.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/eigen/plot_04_lanczos.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_eigen_plot_04_lanczos.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_linalg_eigen_plot_04_lanczos.py:


Lanczos: a few eigenvalues of a very large matrix
======================================================

The 1-D discrete Laplacian ``tridiag(-1, 2, -1)`` of size ``n`` has
eigenvalues ``2 - 2 cos(j pi / (n + 1))``. Lanczos iteration (through
ARPACK) finds a handful of them using only matrix-vector products. In
shift-invert mode it handles a sparse ``n = 100 000`` instance, where a
dense eigensolver would need 80 GB just to store the matrix.

.. GENERATED FROM PYTHON SOURCE LINES 13-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    import scipy.sparse as sp

    from mathematicskit.linalg import lanczos_eigsh


    def laplacian_1d(n):
        return sp.diags([-np.ones(n - 1), 2 * np.ones(n), -np.ones(n - 1)], [-1, 0, 1], format="csc")


    def exact(j, n):
        return 2.0 - 2.0 * np.cos(j * np.pi / (n + 1))









.. GENERATED FROM PYTHON SOURCE LINES 29-31

Largest eigenvalues by plain Lanczos
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-37

.. code-block:: Python


    n = 2000
    result = lanczos_eigsh(laplacian_1d(n), k=4, which="LA")
    for lam, ex in zip(result.eigenvalues, exact(np.arange(n - 3, n + 1), n)):
        print(f"Lanczos {lam:.14f}   exact {ex:.14f}   |diff| {abs(lam - ex):.1e}")





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

 .. code-block:: none

    Lanczos 3.99996056116101   exact 3.99996056116084   |diff| 1.6e-13
    Lanczos 3.99997781562106   exact 3.99997781562108   |diff| 1.8e-14
    Lanczos 3.99999014026616   exact 3.99999014026591   |diff| 2.6e-13
    Lanczos 3.99999753506495   exact 3.99999753506496   |diff| 4.0e-15




.. GENERATED FROM PYTHON SOURCE LINES 38-40

Smallest eigenvalues of n = 100 000, by shift-invert Lanczos
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-56

.. code-block:: Python


    n = 100_000
    result = lanczos_eigsh(laplacian_1d(n), k=4, sigma=0.0, which="LM")
    for lam, ex in zip(result.eigenvalues, exact(np.arange(1, 5), n)):
        print(f"Lanczos {lam:.6e}   exact {ex:.6e}   rel. error {abs(lam - ex) / ex:.1e}")

    fig, ax = plt.subplots(figsize=(6, 4))
    for k in range(3):
        v = result.eigenvectors[:, k]
        ax.plot(np.sign(v[n // 10]) * v, label=f"lambda_{k + 1} = {result.eigenvalues[k]:.3e}")
    ax.set_xlabel("index")
    ax.set_title("Lowest Laplacian eigenvectors: sine modes")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/linalg/eigen/images/sphx_glr_plot_04_lanczos_001.png
   :alt: Lowest Laplacian eigenvectors: sine modes
   :srcset: /api/gallery/linalg/eigen/images/sphx_glr_plot_04_lanczos_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Lanczos 9.869407e-10   exact 9.869408e-10   rel. error 5.1e-08
    Lanczos 3.947763e-09   exact 3.947763e-09   rel. error 5.2e-09
    Lanczos 8.882466e-09   exact 8.882466e-09   rel. error 1.5e-09
    Lanczos 1.579105e-08   exact 1.579105e-08   rel. error 4.1e-09





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

   **Total running time of the script:** (0 minutes 1.068 seconds)


.. _sphx_glr_download_api_gallery_linalg_eigen_plot_04_lanczos.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_lanczos.ipynb <plot_04_lanczos.ipynb>`

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

      :download:`Download Python source code: plot_04_lanczos.py <plot_04_lanczos.py>`

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

      :download:`Download zipped: plot_04_lanczos.zip <plot_04_lanczos.zip>`


.. only:: html

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

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