qmat.nodes
==========

.. py:module:: qmat.nodes

.. autoapi-nested-parse::

   Implement the base :class:`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] <https://doi.org/10.1093/oso/9780198506720.001.0001>`_.

   .. rubric:: Examples

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



Attributes
----------

.. autoapisummary::

   qmat.nodes.NODE_TYPES
   qmat.nodes.QUAD_TYPES


Classes
-------

.. autoapisummary::

   qmat.nodes.NodesGenerator


Module Contents
---------------

.. py:data:: NODE_TYPES
   :value: ['EQUID', 'LEGENDRE', 'CHEBY-1', 'CHEBY-2', 'CHEBY-3', 'CHEBY-4']


   Available types (distributions) for nodes

.. py:data:: QUAD_TYPES
   :value: ['GAUSS', 'RADAU-LEFT', 'RADAU-RIGHT', 'LOBATTO']


   Available quadrature type for each node distribution

.. py:class:: NodesGenerator(nodeType='LEGENDRE', quadType='LOBATTO')



   Class that can be used to generate generic distribution of nodes derived from Gauss quadrature rule.
   Its implementation is fully inspired from `[Gautschi, 2004] <https://doi.org/10.1093/oso/9780198506720.001.0001>`_.

   :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'.


   .. py:attribute:: nodeType
      :value: 'LEGENDRE'


      Type of the node distribution


   .. py:attribute:: quadType
      :value: 'LOBATTO'


      Quadrature type


   .. py:method:: getNodes(nNodes)

      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.
      :rtype: np.1darray



   .. py:method:: 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.



   .. py:method:: 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.



   .. py:method:: getTridiagCoefficients(nNodes)

      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.