.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/numerical_analysis/splines/plot_01_cubic_spline.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_splines_plot_01_cubic_spline.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_splines_plot_01_cubic_spline.py:


Natural and clamped cubic splines
====================================

A cubic spline is a piecewise cubic through the data with continuous
value, first, and second derivatives across nodes. A **natural** spline
sets the curvature to zero at both ends; a **clamped** spline instead
matches prescribed end-slopes. This script fits both to the same data and
compares them against a single high-degree interpolating polynomial,
which oscillates far more between nodes -- the practical motivation for
splines over one global polynomial.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import CubicSpline, LagrangeInterpolant








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Fit natural and clamped splines to the same data
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-43

.. code-block:: Python


    x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0])
    y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -0.9, 0.0])

    natural = CubicSpline(x, y, boundary="natural")
    clamped = CubicSpline(x, y, boundary="clamped", fpa=0.5, fpb=0.5)
    global_poly = LagrangeInterpolant(x, y)

    x_fine = np.linspace(x.min(), x.max(), 400)

    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.plot(x_fine, natural.evaluate(x_fine), color="steelblue", label="natural spline")
    ax.plot(x_fine, clamped.evaluate(x_fine), color="seagreen", label="clamped spline")
    ax.plot(x_fine, global_poly.evaluate(x_fine), "--", color="firebrick", label="degree-6 global polynomial")
    ax.scatter(x, y, color="black", zorder=3, label="nodes")
    ax.set_ylim(-3, 3)
    ax.set_title("Splines stay well-behaved; one global polynomial can oscillate")
    ax.legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/numerical_analysis/splines/images/sphx_glr_plot_01_cubic_spline_001.png
   :alt: Splines stay well-behaved; one global polynomial can oscillate
   :srcset: /api/gallery/numerical_analysis/splines/images/sphx_glr_plot_01_cubic_spline_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 44-46

Natural boundary condition: zero curvature at both ends
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-51

.. code-block:: Python


    second_derivs = natural.second_derivative_at_nodes()
    print("S''(x_0), S''(x_n) for the natural spline:", second_derivs[0], second_derivs[-1])

    plt.show()




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

 .. code-block:: none

    S''(x_0), S''(x_n) for the natural spline: 0.0 8.881784197001252e-16





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

   **Total running time of the script:** (0 minutes 0.038 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_splines_plot_01_cubic_spline.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_cubic_spline.ipynb <plot_01_cubic_spline.ipynb>`

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

      :download:`Download Python source code: plot_01_cubic_spline.py <plot_01_cubic_spline.py>`

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

      :download:`Download zipped: plot_01_cubic_spline.zip <plot_01_cubic_spline.zip>`


.. only:: html

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

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