.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/orthogonal_polynomials/plot_01_legendre_chebyshev.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_special_functions_orthogonal_polynomials_plot_01_legendre_chebyshev.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_orthogonal_polynomials_plot_01_legendre_chebyshev.py:


Legendre and Chebyshev polynomials on [-1, 1]
===================================================

Plots the first Legendre polynomials P_n (weight 1) beside the first
Chebyshev polynomials T_n (weight 1/sqrt(1 - x^2)), checks each family's
orthogonality, and confirms Chebyshev's closed form T_n(cos t) = cos(nt),
whose equal ripples between -1 and 1 are its minimal-oscillation property.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import chebyshev_polynomial, inner_product, legendre_polynomial
    from mathematicskit.special_functions.visualizers.plots import plot_polynomial_family








.. GENERATED FROM PYTHON SOURCE LINES 19-21

The two families side by side
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-29

.. code-block:: Python


    fig, (ax_p, ax_t) = plt.subplots(1, 2, figsize=(10, 4), sharey=True)
    plot_polynomial_family(legendre_polynomial, degrees=[0, 1, 2, 3, 4], x_range=(-1.0, 1.0), ax=ax_p)
    plot_polynomial_family(chebyshev_polynomial, degrees=[0, 1, 2, 3, 4], x_range=(-1.0, 1.0), ax=ax_t)
    ax_p.set_title(r"Legendre $P_n(x)$, weight $1$")
    ax_t.set_title(r"Chebyshev $T_n(x)$, weight $1/\sqrt{1-x^2}$")
    fig.tight_layout()




.. image-sg:: /api/gallery/special_functions/orthogonal_polynomials/images/sphx_glr_plot_01_legendre_chebyshev_001.png
   :alt: Legendre $P_n(x)$, weight $1$, Chebyshev $T_n(x)$, weight $1/\sqrt{1-x^2}$
   :srcset: /api/gallery/special_functions/orthogonal_polynomials/images/sphx_glr_plot_01_legendre_chebyshev_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 30-32

Orthogonality against each family's weight
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-42

.. code-block:: Python


    legendre_ip = inner_product(lambda x: legendre_polynomial(2, x), lambda x: legendre_polynomial(3, x), lambda x: 1.0, -1.0, 1.0)
    legendre_norm = inner_product(lambda x: legendre_polynomial(3, x), lambda x: legendre_polynomial(3, x), lambda x: 1.0, -1.0, 1.0)
    cheb_weight = lambda x: 1.0 / np.sqrt(1.0 - x**2)  # noqa: E731
    chebyshev_ip = inner_product(lambda x: chebyshev_polynomial(2, x), lambda x: chebyshev_polynomial(4, x), cheb_weight, -1.0, 1.0)
    chebyshev_norm = inner_product(lambda x: chebyshev_polynomial(4, x), lambda x: chebyshev_polynomial(4, x), cheb_weight, -1.0, 1.0)

    print(f"<P_2, P_3> = {legendre_ip:.2e}, <P_3, P_3> = {legendre_norm:.6f} (2/7 = {2 / 7:.6f})")
    print(f"<T_2, T_4> = {chebyshev_ip:.2e}, <T_4, T_4> = {chebyshev_norm:.6f} (pi/2 = {np.pi / 2:.6f})")





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

 .. code-block:: none

    <P_2, P_3> = 0.00e+00, <P_3, P_3> = 0.285714 (2/7 = 0.285714)
    <T_2, T_4> = -2.13e-12, <T_4, T_4> = 1.570796 (pi/2 = 1.570796)




.. GENERATED FROM PYTHON SOURCE LINES 43-45

Chebyshev's closed form T_n(cos t) = cos(n t)
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-50

.. code-block:: Python


    t = np.linspace(0.0, np.pi, 400)
    for n in (3, 5, 8):
        err = np.max(np.abs(chebyshev_polynomial(n, np.cos(t)) - np.cos(n * t)))
        print(f"n = {n}: max |T_n(cos t) - cos(nt)| = {err:.1e}")




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

 .. code-block:: none

    n = 3: max |T_n(cos t) - cos(nt)| = 1.2e-15
    n = 5: max |T_n(cos t) - cos(nt)| = 2.2e-15
    n = 8: max |T_n(cos t) - cos(nt)| = 4.3e-15





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

   **Total running time of the script:** (0 minutes 0.084 seconds)


.. _sphx_glr_download_api_gallery_special_functions_orthogonal_polynomials_plot_01_legendre_chebyshev.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_legendre_chebyshev.ipynb <plot_01_legendre_chebyshev.ipynb>`

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

      :download:`Download Python source code: plot_01_legendre_chebyshev.py <plot_01_legendre_chebyshev.py>`

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

      :download:`Download zipped: plot_01_legendre_chebyshev.zip <plot_01_legendre_chebyshev.zip>`


.. only:: html

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

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