qmat.nodes

Implement the base NodesGenerator class that can be used to generate quadrature nodes for many kind of distribution and types. Its implementation is mostly based on the book of [Gautschi, 2004].

Examples

>>> from qmat.nodes import NodesGenerator
>>> gen = NodesGenerator(nodeType="CHEBY-1", quadType="RADAU-RIGHT")
>>> nodes = gen.getNodes(10)
>>> coarse = gen.getNodes(5)

Attributes

`NODE_TYPES`	Available types (distributions) for nodes
`QUAD_TYPES`	Available quadrature type for each node distribution

Classes

NodesGenerator

Class that can be used to generate generic distribution of nodes derived from Gauss quadrature rule.

Module Contents

NODE_TYPES = ['EQUID', 'LEGENDRE', 'CHEBY-1', 'CHEBY-2', 'CHEBY-3', 'CHEBY-4']: Available types (distributions) for nodes

QUAD_TYPES = ['GAUSS', 'RADAU-LEFT', 'RADAU-RIGHT', 'LOBATTO']: Available quadrature type for each node distribution

class NodesGenerator(nodeType='LEGENDRE', quadType='LOBATTO')[source]

Class that can be used to generate generic distribution of nodes derived from Gauss quadrature rule. Its implementation is fully inspired from [Gautschi, 2004].

Parameters:

nodeType (str, optional) – The type of node distribution, can be
- EQUID : equidistant nodes
- LEGENDRE : node distribution from Legendre polynomials
- CHEBY-1 : node distribution from Chebychev polynomials (1st kind)
- CHEBY-2 : node distribution from Chebychev polynomials (2nd kind)
- CHEBY-3 : node distribution from Chebychev polynomials (3rd kind)
- CHEBY-4 : node distribution from Chebychev polynomials (4th kind)
The default is ‘LEGENDRE’.
quadType (str, optional) – The quadrature type, can be
- GAUSS : inner point only, no node at boundary
- RADAU-LEFT : only left boundary as node
- RADAU-RIGHT : only right boundary as node
- LOBATTO : left and right boundary as node
The default is ‘LOBATTO’.

nodeType = 'LEGENDRE': Type of the node distribution

quadType = 'LOBATTO': Quadrature type

getNodes(nNodes)[source]

Computes a given number of quadrature nodes.

Parameters:: nNodes (int) – Number of nodes to compute.
Returns:: nodes – Nodes located in [-1, 1], in increasing order.
Return type:: np.1darray

getOrthogPolyCoefficients(num_coeff)[source]

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.

evalOrthogPoly(t, alpha, beta)[source]

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.

getTridiagCoefficients(nNodes)[source]

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

Parameters:

nNodes (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.