chaospy.quadrature.clenshaw_curtis¶

chaospy.quadrature.clenshaw_curtis(order, domain=(0.0, 1.0), growth=False, segments=1)[source]¶

Generate the quadrature nodes and weights in Clenshaw-Curtis quadrature.

Clenshaw-Curtis quadrature method is a good all-around quadrature method comparable to Gaussian quadrature, but typically limited to finite intervals without a specific weight function. In addition to be quite accurate, the weights and abscissas can be calculated quite fast.

Another thing to note is that Clenshaw-Curtis, with an appropriate growth rule is fully nested. This means, if one applies a method that combines different order of quadrature rules, the number of evaluations can often be reduced as the abscissas can be used across levels.

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 steps 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 appear 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), steps) where steps is the number of samples.
weights (numpy.ndarray):: The quadrature weights with weights.shape == (steps,).

Notes:

Implemented as proposed by Waldvogel [2].

Example:

>>> abscissas, weights = chaospy.quadrature.clenshaw_curtis(4, (0, 1))
>>> abscissas.round(4)
array([[0.    , 0.1464, 0.5   , 0.8536, 1.    ]])
>>> weights.round(4)
array([0.0333, 0.2667, 0.4   , 0.2667, 0.0333])