.. _sphx_glr_api_gallery_linalg_iterative:

Iterative solvers
-----------------

Conjugate gradient (SPD systems) and GMRES (general systems): Krylov
subspace methods that never form a dense factorization.

Also the classical stationary iterations: Jacobi, Gauss-Seidel, and
successive over-relaxation (SOR).


.. 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="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.">

.. only:: html

  .. image:: /api/gallery/linalg/iterative/images/thumb/sphx_glr_plot_01_cg_and_gmres_convergence_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/iterative/plot_01_cg_and_gmres_convergence`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Conjugate gradient and GMRES convergence</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The three classical stationary iterations on the 1-D Poisson matrix tridiag(-1, 2, -1). Gauss-Seidel contracts the error about twice as fast per sweep as Jacobi, and SOR at Young&#x27;s optimal relaxation factor is faster by an order of magnitude.">

.. only:: html

  .. image:: /api/gallery/linalg/iterative/images/thumb/sphx_glr_plot_02_gauss_seidel_and_sor_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/iterative/plot_02_gauss_seidel_and_sor`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Jacobi, Gauss-Seidel, and successive over-relaxation</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/linalg/iterative/plot_01_cg_and_gmres_convergence
   /api/gallery/linalg/iterative/plot_02_gauss_seidel_and_sor