.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/numerical_analysis/approximation/plot_01_bernstein_polynomials.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_numerical_analysis_approximation_plot_01_bernstein_polynomials.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_approximation_plot_01_bernstein_polynomials.py:


Bernstein polynomials and the Weierstrass theorem
=====================================================

Weierstrass proved in 1885 that every continuous function on a closed
interval is a uniform limit of polynomials. Sergei Bernstein's 1912
proof builds the polynomials explicitly: average :math:`f(k/n)` with
binomial weights. This script approximates a function with a kink,
:math:`|x - 1/2|`, and shows the uniform error shrinking, slowly, as the
degree grows.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import bernstein_polynomial








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Bernstein approximants of increasing degree
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-33

.. code-block:: Python


    f = lambda t: np.abs(t - 0.5)
    x = np.linspace(0.0, 1.0, 501)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4.2))
    ax1.plot(x, f(x), "k", lw=2, label="$|x - 1/2|$")
    for n in (4, 16, 64, 256):
        ax1.plot(x, bernstein_polynomial(f, n, x), label=f"$B_{{{n}}}f$")
    ax1.set_title("Bernstein polynomials")
    ax1.legend(fontsize=8)




.. image-sg:: /api/gallery/numerical_analysis/approximation/images/sphx_glr_plot_01_bernstein_polynomials_001.png
   :alt: Bernstein polynomials
   :srcset: /api/gallery/numerical_analysis/approximation/images/sphx_glr_plot_01_bernstein_polynomials_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x119f730e0>



.. GENERATED FROM PYTHON SOURCE LINES 34-36

Uniform error against degree
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-50

.. code-block:: Python


    degrees = np.array([2, 4, 8, 16, 32, 64, 128, 256, 512])
    errors = np.array([np.max(np.abs(bernstein_polynomial(f, n, x) - f(x))) for n in degrees])
    ax2.loglog(degrees, errors, "o-", label="max error")
    ax2.loglog(degrees, 0.4 / np.sqrt(degrees), "--", color="gray", label=r"$\propto n^{-1/2}$")
    ax2.set_xlabel("degree $n$")
    ax2.set_title("Convergence is uniform but slow")
    ax2.legend(fontsize=8)
    fig.tight_layout()

    for n, e in zip(degrees, errors):
        print(f"n = {n:4d}: max |B_n f - f| = {e:.4f}")

    plt.show()




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

 .. code-block:: none

    n =    2: max |B_n f - f| = 0.2500
    n =    4: max |B_n f - f| = 0.1875
    n =    8: max |B_n f - f| = 0.1367
    n =   16: max |B_n f - f| = 0.0982
    n =   32: max |B_n f - f| = 0.0700
    n =   64: max |B_n f - f| = 0.0497
    n =  128: max |B_n f - f| = 0.0352
    n =  256: max |B_n f - f| = 0.0249
    n =  512: max |B_n f - f| = 0.0176





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

   **Total running time of the script:** (0 minutes 0.105 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_approximation_plot_01_bernstein_polynomials.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_bernstein_polynomials.ipynb <plot_01_bernstein_polynomials.ipynb>`

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

      :download:`Download Python source code: plot_01_bernstein_polynomials.py <plot_01_bernstein_polynomials.py>`

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

      :download:`Download zipped: plot_01_bernstein_polynomials.zip <plot_01_bernstein_polynomials.zip>`


.. only:: html

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

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