.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/indicators/plot_b01expressions.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_indicators_plot_b01expressions.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_indicators_plot_b01expressions.py:


Examples of mathematical expressions
====================================

Example of manipulating mathematical expressions and calculation of
derivatives.

:author: Michel Bierlaire, EPFL
:date: Wed Apr 12 21:06:21 2023

.. GENERATED FROM PYTHON SOURCE LINES 13-30

.. code-block:: default

    import numpy as np

    try:
        import matplotlib.pyplot as plt

        can_plot = True
    except ModuleNotFoundError:
        can_plot = False


    from biogeme.expressions import Beta, exp

    # ##
    # We create a simple expression:
    b = Beta('b', 1, None, None, 0)
    expression = exp(-b * b + 1)








.. GENERATED FROM PYTHON SOURCE LINES 31-34

We can calculate its value. Note that, as the expression is
calculated out of Biogeme, the IDs must be prepared. So the
parameter 'prepareIds' is set to True

.. GENERATED FROM PYTHON SOURCE LINES 34-37

.. code-block:: default

    z = expression.getValue_c(prepareIds=True)
    print(f'exp(-b * b + 1) = {z}')





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

 .. code-block:: none

    exp(-b * b + 1) = 1.0




.. GENERATED FROM PYTHON SOURCE LINES 38-41

We can also calculate the value, the first derivative, the second
derivative, and the BHHH, which in this case is the square of the
first derivatives

.. GENERATED FROM PYTHON SOURCE LINES 41-42

.. code-block:: default

    f, g, h, bhhh = expression.getValueAndDerivatives(prepareIds=True)







.. GENERATED FROM PYTHON SOURCE LINES 43-44

.. code-block:: default

    print(f'f = {f}')




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

 .. code-block:: none

    f = 1.0




.. GENERATED FROM PYTHON SOURCE LINES 45-46

.. code-block:: default

    print(f'g = {g}')




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

 .. code-block:: none

    g = [-2.]




.. GENERATED FROM PYTHON SOURCE LINES 47-48

.. code-block:: default

    print(f'h = {h}')




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

 .. code-block:: none

    h = [[2.]]




.. GENERATED FROM PYTHON SOURCE LINES 49-51

.. code-block:: default

    print(f'BHHH = {bhhh}')





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

 .. code-block:: none

    BHHH = [[4.]]




.. GENERATED FROM PYTHON SOURCE LINES 52-55

From the expression, we can create a Python function that takes as
argument the value of the free parameters, and returns the function,
the first and second derivatives.

.. GENERATED FROM PYTHON SOURCE LINES 55-57

.. code-block:: default

    fct = expression.createFunction()








.. GENERATED FROM PYTHON SOURCE LINES 58-59

We can use the function for different values of the parameter

.. GENERATED FROM PYTHON SOURCE LINES 59-65

.. code-block:: default

    beta = 2
    f, g, h = fct(beta)
    print(f'f({beta}) = {f}')
    print(f'g({beta}) = {g}')
    print(f'h({beta}) = {h}')





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

 .. code-block:: none

    f(2) = 0.049787068367863944
    g(2) = [-0.19914827]
    h(2) = [[0.69701896]]




.. GENERATED FROM PYTHON SOURCE LINES 66-72

.. code-block:: default

    beta = 3
    f, g, h = fct(beta)
    print(f'f({beta}) = {f}')
    print(f'g({beta}) = {g}')
    print(f'h({beta}) = {h}')





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

 .. code-block:: none

    f(3) = 0.00033546262790251185
    g(3) = [-0.00201278]
    h(3) = [[0.01140573]]




.. GENERATED FROM PYTHON SOURCE LINES 73-95

.. code-block:: default

    if can_plot:
        # We can also use it to plot the function and its derivatives
        x = np.arange(-2, 2, 0.01)

        # The value of the function is element [0].
        f = [fct(xx)[0] for xx in x]

        # The gradient is element [1]. As it contains only one entry [0],
        # we convert it into float.
        g = [float(fct([xx])[1][0]) for xx in x]

        # The hessian is element [2]. As it contains only one entry
        # [0][0], we convert it into float.
        h = [float(fct([xx])[2][0][0]) for xx in x]

        ax = plt.gca()
        ax.plot(x, f, label="f(x)")
        ax.plot(x, g, label="f'(x)")
        ax.plot(x, h, label='f"(x)')
        ax.legend()

        plt.show()



.. image-sg:: /auto_examples/indicators/images/sphx_glr_plot_b01expressions_001.png
   :alt: plot b01expressions
   :srcset: /auto_examples/indicators/images/sphx_glr_plot_b01expressions_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.081 seconds)


.. _sphx_glr_download_auto_examples_indicators_plot_b01expressions.py:

.. only:: html

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




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

      :download:`Download Python source code: plot_b01expressions.py <plot_b01expressions.py>`

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

      :download:`Download Jupyter notebook: plot_b01expressions.ipynb <plot_b01expressions.ipynb>`


.. only:: html

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

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