.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/quadrature/plot_07_tanh_sinh.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_07_tanh_sinh.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_quadrature_plot_07_tanh_sinh.py:


Tanh-sinh quadrature: integrating endpoint singularities
==============================================================

Integrates 1/sqrt(x) and log(x) over [0, 1], whose integrands blow up
at 0. The tanh-sinh substitution crowds nodes toward the endpoints with
double-exponentially small weights, and converges far faster than
Gauss-Legendre quadrature on these integrands.

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.calculus import GaussianQuadrature, TanhSinhQuadrature








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Where the nodes go
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-27

.. code-block:: Python


    rule = TanhSinhQuadrature(h=0.5, t_max=3.0)
    t = rule.h * np.arange(-6, 7)
    nodes = 0.5 + 0.5 * np.tanh(0.5 * np.pi * np.sinh(t))
    print("tanh-sinh nodes on [0, 1] (h = 0.5):")
    print("  " + ", ".join(f"{x:.3g}" for x in nodes))





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

 .. code-block:: none

    tanh-sinh nodes on [0, 1] (h = 0.5):
      2.15e-14, 5.56e-09, 1.13e-05, 0.00124, 0.0243, 0.163, 0.5, 0.837, 0.976, 0.999, 1, 1, 1




.. GENERATED FROM PYTHON SOURCE LINES 28-30

Convergence on singular integrands
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-48

.. code-block:: Python


    cases = {r"$1/\sqrt{x}$": (lambda x: 1 / np.sqrt(x), 2.0), r"$\log x$": (np.log, -1.0)}
    hs = [1.0, 0.5, 0.25, 0.125, 0.0625]
    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    for ax, (name, (f, exact)) in zip(axes, cases.items()):
        evals, errors = [], []
        for h in hs:
            result = TanhSinhQuadrature(h=h).integrate(f, 0.0, 1.0)
            evals.append(result.n_evaluations)
            errors.append(max(abs(result.value - exact), 1e-17))
        gauss_n = [4, 8, 16, 32, 64, 128]
        gauss = [abs(GaussianQuadrature(n).integrate(f, 0.0, 1.0).value - exact) for n in gauss_n]
        ax.loglog(evals, errors, "o-", label="tanh-sinh")
        ax.loglog(gauss_n, gauss, "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_07_tanh_sinh_001.png
   :alt: $1/\sqrt{x}$, $\log x$
   :srcset: /api/gallery/calculus/quadrature/images/sphx_glr_plot_07_tanh_sinh_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 0.078 seconds)


.. _sphx_glr_download_api_gallery_calculus_quadrature_plot_07_tanh_sinh.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_07_tanh_sinh.ipynb <plot_07_tanh_sinh.ipynb>`

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

      :download:`Download Python source code: plot_07_tanh_sinh.py <plot_07_tanh_sinh.py>`

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

      :download:`Download zipped: plot_07_tanh_sinh.zip <plot_07_tanh_sinh.zip>`


.. only:: html

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

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