.. _sphx_glr_api_gallery_linalg_lu:

LU decomposition
----------------

Elimination as the *Nine Chapters* did it, and Gauss's elimination
recorded as a partial-pivoted LU decomposition for solving systems and
computing determinants from a single factorization.


.. 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="The first problem of Chapter 8 (Fangcheng, &quot;rectangular arrays&quot;) of the Nine Chapters on the Mathematical Art: 3 bundles of top-grade, 2 of medium-grade and 1 of low-grade grain yield 39 dou; 2, 3 and 1 bundles yield 34; 1, 2 and 3 bundles yield 26. How much does one bundle of each grade yield?">

.. only:: html

  .. image:: /api/gallery/linalg/lu/images/thumb/sphx_glr_plot_01_nine_chapters_elimination_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/lu/plot_01_nine_chapters_elimination`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Elimination in the Nine Chapters: the three grades of grain</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Gaussian elimination, with its multipliers stored in L and the eliminated rows in U, factors P A = L U. The script reuses the factorization to solve multiple right-hand sides and compute the determinant, both far cheaper than refactoring from scratch each time.">

.. only:: html

  .. image:: /api/gallery/linalg/lu/images/thumb/sphx_glr_plot_02_lu_decomposition_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/lu/plot_02_lu_decomposition`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">LU decomposition with partial pivoting</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/linalg/lu/plot_01_nine_chapters_elimination
   /api/gallery/linalg/lu/plot_02_lu_decomposition