.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/numerical_analysis/polynomials/plot_02_wilkinson_polynomial.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_polynomials_plot_02_wilkinson_polynomial.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_numerical_analysis_polynomials_plot_02_wilkinson_polynomial.py:


Wilkinson's perfidious polynomial
====================================

:math:`w(x) = (x - 1)(x - 2) \cdots (x - 20)` has well separated integer
roots, yet James Wilkinson found that subtracting :math:`2^{-23}` from
its :math:`x^{19}` coefficient, :math:`-210`, sends ten of the roots far
into the complex plane. The root condition numbers explain why: the
roots near 16 amplify a relative change in that coefficient by a
factor of about :math:`3 \times 10^{10}`.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import root_condition_numbers, wilkinson_polynomial








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Roots before and after a tiny perturbation
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-36

.. code-block:: Python


    w = wilkinson_polynomial(20)
    perturbed = w.copy()
    perturbed[1] -= 2.0**-23
    roots = np.roots(perturbed)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4.2))
    ax1.scatter(np.arange(1, 21), np.zeros(20), color="black", label="roots of $w$")
    ax1.scatter(roots.real, roots.imag, color="firebrick", marker="x", label=r"$a_{19} \to a_{19} - 2^{-23}$")
    ax1.set_xlabel("real part")
    ax1.set_ylabel("imaginary part")
    ax1.set_title("Wilkinson's perturbation experiment")
    ax1.legend(fontsize=8)




.. image-sg:: /api/gallery/numerical_analysis/polynomials/images/sphx_glr_plot_02_wilkinson_polynomial_001.png
   :alt: Wilkinson's perturbation experiment
   :srcset: /api/gallery/numerical_analysis/polynomials/images/sphx_glr_plot_02_wilkinson_polynomial_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1196e3b60>



.. GENERATED FROM PYTHON SOURCE LINES 37-39

Condition number of each root with respect to :math:`a_{19}`
---------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-51

.. code-block:: Python


    kappa = [root_condition_numbers(w, float(r))[1] for r in range(1, 21)]
    ax2.semilogy(np.arange(1, 21), kappa, "o-")
    ax2.set_xlabel("root $r$")
    ax2.set_ylabel(r"$\kappa_{19}(r) = |a_{19}|\, r^{18} / |w'(r)|$")
    ax2.set_title("Relative sensitivity of each root")
    fig.tight_layout()

    print("roots with |imag| > 0.5:", np.sort_complex(roots[np.abs(roots.imag) > 0.5]).round(3))
    print(f"largest condition number: {max(kappa):.2e} at r = {1 + int(np.argmax(kappa))}")

    plt.show()




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

 .. code-block:: none

    roots with |imag| > 0.5: [10.095-0.64j  10.095+0.64j  11.793-1.651j 11.793+1.651j 13.992-2.519j
     13.992+2.519j 16.731-2.813j 16.731+2.813j 19.502-1.94j  19.502+1.94j ]
    largest condition number: 3.16e+10 at r = 16





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

   **Total running time of the script:** (0 minutes 0.072 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_polynomials_plot_02_wilkinson_polynomial.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_wilkinson_polynomial.ipynb <plot_02_wilkinson_polynomial.ipynb>`

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

      :download:`Download Python source code: plot_02_wilkinson_polynomial.py <plot_02_wilkinson_polynomial.py>`

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

      :download:`Download zipped: plot_02_wilkinson_polynomial.zip <plot_02_wilkinson_polynomial.zip>`


.. only:: html

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

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