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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.double_exponential.de_exp_sinh"></a><a class="link" href="de_exp_sinh.html" title="exp_sinh">exp_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">exp_sinh</span>
  31 <span class="special">{</span>
  32 <span class="keyword">public</span><span class="special">:</span>
  33     <span class="identifier">exp_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> <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>
  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="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">&gt;</span>
  42     <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>
  43                    <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>
  44                    <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>
  45                    <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>
  46                    <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>
  47 <span class="special">};</span>
  48 </pre>
  49 <p>
  50         For half-infinite intervals, the <code class="computeroutput"><span class="identifier">exp</span><span class="special">-</span><span class="identifier">sinh</span></code> quadrature
  51         is provided:
  52       </p>
  53 <pre class="programlisting"><span class="identifier">exp_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>
  54 <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="number">3</span><span class="special">*</span><span class="identifier">x</span><span class="special">);</span> <span class="special">};</span>
  55 <span class="keyword">double</span> <span class="identifier">termination</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="keyword">double</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">());</span>
  56 <span class="keyword">double</span> <span class="identifier">error</span><span class="special">;</span>
  57 <span class="keyword">double</span> <span class="identifier">L1</span><span class="special">;</span>
  58 <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="identifier">termination</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>
  59 </pre>
  60 <p>
  61         The native integration range of this integrator is (0, &#8734;), but we also support
  62         /(a, &#8734;), (-&#8734;, 0)/ and /(-&#8734;, b)/ via argument transformations.
  63       </p>
  64 <p>
  65         Endpoint singularities and complex-valued integrands are supported by <code class="computeroutput"><span class="identifier">exp</span><span class="special">-</span><span class="identifier">sinh</span></code>.
  66       </p>
  67 <p>
  68         For example, the modified Bessel function K can be represented via:
  69       </p>
  70 <div class="blockquote"><blockquote class="blockquote"><p>
  71           <span class="inlinemediaobject"><img src="../../../equations/complex_bessel_k_integral.svg"></span>
  72
  73         </p></blockquote></div>
  74 <p>
  75         Which we can code up as:
  76       </p>
  77 <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>
  78 <span class="identifier">Complex</span> <span class="identifier">bessel_K</span><span class="special">(</span><span class="identifier">Complex</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">Complex</span> <span class="identifier">z</span><span class="special">)</span>
  79 <span class="special">{</span>
  80    <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>
  81    <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cosh</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>
  82    <span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[&amp;,</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">z</span><span class="special">](</span><span class="identifier">value_type</span> <span class="identifier">t</span><span class="special">)</span>
  83    <span class="special">{</span>
  84       <span class="identifier">value_type</span> <span class="identifier">ct</span> <span class="special">=</span> <span class="identifier">cosh</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span>
  85       <span class="keyword">if</span> <span class="special">(</span><span class="identifier">ct</span> <span class="special">&gt;</span> <span class="identifier">log</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">value_type</span><span class="special">&gt;::</span><span class="identifier">max</span><span class="special">()))</span>
  86          <span class="keyword">return</span> <span class="identifier">Complex</span><span class="special">(</span><span class="number">0</span><span class="special">);</span>
  87       <span class="keyword">return</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">z</span> <span class="special">*</span> <span class="identifier">ct</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">cosh</span><span class="special">(</span><span class="identifier">alpha</span> <span class="special">*</span> <span class="identifier">t</span><span class="special">);</span>
  88    <span class="special">};</span>
  89    <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">exp_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>
  90    <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>
  91 <span class="special">}</span>
  92 </pre>
  93 <p>
  94         The only wrinkle in the above code is the need to check for large <code class="computeroutput"><span class="identifier">cosh</span><span class="special">(</span><span class="identifier">t</span><span class="special">)</span></code> in which case we assume that <code class="computeroutput"><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span>
  95         <span class="identifier">cosh</span><span class="special">(</span><span class="identifier">t</span><span class="special">))</span></code> tends
  96         to zero faster than <code class="computeroutput"><span class="identifier">cosh</span><span class="special">(</span><span class="identifier">alpha</span> <span class="identifier">x</span><span class="special">)</span></code> tends
  97         to infinity and return <code class="computeroutput"><span class="number">0</span></code>. Without
  98         that check we end up with <code class="computeroutput"><span class="number">0</span> <span class="special">*</span> <span class="identifier">Infinity</span></code>
  99         as the result (a NaN).
 100       </p>
 101 </div>
 102 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 103 <td align="left"></td>
 104 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 105       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 106       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 107       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 108       Daryle Walker and Xiaogang Zhang<p>
 109         Distributed under the Boost Software License, Version 1.0. (See accompanying
 110         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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