qmat.qdelta.algebraic
=====================

.. py:module:: qmat.qdelta.algebraic

.. autoapi-nested-parse::

   Submodule for QDelta coefficients based on algebraic approaches,
   in particular the `LU trick` from `[Weiser, 2014]`_.

   .. rubric:: Examples

   >>> from qmat.qcoeff.collocation import Collocation
   >>> coll = Collocation(nNodes=4, nodeType="LEGENDRE", quadType="RADAU-RIGHT")
   >>>
   >>> from qmat import genQDeltaCoeffs
   >>> QDelta = genQDeltaCoeffs("LU", Q=coll.Q)
   >>>
   >>> from qmat.qdelta.algebraic import LDU
   >>> gen = LDU(Q=coll.Q)
   >>> QDelta = gen.getQDelta()



Classes
-------

.. autoapisummary::

   qmat.qdelta.algebraic.PIC
   qmat.qdelta.algebraic.Exact
   qmat.qdelta.algebraic.LU
   qmat.qdelta.algebraic.LU2
   qmat.qdelta.algebraic.QPar
   qmat.qdelta.algebraic.GS
   qmat.qdelta.algebraic.LDU


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

.. py:class:: PIC(Q, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.algebraic.PIC
      :parts: 1


   Picard approximation (zeros)


   .. py:attribute:: aliases
      :value: ['Picard']



   .. py:method:: computeQDelta(k=None)

      Compute and returns the :math:`Q_\Delta` matrix, has to be implemented
      in the specialized class.

      :Parameters: **k** (*int, optional*) -- Iteration number of the approximation. The default is None.

      :returns: **QDelta**
      :rtype: np.ndarray



.. py:class:: Exact(Q, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.algebraic.Exact
      :parts: 1


   Takes Q (exact approximation)


   .. py:attribute:: aliases
      :value: ['EXACT']



   .. py:method:: computeQDelta(k=None)

      Compute and returns the :math:`Q_\Delta` matrix, has to be implemented
      in the specialized class.

      :Parameters: **k** (*int, optional*) -- Iteration number of the approximation. The default is None.

      :returns: **QDelta**
      :rtype: np.ndarray



.. py:class:: LU(Q, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.algebraic.LU
      :parts: 1


   LU approximation from `[Weiser, 2014] <https://link.springer.com/article/10.1007/s10543-014-0540-y>`_


   .. py:method:: computeQDelta(k=None)

      Compute and returns the :math:`Q_\Delta` matrix, has to be implemented
      in the specialized class.

      :Parameters: **k** (*int, optional*) -- Iteration number of the approximation. The default is None.

      :returns: **QDelta**
      :rtype: np.ndarray



.. py:class:: LU2(Q, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.algebraic.LU2
      :parts: 1


   LU approximation from `[Weiser, 2014]`_ multiplied by 2


   .. py:method:: computeQDelta(k=None)

      Compute and returns the :math:`Q_\Delta` matrix, has to be implemented
      in the specialized class.

      :Parameters: **k** (*int, optional*) -- Iteration number of the approximation. The default is None.

      :returns: **QDelta**
      :rtype: np.ndarray



.. py:class:: QPar(Q, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.algebraic.QPar
      :parts: 1


   Approximation using diagonal of Q


   .. py:attribute:: aliases
      :value: ['Qpar', 'Qdiag']



   .. py:method:: computeQDelta(k=None)

      Compute and returns the :math:`Q_\Delta` matrix, has to be implemented
      in the specialized class.

      :Parameters: **k** (*int, optional*) -- Iteration number of the approximation. The default is None.

      :returns: **QDelta**
      :rtype: np.ndarray



.. py:class:: GS(Q, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.algebraic.GS
      :parts: 1


   Approximation using lower triangular part of Q


   .. py:attribute:: aliases
      :value: ['GaussSeidel']



   .. py:method:: computeQDelta(k=None)

      Compute and returns the :math:`Q_\Delta` matrix, has to be implemented
      in the specialized class.

      :Parameters: **k** (*int, optional*) -- Iteration number of the approximation. The default is None.

      :returns: **QDelta**
      :rtype: np.ndarray



.. py:class:: LDU(Q, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.algebraic.LDU
      :parts: 1


   Diagonal approximation using LDU factorization


   .. py:method:: computeQDelta(k=None)

      Compute and returns the :math:`Q_\Delta` matrix, has to be implemented
      in the specialized class.

      :Parameters: **k** (*int, optional*) -- Iteration number of the approximation. The default is None.

      :returns: **QDelta**
      :rtype: np.ndarray