.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/quadrature/plot_04_euler_maclaurin.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_calculus_quadrature_plot_04_euler_maclaurin.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_quadrature_plot_04_euler_maclaurin.py:


Euler-Maclaurin: correcting the trapezoidal rule
======================================================

Adds Euler-Maclaurin endpoint corrections, built from Bernoulli numbers
and odd derivatives at the endpoints, to the trapezoidal rule. Each
extra term raises the convergence order by two.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.calculus import euler_maclaurin_trapezoid








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Corrections for the integral of exp over [0, 1]
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python


    exact = np.e - 1
    for m in range(4):
        value = euler_maclaurin_trapezoid(np.exp, 0.0, 1.0, 8, [np.exp] * m).value
        print(f"{m} correction terms, n = 8: error = {abs(value - exact):.2e}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    0 correction terms, n = 8: error = 2.24e-03
    1 correction terms, n = 8: error = 5.82e-07
    2 correction terms, n = 8: error = 2.17e-10
    3 correction terms, n = 8: error = 8.46e-14




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Convergence with 0, 1, and 2 correction terms
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-38

.. code-block:: Python


    ns = np.array([2, 4, 8, 16, 32, 64])
    fig, ax = plt.subplots()
    for m in range(3):
        errors = [abs(euler_maclaurin_trapezoid(np.exp, 0.0, 1.0, int(n), [np.exp] * m).value - exact) for n in ns]
        ax.loglog(ns, errors, "o-", label=f"{m} corrections: O(h^{2 * m + 2})")
    ax.set_xlabel("n")
    ax.set_ylabel("|error|")
    ax.legend()
    ax.set_title("Euler-Maclaurin corrections to the trapezoidal rule")



.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_04_euler_maclaurin_001.png
   :alt: Euler-Maclaurin corrections to the trapezoidal rule
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_04_euler_maclaurin_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'Euler-Maclaurin corrections to the trapezoidal rule')




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

   **Total running time of the script:** (0 minutes 0.054 seconds)


.. _sphx_glr_download_api_gallery_calculus_quadrature_plot_04_euler_maclaurin.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_euler_maclaurin.ipynb <plot_04_euler_maclaurin.ipynb>`

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

      :download:`Download Python source code: plot_04_euler_maclaurin.py <plot_04_euler_maclaurin.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_euler_maclaurin.zip <plot_04_euler_maclaurin.zip>`


.. only:: html

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

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