.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/cholesky/plot_01_cholesky_solve.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_cholesky_plot_01_cholesky_solve.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_cholesky_plot_01_cholesky_solve.py:


Cholesky decomposition for SPD systems
==========================================

Cholesky factors a symmetric positive-definite ``A = L L^T`` at roughly
half the cost of LU, and the factorization succeeding at all (every
``sqrt`` argument staying positive) is itself a certificate of
positive-definiteness.

.. GENERATED FROM PYTHON SOURCE LINES 12-16

.. code-block:: Python

    import numpy as np

    from mathematicskit.linalg import cholesky_decompose, cholesky_solve, is_symmetric_positive_definite, random_spd_matrix








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Factor and solve
--------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-30

.. code-block:: Python


    A = random_spd_matrix(5, seed=1)
    result = cholesky_decompose(A)
    print("L @ L.T == A:", np.allclose(result.L @ result.L.T, A))

    x_true = np.arange(1.0, 6.0)
    b = A @ x_true
    x = cholesky_solve(result, b)
    print("recovered x:", np.round(x, 8))
    print("matches x_true:", np.allclose(x, x_true, atol=1e-8))





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

 .. code-block:: none

    L @ L.T == A: True
    recovered x: [1. 2. 3. 4. 5.]
    matches x_true: True




.. GENERATED FROM PYTHON SOURCE LINES 31-33

Positive-definiteness as a byproduct of the factorization succeeding
--------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-36

.. code-block:: Python


    print("is_symmetric_positive_definite(A):", is_symmetric_positive_definite(A))
    print("is_symmetric_positive_definite(non-SPD):", is_symmetric_positive_definite(np.array([[1.0, 2.0], [2.0, 1.0]])))




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

 .. code-block:: none

    is_symmetric_positive_definite(A): True
    is_symmetric_positive_definite(non-SPD): False





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

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


.. _sphx_glr_download_api_gallery_linalg_cholesky_plot_01_cholesky_solve.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_cholesky_solve.ipynb <plot_01_cholesky_solve.ipynb>`

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

      :download:`Download Python source code: plot_01_cholesky_solve.py <plot_01_cholesky_solve.py>`

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

      :download:`Download zipped: plot_01_cholesky_solve.zip <plot_01_cholesky_solve.zip>`


.. only:: html

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

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