.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/numerical_analysis/interpolation/plot_02_hermite_interpolation.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_interpolation_plot_02_hermite_interpolation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_interpolation_plot_02_hermite_interpolation.py:


Hermite interpolation: matching slopes as well as values
===========================================================

Charles Hermite's 1878 interpolant matches both :math:`f(x_i)` and
:math:`f'(x_i)`, so :math:`n + 1` nodes fix a polynomial of degree
:math:`2n + 1`. On the same few nodes it follows :math:`\sin` far more
closely than the Lagrange interpolant, which knows only the values.

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import HermiteInterpolant, LagrangeInterpolant








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Four nodes on one period of :math:`\sin x`
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-45

.. code-block:: Python


    x = np.linspace(0.0, 2.0 * np.pi, 4)
    hermite = HermiteInterpolant(x, np.sin(x), dydx=np.cos(x))
    lagrange = LagrangeInterpolant(x, np.sin(x))

    t = np.linspace(0.0, 2.0 * np.pi, 400)
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4.2))
    ax1.plot(t, np.sin(t), "k", lw=2, label=r"$\sin x$")
    ax1.plot(t, hermite(t), "--", color="firebrick", label="Hermite (degree 7)")
    ax1.plot(t, lagrange(t), ":", color="steelblue", label="Lagrange (degree 3)")
    ax1.scatter(x, np.sin(x), color="black", zorder=3)
    for xi in x:
        ax1.plot([xi - 0.4, xi + 0.4], [np.sin(xi) - 0.4 * np.cos(xi), np.sin(xi) + 0.4 * np.cos(xi)], color="gray", lw=1)
    ax1.set_title("Values and slopes at the nodes")
    ax1.legend(fontsize=8)

    ax2.semilogy(t, np.abs(hermite(t) - np.sin(t)) + 1e-17, color="firebrick", label="Hermite")
    ax2.semilogy(t, np.abs(lagrange(t) - np.sin(t)) + 1e-17, color="steelblue", label="Lagrange")
    ax2.set_title("Absolute error")
    ax2.legend(fontsize=8)
    fig.tight_layout()

    print(f"max error: Hermite {np.max(np.abs(hermite(t) - np.sin(t))):.2e}, Lagrange {np.max(np.abs(lagrange(t) - np.sin(t))):.2e}")

    plt.show()



.. image-sg:: /api/gallery/numerical_analysis/interpolation/images/sphx_glr_plot_02_hermite_interpolation_001.png
   :alt: Values and slopes at the nodes, Absolute error
   :srcset: /api/gallery/numerical_analysis/interpolation/images/sphx_glr_plot_02_hermite_interpolation_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max error: Hermite 1.88e-03, Lagrange 2.55e-01





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

   **Total running time of the script:** (0 minutes 0.081 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_interpolation_plot_02_hermite_interpolation.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_hermite_interpolation.ipynb <plot_02_hermite_interpolation.ipynb>`

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

      :download:`Download Python source code: plot_02_hermite_interpolation.py <plot_02_hermite_interpolation.py>`

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

      :download:`Download zipped: plot_02_hermite_interpolation.zip <plot_02_hermite_interpolation.zip>`


.. only:: html

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

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