.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_example/plot_general_fd_convergence.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_example_plot_general_fd_convergence.py>`
        to download the full example code.

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

.. _sphx_glr_auto_example_plot_general_fd_convergence.py:


Convergence of the generalized finite differences formulas
==========================================================

This example shows the convergence properties of the generalized finite
difference (F.D.) formulas.

The coefficients of the generalized finite difference formula are 
computed so that they approximate the derivative :math:`f^{(d)}(x)`
with order :math:`p`.
More precisely, in exact arithmetic, we have:

.. math::

    f^{(d)}(x) = \frac{d!}{h^d} \sum_{i = i_{\min}}^{i_\max} c_i f(x + h i)
        + R(x; h)

where :math:`h > 0` is the step, :math:`x` is the point where the derivative
is to be computed, :math:`d` is the differentiation order and :math:`p` is the 
order of accuracy of the formula, 
The remainder of the Taylor expansion is:

.. math::

    R(x; h) = - \frac{d!}{(d + p)!} b_{d + p} f^{(d + p)}(\xi) h^p.

where :math:`\xi \in (x, x + h)` and
the coefficient :math:`b_{d + p}` is defined by the equation:

.. math::

    b_{d + p} = \sum_{i = i_{\min}}^{i_\max} i^{d + p} c_i.

The goal of this example is to show the actual behaviour of the 
remainder in floating point arithmetic for particular 
values of :math:`d` and :math:`p`.

References
----------
- M. Baudin (2023). Méthodes numériques. Dunod.

.. GENERATED FROM PYTHON SOURCE LINES 47-52

.. code-block:: Python

    import numpy as np
    import numericalderivative as nd
    import math
    import pylab as pl








.. GENERATED FROM PYTHON SOURCE LINES 53-58

We consider the sinus function and we want to compute its
first derivative i.e. :math:`d = 1`.
We consider an order :math:`p = 2` formula.
Since :math:`d + p = 3`, the properties of the central finite difference formula
depends on the third derivative of the function.

.. GENERATED FROM PYTHON SOURCE LINES 60-67

.. code-block:: Python

    problem = nd.SinProblem()
    name = problem.get_name()
    first_derivative = problem.get_first_derivative()
    third_derivative = problem.get_third_derivative()
    function = problem.get_function()
    x = problem.get_x()








.. GENERATED FROM PYTHON SOURCE LINES 68-70

We create the generalized central finite difference formula using
:class:`~numericalderivative.GeneralFiniteDifference`.

.. GENERATED FROM PYTHON SOURCE LINES 72-83

.. code-block:: Python

    formula_accuracy = 2
    differentiation_order = 1
    formula = nd.GeneralFiniteDifference(
        function,
        x,
        differentiation_order,
        formula_accuracy,
        direction="central",
    )









.. GENERATED FROM PYTHON SOURCE LINES 84-93

Compute the absolute error of approximate derivative i.e.
:math:`\left|f^{(d)}(x) - \widetilde{f}^{(d)}(x)\right|`
where :math:`f^{(d)}(x)` is the exact order :math:`d` derivative
of the function :math:`f` and :math:`\widetilde{f}^{(d)}(x)`
is the approximation from the finite difference formula.
Moreover, compute the absolute value of the remainder of the
Taylor expansion, i.e. :math:`|R(x; h)|`.
This requires to compute the constant :math:`b_{d + p}`,
which is performed by :meth:`~numericalderivative.GeneralFiniteDifference.compute_b_constant`.

.. GENERATED FROM PYTHON SOURCE LINES 95-115

.. code-block:: Python

    first_derivative_exact = first_derivative(x)
    b_constant = formula.compute_b_constant()
    scaled_b_parameter = (
        b_constant
        * math.factorial(differentiation_order)
        / math.factorial(differentiation_order + formula_accuracy)
    )
    number_of_steps = 50
    step_array = np.logspace(-15, 1, number_of_steps)
    abs_error_array = np.zeros((number_of_steps))
    remainder_array = np.zeros((number_of_steps))
    for i in range(number_of_steps):
        step = step_array[i]
        derivative_approx = formula.compute(step)
        abs_error_array[i] = abs(derivative_approx - first_derivative_exact)
        remainder_array[i] = (
            abs(scaled_b_parameter * third_derivative(x)) * step**formula_accuracy
        )









.. GENERATED FROM PYTHON SOURCE LINES 116-129

.. code-block:: Python

    pl.figure()
    pl.title(
        f"Derivative of {name} at x={x}, "
        f"d={differentiation_order}, p={formula_accuracy}"
    )
    pl.plot(step_array, abs_error_array, "o--", label="Abs. error")
    pl.plot(step_array, remainder_array, "^:", label="Abs. remainder")
    pl.xlabel("Step h")
    pl.ylabel("Absolute error")
    pl.xscale("log")
    pl.yscale("log")
    _ = pl.legend()




.. image-sg:: /auto_example/images/sphx_glr_plot_general_fd_convergence_001.png
   :alt: Derivative of sin at x=1.0, d=1, p=2
   :srcset: /auto_example/images/sphx_glr_plot_general_fd_convergence_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 130-134

We see that there is a good agreement between the model and the
actual error when the step size is sufficiently large.
When the step is close to zero however, the rounding errors increase the
absolute error and the model does not fit anymore.


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

   **Total running time of the script:** (0 minutes 0.210 seconds)


.. _sphx_glr_download_auto_example_plot_general_fd_convergence.py:

.. only:: html

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

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

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

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

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

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

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