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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.double_exponential.de_overview"></a><a class="link" href="de_overview.html" title="Overview">Overview</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.double_exponential.de_overview.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.double_exponential.de_overview.synopsis"></a></span><a class="link" href="de_overview.html#math_toolkit.double_exponential.de_overview.synopsis">Synopsis</a>
  32       </h5>
  33 <pre class="programlisting">    <span class="preprocessor">#include</span> <span class="special">&lt;</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">tanh_sinh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34     <span class="preprocessor">#include</span> <span class="special">&lt;</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">exp_sinh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  35     <span class="preprocessor">#include</span> <span class="special">&lt;</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">sinh_sinh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  36
  37     <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>
  38
  39     <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>
  40     <span class="keyword">class</span> <span class="identifier">tanh_sinh</span>
  41     <span class="special">{</span>
  42     <span class="keyword">public</span><span class="special">:</span>
  43         <span class="identifier">tanh_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">15</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">min_complement</span> <span class="special">=</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">min_value</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;()</span> <span class="special">*</span> <span class="number">4</span><span class="special">)</span>
  44
  45         <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>
  46         <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>
  47                        <span class="identifier">Real</span> <span class="identifier">tolerance</span> <span class="special">=</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;(),</span>
  48                        <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>
  49                        <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>
  50                        <span class="identifier">std</span><span class="special">::</span><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>
  51
  52         <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>
  53         <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>
  54                        <span class="identifier">tolerance</span> <span class="special">=</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;(),</span>
  55                        <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>
  56                        <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>
  57                        <span class="identifier">std</span><span class="special">::</span><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>
  58
  59     <span class="special">};</span>
  60
  61     <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>
  62     <span class="keyword">class</span> <span class="identifier">exp_sinh</span>
  63     <span class="special">{</span>
  64     <span class="keyword">public</span><span class="special">:</span>
  65         <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>
  66
  67         <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>
  68         <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>
  69                        <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>
  70                        <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>
  71                        <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>
  72                        <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>
  73         <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>
  74         <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>
  75                        <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>
  76                        <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>
  77                        <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>
  78                        <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>
  79     <span class="special">};</span>
  80
  81     <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>
  82     <span class="keyword">class</span> <span class="identifier">sinh_sinh</span>
  83     <span class="special">{</span>
  84     <span class="keyword">public</span><span class="special">:</span>
  85         <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>
  86
  87         <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>
  88         <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>
  89                        <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>
  90                        <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>
  91                        <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>
  92                        <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>
  93     <span class="special">};</span>
  94
  95 <span class="special">}}</span>
  96 </pre>
  97 <p>
  98         These three integration routines provide robust general purpose quadrature,
  99         each having a "native" range over which quadrature is performed.
 100         For example, the <code class="computeroutput"><span class="identifier">sinh_sinh</span></code>
 101         quadrature integrates over the entire real line, the <code class="computeroutput"><span class="identifier">tanh_sinh</span></code>
 102         over (-1, 1), and the <code class="computeroutput"><span class="identifier">exp_sinh</span></code>
 103         over (0, &#8734;). The latter integrators also have auxilliary ranges which are
 104         handled via a change of variables on the function being integrated, so that
 105         the <code class="computeroutput"><span class="identifier">tanh_sinh</span></code> can handle
 106         integration over <span class="emphasis"><em>(a, b)</em></span>, and <code class="computeroutput"><span class="identifier">exp_sinh</span></code>
 107         over /(a, &#8734;) and(-&#8734;, b)/.
 108       </p>
 109 <p>
 110         Like the other quadrature routines in Boost, these routines support both
 111         real and complex-valued integrands.
 112       </p>
 113 <p>
 114         The <code class="computeroutput"><span class="identifier">integrate</span></code> methods which
 115         do not specify a range always integrate over the native range of the method,
 116         and generally are the most efficient and produce the smallest code, on the
 117         other hand the methods which do specify the bounds of integration are the
 118         most general, and use argument transformations which are generally very robust.
 119         The following table summarizes the ranges supported by each method:
 120       </p>
 121 <div class="informaltable"><table class="table">
 122 <colgroup>
 123 <col>
 124 <col>
 125 <col>
 126 <col>
 127 </colgroup>
 128 <thead><tr>
 129 <th>
 130                 <p>
 131                   Integrator
 132                 </p>
 133               </th>
 134 <th>
 135                 <p>
 136                   Native range
 137                 </p>
 138               </th>
 139 <th>
 140                 <p>
 141                   Other supported ranges
 142                 </p>
 143               </th>
 144 <th>
 145                 <p>
 146                   Comments
 147                 </p>
 148               </th>
 149 </tr></thead>
 150 <tbody>
 151 <tr>
 152 <td>
 153                 <p>
 154                   tanh_sinh
 155                 </p>
 156               </td>
 157 <td>
 158                 <p>
 159                   (-1,1)
 160                 </p>
 161               </td>
 162 <td>
 163                 <p>
 164                   (a,b)<br> (a,&#8734;)<br> (-&#8734;,b)<br> (-&#8734;,&#8734;)
 165                 </p>
 166               </td>
 167 <td>
 168                 <p>
 169                   Special care is taken for endpoints at or near zero to ensure that
 170                   abscissa values are calculated without the loss of precision that
 171                   would normally occur. Likewise when transforming to an infinite
 172                   endpoint, the additional information which tanh_sinh has internally
 173                   on abscissa values is used to ensure no loss of precision during
 174                   the transformation.
 175                 </p>
 176               </td>
 177 </tr>
 178 <tr>
 179 <td>
 180                 <p>
 181                   exp_sinh
 182                 </p>
 183               </td>
 184 <td>
 185                 <p>
 186                   (0,&#8734;)
 187                 </p>
 188               </td>
 189 <td>
 190                 <p>
 191                   (a,&#8734;)<br> (-&#8734;,0)<br> (-&#8734;,b)
 192                 </p>
 193               </td>
 194 <td>
 195               </td>
 196 </tr>
 197 <tr>
 198 <td>
 199                 <p>
 200                   sinh_sinh
 201                 </p>
 202               </td>
 203 <td>
 204                 <p>
 205                   (-&#8734;,&#8734;)
 206                 </p>
 207               </td>
 208 <td>
 209               </td>
 210 <td>
 211               </td>
 212 </tr>
 213 </tbody>
 214 </table></div>
 215 </div>
 216 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 217 <td align="left"></td>
 218 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 219       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 220       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 221       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 222       Daryle Walker and Xiaogang Zhang<p>
 223         Distributed under the Boost Software License, Version 1.0. (See accompanying
 224         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>)
 225       </p>
 226 </div></td>
 227 </tr></table>
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