.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/numerical_analysis/root_finding/plot_03_halley_method.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_root_finding_plot_03_halley_method.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_root_finding_plot_03_halley_method.py:


Halley's method: cubic convergence
=====================================

Edmond Halley's 1694 iteration uses the second derivative as well as the
first, stepping to the root of an osculating hyperbola instead of a
tangent line. Near a simple root it *triples* the number of correct
digits per step, against Newton's doubling. This script solves
:math:`x^3 = 2` with both methods from the same starting guess and
compares their error histories and empirical convergence orders.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import Halley, NewtonRaphson, estimate_convergence_order








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Newton vs. Halley on :math:`f(x) = x^3 - 2`
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-42

.. code-block:: Python


    f = lambda x: x**3 - 2.0
    fp = lambda x: 3.0 * x**2
    fpp = lambda x: 6.0 * x
    root = 2.0 ** (1.0 / 3.0)

    newton = NewtonRaphson(f, fp, x0=3.0, tol=1e-15).solve()
    halley = Halley(f, fp, fpp, x0=3.0, tol=1e-15).solve()

    fig, ax = plt.subplots(figsize=(7, 4.5))
    for result, color, label in ((newton, "steelblue", "Newton (order 2)"), (halley, "firebrick", "Halley (order 3)")):
        err = np.abs(result.history - root)
        err = np.where(err == 0.0, np.finfo(float).eps * root, err)
        ax.semilogy(err, "o-", color=color, label=label)
    ax.set_xlabel("iteration $n$")
    ax.set_ylabel(r"$|x_n - 2^{1/3}|$")
    ax.set_title("Halley's method converges in fewer steps")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/numerical_analysis/root_finding/images/sphx_glr_plot_03_halley_method_001.png
   :alt: Halley's method converges in fewer steps
   :srcset: /api/gallery/numerical_analysis/root_finding/images/sphx_glr_plot_03_halley_method_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 43-45

Empirical convergence orders
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-50

.. code-block:: Python


    print(f"Newton: {newton.iterations} iterations, order ~ {estimate_convergence_order(newton.history[:-1], root):.2f}")
    print(f"Halley: {halley.iterations} iterations, order ~ {estimate_convergence_order(halley.history[:-1], root):.2f}")

    plt.show()




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

 .. code-block:: none

    Newton: 8 iterations, order ~ 1.99
    Halley: 5 iterations, order ~ 2.86





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

   **Total running time of the script:** (0 minutes 0.053 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_root_finding_plot_03_halley_method.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_halley_method.ipynb <plot_03_halley_method.ipynb>`

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

      :download:`Download Python source code: plot_03_halley_method.py <plot_03_halley_method.py>`

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

      :download:`Download zipped: plot_03_halley_method.zip <plot_03_halley_method.zip>`


.. only:: html

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

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