def integration_rule(self):
'''
Return the integration rule associated with the measure
Returns
-------
tensap.IntegrationRule
The integration rule associated with the measure
'''
return tensap.IntegrationRule(self.values, self.weights)
def integration_rule(self):
'''
Return the integration rule object associated with the discrete random
variable.
Returns
-------
tensap.IntegrationRule
The integration rule object associated with the discrete random
variable.
'''
return tensap.IntegrationRule(self.values, self.probabilities)
def gauss_integration_rule(self, nb_pts):
'''
Return the nb_pts-points gauss integration rule associated with the
measure of self, using Golub-Welsch algorithm.
Parameters
----------
nb_pts : int
The number of integration points.
Returns
-------
tensap.IntegrationRule
The integration rule associated with the measure of self.
'''
poly = self.orthonormal_polynomials(nb_pts + 1)
if isinstance(poly, tensap.ShiftedOrthonormalPolynomials):
shift = poly.shift
scaling = poly.scaling
poly = poly.polynomials
flag = True
else:
flag = False
coef = poly._recurrence_coefficients
if coef.shape[1] < nb_pts:
coef = poly.recurrence(poly.measure, nb_pts - 1)
else:
coef = coef[:, :nb_pts]
# Jacobi matrix
if nb_pts == 1:
jacobi_matrix = np.diag(coef[0, :])
else:
jacobi_matrix = np.diag(coef[0, :]) + \
np.diag(np.sqrt(coef[1, 1:]), -1) + \
np.diag(np.sqrt(coef[1, 1:]), 1)
# Quadrature points are the eigenvalues of the Jacobi matrix, weights
# are deduced from the eigenvectors
eig_values, eig_vectors = np.linalg.eig(jacobi_matrix)
points = np.sort(eig_values)
ind = np.argsort(eig_values)
eig_vectors = eig_vectors[:, ind]
weights = eig_vectors[0, :]**2 / np.sqrt(np.sum(eig_vectors**2, 0))
if flag:
points = shift + scaling * points
return tensap.IntegrationRule(points, weights)