.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/numerical_analysis/approximation/plot_02_pade_approximants.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_numerical_analysis_approximation_plot_02_pade_approximants.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_approximation_plot_02_pade_approximants.py:


Padé approximants beyond the radius of convergence
=====================================================

Henri Padé's 1892 thesis organized rational approximations
:math:`p(x)/q(x)` that match a function's Taylor series as far as
possible. The Taylor series of :math:`\log(1 + x)` diverges for
:math:`x > 1`, but the Padé approximants built from *the same*
coefficients keep converging well beyond it.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import PadeApproximant








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Taylor polynomial vs. Padé approximant of :math:`\log(1 + x)`
----------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-47

.. code-block:: Python


    N = 8
    c = np.array([0.0] + [(-1.0) ** (k + 1) / k for k in range(1, N + 1)])
    taylor = np.polynomial.Polynomial(c)
    pade = PadeApproximant(c, N // 2, N // 2)

    x = np.linspace(0.0, 4.0, 400)
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4.2))
    ax1.plot(x, np.log1p(x), "k", lw=2, label=r"$\log(1 + x)$")
    ax1.plot(x, taylor(x), "--", color="steelblue", label=f"Taylor, degree {N}")
    ax1.plot(x, pade(x), ":", color="firebrick", lw=2, label=f"Padé [{N // 2}/{N // 2}]")
    ax1.axvline(1.0, color="gray", lw=0.8)
    ax1.set_ylim(-1, 3)
    ax1.set_title("Same 9 Taylor coefficients")
    ax1.legend(fontsize=8)

    ax2.semilogy(x, np.abs(taylor(x) - np.log1p(x)) + 1e-17, color="steelblue", label="Taylor")
    ax2.semilogy(x, np.abs(pade(x) - np.log1p(x)) + 1e-17, color="firebrick", label="Padé")
    ax2.axvline(1.0, color="gray", lw=0.8)
    ax2.set_title("Absolute error (radius of convergence at $x = 1$)")
    ax2.legend(fontsize=8)
    fig.tight_layout()

    print(f"at x = 3: Taylor error {abs(taylor(3.0) - np.log(4.0)):.2e}, Padé error {abs(pade(3.0) - np.log(4.0)):.2e}")

    plt.show()



.. image-sg:: /api/gallery/numerical_analysis/approximation/images/sphx_glr_plot_02_pade_approximants_001.png
   :alt: Same 9 Taylor coefficients, Absolute error (radius of convergence at $x = 1$)
   :srcset: /api/gallery/numerical_analysis/approximation/images/sphx_glr_plot_02_pade_approximants_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    at x = 3: Taylor error 5.95e+02, Padé error 2.97e-04





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

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_approximation_plot_02_pade_approximants.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_pade_approximants.ipynb <plot_02_pade_approximants.ipynb>`

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

      :download:`Download Python source code: plot_02_pade_approximants.py <plot_02_pade_approximants.py>`

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

      :download:`Download zipped: plot_02_pade_approximants.zip <plot_02_pade_approximants.zip>`


.. only:: html

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

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