.. _sphx_glr_api_gallery_calculus_quadrature:

Quadrature
----------

Riemann sums, Simpson's rule, Gauss-Legendre, Euler-Maclaurin, Romberg,
Clenshaw-Curtis, tanh-sinh, and QUADPACK-style adaptive quadrature.


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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="Gauss let a quadrature rule choose its sample points as well as its weights. The optimal n nodes turn out to be the roots of the Legendre polynomial P_n, and the resulting rule integrates every polynomial of degree up to 2n-1 exactly -- about twice the degree an equally spaced rule with the same number of points manages. This example shows the nodes and weights from legendre_nodes_and_weights, checks the degree of exactness of GaussianQuadrature, and compares its convergence with the equally spaced trapezoidal rule.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_01_gauss_legendre_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_01_gauss_legendre`

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      <div class="sphx-glr-thumbnail-title">Gaussian quadrature: optimal nodes are exact to degree 2n-1</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Simpson&#x27;s rule integrates cubics exactly and converges at fourth order. This example compares it with the trapezoidal rule on a smooth integrand and shows the parabolic arcs it fits.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_02_simpson_parabolas_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_02_simpson_parabolas`

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      <div class="sphx-glr-thumbnail-title">Simpson's rule: parabolas through pairs of panels</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Draws left, right, and midpoint Riemann sums for the same integrand and shows their convergence: first order for the endpoint rules, second order for the midpoint rule.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_03_riemann_sums_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_03_riemann_sums`

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      <div class="sphx-glr-thumbnail-title">Riemann sums: the definition of the integral</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Adds Euler-Maclaurin endpoint corrections, built from Bernoulli numbers and odd derivatives at the endpoints, to the trapezoidal rule. Each extra term raises the convergence order by two.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_04_euler_maclaurin_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_04_euler_maclaurin`

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      <div class="sphx-glr-thumbnail-title">Euler-Maclaurin: correcting the trapezoidal rule</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Prints the Romberg table for the integral of 4/(1 + x^2) over [0, 1], which equals pi. Each column applies one more Richardson extrapolation to the halved trapezoidal rules in the first column.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_05_romberg_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_05_romberg`

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      <div class="sphx-glr-thumbnail-title">Romberg integration: extrapolating the trapezoidal rule</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Compares Clenshaw-Curtis quadrature, which integrates the Chebyshev interpolant, with Gauss-Legendre quadrature on smooth integrands. The two converge at nearly the same rate.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_06_clenshaw_curtis_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_06_clenshaw_curtis`

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      <div class="sphx-glr-thumbnail-title">Clenshaw-Curtis vs. Gauss-Legendre quadrature</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Integrates 1/sqrt(x) and log(x) over [0, 1], whose integrands blow up at 0. The tanh-sinh substitution crowds nodes toward the endpoints with double-exponentially small weights, and converges far faster than Gauss-Legendre quadrature on these integrands.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_07_tanh_sinh_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_07_tanh_sinh`

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      <div class="sphx-glr-thumbnail-title">Tanh-sinh quadrature: integrating endpoint singularities</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="QUADPACK&#x27;s adaptive strategy estimates each subinterval&#x27;s integral two ways and splits only those subintervals where the estimates disagree. AdaptiveQuadrature wraps scipy.integrate.quad (QUADPACK&#x27;s QAGS). Recording every point at which it samples a sharply peaked integrand shows the evaluations crowding around the peak, and a fixed equally spaced rule needs far more evaluations to match its accuracy.">

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  .. image:: /api/gallery/calculus/quadrature/images/thumb/sphx_glr_plot_08_adaptive_quadpack_thumb.png
    :alt:

  :doc:`/api/gallery/calculus/quadrature/plot_08_adaptive_quadpack`

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      <div class="sphx-glr-thumbnail-title">Adaptive quadrature (QUADPACK): subdividing where it is needed</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/calculus/quadrature/plot_01_gauss_legendre
   /api/gallery/calculus/quadrature/plot_02_simpson_parabolas
   /api/gallery/calculus/quadrature/plot_03_riemann_sums
   /api/gallery/calculus/quadrature/plot_04_euler_maclaurin
   /api/gallery/calculus/quadrature/plot_05_romberg
   /api/gallery/calculus/quadrature/plot_06_clenshaw_curtis
   /api/gallery/calculus/quadrature/plot_07_tanh_sinh
   /api/gallery/calculus/quadrature/plot_08_adaptive_quadpack