.. 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_01_lagrange_newton.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_01_lagrange_newton.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_interpolation_plot_01_lagrange_newton.py:


Lagrange vs. Newton divided-difference interpolation
======================================================

There is exactly one polynomial of degree :math:`\leq n` through
:math:`n+1` distinct points, so :class:`~mathematicskit.numerical_analysis.systems.interpolation.LagrangeInterpolant`
and :class:`~mathematicskit.numerical_analysis.systems.interpolation.NewtonDividedDifference`
-- built from completely different representations (a sum of basis
polynomials vs. nested divided differences) -- must produce identical
curves. Both are also exact for any polynomial up to the interpolation
degree, which this script verifies directly.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import LagrangeInterpolant, NewtonDividedDifference








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Same data, two representations
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-44

.. code-block:: Python


    x = np.array([0.0, 1.0, 2.0, 3.0, 4.0])
    y = np.sin(x) + 0.5 * x

    p_lagrange = LagrangeInterpolant(x, y)
    p_newton = NewtonDividedDifference(x, y)

    x_fine = np.linspace(x.min(), x.max(), 300)
    y_lagrange = p_lagrange.evaluate(x_fine)
    y_newton = p_newton.evaluate(x_fine)

    print("max |Lagrange - Newton| over the fine grid:", np.max(np.abs(y_lagrange - y_newton)))

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(x_fine, y_lagrange, color="steelblue", lw=2, label="Lagrange")
    ax.plot(x_fine, y_newton, "--", color="firebrick", lw=1.5, label="Newton divided-difference")
    ax.scatter(x, y, color="black", zorder=3, label="nodes")
    ax.set_title("Two representations of the same interpolating polynomial")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/numerical_analysis/interpolation/images/sphx_glr_plot_01_lagrange_newton_001.png
   :alt: Two representations of the same interpolating polynomial
   :srcset: /api/gallery/numerical_analysis/interpolation/images/sphx_glr_plot_01_lagrange_newton_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |Lagrange - Newton| over the fine grid: 8.881784197001252e-16




.. GENERATED FROM PYTHON SOURCE LINES 45-49

Exactness on a polynomial up to the interpolation degree
-----------------------------------------------------------
A degree-4 polynomial through 5 nodes should be reproduced to
floating-point precision.

.. GENERATED FROM PYTHON SOURCE LINES 49-58

.. code-block:: Python


    x5 = np.linspace(0.0, 4.0, 5)
    y5 = 2.0 * x5**4 - 3.0 * x5**3 + x5 - 1.0
    p5 = NewtonDividedDifference(x5, y5)
    x_test = np.linspace(0.0, 4.0, 50)
    expected = 2.0 * x_test**4 - 3.0 * x_test**3 + x_test - 1.0
    print("max error vs. exact degree-4 polynomial:", np.max(np.abs(p5.evaluate(x_test) - expected)))

    plt.show()




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

 .. code-block:: none

    max error vs. exact degree-4 polynomial: 5.684341886080802e-14





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

   **Total running time of the script:** (0 minutes 0.044 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_interpolation_plot_01_lagrange_newton.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_lagrange_newton.ipynb <plot_01_lagrange_newton.ipynb>`

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

      :download:`Download Python source code: plot_01_lagrange_newton.py <plot_01_lagrange_newton.py>`

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

      :download:`Download zipped: plot_01_lagrange_newton.zip <plot_01_lagrange_newton.zip>`


.. only:: html

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

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