.. _sphx_glr_api_gallery_linalg_qr:

QR decomposition
----------------

Householder reflections vs. classical/modified Gram-Schmidt, and the
orthogonality each preserves for an ill-conditioned matrix.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="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.">

.. only:: html

  .. image:: /api/gallery/linalg/qr/images/thumb/sphx_glr_plot_01_householder_vs_gram_schmidt_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/qr/plot_01_householder_vs_gram_schmidt`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Householder vs. Gram-Schmidt QR</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/linalg/qr/plot_01_householder_vs_gram_schmidt