.. _sphx_glr_api_gallery_calculus_finite_differences:

Finite differences
-------------------

Richardson extrapolation, Fermat's adequality, and the complex-step
derivative.


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    <div class="sphx-glr-thumbcontainer" tooltip="Richardson&#x27;s idea: if an estimate D(h) has error c_1 h^2 + c_2 h^4 + \dots, then combining D(h) and D(h/2) as (4D(h/2) - D(h))/3 cancels the h^2 term. Repeating the combination on successively halved steps removes one error order per level. This example applies richardson_extrapolation to a central difference and shows it reaching near machine precision from a coarse starting step, where no plain difference quotient can.">

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  .. image:: /api/gallery/calculus/finite_differences/images/thumb/sphx_glr_plot_01_richardson_extrapolation_thumb.png
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  :doc:`/api/gallery/calculus/finite_differences/plot_01_richardson_extrapolation`

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      <div class="sphx-glr-thumbnail-title">Richardson extrapolation: cancelling error terms at a fixed step</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Fermat found the maximum of a function by setting a difference quotient (f(x+h) - f(x))/h equal to zero and then letting h vanish. This example locates the maximum of Fermat&#x27;s own problem, the largest product x(a - x), and shows the O(h) error of the forward difference.">

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  .. image:: /api/gallery/calculus/finite_differences/images/thumb/sphx_glr_plot_02_fermat_adequality_thumb.png
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  :doc:`/api/gallery/calculus/finite_differences/plot_02_fermat_adequality`

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      <div class="sphx-glr-thumbnail-title">Fermat's adequality: maxima from a difference quotient</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Compares the central difference with the complex-step derivative Im f(x + ih)/h as the step shrinks. The central difference is ruined by subtractive cancellation, while the complex step stays accurate to machine precision at any tiny step.">

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  .. image:: /api/gallery/calculus/finite_differences/images/thumb/sphx_glr_plot_03_complex_step_thumb.png
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  :doc:`/api/gallery/calculus/finite_differences/plot_03_complex_step`

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      <div class="sphx-glr-thumbnail-title">The complex-step derivative: no cancellation</div>
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.. toctree::
   :hidden:

   /api/gallery/calculus/finite_differences/plot_01_richardson_extrapolation
   /api/gallery/calculus/finite_differences/plot_02_fermat_adequality
   /api/gallery/calculus/finite_differences/plot_03_complex_step