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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.double_exponential.de_exp_sinh"></a><a class="link" href="de_exp_sinh.html" title="exp_sinh">exp_sinh</a>
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</h3></div></div></div>
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<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>
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<span class="keyword">class</span> <span class="identifier">exp_sinh</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<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>
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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>
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<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>
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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">Real</span><span class="special">*</span> <span class="identifier">error</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="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>
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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>
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<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>
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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">Real</span><span class="special">*</span> <span class="identifier">error</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="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>
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<span class="special">};</span>
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</pre>
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<p>
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For half-infinite intervals, the exp-sinh quadrature is provided:
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</p>
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<pre class="programlisting"><span class="identifier">exp_sinh</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">integrator</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="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>
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<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"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
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<span class="keyword">double</span> <span class="identifier">error</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">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">&</span><span class="identifier">error</span><span class="special">,</span> <span class="special">&</span><span class="identifier">L1</span><span class="special">);</span>
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</pre>
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<p>
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The native integration range of this integrator is (0, ∞), but we also support
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(a, ∞), (-∞, 0) and (-∞, b) via argument transformations.
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</p>
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<p>
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Endpoint singularities and complex-valued integrands are supported by exp-sinh.
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</p>
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<p>
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For example, the modified Bessel function K can be represented via:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/complex_bessel_k_integral.svg"></span>
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</p>
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<p>
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Which we can code up as:
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</p>
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<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>
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<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>
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<span class="special">{</span>
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<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>
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<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>
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<span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[&,</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>
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<span class="special">{</span>
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<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>
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<span class="keyword">if</span> <span class="special">(</span><span class="identifier">ct</span> <span class="special">></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"><</span><span class="identifier">value_type</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()))</span>
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<span class="keyword">return</span> <span class="identifier">Complex</span><span class="special">(</span><span class="number">0</span><span class="special">);</span>
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<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>
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<span class="special">};</span>
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<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">value_type</span><span class="special">></span> <span class="identifier">integrator</span><span class="special">;</span>
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<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>
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<span class="special">}</span>
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</pre>
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<p>
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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>
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<span class="identifier">cosh</span><span class="special">(</span><span class="identifier">t</span><span class="special">))</span></code> tends
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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
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to infinity and return <code class="computeroutput"><span class="number">0</span></code>. Without
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that check we end up with <code class="computeroutput"><span class="number">0</span> <span class="special">*</span> <span class="identifier">Infinity</span></code>
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as the result (a NaN).
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</p>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014, 2017 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, Johan Råde, Gautam
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Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
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and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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