.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/eigen/plot_05_francis_qr_algorithm.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_05_francis_qr_algorithm.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_eigen_plot_05_francis_qr_algorithm.py:


The QR algorithm and the Schur form
========================================

Factor ``A_k = Q_k R_k`` and multiply back in reverse order,
``A_{k+1} = R_k Q_k = Q_k^T A_k Q_k``. Every iterate is similar to
``A``, and the subdiagonal entries decay like
``|lambda_{i+1} / lambda_i|^k``, so the iterates converge to an upper
triangular Schur form with the eigenvalues on the diagonal. Production
codes add Hessenberg reduction and Francis's shifts; the result is what
:func:`scipy.linalg.schur` returns.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.linalg import hessenberg_reduce, householder_qr, schur_decompose








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Unshifted QR iteration
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-39

.. code-block:: Python


    rng = np.random.default_rng(7)
    lams = np.array([9.0, 6.0, 4.0, 2.5, 1.0])
    S = rng.normal(size=(5, 5)) + 3.0 * np.eye(5)
    A = S @ np.diag(lams) @ np.linalg.inv(S)  # nonsymmetric, eigenvalues lams
    H, _ = hessenberg_reduce(A)

    Ak = H.copy()
    subdiag = []
    for _ in range(60):
        qr = householder_qr(Ak)
        Ak = qr.R @ qr.Q
        subdiag.append(np.abs(np.diag(Ak, -1)))
    subdiag = np.array(subdiag)
    print("diag(A_60):   ", np.round(np.diag(Ak), 8))





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

 .. code-block:: none

    diag(A_60):    [9.  6.  4.  2.5 1. ]




.. GENERATED FROM PYTHON SOURCE LINES 40-42

Compare with LAPACK's Schur decomposition
-----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-60

.. code-block:: Python


    schur = schur_decompose(A)
    print("diag(T) Schur:", np.round(np.sort(np.diag(schur.T))[::-1], 8))
    print("T below the diagonal is zero:", np.allclose(np.tril(schur.T, -1), 0.0))
    print(f"||Z T Z^T - A|| = {np.linalg.norm(schur.Z @ schur.T @ schur.Z.T - A):.1e}")

    fig, ax = plt.subplots(figsize=(6, 4))
    k = np.arange(1, len(subdiag) + 1)
    for i in range(4):
        ax.semilogy(k, subdiag[:, i], label=f"|a_{i + 2},{i + 1}|")
        ax.semilogy(k, subdiag[0, i] * (lams[i + 1] / lams[i]) ** (k - 1), "k:", lw=0.8)
    ax.set_xlabel("QR step k")
    ax.set_ylabel("subdiagonal magnitude")
    ax.set_title("Subdiagonal decay ~ (lambda_{i+1}/lambda_i)^k")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/linalg/eigen/images/sphx_glr_plot_05_francis_qr_algorithm_001.png
   :alt: Subdiagonal decay ~ (lambda_{i+1}/lambda_i)^k
   :srcset: /api/gallery/linalg/eigen/images/sphx_glr_plot_05_francis_qr_algorithm_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    diag(T) Schur: [9.  6.  4.  2.5 1. ]
    T below the diagonal is zero: True
    ||Z T Z^T - A|| = 2.8e-14





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

   **Total running time of the script:** (0 minutes 0.075 seconds)


.. _sphx_glr_download_api_gallery_linalg_eigen_plot_05_francis_qr_algorithm.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_05_francis_qr_algorithm.ipynb <plot_05_francis_qr_algorithm.ipynb>`

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

      :download:`Download Python source code: plot_05_francis_qr_algorithm.py <plot_05_francis_qr_algorithm.py>`

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

      :download:`Download zipped: plot_05_francis_qr_algorithm.zip <plot_05_francis_qr_algorithm.zip>`


.. only:: html

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

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