:orphan:

Examples
========

This gallery walks through every public feature of ``mathematicskit.calculus``:
finite-difference differentiation and Richardson extrapolation,
quadrature rules, forward- and reverse-mode automatic differentiation,
and Taylor series.

Each script in this gallery is self-contained and can be run directly with
``python examples/calculus/<section>/<script>.py``.

Sections
--------

- **exhaustion** -- Archimedes' polygon bounds on pi.
- **newton_leibniz** -- the derivative and the integral as limits of
  difference quotients and sums.
- **finite_differences** -- Richardson extrapolation, Fermat's
  adequality, and the complex-step derivative.
- **quadrature** -- Riemann sums, Simpson's rule, Gauss-Legendre,
  Euler-Maclaurin, Romberg, Clenshaw-Curtis, tanh-sinh, and adaptive
  (QUADPACK) quadrature.
- **dual_numbers** -- forward-mode automatic differentiation with no
  finite-difference truncation error.
- **autodiff** -- reverse-mode (backpropagation-style) automatic
  differentiation for multivariable gradients.
- **taylor_series** -- Taylor's theorem: Taylor polynomials, the
  Lagrange remainder bound, and radius of convergence.


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Reverse-mode autodiff
---------------------

A small backpropagation-style engine for multivariable gradients.


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    <div class="sphx-glr-thumbcontainer" tooltip="A minimal backpropagation-style engine: one forward pass builds a computation graph, then one backward pass computes the gradient with respect to every input in a single traversal -- the technique underlying modern deep-learning frameworks, applied here to a small analytic function.">

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  .. image:: /api/gallery/calculus/autodiff/images/thumb/sphx_glr_plot_01_reverse_mode_gradients_thumb.png
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  :doc:`/api/gallery/calculus/autodiff/plot_01_reverse_mode_gradients`

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      <div class="sphx-glr-thumbnail-title">Reverse-mode automatic differentiation for multivariable gradients</div>
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Dual numbers
------------

Forward-mode automatic differentiation: exact derivatives with no
finite-difference truncation error.


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    <div class="sphx-glr-thumbcontainer" tooltip="Dual numbers compute a function&#x27;s value and its exact derivative together, with no finite-difference step-size tradeoff between truncation error (large h) and floating-point cancellation (small h).">

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  .. image:: /api/gallery/calculus/dual_numbers/images/thumb/sphx_glr_plot_01_forward_mode_autodiff_thumb.png
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  :doc:`/api/gallery/calculus/dual_numbers/plot_01_forward_mode_autodiff`

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      <div class="sphx-glr-thumbnail-title">Forward-mode automatic differentiation via dual numbers</div>
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Method of exhaustion
--------------------

Archimedes' polygon bounds on pi, the oldest convergent approximation
scheme.


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    <div class="sphx-glr-thumbcontainer" tooltip="Doubles the sides of inscribed and circumscribed regular polygons from 6 to 96, reproducing Archimedes&#x27; bounds 3 10/71 &lt; pi &lt; 3 1/7, and shows that the gap shrinks by a factor of four with each doubling.">

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  .. image:: /api/gallery/calculus/exhaustion/images/thumb/sphx_glr_plot_01_archimedes_pi_thumb.png
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  :doc:`/api/gallery/calculus/exhaustion/plot_01_archimedes_pi`

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      <div class="sphx-glr-thumbnail-title">Archimedes' method of exhaustion: squeezing pi</div>
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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>
    </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>
    </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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Newton and Leibniz
------------------

The two central facts of the calculus of Newton and Leibniz: the
derivative is a limiting difference quotient, and the definite integral
is a limiting sum.


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    <div class="sphx-glr-thumbcontainer" tooltip="The calculus of Newton and Leibniz rests on two limits: the derivative dy/dx is the limit of a difference quotient, and the integral \int f\,dx is the limit of a sum of thin strips. This example watches secant slopes (via central_difference) close in on the tangent slope, and trapezoidal sums (via TrapezoidalRule) close in on the area under the curve -- and then checks that the two operations undo each other, as the fundamental theorem of calculus says they must.">

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  .. image:: /api/gallery/calculus/newton_leibniz/images/thumb/sphx_glr_plot_01_derivative_and_integral_as_limits_thumb.png
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  :doc:`/api/gallery/calculus/newton_leibniz/plot_01_derivative_and_integral_as_limits`

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      <div class="sphx-glr-thumbnail-title">Newton and Leibniz: the derivative and the integral as limits</div>
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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 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
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  :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>
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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
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  :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>
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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
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  :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>
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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
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  :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>
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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
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  :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>
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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
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  :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>
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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
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  :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>
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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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      <div class="sphx-glr-thumbnail-title">Adaptive quadrature (QUADPACK): subdividing where it is needed</div>
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Taylor series
-------------

Maclaurin series for standard functions, their convergence, and their
radius of convergence.


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    <div class="sphx-glr-thumbcontainer" tooltip="Taylor&#x27;s theorem expands a function in powers of x - x_0 built from its derivatives at one point, with a Lagrange remainder |R_n(x)| \le M|x-x_0|^{n+1}/(n+1)!. This example draws the Taylor (Maclaurin, x_0 = 0) polynomials of \sin closing in on the function, checks the true error against taylor_remainder_bound, and shows what happens outside a series&#x27; radius of convergence, where adding terms no longer helps.">

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  .. image:: /api/gallery/calculus/taylor_series/images/thumb/sphx_glr_plot_01_maclaurin_series_thumb.png
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  :doc:`/api/gallery/calculus/taylor_series/plot_01_maclaurin_series`

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      <div class="sphx-glr-thumbnail-title">Taylor's theorem: polynomials, remainder bound, and convergence</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/calculus/autodiff/index.rst
   /api/gallery/calculus/dual_numbers/index.rst
   /api/gallery/calculus/exhaustion/index.rst
   /api/gallery/calculus/finite_differences/index.rst
   /api/gallery/calculus/newton_leibniz/index.rst
   /api/gallery/calculus/quadrature/index.rst
   /api/gallery/calculus/taylor_series/index.rst



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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_