.. _sphx_glr_api_gallery_numerical_analysis_chebyshev:

Runge's phenomenon and Chebyshev nodes
--------------------------------------

Why node placement matters: equally spaced interpolation nodes can make
the error diverge as the polynomial degree grows (Runge's 1901 example),
while Chebyshev nodes keep the error shrinking.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Carl Runge&#x27;s 1901 example f(x) = 1/(1+25x^2) is smooth on [-1, 1], yet interpolating it at more equally spaced nodes makes the fit worse: oscillations near the endpoints grow without bound as the degree increases. This script shows the overshoot for growing degree, the maximum error blowing up, and the exponentially growing Lebesgue constant of equally spaced nodes that causes it.">

.. only:: html

  .. image:: /api/gallery/numerical_analysis/chebyshev/images/thumb/sphx_glr_plot_01_runge_phenomenon_thumb.png
    :alt:

  :doc:`/api/gallery/numerical_analysis/chebyshev/plot_01_runge_phenomenon`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Runge's phenomenon: equally spaced interpolation diverges</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Chebyshev&#x27;s least-deviation polynomials suggest placing interpolation nodes at the extrema of T_n(x) = \cos(n \arccos x), which cluster toward the ends of [-1, 1]. At these nodes the Lebesgue constant grows only logarithmically, so the interpolant of Runge&#x27;s function 1/(1+25x^2) converges as the degree rises. This script shows the node clustering, the degree-20 Chebyshev interpolant, and its error shrinking geometrically.">

.. only:: html

  .. image:: /api/gallery/numerical_analysis/chebyshev/images/thumb/sphx_glr_plot_02_chebyshev_nodes_thumb.png
    :alt:

  :doc:`/api/gallery/numerical_analysis/chebyshev/plot_02_chebyshev_nodes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Chebyshev nodes: the cure for Runge's phenomenon</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/numerical_analysis/chebyshev/plot_01_runge_phenomenon
   /api/gallery/numerical_analysis/chebyshev/plot_02_chebyshev_nodes