.. _sphx_glr_api_gallery_numerical_analysis_approximation:

Function approximation
----------------------

Approximating a function over a whole interval: Bernstein polynomials and
the Weierstrass theorem, Padé rational approximants, and Remez's best
uniform (minimax) polynomials.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Weierstrass proved in 1885 that every continuous function on a closed interval is a uniform limit of polynomials. Sergei Bernstein&#x27;s 1912 proof builds the polynomials explicitly: average f(k/n) with binomial weights. This script approximates a function with a kink, |x - 1/2|, and shows the uniform error shrinking, slowly, as the degree grows.">

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  .. image:: /api/gallery/numerical_analysis/approximation/images/thumb/sphx_glr_plot_01_bernstein_polynomials_thumb.png
    :alt:

  :doc:`/api/gallery/numerical_analysis/approximation/plot_01_bernstein_polynomials`

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      <div class="sphx-glr-thumbnail-title">Bernstein polynomials and the Weierstrass theorem</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Henri Padé&#x27;s 1892 thesis organized rational approximations p(x)/q(x) that match a function&#x27;s Taylor series as far as possible. The Taylor series of \log(1 + x) diverges for x &gt; 1, but the Padé approximants built from the same coefficients keep converging well beyond it.">

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  .. image:: /api/gallery/numerical_analysis/approximation/images/thumb/sphx_glr_plot_02_pade_approximants_thumb.png
    :alt:

  :doc:`/api/gallery/numerical_analysis/approximation/plot_02_pade_approximants`

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      <div class="sphx-glr-thumbnail-title">Padé approximants beyond the radius of convergence</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Chebyshev showed that the polynomial of degree n closest to f in the maximum norm is characterized by an error curve that reaches its peak magnitude n + 2 times with alternating sign. Evgeny Remez&#x27;s 1934 exchange algorithm finds it by repeatedly levelling the error on a set of reference points. This script computes the best degree-6 approximation to e^x \sin(3x) on [-1, 1] and compares its error with Chebyshev interpolation of the same degree.">

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  .. image:: /api/gallery/numerical_analysis/approximation/images/thumb/sphx_glr_plot_03_remez_minimax_thumb.png
    :alt:

  :doc:`/api/gallery/numerical_analysis/approximation/plot_03_remez_minimax`

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      <div class="sphx-glr-thumbnail-title">Remez's algorithm and best uniform approximation</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/numerical_analysis/approximation/plot_01_bernstein_polynomials
   /api/gallery/numerical_analysis/approximation/plot_02_pade_approximants
   /api/gallery/numerical_analysis/approximation/plot_03_remez_minimax