.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/lu/plot_02_lu_decomposition.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_lu_plot_02_lu_decomposition.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_lu_plot_02_lu_decomposition.py:


LU decomposition with partial pivoting
==========================================

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.

.. GENERATED FROM PYTHON SOURCE LINES 12-16

.. code-block:: Python

    import numpy as np

    from mathematicskit.linalg import lu_decompose, lu_det, lu_solve








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Factor once, solve for several right-hand sides
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: Python


    A = np.array([[2.0, 1.0, 1.0], [4.0, 3.0, 3.0], [8.0, 7.0, 9.0]])
    result = lu_decompose(A)
    print("P @ A == L @ U:", np.allclose(result.P @ A, result.L @ result.U))

    for b in (np.array([5.0, 12.0, 24.0]), np.array([1.0, 0.0, -1.0])):
        x = lu_solve(result, b)
        print(f"b={b} -> x={np.round(x, 6)}, residual={np.linalg.norm(A @ x - b):.2e}")





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

 .. code-block:: none

    P @ A == L @ U: True
    b=[ 5. 12. 24.] -> x=[ 1.5  3.  -1. ], residual=8.88e-16
    b=[ 1.  0. -1.] -> x=[ 1.5 -2.5  0.5], residual=4.97e-16




.. GENERATED FROM PYTHON SOURCE LINES 29-31

Determinant via the pivoted triangular factors
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-35

.. code-block:: Python


    det_lu = lu_det(A)
    det_numpy = np.linalg.det(A)
    print(f"det via LU: {det_lu:.10f}, numpy.linalg.det: {det_numpy:.10f}")




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

 .. code-block:: none

    det via LU: 4.0000000000, numpy.linalg.det: 4.0000000000





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

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


.. _sphx_glr_download_api_gallery_linalg_lu_plot_02_lu_decomposition.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_lu_decomposition.ipynb <plot_02_lu_decomposition.ipynb>`

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

      :download:`Download Python source code: plot_02_lu_decomposition.py <plot_02_lu_decomposition.py>`

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

      :download:`Download zipped: plot_02_lu_decomposition.zip <plot_02_lu_decomposition.zip>`


.. only:: html

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

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