.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/quadrature/plot_06_clenshaw_curtis.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_06_clenshaw_curtis.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_quadrature_plot_06_clenshaw_curtis.py:


Clenshaw-Curtis vs. Gauss-Legendre quadrature
===================================================

Compares Clenshaw-Curtis quadrature, which integrates the Chebyshev
interpolant, with Gauss-Legendre quadrature on smooth integrands. The
two converge at nearly the same rate.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.calculus import ClenshawCurtisQuadrature, GaussianQuadrature, clenshaw_curtis_nodes_and_weights








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Nodes and weights
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python


    x, w = clenshaw_curtis_nodes_and_weights(8)
    print("Clenshaw-Curtis n = 8")
    for xk, wk in zip(x, w):
        print(f"  x = {xk:+.6f}, w = {wk:.6f}")





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

 .. code-block:: none

    Clenshaw-Curtis n = 8
      x = +1.000000, w = 0.015873
      x = +0.923880, w = 0.146219
      x = +0.707107, w = 0.279365
      x = +0.382683, w = 0.361718
      x = +0.000000, w = 0.393651
      x = -0.382683, w = 0.361718
      x = -0.707107, w = 0.279365
      x = -0.923880, w = 0.146219
      x = -1.000000, w = 0.015873




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Convergence on two smooth integrands
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-41

.. code-block:: Python


    cases = {r"$e^x$": (np.exp, np.e - 1 / np.e), r"$1/(1+16x^2)$": (lambda t: 1 / (1 + 16 * t**2), np.arctan(4) / 2)}
    points = np.arange(2, 40, 2)
    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    for ax, (name, (f, exact)) in zip(axes, cases.items()):
        cc = [max(abs(ClenshawCurtisQuadrature(n - 1).integrate(f, -1.0, 1.0).value - exact), 1e-17) for n in points]
        gl = [max(abs(GaussianQuadrature(n).integrate(f, -1.0, 1.0).value - exact), 1e-17) for n in points]
        ax.semilogy(points, cc, "o-", label="Clenshaw-Curtis")
        ax.semilogy(points, gl, "s-", label="Gauss-Legendre")
        ax.set_xlabel("function evaluations")
        ax.set_title(name)
        ax.legend()
    axes[0].set_ylabel("|error|")



.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_06_clenshaw_curtis_001.png
   :alt: $e^x$, $1/(1+16x^2)$
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_06_clenshaw_curtis_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(70.72222222222221, 0.5, '|error|')




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

   **Total running time of the script:** (0 minutes 0.086 seconds)


.. _sphx_glr_download_api_gallery_calculus_quadrature_plot_06_clenshaw_curtis.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_06_clenshaw_curtis.ipynb <plot_06_clenshaw_curtis.ipynb>`

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

      :download:`Download Python source code: plot_06_clenshaw_curtis.py <plot_06_clenshaw_curtis.py>`

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

      :download:`Download zipped: plot_06_clenshaw_curtis.zip <plot_06_clenshaw_curtis.zip>`


.. only:: html

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

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