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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.fourier_integrals"></a><a class="link" href="fourier_integrals.html" title="Fourier Integrals">Fourier Integrals</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.fourier_integrals.h0"></a>
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<span class="phrase"><a name="math_toolkit.fourier_integrals.synopsis"></a></span><a class="link" href="fourier_integrals.html#math_toolkit.fourier_integrals.synopsis">Synopsis</a>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</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">quadrature</span><span class="special">/</span><span class="identifier">ooura_fourier_integrals</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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> <span class="keyword">namespace</span> <span class="identifier">quadrature</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">Real</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">ooura_fourier_sin</span> <span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">ooura_fourier_sin</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span> <span class="identifier">relative_error_tolerance</span> <span class="special">=</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>(),</span> <span class="identifier">size_t</span> <span class="identifier">levels</span> <span class="special">=</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">Real</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">omega</span><span class="special">);</span>
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<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">Real</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">ooura_fourier_cos</span> <span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">ooura_fourier_cos</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span> <span class="identifier">relative_error_tolerance</span> <span class="special">=</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>(),</span> <span class="identifier">size_t</span> <span class="identifier">levels</span> <span class="special">=</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">Real</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">omega</span><span class="special">);</span>
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<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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Ooura's method for Fourier integrals computes
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</p>
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<p>
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∫<sub>0</sub><sup>∞</sup> f(t)sin(ω t) dt
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</p>
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<p>
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and
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</p>
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<p>
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∫<sub>0</sub><sup>∞</sup> f(t)cos(ω t) dt
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</p>
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<p>
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by a double exponentially decaying transformation. These integrals arise when
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computing continuous Fourier transform of odd and even functions, respectively.
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Oscillatory integrals are known to cause trouble for standard quadrature methods,
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so these routines are provided to cope with the most common oscillatory use
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case.
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</p>
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<p>
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The basic usage is shown below:
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</p>
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<pre class="programlisting"><span class="keyword">using</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">ooura_fourier_sin</span><span class="special">;</span>
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<span class="keyword">auto</span> <span class="identifier">integrator</span> <span class="special">=</span> <span class="identifier">ooura_fourier_sin</span><span class="special"><</span><span class="keyword">double</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="number">1</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">omega</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span>
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<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">Is</span><span class="special">,</span> <span class="identifier">relative_error</span><span class="special">]</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">omega</span><span class="special">);</span>
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</pre>
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<p>
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The computed value should be π/2. Note that this integrator is more insistent
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about examining the error estimate, than (say) tanh-sinh, which just returns
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the value of the integral.
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</p>
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<p>
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A classical cosine transform is presented below:
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</p>
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<pre class="programlisting"><span class="keyword">using</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">ooura_fourier_cos</span><span class="special">;</span>
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<span class="keyword">auto</span> <span class="identifier">integrator</span> <span class="special">=</span> <span class="identifier">ooura_fourier_cos</span><span class="special"><</span><span class="keyword">double</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="number">1</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="number">1</span><span class="special">);</span> <span class="special">};</span>
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<span class="keyword">double</span> <span class="identifier">omega</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span>
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<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">Ic</span><span class="special">,</span> <span class="identifier">relative_error</span><span class="special">]</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">omega</span><span class="special">);</span>
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</pre>
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<p>
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The value of <code class="computeroutput"><span class="identifier">Ic</span></code> should be π/2e.
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</p>
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<p>
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The integrator precomputes nodes and weights, and hence can be reused for many
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different frequencies with good efficiency. The integrator is pimpl'd and hence
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can be shared between threads without a memcpy of the nodes and weights.
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</p>
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<p>
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Ooura and Mori's paper identifies criteria for rapid convergence based on the
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position of the poles of the integrand in the complex plane. If these poles
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are too close to the real axis the convergence is slow. It is not trivial to
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predict the convergence rate a priori, so if you are interested in figuring
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out if the convergence is rapid compile with <code class="computeroutput"><span class="special">-</span><span class="identifier">DBOOST_MATH_INSTRUMENT_OOURA</span></code> and some amount
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of printing will give you a good idea of how well this method is performing.
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</p>
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<h4>
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<a name="math_toolkit.fourier_integrals.h1"></a>
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<span class="phrase"><a name="math_toolkit.fourier_integrals.references"></a></span><a class="link" href="fourier_integrals.html#math_toolkit.fourier_integrals.references">References</a>
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</h4>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
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Ooura, Takuya, and Masatake Mori, <span class="emphasis"><em>A robust double exponential
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formula for Fourier-type integrals.</em></span> Journal of computational
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and applied mathematics 112.1-2 (1999): 229-241.
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</li></ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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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, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker 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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