.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/quadrature/plot_03_riemann_sums.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_03_riemann_sums.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_quadrature_plot_03_riemann_sums.py:


Riemann sums: the definition of the integral
==================================================

Draws left, right, and midpoint Riemann sums for the same integrand and
shows their convergence: first order for the endpoint rules, second
order for the midpoint rule.

.. GENERATED FROM PYTHON SOURCE LINES 11-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.calculus import RiemannSum
    from mathematicskit.calculus.visualizers.plots import plot_quadrature_convergence








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Three ways to sample each subinterval
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-33

.. code-block:: Python


    f, a, b, n = np.exp, 0.0, 1.0, 8
    exact = np.e - 1
    fig, axes = plt.subplots(1, 3, figsize=(11, 3.5), sharey=True)
    grid = np.linspace(a, b, 200)
    h = (b - a) / n
    for ax, rule, offset in zip(axes, ("left", "right", "midpoint"), (0.0, 1.0, 0.5)):
        ax.plot(grid, f(grid), "k")
        left_edges = a + h * np.arange(n)
        ax.bar(left_edges, f(left_edges + offset * h), width=h, align="edge", alpha=0.4, edgecolor="k")
        value = RiemannSum(n, rule).integrate(f, a, b).value
        ax.set_title(f"{rule}: {value:.4f} (exact {exact:.4f})")




.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_03_riemann_sums_001.png
   :alt: left: 1.6131 (exact 1.7183), right: 1.8279 (exact 1.7183), midpoint: 1.7172 (exact 1.7183)
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_03_riemann_sums_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 34-36

Convergence orders
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-42

.. code-block:: Python


    ns = [4, 8, 16, 32, 64, 128, 256]
    ax = None
    for rule in ("left", "right", "midpoint"):
        ax = plot_quadrature_convergence(lambda n, r=rule: RiemannSum(n, r), f, a, b, exact, ns, ax=ax, label=rule)
    ax.set_title("Riemann sums: O(h) at the endpoints, O(h^2) at midpoints")



.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_03_riemann_sums_002.png
   :alt: Riemann sums: O(h) at the endpoints, O(h^2) at midpoints
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_03_riemann_sums_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Riemann sums: O(h) at the endpoints, O(h^2) at midpoints')




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

   **Total running time of the script:** (0 minutes 0.097 seconds)


.. _sphx_glr_download_api_gallery_calculus_quadrature_plot_03_riemann_sums.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_riemann_sums.ipynb <plot_03_riemann_sums.ipynb>`

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

      :download:`Download Python source code: plot_03_riemann_sums.py <plot_03_riemann_sums.py>`

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

      :download:`Download zipped: plot_03_riemann_sums.zip <plot_03_riemann_sums.zip>`


.. only:: html

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

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