qmat.qdelta.timestepping
========================

.. py:module:: qmat.qdelta.timestepping

.. autoapi-nested-parse::

   Submodule for QDelta coefficients based on time-stepping scheme.
   Allows to build equivalent SDC sweeps as those introduced in the original
   paper from `[Dutt, Greengard & Rokhlin, 2000] <https://link.springer.com/article/10.1023/A:1022338906936>`_.

   .. rubric:: Examples

   >>> from qmat.qcoeff.collocation import Collocation
   >>> coll = Collocation(nNodes=4, nodeType="LEGENDRE", quadType="RADAU-RIGHT")
   >>>
   >>> from qmat import genQDeltaCoeffs
   >>> QDelta = genQDeltaCoeffs("IE", nodes=coll.nodes)
   >>>
   >>> from qmat.qdelta.timestepping import TRAP
   >>> gen = TRAP(nodes=coll.nodes)
   >>> SDelta, dTau = gen.genCoeffs(form="N2N", dTau=True)



Classes
-------

.. autoapisummary::

   qmat.qdelta.timestepping.TimeStepping
   qmat.qdelta.timestepping.BE
   qmat.qdelta.timestepping.FE
   qmat.qdelta.timestepping.TRAP
   qmat.qdelta.timestepping.BEPAR
   qmat.qdelta.timestepping.TRAPAR


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

.. py:class:: TimeStepping(nodes, tLeft=0, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.timestepping.TimeStepping
      :parts: 1


   Base class for time-stepping based :math:`Q_\Delta` approximations

   :Parameters: * **nodes** (*list-like*) -- Normalized nodes in increasing order.
                * **tLeft** (*float, optional*) -- Left bound for the nodes. The default is 0.
                * **\*\*kwargs** -- Additional parameters given in a generic call, ignored by this class.


   .. py:attribute:: deltas
      :type:  numpy.ndarray

      Differences between nodes


   .. py:attribute:: nodes
      :type:  numpy.ndarray

      Array of normalized nodes


   .. py:attribute:: tLeft
      :type:  float
      :value: 0


      Left bound for the nodes


   .. py:method:: extractParams(qGen: qmat.qdelta.QGenerator) -> dict
      :staticmethod:


      Extract from a :math:`Q`-generator object all parameters
      required to instantiate the :math:`Q_\Delta`-generator



   .. py:property:: size
      :type: int


      Dimension of the approximated :math:`Q`-coefficients (number of nodes)


.. py:class:: BE(nodes, tLeft=0, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.timestepping.BE
      :parts: 1


   Approximation based on Backward Euler steps between the nodes


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



   .. 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:: FE(nodes, tLeft=0, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.timestepping.FE
      :parts: 1


   Approximation based on Forward Euler steps between the nodes


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



   .. 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:property:: dTau
      :type: numpy.ndarray


      The :math:`\delta_\tau` coefficients associated to :math:`Q_\Delta`


.. py:class:: TRAP(nodes, tLeft=0, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.timestepping.TRAP
      :parts: 1


   Approximation based on Trapezoidal Rule between the nodes


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



   .. 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:property:: dTau
      :type: numpy.ndarray


      The :math:`\delta_\tau` coefficients associated to :math:`Q_\Delta`


.. py:class:: BEPAR(nodes, tLeft=0, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.timestepping.BEPAR
      :parts: 1


   Approximation based on parallel Backward Euler steps from zero to nodes


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



   .. 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:: TRAPAR(nodes, tLeft=0, **kwargs)


   .. autoapi-inheritance-diagram:: qmat.qdelta.timestepping.TRAPAR
      :parts: 1


   Approximation based on parallel Trapezoidal Rule from zero to nodes


   .. 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:property:: dTau
      :type: numpy.ndarray


      The :math:`\delta_\tau` coefficients associated to :math:`Q_\Delta`