.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/qr/plot_01_householder_vs_gram_schmidt.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_qr_plot_01_householder_vs_gram_schmidt.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_qr_plot_01_householder_vs_gram_schmidt.py:


Householder vs. Gram-Schmidt QR
====================================

Both factor ``A = Q R``, but Householder reflections stay orthogonal to
machine precision even for ill-conditioned ``A``, while classical
Gram-Schmidt can lose orthogonality badly -- modified Gram-Schmidt is a
partial (but not complete) remedy. This is the textbook motivation for
preferring Householder in production numerical software.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import numpy as np

    from mathematicskit.linalg import gram_schmidt_qr, householder_qr
    from mathematicskit.linalg.systems.qr import orthogonality_error








.. GENERATED FROM PYTHON SOURCE LINES 19-21

A well-conditioned matrix: all three methods agree
--------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-31

.. code-block:: Python


    rng = np.random.default_rng(0)
    A = rng.uniform(-2, 2, size=(6, 4))
    for label, result in (
        ("householder", householder_qr(A)),
        ("gram_schmidt (modified)", gram_schmidt_qr(A, modified=True)),
        ("gram_schmidt (classical)", gram_schmidt_qr(A, modified=False)),
    ):
        print(f"{label:>28s}: ||QR - A|| = {np.linalg.norm(result.Q @ result.R - A):.2e}, orthogonality error = {orthogonality_error(result.Q):.2e}")





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

 .. code-block:: none

                     householder: ||QR - A|| = 2.36e-15, orthogonality error = 4.44e-16
         gram_schmidt (modified): ||QR - A|| = 3.38e-16, orthogonality error = 2.22e-16
        gram_schmidt (classical): ||QR - A|| = 5.12e-16, orthogonality error = 2.22e-16




.. GENERATED FROM PYTHON SOURCE LINES 32-36

An ill-conditioned matrix: orthogonality breaks down differently
---------------------------------------------------------------------
Nearly-collinear columns (Trefethen & Bau's classic example) expose the
gap between the three methods.

.. GENERATED FROM PYTHON SOURCE LINES 36-45

.. code-block:: Python


    eps = 1e-8
    A_ill = np.array([[1.0, 1.0, 1.0], [eps, 0.0, 0.0], [0.0, eps, 0.0], [0.0, 0.0, eps]])
    for label, result in (
        ("householder", householder_qr(A_ill)),
        ("gram_schmidt (modified)", gram_schmidt_qr(A_ill, modified=True)),
        ("gram_schmidt (classical)", gram_schmidt_qr(A_ill, modified=False)),
    ):
        print(f"{label:>28s}: orthogonality error = {orthogonality_error(result.Q):.3e}")




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

 .. code-block:: none

                     householder: orthogonality error = 6.661e-16
         gram_schmidt (modified): orthogonality error = 7.071e-09
        gram_schmidt (classical): orthogonality error = 5.000e-01





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

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_linalg_qr_plot_01_householder_vs_gram_schmidt.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_householder_vs_gram_schmidt.ipynb <plot_01_householder_vs_gram_schmidt.ipynb>`

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

      :download:`Download Python source code: plot_01_householder_vs_gram_schmidt.py <plot_01_householder_vs_gram_schmidt.py>`

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

      :download:`Download zipped: plot_01_householder_vs_gram_schmidt.zip <plot_01_householder_vs_gram_schmidt.zip>`


.. only:: html

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

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