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