad.S2(method_specs, polar_sampling='cos')#

Two-dimensional unit sphere.

This class provides an interface to construct quadrature methods for the integral

\[\int_{0}^{\pi} \int_{0}^{2\pi} f(\theta, \phi) \, \sin(\theta) \, d\phi \, d\theta\]

The innermost integral is evaluated first.

The default coordinates of this geometry are the spherical polar angles [3] in the order \((\theta, \phi)\).

Parameters:
method_specslist of tuples

Specification of the quadrature methods. Each entry of the list contains a tuple corresponding to a quadrature method and its options.

The first entry of each tuple needs to be a string denoting the quadrature method. See this list for an overview of available methods. The other entries of the tuple need to be keyword arguments for the individual methods. At least, size needs to be given, or degree if applicable. Other options are tabulated here.

Individual quadrature methods are applied to the default coordinates in the order mentioned above. See below, for examples.

polar_sampling{‘angle’, ‘cos’}, optional

Controls if the second integral is sampled along \(\theta\) or \(\cos\theta\).

References

Examples

A quadrature composed of one-dimensional quadratures. Gauss-Legendre quadrature is applied to the first coordinate, i.e., \(\theta\). The composite trapezoidal rule is applied to the second coordinate, \(\phi\):

>>> openquad.S2([
...     ('GaussLegendre', dict(degree=6)),
...     ('trapezoid', dict(size=7)),
... ])

A quadrature composed of a two-dimensional \(\mathrm{S}^2\) Gauss quadrature:

>>> openquad.S2([
...     ('S2-Gauss-LebedevLaikov', dict(degree=5)),
... ])
Attributes:
dimint

Dimension of the geometry.

sizeint

Total number of sampling points.

shapetuple of int

The shape of points, (dim, size).

pointsndarray

Sampling points in the default coordinates. The array has shape (2, size).

weightsndarray

Quadrature weights. The array has shape (size,).

anglesndarray

Equivalent to points

xyzndarray

Sampling points in cartesian coordinates, \((x, y, z)\). This array has shape (3, size).

integrate(f, f_kwargs={}, axis=-1)#

Integrate the given data or callable.

Parameters:
fcallable or array_like

Representation of the integrand. Either an array-like or an callable.

f_kwargsdict, optional

Additional keyword arguments to pass to f, if f is a callable.

axissequence of ints, optional

Axes along which to perform the integration. The number of axes provided must match the number of integration methods hold by the quadrature object. By default, start with the last axis of f.

Caution

Experimental feature

Returns:
resultscalar or ndarray

The result of the integration. If the dimension of f equals the number of integration methods, the result is a scalar.

savetxt(filename, weights=True)#

Save sampling points and weights to a text file.

This function wraps numpy.savetxt().

Parameters:
filenamefilename, file handle or pathlib.Path

Name of the text file.

weightslogical, optional

If False, only save sampling points and not the quadrature weights.

angles#
property dim#

int: Dimension of this geometry.

points#
weights#
xyz#
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