.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/quadrature/plot_05_romberg.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_05_romberg.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_quadrature_plot_05_romberg.py:


Romberg integration: extrapolating the trapezoidal rule
=============================================================

Prints the Romberg table for the integral of 4/(1 + x^2) over [0, 1],
which equals pi. Each column applies one more Richardson extrapolation
to the halved trapezoidal rules in the first column.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.calculus import RombergQuadrature








.. GENERATED FROM PYTHON SOURCE LINES 17-19

The Romberg table
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-30

.. code-block:: Python



    def f(x):
        return 4.0 / (1.0 + x**2)


    result = RombergQuadrature(levels=5).integrate(f, 0.0, 1.0)
    for i, row in enumerate(result.extra["table"]):
        print(f"n = {2**i:3d}: " + "  ".join(f"{v:.10f}" for v in row))
    print(f"estimate {result.value:.14f}, pi = {np.pi:.14f}")





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

 .. code-block:: none

    n =   1: 3.0000000000
    n =   2: 3.1000000000  3.1333333333
    n =   4: 3.1311764706  3.1415686275  3.1421176471
    n =   8: 3.1389884945  3.1415925025  3.1415940941  3.1415857838
    n =  16: 3.1409416120  3.1415926512  3.1415926611  3.1415926384  3.1415926653
    n =  32: 3.1414298932  3.1415926536  3.1415926537  3.1415926536  3.1415926536  3.1415926536
    estimate 3.14159265363824, pi = 3.14159265358979




.. GENERATED FROM PYTHON SOURCE LINES 31-33

First column against the diagonal
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-44

.. code-block:: Python


    table = RombergQuadrature(levels=8).integrate(f, 0.0, 1.0).extra["table"]
    ns = [2**i for i in range(len(table))]
    fig, ax = plt.subplots()
    ax.semilogy(ns, [abs(row[0] - np.pi) for row in table], "o-", label="trapezoidal (first column)")
    ax.semilogy(ns, [max(abs(row[-1] - np.pi), 1e-17) for row in table], "s-", label="Romberg (diagonal)")
    ax.set_xscale("log", base=2)
    ax.set_xlabel("subintervals")
    ax.set_ylabel("|error|")
    ax.legend()
    ax.set_title("Romberg (1955)")



.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_05_romberg_001.png
   :alt: Romberg (1955)
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_05_romberg_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Romberg (1955)')




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

   **Total running time of the script:** (0 minutes 0.045 seconds)


.. _sphx_glr_download_api_gallery_calculus_quadrature_plot_05_romberg.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_05_romberg.ipynb <plot_05_romberg.ipynb>`

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

      :download:`Download Python source code: plot_05_romberg.py <plot_05_romberg.py>`

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

      :download:`Download zipped: plot_05_romberg.zip <plot_05_romberg.zip>`


.. only:: html

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

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