.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/integrators/plot_05_yoshida_composition.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_ode_dynamics_integrators_plot_05_yoshida_composition.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_ode_dynamics_integrators_plot_05_yoshida_composition.py:


Yoshida's symplectic composition: fourth order from leapfrog
============================================================

Yoshida (1990) raised the order of a symmetric method by *composing* it
with itself. Three leapfrog substeps of sizes :math:`w_1 h, w_0 h, w_1 h`
with

.. math::

    w_1 = \frac{1}{2 - 2^{1/3}},\qquad w_0 = -\frac{2^{1/3}}{2 - 2^{1/3}}

cancel the :math:`h^3` error term. The result is fourth order and, being
built from symplectic steps, still symplectic. The middle substep runs
*backwards* in time (:math:`w_0 < 0`).

.. GENERATED FROM PYTHON SOURCE LINES 19-39

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.integrators import leapfrog_integrate, njit, yoshida4_integrate


    @njit
    def spring(q, t, params):
        return -q


    t_end = 10.0
    ns = 2 ** np.arange(5, 12)
    hs = t_end / ns
    errors = {"leapfrog": [], "Yoshida 4th order": []}
    for h, n in zip(hs, ns, strict=False):
        for label, method in (("leapfrog", leapfrog_integrate), ("Yoshida 4th order", yoshida4_integrate)):
            _, qs, _ = method(spring, np.array([1.0]), np.array([0.0]), 0.0, h, n, np.zeros(1))
            errors[label].append(abs(qs[-1, 0] - np.cos(t_end)))








.. GENERATED FROM PYTHON SOURCE LINES 40-42

Error vs step size
------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-56

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 4.5))
    for (label, err), cost in zip(errors.items(), (1, 3), strict=False):
        ax.loglog(hs, err, "o-", label=f"{label} ({cost} force evaluation(s) per step)")
        print(f"{label}: error ratio for halved h = {err[-2] / err[-1]:.1f}")
    ax.loglog(hs, 0.05 * hs**2, "k--", lw=0.8, label=r"$O(h^2)$")
    ax.loglog(hs, 0.01 * hs**4, "k:", lw=0.8, label=r"$O(h^4)$")
    ax.set_xlabel("h")
    ax.set_ylabel("|q(10) - cos 10|")
    ax.set_title("Yoshida composition: slope 4 instead of 2")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_05_yoshida_composition_001.png
   :alt: Yoshida composition: slope 4 instead of 2
   :srcset: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_05_yoshida_composition_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    leapfrog: error ratio for halved h = 4.0
    Yoshida 4th order: error ratio for halved h = 16.0





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

   **Total running time of the script:** (0 minutes 2.103 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_integrators_plot_05_yoshida_composition.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/ode_dynamics/integrators/plot_05_yoshida_composition.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_yoshida_composition.ipynb <plot_05_yoshida_composition.ipynb>`

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

      :download:`Download Python source code: plot_05_yoshida_composition.py <plot_05_yoshida_composition.py>`

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

      :download:`Download zipped: plot_05_yoshida_composition.zip <plot_05_yoshida_composition.zip>`


.. only:: html

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

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