Quadrature methods

Currently, the following quadrature methods are implemented, sorted according to their geometry.

The first entry denotes the input string for the method specification, cf. the method_spec parameter in openquad.Rn.

\(\mathbb{R}^1\) methods

• GaussLegendre: Gauss-Legendre quadrature

• GaussLegendreLobatto: Gauss-Lobatto-Legendre quadrature

• Trapezoid: Composite trapezoidal rule, see scipy.integrate.trapezoid()

• Simpson: Composite Simpson’s rule, see scipy.integrate.simpson()

• Romberg: Romberg’s method, see scipy.integrate.romberg()

• 1d-MonteCarlo:Monte Carlo integration

These methods need the following additional keyword paramters:

• size: float: Number of sampling points.

• degree: float: Degree of exactness. For Gauss methods alternative to size.

• a : float: Lower boundary of the integral.

• b : float: Upper boundary of the integral.

• jacobian : callable, optional: Jacobian to apply to the coordinate.

• periodic : logical, optinal: If True, assume periodic boundary conditions.

\(\mathrm{S}^2\) methods

• S2-Gauss-LebedevLaikov: spherical Gauss quadrature from Lebedev-Laikov

• S2-Gauss-Graef: spherical Gauss qudarature from Manuel Gräf

• S2-Design-Graef: spherical designs from Manuel Gräf

• S2-Design-Womersley: spherical designs from Robert Womersley

• S2-Covering-Fibonacci: near-uniform coverings based on Fibonacci numbers

• S2-Covering-ZCW: near-uniform coverings with the ZCW method

• S2-MonteCarlo: Monte Carlo integration

These methods need the following additional keyword paramters:

• size: float: Number of sampling points.

• degree: float: Degree of exactness. For Gauss methods and spherical desings alternative to size.

\(\mathrm{SO}(3)\) methods

• SO3-Gauss-Graef: spherical Gauss qudarature from Manuel Gräf

• SO3-Chebyshev-Graef: spherical Chebyshev quadrature from Manuel Gräf

• SO3-Covering-Karney: near-uniform coverings from Charles Karney

• SO3-MonteCarlo: Monte Carlo integration

These methods need the following additional keyword paramters:

• size: float: Number of sampling points.

• degree: float: Degree of exactness. For Gauss and Chebyshev methods alternative to size.

Note

The following extensions are planned for the next release:

• Implementing the three-dimensional unit sphere, \(\mathrm{S}^3\).

• Adding spherical desings and coverings from Neil Sloane and Ronald Hardin.

• Providing a way to use \(\mathrm{S}^3\) quadratures for \(\mathrm{SO}(3)\) and vice versa.

• Implementing a class for the unit circle, \(\mathrm{S}^1\).

• and more…