1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>sinh_sinh</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../double_exponential.html" title="Double-exponential quadrature"> 9<link rel="prev" href="de_tanh_sinh_2_arg.html" title="Handling functions with large features near an endpoint with tanh-sinh quadrature"> 10<link rel="next" href="de_exp_sinh.html" title="exp_sinh"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a 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title="sinh_sinh">sinh_sinh</a> 28</h3></div></div></div> 29<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></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"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></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"><</span><span class="identifier">Real</span><span class="special">>::</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">)-></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"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()))</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"><</span><span class="keyword">double</span><span class="special">></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">&</span><span class="identifier">error</span><span class="special">,</span> <span class="special">&</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 (-∞, +∞). 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"><</span><span class="keyword">class</span> <span class="identifier">Complex</span><span class="special">></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"><</span><span class="identifier">value_type</span><span class="special">>();</span> 74 <span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[&,</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"><</span><span class="identifier">value_type</span><span class="special">></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> 79</div> 80<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 81<td align="left"></td> 82<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 83 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 84 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 85 Råde, 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>) 89 </p> 90</div></td> 91</tr></table> 92<hr> 93<div class="spirit-nav"> 94<a accesskey="p" href="de_tanh_sinh_2_arg.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../double_exponential.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="de_exp_sinh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 95</div> 96</body> 97</html> 98
