1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Fourier Integrals</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="../quadrature.html" title="Chapter 13. Quadrature and Differentiation"> 9<link rel="prev" href="double_exponential/de_refes.html" title="References"> 10<link rel="next" href="naive_monte_carlo.html" title="Naive Monte Carlo Integration"> 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 href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="double_exponential/de_refes.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.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="naive_monte_carlo.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.fourier_integrals"></a><a class="link" href="fourier_integrals.html" title="Fourier Integrals">Fourier Integrals</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.fourier_integrals.h0"></a> 31 <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> 32 </h4> 33<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> 34 35<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> 36 37<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 38<span class="keyword">class</span> <span class="identifier">ooura_fourier_sin</span> <span class="special">{</span> 39<span class="keyword">public</span><span class="special">:</span> 40 <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> 41 42 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span> 43 <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> 44 45<span class="special">};</span> 46 47 48<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 49<span class="keyword">class</span> <span class="identifier">ooura_fourier_cos</span> <span class="special">{</span> 50<span class="keyword">public</span><span class="special">:</span> 51 <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> 52 53 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span> 54 <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> 55<span class="special">};</span> 56 57<span class="special">}}}</span> <span class="comment">// namespaces</span> 58</pre> 59<p> 60 Ooura's method for Fourier integrals computes 61 </p> 62<div class="blockquote"><blockquote class="blockquote"><p> 63 <span class="serif_italic">∫<sub>0</sub><sup>∞</sup> f(t)sin(ω t) dt</span> 64 </p></blockquote></div> 65<p> 66 and 67 </p> 68<div class="blockquote"><blockquote class="blockquote"><p> 69 <span class="serif_italic">∫<sub>0</sub><sup>∞</sup> f(t)cos(ω t) dt</span> 70 </p></blockquote></div> 71<p> 72 by a double exponentially decaying transformation. These integrals arise when 73 computing continuous Fourier transform of odd and even functions, respectively. 74 Oscillatory integrals are known to cause trouble for standard quadrature methods, 75 so these routines are provided to cope with the most common oscillatory use 76 case. 77 </p> 78<p> 79 The basic usage is shown below: 80 </p> 81<pre class="programlisting"><span class="identifier">ooura_fourier_sin</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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> 82<span class="comment">// Use the default tolerance root_epsilon and eight levels for type double.</span> 83 84<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> 85<span class="special">{</span> <span class="comment">// Simple reciprocal function for sinc.</span> 86 <span class="keyword">return</span> <span class="number">1</span> <span class="special">/</span> <span class="identifier">x</span><span class="special">;</span> 87<span class="special">};</span> 88 89<span class="keyword">double</span> <span class="identifier">omega</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> 90<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="identifier">result</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> 91<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Integral = "</span> <span class="special"><<</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", relative error estimate "</span> <span class="special"><<</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">second</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 92</pre> 93<p> 94 and compare with the expected value π/2 of the integral. 95 </p> 96<pre class="programlisting"><span class="keyword">constexpr</span> <span class="keyword">double</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="identifier">half_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>();</span> 97<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"pi/2 = "</span> <span class="special"><<</span> <span class="identifier">expected</span> <span class="special"><<</span> <span class="string">", difference "</span> <span class="special"><<</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">first</span> <span class="special">-</span> <span class="identifier">expected</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 98</pre> 99<p> 100 The output is 101 </p> 102<pre class="programlisting"><span class="identifier">integral</span> <span class="special">=</span> <span class="number">1.5707963267948966</span><span class="special">,</span> <span class="identifier">relative</span> <span class="identifier">error</span> <span class="identifier">estimate</span> <span class="number">1.2655356398390254e-11</span> 103<span class="identifier">pi</span><span class="special">/</span><span class="number">2</span> <span class="special">=</span> <span class="number">1.5707963267948966</span><span class="special">,</span> <span class="identifier">difference</span> <span class="number">0</span> 104</pre> 105<div class="note"><table border="0" summary="Note"> 106<tr> 107<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td> 108<th align="left">Note</th> 109</tr> 110<tr><td align="left" valign="top"><p> 111 This integrator is more insistent about examining the error estimate, than 112 (say) tanh-sinh, which just returns the value of the integral. 113 </p></td></tr> 114</table></div> 115<p> 116 With the macro BOOST_MATH_INSTRUMENT_OOURA defined, we can follow the progress: 117 </p> 118<pre class="programlisting"><span class="identifier">ooura_fourier_sin</span> <span class="identifier">with</span> <span class="identifier">relative</span> <span class="identifier">error</span> <span class="identifier">goal</span> <span class="number">1.4901161193847656e-08</span> <span class="special">&</span> <span class="number">8</span> <span class="identifier">levels</span><span class="special">.</span> 119<span class="identifier">h</span> <span class="special">=</span> <span class="number">1.000000000000000</span><span class="special">,</span> <span class="identifier">I_h</span> <span class="special">=</span> <span class="number">1.571890732004545</span> <span class="special">=</span> <span class="number">0x1</span><span class="special">.</span><span class="number">92676e56d</span><span class="number">853500</span><span class="identifier">p</span><span class="special">+</span><span class="number">0</span><span class="special">,</span> <span class="identifier">absolute</span> <span class="identifier">error</span> <span class="identifier">estimate</span> <span class="special">=</span> <span class="identifier">nan</span> 120<span class="identifier">h</span> <span class="special">=</span> <span class="number">0.500000000000000</span><span class="special">,</span> <span class="identifier">I_h</span> <span class="special">=</span> <span class="number">1.570793292491940</span> <span class="special">=</span> <span class="number">0x1</span><span class="special">.</span><span class="number">921f</span><span class="number">825</span><span class="identifier">c076f600p</span><span class="special">+</span><span class="number">0</span><span class="special">,</span> <span class="identifier">absolute</span> <span class="identifier">error</span> <span class="identifier">estimate</span> <span class="special">=</span> <span class="number">1.097439512605325e-03</span> 121<span class="identifier">h</span> <span class="special">=</span> <span class="number">0.250000000000000</span><span class="special">,</span> <span class="identifier">I_h</span> <span class="special">=</span> <span class="number">1.570796326814776</span> <span class="special">=</span> <span class="number">0x1</span><span class="special">.</span><span class="number">921f</span><span class="identifier">b54458acf00p</span><span class="special">+</span><span class="number">0</span><span class="special">,</span> <span class="identifier">absolute</span> <span class="identifier">error</span> <span class="identifier">estimate</span> <span class="special">=</span> <span class="number">3.034322835882008e-06</span> 122<span class="identifier">h</span> <span class="special">=</span> <span class="number">0.125000000000000</span><span class="special">,</span> <span class="identifier">I_h</span> <span class="special">=</span> <span class="number">1.570796326794897</span> <span class="special">=</span> <span class="number">0x1</span><span class="special">.</span><span class="number">921f</span><span class="identifier">b54442d1800p</span><span class="special">+</span><span class="number">0</span><span class="special">,</span> <span class="identifier">absolute</span> <span class="identifier">error</span> <span class="identifier">estimate</span> <span class="special">=</span> <span class="number">1.987898734512328e-11</span> 123<span class="identifier">Integral</span> <span class="special">=</span> <span class="number">1.570796326794897e+00</span><span class="special">,</span> <span class="identifier">relative</span> <span class="identifier">error</span> <span class="identifier">estimate</span> <span class="number">1.265535639839025e-11</span> 124<span class="identifier">pi</span><span class="special">/</span><span class="number">2</span> <span class="special">=</span> <span class="number">1.570796326794897e+00</span><span class="special">,</span> <span class="identifier">difference</span> <span class="number">0.000000000000000e+00</span> 125</pre> 126<p> 127 Working code of this example is at <a href="../../../example/ooura_fourier_integrals_example.cpp" target="_top">ooura_fourier_integrals_example.cpp</a> 128 </p> 129<p> 130 A classical cosine transform is presented below: 131 </p> 132<pre class="programlisting"><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> 133<span class="comment">// Use the default tolerance root_epsilon and eight levels for type double.</span> 134 135<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> 136<span class="special">{</span> <span class="comment">// More complex example function.</span> 137 <span class="keyword">return</span> <span class="number">1</span> <span class="special">/</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> 138<span class="special">};</span> 139 140<span class="keyword">double</span> <span class="identifier">omega</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> 141 142<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">result</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> 143<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Integral = "</span> <span class="special"><<</span> <span class="identifier">result</span> <span class="special"><<</span> <span class="string">", relative error estimate "</span> <span class="special"><<</span> <span class="identifier">relative_error</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 144</pre> 145<p> 146 The value of this integral should be π/(2e) and can be shown : 147 </p> 148<pre class="programlisting"><span class="keyword">constexpr</span> <span class="keyword">double</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="identifier">half_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>()</span> <span class="special">/</span> <span class="identifier">e</span><span class="special"><</span><span class="keyword">double</span><span class="special">>();</span> 149<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"pi/(2e) = "</span> <span class="special"><<</span> <span class="identifier">expected</span> <span class="special"><<</span> <span class="string">", difference "</span> <span class="special"><<</span> <span class="identifier">result</span> <span class="special">-</span> <span class="identifier">expected</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 150</pre> 151<p> 152 or with the macro BOOST_MATH_INSTRUMENT_OOURA defined, we can follow the progress: 153 </p> 154<pre class="programlisting"> 155ooura_fourier_cos with relative error goal 1.4901161193847656e-08 & 8 levels. 156epsilon for type = 2.2204460492503131e-16 157h = 1.000000000000000, I_h = 0.588268622591776 = 0x1.2d318b7e96dbe00p-1, absolute error estimate = nan 158h = 0.500000000000000, I_h = 0.577871642184837 = 0x1.27decab8f07b200p-1, absolute error estimate = 1.039698040693926e-02 159h = 0.250000000000000, I_h = 0.577863671186883 = 0x1.27ddbf42969be00p-1, absolute error estimate = 7.970997954576120e-06 160h = 0.125000000000000, I_h = 0.577863674895461 = 0x1.27ddbf6271dc000p-1, absolute error estimate = 3.708578555361441e-09 161Integral = 5.778636748954611e-01, relative error estimate 6.417739540441515e-09 162pi/(2e) = 5.778636748954609e-01, difference 2.220446049250313e-16 163 164</pre> 165<p> 166 Working code of this example is at <a href="../../../example/ooura_fourier_integrals_cosine_example.cpp" target="_top">ooura_fourier_integrals_consine_example.cpp</a> 167 </p> 168<h6> 169<a name="math_toolkit.fourier_integrals.h1"></a> 170 <span class="phrase"><a name="math_toolkit.fourier_integrals.performance"></a></span><a class="link" href="fourier_integrals.html#math_toolkit.fourier_integrals.performance">Performance</a> 171 </h6> 172<p> 173 The integrator precomputes nodes and weights, and hence can be reused for many 174 different frequencies with good efficiency. The integrator is pimpl'd and hence 175 can be shared between threads without a <code class="computeroutput"><span class="identifier">memcpy</span></code> 176 of the nodes and weights. 177 </p> 178<p> 179 Ooura and Mori's paper identifies criteria for rapid convergence based on the 180 position of the poles of the integrand in the complex plane. If these poles 181 are too close to the real axis the convergence is slow. It is not trivial to 182 predict the convergence rate a priori, so if you are interested in figuring 183 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 184 of printing will give you a good idea of how well this method is performing. 185 </p> 186<h6> 187<a name="math_toolkit.fourier_integrals.h2"></a> 188 <span class="phrase"><a name="math_toolkit.fourier_integrals.multi_precision"></a></span><a class="link" href="fourier_integrals.html#math_toolkit.fourier_integrals.multi_precision">Higher 189 precision</a> 190 </h6> 191<p> 192 It is simple to extend to higher precision using <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>. 193 </p> 194<pre class="programlisting"><span class="comment">// Use the default parameters for tolerance root_epsilon and eight levels for a type of 8 bytes.</span> 195<span class="comment">//auto integrator = ooura_fourier_cos<Real>();</span> 196<span class="comment">// Decide on a (tight) tolerance.</span> 197<span class="keyword">const</span> <span class="identifier">Real</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="number">2</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> 198<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="identifier">Real</span><span class="special">>(</span><span class="identifier">tol</span><span class="special">,</span> <span class="number">8</span><span class="special">);</span> <span class="comment">// Loops or gets worse for more than 8.</span> 199 200<span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> 201<span class="special">{</span> <span class="comment">// More complex example function.</span> 202 <span class="keyword">return</span> <span class="number">1</span> <span class="special">/</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> 203<span class="special">};</span> 204 205<span class="keyword">double</span> <span class="identifier">omega</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> 206<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">result</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> 207</pre> 208<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Integral = "</span> <span class="special"><<</span> <span class="identifier">result</span> <span class="special"><<</span> <span class="string">", relative error estimate "</span> <span class="special"><<</span> <span class="identifier">relative_error</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 209 210<span class="keyword">const</span> <span class="identifier">Real</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="identifier">half_pi</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()</span> <span class="special">/</span> <span class="identifier">e</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>();</span> <span class="comment">// Expect integral = 1/(2e)</span> 211<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"pi/(2e) = "</span> <span class="special"><<</span> <span class="identifier">expected</span> <span class="special"><<</span> <span class="string">", difference "</span> <span class="special"><<</span> <span class="identifier">result</span> <span class="special">-</span> <span class="identifier">expected</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 212</pre> 213<p> 214 with output: 215 </p> 216<pre class="programlisting"> 217Integral = 0.5778636748954608589550465916563501587, relative error estimate 4.609814684522163895264277312610830278e-17 218pi/(2e) = 0.5778636748954608659545328919193707407, difference -6.999486300263020581921171645255733758e-18 219 220</pre> 221<p> 222 And with diagnostics on: 223 </p> 224<pre class="programlisting"> 225ooura_fourier_cos with relative error goal 3.851859888774471706111955885169854637e-34 & 15 levels. 226epsilon for type = 1.925929944387235853055977942584927319e-34 227h = 1.000000000000000000000000000000000, I_h = 0.588268622591776615359568690603776 = 0.5882686225917766153595686906037760, absolute error estimate = nan 228h = 0.500000000000000000000000000000000, I_h = 0.577871642184837461311756940493259 = 0.5778716421848374613117569404932595, absolute error estimate = 1.039698040693915404781175011051656e-02 229h = 0.250000000000000000000000000000000, I_h = 0.577863671186882539559996800783122 = 0.5778636711868825395599968007831220, absolute error estimate = 7.970997954921751760139710137450075e-06 230h = 0.125000000000000000000000000000000, I_h = 0.577863674895460885593491133506723 = 0.5778636748954608855934911335067232, absolute error estimate = 3.708578346033494332723601147051768e-09 231h = 0.062500000000000000000000000000000, I_h = 0.577863674895460858955046591656350 = 0.5778636748954608589550465916563502, absolute error estimate = 2.663844454185037302771663314961535e-17 232h = 0.031250000000000000000000000000000, I_h = 0.577863674895460858955046591656348 = 0.5778636748954608589550465916563484, absolute error estimate = 1.733336949948512267750380148326435e-33 233h = 0.015625000000000000000000000000000, I_h = 0.577863674895460858955046591656348 = 0.5778636748954608589550465916563479, absolute error estimate = 4.814824860968089632639944856462318e-34 234h = 0.007812500000000000000000000000000, I_h = 0.577863674895460858955046591656347 = 0.5778636748954608589550465916563473, absolute error estimate = 6.740754805355325485695922799047246e-34 235h = 0.003906250000000000000000000000000, I_h = 0.577863674895460858955046591656347 = 0.5778636748954608589550465916563475, absolute error estimate = 1.925929944387235853055977942584927e-34 236Integral = 5.778636748954608589550465916563475e-01, relative error estimate 3.332844800697411177051445985473052e-34 237pi/(2e) = 5.778636748954608589550465916563481e-01, difference -6.740754805355325485695922799047246e-34 238 239</pre> 240<p> 241 Working code of this example is at <a href="../../../example/ooura_fourier_integrals_multiprecision_example.cpp" target="_top">ooura_fourier_integrals_multiprecision_example.cpp</a> 242 </p> 243<p> 244 For more examples of other functions and tests, see the full test suite at 245 <a href="../../../test/ooura_fourier_integral_test.cpp" target="_top">ooura_fourier_integral_test.cpp</a>. 246 </p> 247<p> 248 Ngyen and Nuyens make use of <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a> 249 in their extension to multiple dimensions, showing relative errors reducing 250 to ≅ 10<sup>-2000</sup>! 251 </p> 252<h6> 253<a name="math_toolkit.fourier_integrals.h3"></a> 254 <span class="phrase"><a name="math_toolkit.fourier_integrals.rationale"></a></span><a class="link" href="fourier_integrals.html#math_toolkit.fourier_integrals.rationale">Rationale</a> 255 </h6> 256<p> 257 This implementation is base on Ooura's 1999 paper rather than the later 2005 258 paper. It does not preclude a second future implementation based on the later 259 work. 260 </p> 261<p> 262 Some of the features of the Ooura's 2005 paper that were less appealing were: 263 </p> 264<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 265<li class="listitem"> 266 The advance of that paper is that one can compute <span class="emphasis"><em>both</em></span> 267 the Fourier sine transform and Fourier cosine transform in a single shot. 268 But there are examples, like sinc integral, where the Fourier sine would 269 converge, but the Fourier cosine would diverge. 270 </li> 271<li class="listitem"> 272 It would force users to live in the complex plane, when many potential 273 applications only need real. 274 </li> 275</ul></div> 276<h5> 277<a name="math_toolkit.fourier_integrals.h4"></a> 278 <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> 279 </h5> 280<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 281<li class="listitem"> 282 Ooura, Takuya, and Masatake Mori, <span class="emphasis"><em>A robust double exponential 283 formula for Fourier-type integrals.</em></span> Journal of computational 284 and applied mathematics, 112.1-2 (1999): 229-241. 285 </li> 286<li class="listitem"> 287 Ooura, Takuya, <span class="emphasis"><em>A Double Exponential Formula for the Fourier Transforms.</em></span> 288 Publ. RIMS, Kyoto Univ., 41 (2005), 971-977. <a href="https://pdfs.semanticscholar.org/16ec/a5d76fd6b3d7acaaff0b2a6e8a70caa70190.pdf" target="_top">https://pdfs.semanticscholar.org/16ec/a5d76fd6b3d7acaaff0b2a6e8a70caa70190.pdf</a> 289 </li> 290<li class="listitem"> 291 Khatibi, Arezoo and Khatibi, Omid,<span class="emphasis"><em>Criteria for the Application 292 of Double Exponential Transformation.</em></span> (2017) <a href="https://arxiv.org/pdf/1704.05752.pdf" target="_top">1704.05752.pdf</a>. 293 </li> 294<li class="listitem"> 295 Nguyen, Dong T.P. and Nuyens, Dirk, <span class="emphasis"><em>Multivariate integration 296 over Reals with exponential rate of convergence.</em></span> (2016) <a href="https://core.ac.uk/download/pdf/80799199.pdf" target="_top">https://core.ac.uk/download/pdf/80799199.pdf</a>. 297 </li> 298</ul></div> 299</div> 300<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 301<td align="left"></td> 302<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 303 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 304 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 305 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 306 Daryle Walker and Xiaogang Zhang<p> 307 Distributed under the Boost Software License, Version 1.0. (See accompanying 308 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>) 309 </p> 310</div></td> 311</tr></table> 312<hr> 313<div class="spirit-nav"> 314<a accesskey="p" href="double_exponential/de_refes.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.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="naive_monte_carlo.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 315</div> 316</body> 317</html> 318
