.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/number_theory/continued_fractions/plot_01_approximating_pi.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_number_theory_continued_fractions_plot_01_approximating_pi.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_number_theory_continued_fractions_plot_01_approximating_pi.py:


Approximating pi by continued fractions
==============================================

Expands pi as a continued fraction and shows how quickly its
convergents (355/113 famously accurate to 7 digits) approach the true
value.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import numpy as np

    from mathematicskit.number_theory import best_rational_approximation, continued_fraction_expansion
    from mathematicskit.number_theory.visualizers.plots import plot_convergent_errors








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Expand pi and inspect its convergents
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python


    result = continued_fraction_expansion(np.pi, max_terms=8)
    print("terms:", result.terms)
    for p, q in result.convergents:
        print(f"  {p}/{q} = {p / q:.10f}  (error {abs(p / q - np.pi):.2e})")





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

 .. code-block:: none

    terms: [3, 7, 15, 1, 292, 1, 1, 1]
      3/1 = 3.0000000000  (error 1.42e-01)
      22/7 = 3.1428571429  (error 1.26e-03)
      333/106 = 3.1415094340  (error 8.32e-05)
      355/113 = 3.1415929204  (error 2.67e-07)
      103993/33102 = 3.1415926530  (error 5.78e-10)
      104348/33215 = 3.1415926539  (error 3.32e-10)
      208341/66317 = 3.1415926535  (error 1.22e-10)
      312689/99532 = 3.1415926536  (error 2.91e-11)




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Best approximation under a denominator bound
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-32

.. code-block:: Python


    p, q = best_rational_approximation(np.pi, max_denominator=1000)
    print(f"\nbest approximation with denominator <= 1000: {p}/{q}")





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

 .. code-block:: none


    best approximation with denominator <= 1000: 355/113




.. GENERATED FROM PYTHON SOURCE LINES 33-35

Plot the convergent errors
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-37

.. code-block:: Python


    plot_convergent_errors(result, x=np.pi)



.. image-sg:: /api/gallery/number_theory/continued_fractions/images/sphx_glr_plot_01_approximating_pi_001.png
   :alt: Continued-fraction convergent error
   :srcset: /api/gallery/number_theory/continued_fractions/images/sphx_glr_plot_01_approximating_pi_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <Axes: title={'center': 'Continued-fraction convergent error'}, xlabel='denominator q', ylabel='|p/q - x|'>




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

   **Total running time of the script:** (0 minutes 0.029 seconds)


.. _sphx_glr_download_api_gallery_number_theory_continued_fractions_plot_01_approximating_pi.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_approximating_pi.ipynb <plot_01_approximating_pi.ipynb>`

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

      :download:`Download Python source code: plot_01_approximating_pi.py <plot_01_approximating_pi.py>`

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

      :download:`Download zipped: plot_01_approximating_pi.zip <plot_01_approximating_pi.zip>`


.. only:: html

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

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