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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_barycentric_interpolation.py"
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.. only:: html

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

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

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

.. _sphx_glr_auto_examples_plot_barycentric_interpolation.py:


Multivariate Lagrange Interpolation
-----------------------------------
For smooth function Lagrange polynomials can be used to
interpolate univariate functions evaluated on a tensor-product grid.
Pyapprox uses the Barycentric formulation of Lagrange interpolation
which is more efficient and stable than traditional Lagrange interpolation.

We must define the univariate grids that we will use to construct the
tensor product grid. While technically Lagrange interpolation can be used with
any 1D grids, it is better to use points well suited to polynomial
interpolation. Here we use the samples of a Gaussian quadrature rule.

.. GENERATED FROM PYTHON SOURCE LINES 14-29

.. code-block:: default

    from scipy import stats
    import numpy as np
    import matplotlib.pyplot as plt
    from functools import partial
    from pyapprox import surrogates
    from pyapprox.analysis import visualize
    from pyapprox import variables
    from pyapprox import util

    degree = 10
    marginals = [stats.uniform(-1, 2), stats.uniform(-1, 2)]
    grid_samples_1d = [surrogates.get_gauss_quadrature_rule_from_marginal(
        rv, degree+1)(degree+1)[0] for rv in marginals]









.. GENERATED FROM PYTHON SOURCE LINES 30-32

Now lets define the function we want to interpolate, e.g.
:math:`f(\rv)=\rv_1^2+\rv_2^2`

.. GENERATED FROM PYTHON SOURCE LINES 32-52

.. code-block:: default

    def fun(samples):
        return np.sum(samples**2, axis=0)[:, None]


    ##%
    #Now we will use partial to create a callable function that just takes
    #the samples at which we want to evaluate the interpolant
    #This function will evaluate fun on a tensor product grid internally
    interp_fun = partial(
        surrogates.tensor_product_barycentric_lagrange_interpolation,
        grid_samples_1d, fun)
    variable = variables.IndependentMarginalsVariable(marginals)
    X, Y, Z = visualize.get_meshgrid_function_data_from_variable(
        interp_fun, variable, 50)
    fig, ax = plt.subplots(1, 1, figsize=(8, 6))
    ax.contourf(X, Y, Z, levels=np.linspace(Z.min(), Z.max(), 20))
    util.plot_2d_samples(
        util.cartesian_product(grid_samples_1d), ax, marker='o', c='r')
    plt.show()




.. image-sg:: /auto_examples/images/sphx_glr_plot_barycentric_interpolation_001.png
   :alt: plot barycentric interpolation
   :srcset: /auto_examples/images/sphx_glr_plot_barycentric_interpolation_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 53-56

Barycentric interpolation can be used for any number of variables. However,
the number of evaluations of the target function grows exponentially with
the number of variables


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

   **Total running time of the script:** ( 0 minutes  0.045 seconds)


.. _sphx_glr_download_auto_examples_plot_barycentric_interpolation.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_barycentric_interpolation.py <plot_barycentric_interpolation.py>`

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

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


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

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