.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/quadrature/plot_02_simpson_parabolas.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_calculus_quadrature_plot_02_simpson_parabolas.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_quadrature_plot_02_simpson_parabolas.py:


Simpson's rule: parabolas through pairs of panels
=======================================================

Simpson's rule integrates cubics exactly and converges at fourth order.
This example compares it with the trapezoidal rule on a smooth
integrand and shows the parabolic arcs it fits.

.. GENERATED FROM PYTHON SOURCE LINES 11-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.integrate import quad

    from mathematicskit.calculus import SimpsonsRule, TrapezoidalRule
    from mathematicskit.calculus.visualizers.plots import plot_quadrature_convergence








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Exact for cubics
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-25

.. code-block:: Python


    cubic = SimpsonsRule(n=2).integrate(lambda x: x**3 - 2 * x + 1, 0.0, 2.0)
    print(f"Simpson with 2 panels on x^3 - 2x + 1 over [0, 2]: {cubic.value} (exact 2)")





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

 .. code-block:: none

    Simpson with 2 panels on x^3 - 2x + 1 over [0, 2]: 2.0 (exact 2)




.. GENERATED FROM PYTHON SOURCE LINES 26-28

The parabolic arcs
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-43

.. code-block:: Python


    f, a, b, n = (lambda x: np.exp(-x) * np.sin(3 * x) + 1), 0.0, 3.0, 6
    xs = np.linspace(a, b, n + 1)
    fig, ax = plt.subplots()
    grid = np.linspace(a, b, 400)
    ax.plot(grid, f(grid), "k", label="f")
    for i in range(0, n, 2):
        seg = xs[i : i + 3]
        coeffs = np.polyfit(seg, f(seg), 2)
        local = np.linspace(seg[0], seg[-1], 50)
        ax.fill_between(local, np.polyval(coeffs, local), alpha=0.3)
    ax.plot(xs, f(xs), "o")
    ax.set_title("Simpson's rule: one parabola per pair of panels")
    ax.legend()




.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_02_simpson_parabolas_001.png
   :alt: Simpson's rule: one parabola per pair of panels
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_02_simpson_parabolas_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1192fb770>



.. GENERATED FROM PYTHON SOURCE LINES 44-46

Fourth-order convergence
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-51

.. code-block:: Python


    exact, _ = quad(f, a, b, epsabs=1e-14)
    ns = [2, 4, 8, 16, 32, 64, 128]
    ax = plot_quadrature_convergence(lambda n: TrapezoidalRule(n), f, a, b, exact, ns, label="trapezoidal O(h^2)")
    plot_quadrature_convergence(lambda n: SimpsonsRule(n), f, a, b, exact, ns, ax=ax, label="Simpson O(h^4)")



.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_02_simpson_parabolas_002.png
   :alt: Quadrature convergence
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_02_simpson_parabolas_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <Axes: title={'center': 'Quadrature convergence'}, xlabel='n', ylabel='|error|'>




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

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_calculus_quadrature_plot_02_simpson_parabolas.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_simpson_parabolas.ipynb <plot_02_simpson_parabolas.ipynb>`

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

      :download:`Download Python source code: plot_02_simpson_parabolas.py <plot_02_simpson_parabolas.py>`

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

      :download:`Download zipped: plot_02_simpson_parabolas.zip <plot_02_simpson_parabolas.zip>`


.. only:: html

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

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