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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.double_exponential.de_sinh_sinh"></a><a class="link" href="de_sinh_sinh.html" title="sinh_sinh">sinh_sinh</a>
  28 </h3></div></div></div>
  29 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  30 <span class="keyword">class</span> <span class="identifier">sinh_sinh</span>
  31 <span class="special">{</span>
  32 <span class="keyword">public</span><span class="special">:</span>
  33     <span class="identifier">sinh_sinh</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">max_refinements</span> <span class="special">=</span> <span class="number">9</span><span class="special">);</span>
  34
  35     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">&gt;</span>
  36     <span class="keyword">auto</span> <span class="identifier">integrate</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  37                    <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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()),</span>
  38                    <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">error</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">,</span>
  39                    <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>
  40                    <span class="identifier">size_t</span><span class="special">*</span> <span class="identifier">levels</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">)-&gt;</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special">&lt;</span><span class="identifier">F</span><span class="special">&gt;()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;()))</span> <span class="keyword">const</span><span class="special">;;</span>
  41 <span class="special">};</span>
  42 </pre>
  43 <p>
  44         The sinh-sinh quadrature allows computation over the entire real line, and
  45         is called as follows:
  46       </p>
  47 <pre class="programlisting"><span class="identifier">sinh_sinh</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">integrator</span><span class="special">;</span>
  48 <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="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">);</span> <span class="special">};</span>
  49 <span class="keyword">double</span> <span class="identifier">error</span><span class="special">;</span>
  50 <span class="keyword">double</span> <span class="identifier">L1</span><span class="special">;</span>
  51 <span class="keyword">double</span> <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">integrator</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">error</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">L1</span><span class="special">);</span>
  52 </pre>
  53 <p>
  54         Note that the limits of integration are understood to be (-&#8734;, +&#8734;).
  55       </p>
  56 <p>
  57         Complex valued integrands are supported as well, for example the <a href="https://en.wikipedia.org/wiki/Dirichlet_eta_function" target="_top">Dirichlet
  58         Eta function</a> can be represented via:
  59       </p>
  60 <div class="blockquote"><blockquote class="blockquote"><p>
  61           <span class="inlinemediaobject"><img src="../../../equations/complex_eta_integral.svg"></span>
  62
  63         </p></blockquote></div>
  64 <p>
  65         which we can directly code up as:
  66       </p>
  67 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Complex</span><span class="special">&gt;</span>
  68 <span class="identifier">Complex</span> <span class="identifier">eta</span><span class="special">(</span><span class="identifier">Complex</span> <span class="identifier">s</span><span class="special">)</span>
  69 <span class="special">{</span>
  70    <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">Complex</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">value_type</span><span class="special">;</span>
  71    <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span><span class="special">;</span>  <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special">;</span>
  72    <span class="identifier">Complex</span> <span class="identifier">i</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
  73    <span class="identifier">value_type</span> <span class="identifier">pi</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">pi</span><span class="special">&lt;</span><span class="identifier">value_type</span><span class="special">&gt;();</span>
  74    <span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[&amp;,</span> <span class="identifier">s</span><span class="special">,</span> <span class="identifier">i</span><span class="special">](</span><span class="identifier">value_type</span> <span class="identifier">t</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">pow</span><span class="special">(</span><span class="number">0.5</span> <span class="special">+</span> <span class="identifier">i</span> <span class="special">*</span> <span class="identifier">t</span><span class="special">,</span> <span class="special">-</span><span class="identifier">s</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">exp</span><span class="special">(</span><span class="identifier">pi</span> <span class="special">*</span> <span class="identifier">t</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">pi</span> <span class="special">*</span> <span class="identifier">t</span><span class="special">));</span> <span class="special">};</span>
  75    <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">sinh_sinh</span><span class="special">&lt;</span><span class="identifier">value_type</span><span class="special">&gt;</span> <span class="identifier">integrator</span><span class="special">;</span>
  76    <span class="keyword">return</span> <span class="identifier">integrator</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f</span><span class="special">);</span>
  77 <span class="special">}</span>
  78 </pre>
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  82 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
  83       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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  85       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
  86       Daryle Walker and Xiaogang Zhang<p>
  87         Distributed under the Boost Software License, Version 1.0. (See accompanying
  88         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>)
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