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- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.trapezoidal"></a><a class="link" href="trapezoidal.html" title="Trapezoidal Quadrature">Trapezoidal Quadrature</a>
- </h2></div></div></div>
- <h4>
- <a name="math_toolkit.trapezoidal.h0"></a>
- <span class="phrase"><a name="math_toolkit.trapezoidal.synopsis"></a></span><a class="link" href="trapezoidal.html#math_toolkit.trapezoidal.synopsis">Synopsis</a>
- </h4>
- <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">quadrature</span><span class="special">/</span><span class="identifier">trapezoidal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
- <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">quadrature</span> <span class="special">{</span>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
- <span class="keyword">auto</span> <span class="identifier">trapezoidal</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span>
- <span class="identifier">Real</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()),</span>
- <span class="identifier">size_t</span> <span class="identifier">max_refinements</span> <span class="special">=</span> <span class="number">12</span><span class="special">,</span>
- <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">error_estimate</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">,</span>
- <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">L1</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <span class="keyword">auto</span> <span class="identifier">trapezoidal</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">tol</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">max_refinements</span><span class="special">,</span>
- <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">error_estimate</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">L1</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span>
- <span class="special">}}}</span> <span class="comment">// namespaces</span>
- </pre>
- <h4>
- <a name="math_toolkit.trapezoidal.h1"></a>
- <span class="phrase"><a name="math_toolkit.trapezoidal.description"></a></span><a class="link" href="trapezoidal.html#math_toolkit.trapezoidal.description">Description</a>
- </h4>
- <p>
- The functional <code class="computeroutput"><span class="identifier">trapezoidal</span></code>
- calculates the integral of a function <span class="emphasis"><em>f</em></span> using the surprisingly
- simple trapezoidal rule. If we assume only that the integrand is twice continuously
- differentiable, we can prove that the error of the composite trapezoidal rule
- is 𝑶(h<sup>2</sup>). Hence halving the interval only cuts the error by about a fourth,
- which in turn implies that we must evaluate the function many times before
- an acceptable accuracy can be achieved.
- </p>
- <p>
- However, the trapezoidal rule has an astonishing property: If the integrand
- is periodic, and we integrate it over a period, then the trapezoidal rule converges
- faster than any power of the step size <span class="emphasis"><em>h</em></span>. This can be
- seen by examination of the <a href="https://en.wikipedia.org/wiki/Euler-Maclaurin_formula" target="_top">Euler-Maclaurin
- summation formula</a>, which relates a definite integral to its trapezoidal
- sum and error terms proportional to the derivatives of the function at the
- endpoints and the Bernoulli numbers. If the derivatives at the endpoints are
- the same or vanish, then the error very nearly vanishes. Hence the trapezoidal
- rule is essentially optimal for periodic integrands.
- </p>
- <p>
- Other classes of integrands which are integrated efficiently by this method
- are the C<sub>0</sub><sup>∞</sup>(∝) <a href="https://en.wikipedia.org/wiki/Bump_function" target="_top">bump
- functions</a> and bell-shaped integrals over the infinite interval. For
- details, see <a href="http://epubs.siam.org/doi/pdf/10.1137/130932132" target="_top">Trefethen's</a>
- SIAM review.
- </p>
- <p>
- In its simplest form, an integration can be performed by the following code
- </p>
- <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quadrature</span><span class="special">::</span><span class="identifier">trapezoidal</span><span class="special">;</span>
- <span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="number">1</span><span class="special">/(</span><span class="number">5</span> <span class="special">-</span> <span class="number">4</span><span class="special">*</span><span class="identifier">cos</span><span class="special">(</span><span class="identifier">x</span><span class="special">));</span> <span class="special">};</span>
- <span class="keyword">double</span> <span class="identifier">I</span> <span class="special">=</span> <span class="identifier">trapezoidal</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">0.0</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">two_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>());</span>
- </pre>
- <p>
- The integrand must accept a real number argument, but can return a complex
- number. This is useful for contour integrals (which are manifestly periodic)
- and high-order numerical differentiation of analytic functions. An example
- using the integral definition of the complex Bessel function is shown here:
- </p>
- <pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">bessel_integrand</span> <span class="special">=</span> <span class="special">[&</span><span class="identifier">n</span><span class="special">,</span> <span class="special">&</span><span class="identifier">z</span><span class="special">](</span><span class="keyword">double</span> <span class="identifier">theta</span><span class="special">)-></span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span>
- <span class="special">{</span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">z</span><span class="special">{</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">};</span>
- <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cos</span><span class="special">;</span>
- <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span><span class="special">;</span>
- <span class="keyword">return</span> <span class="identifier">cos</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">sin</span><span class="special">(</span><span class="identifier">theta</span><span class="special">)</span> <span class="special">-</span> <span class="number">2</span><span class="special">*</span><span class="identifier">theta</span><span class="special">)/</span><span class="identifier">pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>();</span>
- <span class="special">};</span>
- <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quadrature</span><span class="special">::</span><span class="identifier">trapezoidal</span><span class="special">;</span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">Jnz</span> <span class="special">=</span> <span class="identifier">trapezoidal</span><span class="special">(</span><span class="identifier">bessel_integrand</span><span class="special">,</span> <span class="number">0.0</span><span class="special">,</span> <span class="identifier">pi</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>());</span>
- </pre>
- <p>
- Other special functions which are efficiently evaluated in the complex plane
- by trapezoidal quadrature are modified Bessel functions and the complementary
- error function. Another application of complex-valued trapezoidal quadrature
- is computation of high-order numerical derivatives; see Lyness and Moler for
- details.
- </p>
- <p>
- Since the routine is adaptive, step sizes are halved continuously until a tolerance
- is reached. In order to control this tolerance, simply call the routine with
- an additional argument
- </p>
- <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">I</span> <span class="special">=</span> <span class="identifier">trapezoidal</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">0.0</span><span class="special">,</span> <span class="identifier">two_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> <span class="number">1e-6</span><span class="special">);</span>
- </pre>
- <p>
- The routine stops when successive estimates of the integral <code class="computeroutput"><span class="identifier">I1</span></code>
- and <code class="computeroutput"><span class="identifier">I0</span></code> differ by less than
- the tolerance multiplied by the estimated L<sub>1</sub> norm of the function. A good choice
- for the tolerance is √ε, which is the default. If the integrand is periodic,
- then the number of correct digits should double on each interval halving. Hence,
- once the integration routine has estimated that the error is √ε, then the actual
- error should be ~ε. If the integrand is <span class="bold"><strong>not</strong></span>
- periodic, then reducing the error to √ε takes much longer, but is nonetheless
- possible without becoming a major performance bug.
- </p>
- <p>
- A question arises as to what to do when successive estimates never pass below
- the tolerance threshold. The stepsize would be halved repeatedly, generating
- an exponential explosion in function evaluations. As such, you may pass an
- optional argument <code class="computeroutput"><span class="identifier">max_refinements</span></code>
- which controls how many times the interval may be halved before giving up.
- By default, this maximum number of refinement steps is 12, which should never
- be hit in double precision unless the function is not periodic. However, for
- higher-precision types, it may be of interest to allow the algorithm to compute
- more refinements:
- </p>
- <pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">max_refinements</span> <span class="special">=</span> <span class="number">15</span><span class="special">;</span>
- <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">I</span> <span class="special">=</span> <span class="identifier">trapezoidal</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">0.0L</span><span class="special">,</span> <span class="identifier">two_pi</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">>(),</span> <span class="number">1e-9L</span><span class="special">,</span> <span class="identifier">max_refinements</span><span class="special">);</span>
- </pre>
- <p>
- Note that the maximum allowed compute time grows exponentially with <code class="computeroutput"><span class="identifier">max_refinements</span></code>. The routine will not throw
- an exception if the maximum refinements is achieved without the requested tolerance
- being attained. This is because the value calculated is more often than not
- still usable. However, for applications with high-reliability requirements,
- the error estimate should be queried. This is achieved by passing additional
- pointers into the routine:
- </p>
- <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">error_estimate</span><span class="special">;</span>
- <span class="keyword">double</span> <span class="identifier">L1</span><span class="special">;</span>
- <span class="keyword">double</span> <span class="identifier">I</span> <span class="special">=</span> <span class="identifier">trapezoidal</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">0.0</span><span class="special">,</span> <span class="identifier">two_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">max_refinements</span><span class="special">,</span> <span class="special">&</span><span class="identifier">error_estimate</span><span class="special">,</span> <span class="special">&</span><span class="identifier">L1</span><span class="special">);</span>
- <span class="keyword">if</span> <span class="special">(</span><span class="identifier">error_estimate</span> <span class="special">></span> <span class="identifier">tolerance</span><span class="special">*</span><span class="identifier">L1</span><span class="special">)</span>
- <span class="special">{</span>
- <span class="keyword">double</span> <span class="identifier">I</span> <span class="special">=</span> <span class="identifier">some_other_quadrature_method</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">two_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>());</span>
- <span class="special">}</span>
- </pre>
- <p>
- The final argument is the L<sub>1</sub> norm of the integral. This is computed along with
- the integral, and is an essential component of the algorithm. First, the L<sub>1</sub> norm
- establishes a scale against which the error can be measured. Second, the L<sub>1</sub> norm
- can be used to evaluate the stability of the computation. This can be formulated
- in a rigorous manner by defining the <span class="bold"><strong>condition number
- of summation</strong></span>. The condition number of summation is defined by
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="serif_italic"><span class="emphasis"><em>κ(S<sub>n</sub>) := Σ<sub>i</sub><sup>n</sup> |x<sub>i</sub>|/|Σ<sub>i</sub><sup>n</sup> x<sub>i</sub>|</em></span></span>
- </p></blockquote></div>
- <p>
- If this number of ~10<sup>k</sup>, then <span class="emphasis"><em>k</em></span> additional digits are expected
- to be lost in addition to digits lost due to floating point rounding error.
- As all numerical quadrature methods reduce to summation, their stability is
- controlled by the ratio ∫ |f| dt/|∫ f dt |, which is easily seen
- to be equivalent to condition number of summation when evaluated numerically.
- Hence both the error estimate and the condition number of summation should
- be analyzed in applications requiring very high precision and reliability.
- </p>
- <p>
- As an example, we consider evaluation of Bessel functions by trapezoidal quadrature.
- The Bessel function of the first kind is defined via
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="serif_italic"><span class="emphasis"><em>J<sub>n</sub>(x) = 1/2Π ∫<sub>-Π</sub><sup>Π</sup> cos(n
- t - x sin(t)) dt</em></span></span>
- </p></blockquote></div>
- <p>
- The integrand is periodic, so the Euler-Maclaurin summation formula guarantees
- exponential convergence via the trapezoidal quadrature. Without careful consideration,
- it seems this would be a very attractive method to compute Bessel functions.
- However, we see that for large <span class="emphasis"><em>n</em></span> the integrand oscillates
- rapidly, taking on positive and negative values, and hence the trapezoidal
- sums become ill-conditioned. In double precision, <span class="emphasis"><em>x = 17</em></span>
- and <span class="emphasis"><em>n = 25</em></span> gives a sum which is so poorly conditioned
- that zero correct digits are obtained.
- </p>
- <p>
- The final <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
- be used to control the behaviour of the function: how it handles errors, what
- level of precision to use etc. Refer to the <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy documentation
- for more details</a>.
- </p>
- <p>
- References:
- </p>
- <p>
- Trefethen, Lloyd N., Weideman, J.A.C., <span class="emphasis"><em>The Exponentially Convergent
- Trapezoidal Rule</em></span>, SIAM Review, Vol. 56, No. 3, 2014.
- </p>
- <p>
- Stoer, Josef, and Roland Bulirsch. <span class="emphasis"><em>Introduction to numerical analysis.
- Vol. 12.</em></span>, Springer Science & Business Media, 2013.
- </p>
- <p>
- Higham, Nicholas J. <span class="emphasis"><em>Accuracy and stability of numerical algorithms.</em></span>
- Society for industrial and applied mathematics, 2002.
- </p>
- <p>
- Lyness, James N., and Cleve B. Moler. <span class="emphasis"><em>Numerical differentiation of
- analytic functions.</em></span> SIAM Journal on Numerical Analysis 4.2 (1967):
- 202-210.
- </p>
- <p>
- Gil, Amparo, Javier Segura, and Nico M. Temme. <span class="emphasis"><em>Computing special
- functions by using quadrature rules.</em></span> Numerical Algorithms 33.1-4
- (2003): 265-275.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
- </p>
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