.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/finite_differences/plot_01_richardson_extrapolation.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_calculus_finite_differences_plot_01_richardson_extrapolation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_finite_differences_plot_01_richardson_extrapolation.py:


Richardson extrapolation: cancelling error terms at a fixed step
=================================================================

Richardson's idea: if an estimate :math:`D(h)` has error
:math:`c_1 h^2 + c_2 h^4 + \dots`, then combining :math:`D(h)` and
:math:`D(h/2)` as :math:`(4D(h/2) - D(h))/3` cancels the :math:`h^2`
term. Repeating the combination on successively halved steps removes
one error order per level. This example applies
:func:`~mathematicskit.calculus.richardson_extrapolation` to a central
difference and shows it reaching near machine precision from a coarse
starting step, where no plain difference quotient can.

.. GENERATED FROM PYTHON SOURCE LINES 16-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.calculus import central_difference, richardson_extrapolation

    f = np.exp
    x0 = 1.0
    exact = np.exp(1.0)








.. GENERATED FROM PYTHON SOURCE LINES 26-31

One extrapolation step by hand
------------------------------

Two central differences, each only :math:`O(h^2)`, combine into an
:math:`O(h^4)` estimate at no extra cost.

.. GENERATED FROM PYTHON SOURCE LINES 31-40

.. code-block:: Python


    h = 0.2
    d_h = central_difference(f, x0, h)
    d_h2 = central_difference(f, x0, h / 2)
    combined = (4 * d_h2 - d_h) / 3
    print(f"D(h)   error: {abs(d_h - exact):.3e}")
    print(f"D(h/2) error: {abs(d_h2 - exact):.3e}")
    print(f"(4D(h/2) - D(h))/3 error: {abs(combined - exact):.3e}")





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

 .. code-block:: none

    D(h)   error: 1.816e-02
    D(h/2) error: 4.533e-03
    (4D(h/2) - D(h))/3 error: 9.072e-06




.. GENERATED FROM PYTHON SOURCE LINES 41-47

Each level cancels one more error order
---------------------------------------

Starting from the same coarse :math:`h = 0.2`, every extra level of
extrapolation gains roughly two more orders of accuracy, until
floating-point round-off takes over.

.. GENERATED FROM PYTHON SOURCE LINES 47-53

.. code-block:: Python


    levels = np.arange(1, 7)
    errs = [abs(richardson_extrapolation(f, x0, h=0.2, levels=int(k)).value - exact) for k in levels]
    for k, e in zip(levels, errs):
        print(f"levels={k}: error={e:.3e}")





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

 .. code-block:: none

    levels=1: error=1.816e-02
    levels=2: error=9.072e-06
    levels=3: error=5.397e-10
    levels=4: error=4.885e-15
    levels=5: error=2.709e-14
    levels=6: error=6.972e-14




.. GENERATED FROM PYTHON SOURCE LINES 54-60

Extrapolation beats simply shrinking the step
---------------------------------------------

A plain central difference at step :math:`h` bottoms out near
:math:`10^{-11}` because of cancellation; Richardson reaches that
accuracy while its smallest step is still large.

.. GENERATED FROM PYTHON SOURCE LINES 60-75

.. code-block:: Python


    hs = np.logspace(-1, -7, 25)
    err_ctr = [abs(central_difference(f, x0, hh) - exact) for hh in hs]
    smallest_step = [0.2 / 2 ** (k - 1) for k in levels]

    fig, ax = plt.subplots(figsize=(6, 4.5))
    ax.loglog(hs, err_ctr, "o-", label="central difference alone")
    ax.loglog(smallest_step, np.maximum(errs, 1e-17), "s-", label="Richardson, levels 1..6")
    ax.set_xlabel("smallest step used")
    ax.set_ylabel("|error in f'(1)|")
    ax.set_title("Richardson extrapolation from a coarse step h = 0.2")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/calculus/finite_differences/images/sphx_glr_plot_01_richardson_extrapolation_001.png
   :alt: Richardson extrapolation from a coarse step h = 0.2
   :srcset: /api/gallery/calculus/finite_differences/images/sphx_glr_plot_01_richardson_extrapolation_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.107 seconds)


.. _sphx_glr_download_api_gallery_calculus_finite_differences_plot_01_richardson_extrapolation.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_richardson_extrapolation.ipynb <plot_01_richardson_extrapolation.ipynb>`

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

      :download:`Download Python source code: plot_01_richardson_extrapolation.py <plot_01_richardson_extrapolation.py>`

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      :download:`Download zipped: plot_01_richardson_extrapolation.zip <plot_01_richardson_extrapolation.zip>`


.. only:: html

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

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