.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_numerical_methods_general_methods_plot_estimate_integral_iterated_quadrature.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_numerical_methods_general_methods_plot_estimate_integral_iterated_quadrature.py:


Estimate an integral
====================

In this example we are going to evaluate an integral of the form.

.. math::
   I_f = \int_{a}^{b}\, \int_{l_1(x_0)}^{u_1(x_0)}\, \int_{l_2(x_0, x_1)}^{u_2(x_0,x_1)}\, \int_{l_{n-1}(x_0, \dots, x_{n-2})}^{u_{n-1}(x_0, \dots, x_{n-2})} \, f(x_0, \dots, x_{n-1})\mathrm{d}{x_{n-1}}\dots\mathrm{d}{x_0}

with the iterated quadrature algorithm.


.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    import math as m
    ot.Log.Show(ot.Log.NONE)








define the integrand and the bounds


.. code-block:: default

    a = -m.pi
    b = m.pi
    f = ot.SymbolicFunction(['x', 'y'], ['1+cos(x)*sin(y)'])
    l = [ot.SymbolicFunction(['x'], [' 2+cos(x)'])]
    u = [ot.SymbolicFunction(['x'], ['-2-cos(x)'])]








Draw the graph of the integrand and the bounds


.. code-block:: default

    g = ot.Graph('Integration nodes', 'x', 'y', True, 'topright')
    g.add(f.draw([a,a],[b,b]))
    curve = l[0].draw(a, b).getDrawable(0)
    curve.setLineWidth(2)
    curve.setColor('red')
    g.add(curve)
    curve = u[0].draw(a, b).getDrawable(0)
    curve.setLineWidth(2)
    curve.setColor('red')
    g.add(curve)
    view = viewer.View(g)




.. image:: /auto_numerical_methods/general_methods/images/sphx_glr_plot_estimate_integral_iterated_quadrature_001.png
    :alt: Integration nodes
    :class: sphx-glr-single-img





compute the integral value


.. code-block:: default

    I2 = ot.IteratedQuadrature().integrate(f, a, b, l, u)
    print(I2)
    plt.show()




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

 Out:

 .. code-block:: none

    [-25.1327]





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

   **Total running time of the script:** ( 0 minutes  0.133 seconds)


.. _sphx_glr_download_auto_numerical_methods_general_methods_plot_estimate_integral_iterated_quadrature.py:


.. only :: html

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



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

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



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

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


.. only:: html

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

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