.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_gaussian_quadrature.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_plot_gaussian_quadrature.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_plot_gaussian_quadrature.py:


Univariate Gaussian Quadrature
------------------------------
Univariate Gaussian quadrature can be used to efficiently integrate
smooth one-dimensional functions. While numpy supports Hermite and Legendre
Gaussian qurature, Pyapprox can generate Gaussian quadrature rules for
any continouous random variable implemented in scipy.stats.

To generate a univariate quadrature rule for uniform random variables

.. GENERATED FROM PYTHON SOURCE LINES 11-20

.. code-block:: default

    from pyapprox import surrogates
    from scipy import stats
    degree = 10

    scipy_var = stats.uniform(-1, 2)
    quad_rule = surrogates.get_gauss_quadrature_rule_from_marginal(
        scipy_var, degree+1)
    x_quad, w_quad = quad_rule(5)








.. GENERATED FROM PYTHON SOURCE LINES 21-23

As an example, we can use this quadrature rule to integrate :math:`\rv^2`
with repsect to the uniform PDF on [-1, 1], i.e. 1/2

.. GENERATED FROM PYTHON SOURCE LINES 23-27

.. code-block:: default

    values = x_quad**2
    integral = values.dot(w_quad)
    print(integral)





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

 .. code-block:: none

    0.3333333333333332




.. GENERATED FROM PYTHON SOURCE LINES 28-31

The quadrature recovers the integral value of 1/3 to machine precision. Note
unlike :class:`~numpy.polynomial.legendre.leggauss` we are integrating with
resepect to the uniform PDF.

.. GENERATED FROM PYTHON SOURCE LINES 33-35

The function is also capable of generating rules on different intervals
For example

.. GENERATED FROM PYTHON SOURCE LINES 35-42

.. code-block:: default

    scipy_var = stats.uniform(0, 2)
    x_quad, w_quad = surrogates.get_gauss_quadrature_rule_from_marginal(
        scipy_var, degree+1)(degree)
    values = x_quad**2
    integral = values.dot(w_quad)
    print(integral)





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

 .. code-block:: none

    1.3333333333333346




.. GENERATED FROM PYTHON SOURCE LINES 43-45

Quadrature rules can be created for almost any random variable. Here
we will generate a quadrature rule for an exponential random variable

.. GENERATED FROM PYTHON SOURCE LINES 45-52

.. code-block:: default

    scipy_var = stats.expon()
    x_quad, w_quad = surrogates.get_gauss_quadrature_rule_from_marginal(
        scipy_var, degree+1)(degree)
    values = x_quad**2
    integral = values.dot(w_quad)
    print(integral)





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

 .. code-block:: none

    2.000000000000001




.. GENERATED FROM PYTHON SOURCE LINES 53-55

For interest, we plot the quadrature rule against the PDF of the exponential
variable

.. GENERATED FROM PYTHON SOURCE LINES 55-64

.. code-block:: default

    import numpy as np
    import matplotlib.pyplot as plt
    from pyapprox.analysis import visualize
    from pyapprox.variables import marginals
    visualize.plot_discrete_measure_1d(x_quad, w_quad)
    vrange = marginals.get_truncated_range(scipy_var, 1-1e-6)
    xx = np.linspace(vrange[0], vrange[1], 101)
    plt.fill_between(xx, 0*xx, scipy_var.pdf(xx), alpha=0.3)
    plt.show()



.. image-sg:: /auto_examples/images/sphx_glr_plot_gaussian_quadrature_001.png
   :alt: plot gaussian quadrature
   :srcset: /auto_examples/images/sphx_glr_plot_gaussian_quadrature_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.098 seconds)


.. _sphx_glr_download_auto_examples_plot_gaussian_quadrature.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_gaussian_quadrature.py <plot_gaussian_quadrature.py>`

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

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


.. only:: html

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

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