chaospy.quadrature.fejer_2¶

chaospy.quadrature.fejer_2(order, domain=(0, 1), growth=False, segments=1)[source]¶

Generate the quadrature abscissas and weights in Fejér type II quadrature.

Fejér proposed two quadrature rules very similar to chaospy.quadrature.clenshaw_curtis(). The only difference is that the endpoints are removed. That is, Fejér only used the interior extrema of the Chebyshev polynomials, i.e. the true stationary points. This makes this a better method for performing quadrature on infinite intervals, as the evaluation does not contain endpoint values.

Args:

order (int, numpy.ndarray):: Quadrature order.
domain (chaospy.Distribution, numpy.ndarray):: Either distribution or bounding of interval to integrate over.
growth (bool):: If True sets the growth rule for the quadrature rule to only include orders that enhances nested samples.
segments (int):: Split intervals into N subintervals and create a patched quadrature based on the segmented quadrature. Can not be lower than order. If 0 is provided, default to square root of order. Nested samples only exist when the number of segments are fixed.

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:

Implemented as proposed by Waldvogel [2].

Example:

>>> abscissas, weights = chaospy.quadrature.fejer_2(3, (0, 1))
>>> abscissas.round(4)
array([[0.0955, 0.3455, 0.6545, 0.9045]])
>>> weights.round(4)
array([0.1804, 0.2996, 0.2996, 0.1804])