.. 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_03_remez_minimax.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_03_remez_minimax.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_approximation_plot_03_remez_minimax.py:


Remez's algorithm and best uniform approximation
===================================================

Chebyshev showed that the polynomial of degree :math:`n` closest to
:math:`f` in the maximum norm is characterized by an error curve that
reaches its peak magnitude :math:`n + 2` times with alternating sign.
Evgeny Remez's 1934 exchange algorithm finds it by repeatedly levelling
the error on a set of reference points. This script computes the best
degree-6 approximation to :math:`e^x \sin(3x)` on :math:`[-1, 1]` and
compares its error with Chebyshev interpolation of the same degree.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import ChebyshevInterpolant, remez_minimax








.. GENERATED FROM PYTHON SOURCE LINES 21-23

The equioscillating error curve
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-47

.. code-block:: Python


    f = lambda x: np.exp(x) * np.sin(3.0 * x)
    n = 6
    best = remez_minimax(f, n)
    cheb = ChebyshevInterpolant(f, n + 1)  # degree n through n + 1 Chebyshev points

    x = np.linspace(-1.0, 1.0, 1000)
    err_best = f(x) - best.evaluate(x)
    err_cheb = f(x) - cheb(x)

    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.plot(x, err_cheb, color="steelblue", label="Chebyshev interpolant")
    ax.plot(x, err_best, color="firebrick", label="Remez minimax")
    ax.scatter(best.reference, f(best.reference) - best.evaluate(best.reference), color="firebrick", zorder=3, label="reference points")
    ax.axhline(best.max_error, color="gray", ls="--", lw=0.8)
    ax.axhline(-best.max_error, color="gray", ls="--", lw=0.8)
    ax.set_title(f"Degree {n}: the minimax error equioscillates at {n + 2} points")
    ax.legend(fontsize=8)
    fig.tight_layout()

    print(f"Remez: {best.iterations} exchanges, max error {best.max_error:.3e}")
    print(f"Chebyshev interpolation: max error {np.max(np.abs(err_cheb)):.3e}")

    plt.show()



.. image-sg:: /api/gallery/numerical_analysis/approximation/images/sphx_glr_plot_03_remez_minimax_001.png
   :alt: Degree 6: the minimax error equioscillates at 8 points
   :srcset: /api/gallery/numerical_analysis/approximation/images/sphx_glr_plot_03_remez_minimax_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Remez: 4 exchanges, max error 3.860e-03
    Chebyshev interpolation: max error 8.326e-03





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

   **Total running time of the script:** (0 minutes 0.057 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_approximation_plot_03_remez_minimax.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_remez_minimax.ipynb <plot_03_remez_minimax.ipynb>`

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

      :download:`Download Python source code: plot_03_remez_minimax.py <plot_03_remez_minimax.py>`

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

      :download:`Download zipped: plot_03_remez_minimax.zip <plot_03_remez_minimax.zip>`


.. only:: html

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

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