.. _sphx_glr_api_gallery_numerical_analysis_interpolation:

Interpolation
-------------

Lagrange and Newton divided-difference polynomial interpolation: the same
unique interpolating polynomial, built and evaluated two different ways.


.. raw:: html

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


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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="There is exactly one polynomial of degree \leq n through n+1 distinct points, so LagrangeInterpolant and NewtonDividedDifference -- built from completely different representations (a sum of basis polynomials vs. nested divided differences) -- must produce identical curves. Both are also exact for any polynomial up to the interpolation degree, which this script verifies directly.">

.. only:: html

  .. image:: /api/gallery/numerical_analysis/interpolation/images/thumb/sphx_glr_plot_01_lagrange_newton_thumb.png
    :alt:

  :doc:`/api/gallery/numerical_analysis/interpolation/plot_01_lagrange_newton`

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      <div class="sphx-glr-thumbnail-title">Lagrange vs. Newton divided-difference interpolation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Charles Hermite&#x27;s 1878 interpolant matches both f(x_i) and f&#x27;(x_i), so n + 1 nodes fix a polynomial of degree 2n + 1. On the same few nodes it follows \sin far more closely than the Lagrange interpolant, which knows only the values.">

.. only:: html

  .. image:: /api/gallery/numerical_analysis/interpolation/images/thumb/sphx_glr_plot_02_hermite_interpolation_thumb.png
    :alt:

  :doc:`/api/gallery/numerical_analysis/interpolation/plot_02_hermite_interpolation`

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      <div class="sphx-glr-thumbnail-title">Hermite interpolation: matching slopes as well as values</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/numerical_analysis/interpolation/plot_01_lagrange_newton
   /api/gallery/numerical_analysis/interpolation/plot_02_hermite_interpolation