core.Nodes module

exception NodesError

Bases: Exception

Exception class to handle error in NodesGenerator class

class NodesGenerator(node_type='LEGENDRE', quad_type='LOBATTO')

Bases: object

Class that can be used to generate generic distribution of nodes derived from Gauss quadrature rule. Its implementation is fully inspired from a book of W. Gautschi.

node_type

The type of node distribution

Type:

str

quad_type

The quadrature type

Type:

str

evalOrthogPoly(t, alpha, beta)

Evaluates the two higher order orthogonal polynomials corresponding to the given (alpha,beta) coefficients.

Parameters:

• t (float or np.1darray) – The point where to evaluate the orthogonal polynomials.

• alpha (np.1darray) – The alpha coefficients of the three-term recurrence.

• beta (np.1darray) – The beta coefficients of the three-term recurrence.

Returns:

• pi[0] (float or np.1darray) – The second higher order orthogonal polynomial evaluation.

• pi[1] (float or np.1darray) – The higher oder orthogonal polynomial evaluation.

getNodes(num_nodes)

Computes a given number of quadrature nodes.

Parameters:

num_nodes (int) – Number of nodes to compute.

Returns:

nodes – Nodes located in [-1, 1], in increasing order.

Return type:

np.1darray

getOrthogPolyCoefficients(num_coeff)

Produces a given number of analytic three-term recurrence coefficients.

Parameters:

num_coeff (int) – Number of coefficients to compute.

Returns:

• alpha (np.1darray) – The alpha coefficients of the three-term recurrence.

• beta (np.1darray) – The beta coefficients of the three-term recurrence.

getTridiagCoefficients(num_nodes)

Computes recurrence coefficients for the tridiagonal Jacobian matrix, taking into account the quadrature type.

Parameters:

num_nodes (int) – Number of nodes that should be computed from those coefficients.

Returns:

• alpha (np.1darray) – The modified alpha coefficients of the three-term recurrence.

• beta (np.1darray) – The modified beta coefficients of the three-term recurrence.