chaospy.quadrature.patterson

chaospy.quadrature.patterson(order, domain)

Generate Gauss-Patterson quadrature abscissa and weights.

The Gauss-Patterson quadrature is a nested family which begins with the Gauss-Legendre rules of orders 1 and 3, and then successively inserts one new abscissa in each subinterval. Thus, after the second rule, the Gauss-Patterson rules do not have the super-high precision of the Gauss-Legendre rules. They trade this precision in exchange for the advantages of nestedness. This means that Gauss-Patterson rules are only available for orders of 0, 2, 6, 14, 30, 62, 126, 254 or 510.

Args:

order (int):

The quadrature order. Must be in the interval (0, 8).

domain (chaospy.Distribution, numpy.ndarray):

Either distribution or bounding of interval to integrate over.

Returns:

abscissas: (numpy.ndarray):

The quadrature points for where to evaluate the model function with abscissas.shape == (len(dist), N) where N is the number of samples.

weights: (numpy.ndarray):

The quadrature weights with weights.shape == (N,).

Notes:

This code is an adapted of John Burkardt’s implementation in Fortran. The algorithm is taken from “Handbook of Computational Methods for Integration”[6], and “The Optimal Addition of Points to Quadrature Formulae”[7].

Example:

>>> distribution = chaospy.Iid(chaospy.Uniform(0, 1), 2)
>>> abscissas, weights = chaospy.quadrature.patterson(1, distribution)
>>> abscissas.round(3)
array([[0.113, 0.113, 0.113, 0.5  , 0.5  , 0.5  , 0.887, 0.887, 0.887],
       [0.113, 0.5  , 0.887, 0.113, 0.5  , 0.887, 0.113, 0.5  , 0.887]])
>>> weights.round(3)
array([0.077, 0.123, 0.077, 0.123, 0.198, 0.123, 0.077, 0.123, 0.077])