.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/quadrature/plot_08_adaptive_quadpack.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_08_adaptive_quadpack.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_quadrature_plot_08_adaptive_quadpack.py:


Adaptive quadrature (QUADPACK): subdividing where it is needed
================================================================

QUADPACK's adaptive strategy estimates each subinterval's integral two
ways and splits only those subintervals where the estimates disagree.
:class:`~mathematicskit.calculus.AdaptiveQuadrature` wraps
:func:`scipy.integrate.quad` (QUADPACK's ``QAGS``). Recording every point
at which it samples a sharply peaked integrand shows the evaluations
crowding around the peak, and a fixed equally spaced rule needs far more
evaluations to match its accuracy.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.calculus import AdaptiveQuadrature, TrapezoidalRule








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Where does the adaptive routine sample?
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-54

.. code-block:: Python



    def peaked(x):
        return 1.0 / (1.0 + 1000.0 * (x - 0.3) ** 2)


    a, b = 0.0, 1.0
    s = np.sqrt(1000.0)
    exact = (np.arctan(s * (b - 0.3)) - np.arctan(s * (a - 0.3))) / s

    samples = []


    def recorded(x):
        samples.append(x)
        return peaked(x)


    result = AdaptiveQuadrature(tol=1e-10).integrate(recorded, a, b)
    print(f"adaptive: value={result.value:.12f}, |error|={abs(result.value - exact):.1e}, evaluations={result.n_evaluations}")

    grid = np.linspace(a, b, 1000)
    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(6, 5.5), sharex=True, gridspec_kw={"height_ratios": [2, 1]})
    ax1.plot(grid, peaked(grid), "k")
    ax1.plot(samples, peaked(np.asarray(samples)), "|", color="C0", ms=10)
    ax1.set_title("QUADPACK samples cluster where the integrand varies")
    ax2.hist(samples, bins=50, color="C0")
    ax2.set_xlabel("x")
    ax2.set_ylabel("evaluations")
    fig.tight_layout()




.. image-sg:: /api/gallery/calculus/quadrature/images/sphx_glr_plot_08_adaptive_quadpack_001.png
   :alt: QUADPACK samples cluster where the integrand varies
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_08_adaptive_quadpack_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    adaptive: value=0.094597212547, |error|=1.8e-16, evaluations=189




.. GENERATED FROM PYTHON SOURCE LINES 55-57

Matching the accuracy with a fixed, equally spaced rule
-------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 57-63

.. code-block:: Python


    for n in (100, 1000, 10000, 100000):
        err = abs(TrapezoidalRule(n=n).integrate(peaked, a, b).value - exact)
        print(f"trapezoidal, {n + 1:6d} evaluations: |error|={err:.1e}")

    plt.show()




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

 .. code-block:: none

    trapezoidal,    101 evaluations: |error|=6.5e-07
    trapezoidal,   1001 evaluations: |error|=6.5e-09
    trapezoidal,  10001 evaluations: |error|=6.5e-11
    trapezoidal, 100001 evaluations: |error|=6.5e-13





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

   **Total running time of the script:** (0 minutes 0.068 seconds)


.. _sphx_glr_download_api_gallery_calculus_quadrature_plot_08_adaptive_quadpack.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_08_adaptive_quadpack.ipynb <plot_08_adaptive_quadpack.ipynb>`

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

      :download:`Download Python source code: plot_08_adaptive_quadpack.py <plot_08_adaptive_quadpack.py>`

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

      :download:`Download zipped: plot_08_adaptive_quadpack.zip <plot_08_adaptive_quadpack.zip>`


.. only:: html

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

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