chaospy.quadrature.gaussian

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

Create Gaussian quadrature nodes and weights.

Generating Gaussian quadrature by first generating so called three terms recurrence coefficients using one various different algorithms. The coefficients are them converted to abscissas and weights by constructing lower banded Jacobi matrix and calculating eigenvalues and eigenvectors, which can be directly translated to abscissas and weights. Construction of the coefficients is potentially numerically unstable, there for multiple algorithms exists.

Args:

order (int):

The order of the quadrature.

dist (chaospy.Distribution):

The distribution which density will be used as weight function.

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):

Gaussian quadrature abscissas and weights with shapes (len(dist), order+1) and (order+1,) respectively.

Raises:

NotImplementedError:

In the case of analytical three terms recurrence algorithm, error is raised if the distribution does not implement the feature. coefficients.

numpy.linalg.LinAlgError:

For non-canonical random variables, the construction might fail because of illegal numerical operations.

Notes:

Quadrature constructed as outlined by Walter Gautschi [3].

Examples:

>>> distribution = chaospy.Normal(0, 1)
>>> abscissas, weights = chaospy.quadrature.gaussian(5, distribution)
>>> abscissas.round(4)
array([[-3.3243, -1.8892, -0.6167,  0.6167,  1.8892,  3.3243]])
>>> weights.round(4)
array([0.0026, 0.0886, 0.4088, 0.4088, 0.0886, 0.0026])
>>> distribution = chaospy.J(chaospy.Uniform(), chaospy.Normal())
>>> abscissas, weights = chaospy.quadrature.gaussian(2, distribution)
>>> abscissas.round(2)
array([[ 0.11,  0.11,  0.11,  0.5 ,  0.5 ,  0.5 ,  0.89,  0.89,  0.89],
       [-1.73,  0.  ,  1.73, -1.73,  0.  ,  1.73, -1.73,  0.  ,  1.73]])
>>> weights.round(3)
array([0.046, 0.185, 0.046, 0.074, 0.296, 0.074, 0.046, 0.185, 0.046])