.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "_build/auto_examples/convergence/ImplicitRungeKutta_ivp.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download__build_auto_examples_convergence_ImplicitRungeKutta_ivp.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr__build_auto_examples_convergence_ImplicitRungeKutta_ivp.py:


Convergence analysis of implicit Runge-Kutta methods for ODE IVP
================================================================

.. GENERATED FROM PYTHON SOURCE LINES 7-8

Evaluation of relative quadrature error with the following parameters:

.. GENERATED FROM PYTHON SOURCE LINES 8-35

.. code-block:: Python



    eq_names = [
        "y'' = -y"          ,
        "y'' = - exp(y)"    ,
        "y'' = xy"          ,
        "y' = Az; z' = By"  ,
    ]

    method_names = [
        'Gauss'   ,            
        'Radau_IA'   ,            
        'Radau_IIA'   ,            
        'Radau_IB'   ,            
        'Radau_IIB'   ,                   
        'Lobatto_IIIA'   ,            
        'Lobatto_IIIB'   ,            
        'Lobatto_IIIC'   ,            
        'Lobatto_IIIC*'  ,            
        'Lobatto_IIID'  ,            
        'Lobatto_IIIS'  ,            
    ]

    # all_orders = range(2,6)
    all_orders = [4]









.. GENERATED FROM PYTHON SOURCE LINES 118-119

The following plots give the measured relative error as a function of the number of quadrature subintervals

.. GENERATED FROM PYTHON SOURCE LINES 119-124

.. code-block:: Python



    plt.show()





.. image-sg:: /_build/auto_examples/convergence/images/sphx_glr_ImplicitRungeKutta_ivp_001.png
   :alt: Relative error on integrand y'' = -y, Relative error on integrand y'' = - exp(y), Relative error on integrand y'' = xy, Relative error on integrand y' = Az; z' = By
   :srcset: /_build/auto_examples/convergence/images/sphx_glr_ImplicitRungeKutta_ivp_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 171-172

Error as a function of running time

.. GENERATED FROM PYTHON SOURCE LINES 172-177

.. code-block:: Python



    plt.show()





.. image-sg:: /_build/auto_examples/convergence/images/sphx_glr_ImplicitRungeKutta_ivp_002.png
   :alt: Relative error as a function of computational cost for equation y'' = -y, Relative error as a function of computational cost for equation y'' = - exp(y), Relative error as a function of computational cost for equation y'' = xy, Relative error as a function of computational cost for equation y' = Az; z' = By
   :srcset: /_build/auto_examples/convergence/images/sphx_glr_ImplicitRungeKutta_ivp_002.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 2.499 seconds)


.. _sphx_glr_download__build_auto_examples_convergence_ImplicitRungeKutta_ivp.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: ImplicitRungeKutta_ivp.ipynb <ImplicitRungeKutta_ivp.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: ImplicitRungeKutta_ivp.py <ImplicitRungeKutta_ivp.py>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_