.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/classical/plot_01_cramers_rule.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_classical_plot_01_cramers_rule.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_classical_plot_01_cramers_rule.py:


Cramer's rule vs. elimination
==================================

Cramer's rule gives every unknown as a ratio of two determinants,
``x_i = det(A_i) / det(A)``. It is exact in theory but needs ``n + 1``
determinants, each an ``O(n^3)`` factorization, so its cost grows like
``n^4`` against elimination's ``n^3``.

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python

    import time

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.linalg import cramer_solve, lu_solve_system








.. GENERATED FROM PYTHON SOURCE LINES 20-22

A 3x3 system solved by determinants
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-32

.. code-block:: Python


    A = np.array([[2.0, 1.0, -1.0], [-3.0, -1.0, 2.0], [-2.0, 1.0, 2.0]])
    b = np.array([8.0, -11.0, -3.0])
    print("det(A) =", round(np.linalg.det(A), 10))
    for i in range(3):
        A_i = A.copy()
        A_i[:, i] = b
        print(f"det(A_{i}) = {np.linalg.det(A_i):+.10f}")
    print("Cramer solution:", np.round(cramer_solve(A, b), 10))





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

 .. code-block:: none

    det(A) = -1.0
    det(A_0) = -2.0000000000
    det(A_1) = -3.0000000000
    det(A_2) = +1.0000000000
    Cramer solution: [ 2.  3. -1.]




.. GENERATED FROM PYTHON SOURCE LINES 33-35

Cost as n grows
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-59

.. code-block:: Python


    rng = np.random.default_rng(0)
    sizes = [10, 20, 40, 80, 160]
    t_cramer, t_lu = [], []
    for n in sizes:
        M = rng.normal(size=(n, n))
        rhs = rng.normal(size=n)
        t0 = time.perf_counter()
        x_c = cramer_solve(M, rhs)
        t_cramer.append(time.perf_counter() - t0)
        t0 = time.perf_counter()
        x_lu = lu_solve_system(M, rhs)
        t_lu.append(time.perf_counter() - t0)
        print(f"n = {n:3d}: max |x_cramer - x_lu| = {np.max(np.abs(x_c - x_lu)):.1e}")

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.loglog(sizes, t_cramer, "o-", label="Cramer's rule")
    ax.loglog(sizes, t_lu, "s-", label="LU solve")
    ax.set_xlabel("n")
    ax.set_ylabel("time (s)")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/linalg/classical/images/sphx_glr_plot_01_cramers_rule_001.png
   :alt: plot 01 cramers rule
   :srcset: /api/gallery/linalg/classical/images/sphx_glr_plot_01_cramers_rule_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    n =  10: max |x_cramer - x_lu| = 1.6e-14
    n =  20: max |x_cramer - x_lu| = 1.6e-14
    n =  40: max |x_cramer - x_lu| = 2.7e-14
    n =  80: max |x_cramer - x_lu| = 4.5e-13
    n = 160: max |x_cramer - x_lu| = 2.5e-13





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

   **Total running time of the script:** (0 minutes 0.064 seconds)


.. _sphx_glr_download_api_gallery_linalg_classical_plot_01_cramers_rule.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_cramers_rule.ipynb <plot_01_cramers_rule.ipynb>`

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

      :download:`Download Python source code: plot_01_cramers_rule.py <plot_01_cramers_rule.py>`

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

      :download:`Download zipped: plot_01_cramers_rule.zip <plot_01_cramers_rule.zip>`


.. only:: html

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

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