.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/iterative/plot_01_cg_and_gmres_convergence.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_iterative_plot_01_cg_and_gmres_convergence.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_iterative_plot_01_cg_and_gmres_convergence.py:


Conjugate gradient and GMRES convergence
=============================================

Both are Krylov-subspace iterative solvers that never form a dense
factorization of ``A``. Conjugate gradient requires ``A`` symmetric
positive-definite; GMRES works for any nonsingular ``A``. This script
watches both converge and plots the residual-norm history.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.linalg import GMRES, ConjugateGradient, random_spd_matrix
    from mathematicskit.linalg.visualizers.plots import plot_residual_history








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Conjugate gradient on an SPD system
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-27

.. code-block:: Python


    A_spd = random_spd_matrix(20, seed=4)
    b = np.ones(20)
    result_cg = ConjugateGradient(tol=1e-10).solve(A_spd, b)
    print(f"CG converged in {result_cg.iterations} iterations (n=20); final residual {result_cg.residual_history[-1]:.2e}")





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

 .. code-block:: none

    CG converged in 24 iterations (n=20); final residual 1.40e-11




.. GENERATED FROM PYTHON SOURCE LINES 28-30

GMRES on a general nonsymmetric system
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-43

.. code-block:: Python


    rng = np.random.default_rng(5)
    A_gen = rng.uniform(-1, 1, size=(20, 20)) + 20.0 * np.eye(20)
    result_gmres = GMRES(tol=1e-10).solve(A_gen, b)
    print(f"GMRES converged in {result_gmres.iterations} iterations (n=20); final residual {result_gmres.residual_history[-1]:.2e}")

    fig, ax = plt.subplots(figsize=(6, 4))
    plot_residual_history(result_cg, ax=ax)
    plot_residual_history(result_gmres, ax=ax)
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/linalg/iterative/images/sphx_glr_plot_01_cg_and_gmres_convergence_001.png
   :alt: Iterative solver convergence
   :srcset: /api/gallery/linalg/iterative/images/sphx_glr_plot_01_cg_and_gmres_convergence_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    GMRES converged in 11 iterations (n=20); final residual 1.61e-11





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

   **Total running time of the script:** (0 minutes 0.042 seconds)


.. _sphx_glr_download_api_gallery_linalg_iterative_plot_01_cg_and_gmres_convergence.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_cg_and_gmres_convergence.ipynb <plot_01_cg_and_gmres_convergence.ipynb>`

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

      :download:`Download Python source code: plot_01_cg_and_gmres_convergence.py <plot_01_cg_and_gmres_convergence.py>`

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

      :download:`Download zipped: plot_01_cg_and_gmres_convergence.zip <plot_01_cg_and_gmres_convergence.zip>`


.. only:: html

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

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