.. 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_01_euler_method.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_01_euler_method.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_01_euler_method.py:


Euler's method: first-order convergence
=======================================

Euler (1768) replaced the curve by its tangent line over a short step
:math:`h`: :math:`y_{n+1} = y_n + h f(t_n, y_n)`. Evaluating the slope at
the *end* of the step instead gives the implicit (backward) variant
:math:`y_{n+1} = y_n + h f(t_{n+1}, y_{n+1})`. Both have a local error of
:math:`O(h^2)`, which adds up to a global error of :math:`O(h)`: halving
the step halves the error.

.. GENERATED FROM PYTHON SOURCE LINES 14-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.integrators import euler_integrate, implicit_euler_integrate, njit


    @njit
    def decay(state, t, params):
        return -state


    def forward_euler(y0, h, n_steps):
        return euler_integrate(decay, np.array([y0]), 0.0, h, n_steps, np.zeros(1))[1][:, 0]









.. GENERATED FROM PYTHON SOURCE LINES 30-32

Tangent-line steps
------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-46

.. code-block:: Python


    h, n = 0.5, 6
    ts = h * np.arange(n + 1)
    _, ys_back = implicit_euler_integrate(decay, np.array([1.0]), 0.0, h, n, np.zeros(1))
    t_fine = np.linspace(0.0, n * h, 200)

    fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 4))
    ax0.plot(t_fine, np.exp(-t_fine), "k", label=r"exact $e^{-t}$")
    ax0.plot(ts, forward_euler(1.0, h, n), "o-", label="forward Euler (undershoots)")
    ax0.plot(ts, ys_back[:, 0], "s-", label="backward Euler (overshoots)")
    ax0.set_xlabel("t")
    ax0.set_title(f"y' = -y with step h = {h}")
    ax0.legend()




.. image-sg:: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_01_euler_method_001.png
   :alt: y' = -y with step h = 0.5
   :srcset: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_01_euler_method_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x7fd7c2f39e80>



.. GENERATED FROM PYTHON SOURCE LINES 47-49

Global error is proportional to h
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-65

.. code-block:: Python


    hs = 1.0 / 2 ** np.arange(3, 11)
    err_fwd = [abs(forward_euler(1.0, h, round(1 / h))[-1] - np.exp(-1)) for h in hs]
    err_back = [abs(implicit_euler_integrate(decay, np.array([1.0]), 0.0, h, round(1 / h), np.zeros(1))[1][-1, 0] - np.exp(-1)) for h in hs]
    ax1.loglog(hs, err_fwd, "o-", label="forward Euler")
    ax1.loglog(hs, err_back, "s-", label="backward Euler")
    ax1.loglog(hs, 0.2 * hs, "k--", label=r"slope 1: $O(h)$")
    ax1.set_xlabel("h")
    ax1.set_ylabel("|error| at t = 1")
    ax1.set_title("First-order convergence")
    ax1.legend()
    fig.tight_layout()

    print(f"error ratio for halved h: forward {err_fwd[-2] / err_fwd[-1]:.3f}, backward {err_back[-2] / err_back[-1]:.3f}")

    plt.show()




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

 .. code-block:: none

    error ratio for halved h: forward 2.001, backward 1.999





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

   **Total running time of the script:** (0 minutes 5.463 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_integrators_plot_01_euler_method.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_01_euler_method.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_euler_method.ipynb <plot_01_euler_method.ipynb>`

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

      :download:`Download Python source code: plot_01_euler_method.py <plot_01_euler_method.py>`

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

      :download:`Download zipped: plot_01_euler_method.zip <plot_01_euler_method.zip>`


.. only:: html

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

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