"""
Generate Gauss-Patterson quadrature abscissa and weights.
Example usage
-------------
With increasing order::
>>> distribution = chaospy.Beta(2, 2, lower=-1, upper=1)
>>> for order in range(3):
... abscissas, weights = chaospy.generate_quadrature(
... order, distribution, rule="patterson")
... print(abscissas.round(2), weights.round(2))
[[0.]] [1.]
[[-0.77 0. 0.77]] [0.17 0.67 0.17]
[[-0.96 -0.77 -0.43 0. 0.43 0.77 0.96]] [0.01 0.08 0.24 0.34 0.24 0.08 0.01]
"""
try:
from functools import lru_cache
except ImportError: # pragma: no cover
from functools32 import lru_cache
import numpy
import chaospy
from .hypercube import hypercube_quadrature
[docs]def 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":cite:`kythe_handbook_2004`, and "The Optimal Addition of
Points to Quadrature Formulae":cite:`patterson_optimum_1968`.
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])
"""
abscissas, weights = hypercube_quadrature(
patterson_simple,
order=int(order),
domain=domain,
)
return abscissas, weights
@lru_cache(None)
def patterson_simple(order):
assert order < 9
abscissas, weights = PATTERSON_VALUES[int(order)]
return numpy.array(abscissas) * 0.5 + 0.5, numpy.array(weights) / 2.0
PATTERSON_VALUES = {
0: ((0e00,), (2.0e00,)),
1: (
(
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0.0e00,
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),
(
0.555555555555555555556e00,
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0.555555555555555555556e00,
),
),
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(
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),
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),
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),
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