.. 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_06_adams_bashforth.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_06_adams_bashforth.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_06_adams_bashforth.py:


Bashforth and Adams: linear multistep methods
=============================================

John Couch Adams, computing drop shapes for Francis Bashforth's 1883 study
of capillarity, reused the slopes already computed at earlier steps
instead of throwing them away. Integrating the polynomial through the
last :math:`k` slopes gives the order-:math:`k` Adams-Bashforth formula,

.. math::

   y_{n+1} = y_n + h \sum_{j=0}^{k-1} \beta_j f_{n-j},
   \qquad \text{e.g. AB2: } y_{n+1} = y_n + h\bigl(\tfrac32 f_n - \tfrac12 f_{n-1}\bigr),

at the cost of one new evaluation of :math:`f` per step, however high
the order. The price appears in the method's stability region, which
shrinks as the order grows.

.. GENERATED FROM PYTHON SOURCE LINES 21-36

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.integrators import adams_bashforth_boundary_locus, adams_bashforth_integrate, njit, rk4_integrate
    from mathematicskit.ode_dynamics.visualizers import plot_stability_regions


    @njit
    def oscillator(state, t, params):
        out = np.empty(2)
        out[0] = state[1]
        out[1] = -state[0]
        return out









.. GENERATED FROM PYTHON SOURCE LINES 37-43

Order k from one evaluation per step
------------------------------------

Error at :math:`t = 2\pi` for :math:`x'' = -x`, against the number of
right-hand-side evaluations. RK4 needs four per step, so AB4 matches its
order at a quarter of the cost per step.

.. GENERATED FROM PYTHON SOURCE LINES 43-58

.. code-block:: Python


    fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(12, 4.8))
    steps = np.array([50, 100, 200, 400, 800, 1600])
    y0, t_end = np.array([1.0, 0.0]), 2.0 * np.pi
    for order in (1, 2, 3, 4):
        errors = [np.abs(adams_bashforth_integrate(oscillator, y0, 0.0, t_end / n, n, np.zeros(1), order)[1][-1] - y0).max() for n in steps]
        ax0.loglog(steps, errors, "o-", label=f"AB{order}")
        print(f"AB{order}: error ratio for halved h = {errors[-2] / errors[-1]:.2f} (2^{order} = {2**order})")
    rk4_errors = [np.abs(rk4_integrate(oscillator, y0, 0.0, t_end / n, n, np.zeros(1))[1][-1] - y0).max() for n in steps]
    ax0.loglog(4 * steps, rk4_errors, "k^--", label="RK4")
    ax0.set_xlabel("right-hand-side evaluations")
    ax0.set_ylabel(r"max error at $t = 2\pi$")
    ax0.set_title("Adams-Bashforth: order k at one evaluation per step")
    ax0.legend()




.. image-sg:: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_06_adams_bashforth_001.png
   :alt: Adams-Bashforth: order k at one evaluation per step
   :srcset: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_06_adams_bashforth_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    AB1: error ratio for halved h = 2.01 (2^1 = 2)
    AB2: error ratio for halved h = 4.00 (2^2 = 4)
    AB3: error ratio for halved h = 7.99 (2^3 = 8)
    AB4: error ratio for halved h = 15.97 (2^4 = 16)

    <matplotlib.legend.Legend object at 0x7fd7c5ab66f0>



.. GENERATED FROM PYTHON SOURCE LINES 59-69

Stability regions shrink with order
-----------------------------------

Shaded: the absolute-stability regions in the :math:`z = \lambda h`
plane. Dashed: the boundary locus
:math:`z(\theta) = \rho(e^{i\theta})/\sigma(e^{i\theta})`, the curve on
which a root of the characteristic polynomial lies on the unit circle.
The real stability interval is :math:`[-2, 0]` for AB1 (Euler),
:math:`[-1, 0]` for AB2, :math:`[-6/11, 0]` for AB3, and
:math:`[-3/10, 0]` for AB4.

.. GENERATED FROM PYTHON SOURCE LINES 69-85

.. code-block:: Python


    plot_stability_regions(
        ["euler", "adams_bashforth2", "adams_bashforth3", "adams_bashforth4"],
        ax=ax1,
        re_range=(-2.3, 0.6),
        im_range=(-1.4, 1.4),
        labels=["AB1 (Euler)", "AB2", "AB3", "AB4"],
    )
    for order in (1, 2, 3, 4):
        locus = adams_bashforth_boundary_locus(order)
        ax1.plot(locus.real, locus.imag, "--", color=f"C{order - 1}", lw=0.8)
    ax1.set_xlim(-2.3, 0.6)
    ax1.set_ylim(-1.4, 1.4)
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 3.757 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_integrators_plot_06_adams_bashforth.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_06_adams_bashforth.ipynb
        :alt: Launch JupyterLite
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 .. rst-class:: sphx-glr-signature

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