.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/eigen/plot_03_gershgorin_discs.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_linalg_eigen_plot_03_gershgorin_discs.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_eigen_plot_03_gershgorin_discs.py:


Gershgorin discs
=====================

Without computing anything but row sums, Gershgorin's theorem traps
every eigenvalue in a union of discs centered on the diagonal entries.
Shrinking the off-diagonal part shrinks the discs onto the eigenvalues.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.linalg import gershgorin_discs








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Discs and eigenvalues
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-44

.. code-block:: Python


    rng = np.random.default_rng(3)
    D = np.diag([-4.0, 1.0, 3.0 + 2.0j, 3.0 - 2.0j, 7.0])
    E = rng.normal(size=(5, 5)) * 0.6

    fig, axes = plt.subplots(1, 2, figsize=(9, 4), sharex=True, sharey=True)
    for ax, eps in zip(axes, (1.0, 0.3)):
        A = D + eps * E
        discs = gershgorin_discs(A)
        eigs = np.linalg.eigvals(A)
        for c, r in zip(discs.centers, discs.radii):
            ax.add_patch(plt.Circle((c.real, c.imag), r, alpha=0.2))
            ax.add_patch(plt.Circle((c.real, c.imag), r, fill=False))
        ax.plot(eigs.real, eigs.imag, "k*", ms=9, label="eigenvalues")
        ax.plot(discs.centers.real, discs.centers.imag, "r+", ms=9, label="a_kk")
        ax.set_title(f"A = D + {eps} E")
        ax.set_aspect("equal")
        ax.set_xlabel("Re")
        print(f"eps = {eps}: all eigenvalues inside the discs? {all(discs.contains(z) for z in eigs)}; max radius {discs.radii.max():.3f}")
    axes[0].set_ylabel("Im")
    axes[0].legend(loc="lower left")
    axes[0].autoscale_view()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/linalg/eigen/images/sphx_glr_plot_03_gershgorin_discs_001.png
   :alt: A = D + 1.0 E, A = D + 0.3 E
   :srcset: /api/gallery/linalg/eigen/images/sphx_glr_plot_03_gershgorin_discs_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    eps = 1.0: all eigenvalues inside the discs? True; max radius 2.781
    eps = 0.3: all eigenvalues inside the discs? True; max radius 0.834





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

   **Total running time of the script:** (0 minutes 0.048 seconds)


.. _sphx_glr_download_api_gallery_linalg_eigen_plot_03_gershgorin_discs.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_gershgorin_discs.ipynb <plot_03_gershgorin_discs.ipynb>`

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

      :download:`Download Python source code: plot_03_gershgorin_discs.py <plot_03_gershgorin_discs.py>`

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

      :download:`Download zipped: plot_03_gershgorin_discs.zip <plot_03_gershgorin_discs.zip>`


.. only:: html

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

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