.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "_build/auto_examples/convergence/Integration_on_segment.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_Integration_on_segment.py>`
        to download the full example code

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

.. _sphx_glr__build_auto_examples_convergence_Integration_on_segment.py:


Convergence analysis of integration methods on segment
======================================================

.. GENERATED FROM PYTHON SOURCE LINES 7-8

Evaluation of relative quadrature error with the following parameters:

.. GENERATED FROM PYTHON SOURCE LINES 8-27

.. code-block:: Python



    fun_names = [
        "exp",
    ]

    methods = [
        'Gauss',
        'Radau_I',
        'Radau_II',
        'Lobatto_III',
    ]

    # all_nsteps = range(2,11)
    all_nsteps = [5]
    refinement_lvl = np.array(range(1,100))
    # refinement_lvl = np.array([2**i for i in range(18)])









.. GENERATED FROM PYTHON SOURCE LINES 93-94

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

.. GENERATED FROM PYTHON SOURCE LINES 94-98

.. code-block:: Python



    plt.show()




.. image-sg:: /_build/auto_examples/convergence/images/sphx_glr_Integration_on_segment_001.png
   :alt: Absolute error on integrand exp
   :srcset: /_build/auto_examples/convergence/images/sphx_glr_Integration_on_segment_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 142-144

The following plots give the measured convergence rate as a function of the number of quadrature subintervals.
The dotted lines are theoretical convergence rates.

.. GENERATED FROM PYTHON SOURCE LINES 144-149

.. code-block:: Python




    plt.show()




.. image-sg:: /_build/auto_examples/convergence/images/sphx_glr_Integration_on_segment_002.png
   :alt: Approximate convergence rate on integrand exp
   :srcset: /_build/auto_examples/convergence/images/sphx_glr_Integration_on_segment_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 212-218

We can see 3 distinct phases on these plots:

* A first pre-convergence phase, where the convergence rate is growing towards its theoretical value. the end of the pre-convergence phase occurs for a number of sub-intervals roughtly independant of the convergence order of the quadrature method.
* A steady convergence phase where the convergence remains close to the theoretical value
* A final phase, where the relative error stagnates arround 1e-15. The value of the integral is computed with maximal accuracy given floating point precision. The approximation of the convergence rate is dominated by seemingly random floating point errors.



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

   **Total running time of the script:** (0 minutes 0.666 seconds)


.. _sphx_glr_download__build_auto_examples_convergence_Integration_on_segment.py:

.. only:: html

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

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

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

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

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


.. only:: html

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

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