chaospy.quadrature.lobatto

chaospy.quadrature.lobatto(order, dist, recurrence_algorithm='stieltjes', rule='fejer_2', tolerance=1e-10, scaling=3, n_max=5000)

Generate the abscissas and weights in Gauss-Loboto quadrature.

Also known as Lobatto quadrature, named after Dutch mathematician Rehuel Lobatto. It is similar to Gaussian quadrature with the following differences:

• The integration points include the end points of the integration interval.

• It is accurate for polynomials up to degree \(2n–3\), where \(n\) is the number of integration points.

Args:

order (int):

Quadrature order.

dist (chaospy.distributions.baseclass.Distribution):

The distribution weights to be used to create higher order nodes from.

recurrence_algorithm (str):

Name of the algorithm used to generate abscissas and weights.

rule (str):

In the case of lanczos or stieltjes, defines the proxy-integration scheme.

tolerance (float):

The allowed relative error in norm between two quadrature orders before method assumes convergence.

scaling (float):

A multiplier the adaptive order increases with for each step quadrature order is not converged. Use 0 to indicate unit increments.

n_max (int):

The allowed number of quadrature points to use in approximation.

Returns:

(numpy.ndarray, numpy.ndarray):

abscissas:

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

weights:

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

Example:

>>> distribution = chaospy.Uniform(-1, 1)
>>> abscissas, weights = chaospy.quadrature.lobatto(4, distribution)
>>> abscissas.round(3)
array([[-1.   , -0.872, -0.592, -0.209,  0.209,  0.592,  0.872,  1.   ]])
>>> weights.round(3)
array([0.018, 0.105, 0.171, 0.206, 0.206, 0.171, 0.105, 0.018])