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    24. 

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    25. 

  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
    27. 

  27. <a name="math_toolkit.fourier_integrals"></a><a class="link" href="fourier_integrals.html" title="Fourier Integrals">Fourier Integrals</a>
    28. 

  28. </h2></div></div></div>
    29. 

  29. <h4>
    30. 

  30. <a name="math_toolkit.fourier_integrals.h0"></a>
    31. 

  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. 

  32.     </h4>
    33. 

  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">ooura_fourier_integrals</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    34. 

  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. 

  36. 

  37. <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>
    38. 

  38. <span class="keyword">class</span> <span class="identifier">ooura_fourier_sin</span> <span class="special">{</span>
    39. 

  39. <span class="keyword">public</span><span class="special">:</span>
    40. 

  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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;(),</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. 

  41. 

  42.     <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>
    43. 

  43.     <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="keyword">const</span> <span class="special">&amp;</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. 

  44. 

  45. <span class="special">};</span>
    46. 

  46. 

  47. 

  48. <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>
    49. 

  49. <span class="keyword">class</span> <span class="identifier">ooura_fourier_cos</span> <span class="special">{</span>
    50. 

  50. <span class="keyword">public</span><span class="special">:</span>
    51. 

  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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;(),</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. 

  52. 

  53.     <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>
    54. 

  54.     <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="keyword">const</span> <span class="special">&amp;</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. 

  55. <span class="special">};</span>
    56. 

  56. 

  57. <span class="special">}}}</span> <span class="comment">// namespaces</span>
    58. 

  58. </pre>
    59. 

  59. <p>
    60. 

  60.       Ooura's method for Fourier integrals computes
    61. 

  61.     </p>
    62. 

  62. <div class="blockquote"><blockquote class="blockquote"><p>
    63. 

  63.         <span class="serif_italic">&#8747;<sub>0</sub><sup>&#8734;</sup> f(t)sin(&#969; t) dt</span>
    64. 

  64.       </p></blockquote></div>
    65. 

  65. <p>
    66. 

  66.       and
    67. 

  67.     </p>
    68. 

  68. <div class="blockquote"><blockquote class="blockquote"><p>
    69. 

  69.         <span class="serif_italic">&#8747;<sub>0</sub><sup>&#8734;</sup> f(t)cos(&#969; t) dt</span>
    70. 

  70.       </p></blockquote></div>
    71. 

  71. <p>
    72. 

  72.       by a double exponentially decaying transformation. These integrals arise when
    73. 

  73.       computing continuous Fourier transform of odd and even functions, respectively.
    74. 

  74.       Oscillatory integrals are known to cause trouble for standard quadrature methods,
    75. 

  75.       so these routines are provided to cope with the most common oscillatory use
    76. 

  76.       case.
    77. 

  77.     </p>
    78. 

  78. <p>
    79. 

  79.       The basic usage is shown below:
    80. 

  80.     </p>
    81. 

  81. <pre class="programlisting"><span class="identifier">ooura_fourier_sin</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">integrator</span> <span class="special">=</span> <span class="identifier">ooura_fourier_sin</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;();</span>
    82. 

  82. <span class="comment">// Use the default tolerance root_epsilon and eight levels for type double.</span>
    83. 

  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. 

  85. <span class="special">{</span> <span class="comment">// Simple reciprocal function for sinc.</span>
    86. 

  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. 

  87. <span class="special">};</span>
    88. 

  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. 

  90. <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">double</span><span class="special">&gt;</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. 

  91. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Integral = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">first</span> <span class="special">&lt;&lt;</span> <span class="string">", relative error estimate "</span> <span class="special">&lt;&lt;</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">second</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    92. 

  92. </pre>
    93. 

  93. <p>
    94. 

  94.       and compare with the expected value &#960;/2 of the integral.
    95. 

  95.     </p>
    96. 

  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">&lt;</span><span class="keyword">double</span><span class="special">&gt;();</span>
    97. 

  97. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"pi/2 =     "</span> <span class="special">&lt;&lt;</span> <span class="identifier">expected</span> <span class="special">&lt;&lt;</span> <span class="string">", difference "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    98. 

  98. </pre>
    99. 

  99. <p>
    100. 

  100.       The output is
    101. 

  101.     </p>
    102. 

  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. 

  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. 

  104. </pre>
    105. 

  105. <div class="note"><table border="0" summary="Note">
    106. 

  106. <tr>
    107. 

  107. <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
    108. 

  108. <th align="left">Note</th>
    109. 

  109. </tr>
    110. 

  110. <tr><td align="left" valign="top"><p>
    111. 

  111.         This integrator is more insistent about examining the error estimate, than
    112. 

  112.         (say) tanh-sinh, which just returns the value of the integral.
    113. 

  113.       </p></td></tr>
    114. 

  114. </table></div>
    115. 

  115. <p>
    116. 

  116.       With the macro BOOST_MATH_INSTRUMENT_OOURA defined, we can follow the progress:
    117. 

  117.     </p>
    118. 

  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">&amp;</span> <span class="number">8</span> <span class="identifier">levels</span><span class="special">.</span>
    119. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  125. </pre>
    126. 

  126. <p>
    127. 

  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. 

  128.     </p>
    129. 

  129. <p>
    130. 

  130.       A classical cosine transform is presented below:
    131. 

  131.     </p>
    132. 

  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">&lt;</span><span class="keyword">double</span><span class="special">&gt;();</span>
    133. 

  133. <span class="comment">// Use the default tolerance root_epsilon and eight levels for type double.</span>
    134. 

  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. 

  136. <span class="special">{</span> <span class="comment">// More complex example function.</span>
    137. 

  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. 

  138. <span class="special">};</span>
    139. 

  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. 

  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. 

  143. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Integral = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">result</span> <span class="special">&lt;&lt;</span> <span class="string">", relative error estimate "</span> <span class="special">&lt;&lt;</span> <span class="identifier">relative_error</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    144. 

  144. </pre>
    145. 

  145. <p>
    146. 

  146.       The value of this integral should be &#960;/(2e) and can be shown :
    147. 

  147.     </p>
    148. 

  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">&lt;</span><span class="keyword">double</span><span class="special">&gt;()</span> <span class="special">/</span> <span class="identifier">e</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;();</span>
    149. 

  149. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"pi/(2e) =  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">expected</span> <span class="special">&lt;&lt;</span> <span class="string">", difference "</span> <span class="special">&lt;&lt;</span> <span class="identifier">result</span> <span class="special">-</span> <span class="identifier">expected</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    150. 

  150. </pre>
    151. 

  151. <p>
    152. 

  152.       or with the macro BOOST_MATH_INSTRUMENT_OOURA defined, we can follow the progress:
    153. 

  153.     </p>
    154. 

  154. <pre class="programlisting">
    155. 

  155. ooura_fourier_cos with relative error goal 1.4901161193847656e-08 &amp; 8 levels.
    156. 

  156. epsilon for type = 2.2204460492503131e-16
    157. 

  157. h = 1.000000000000000, I_h = 0.588268622591776 = 0x1.2d318b7e96dbe00p-1, absolute error estimate = nan
    158. 

  158. h = 0.500000000000000, I_h = 0.577871642184837 = 0x1.27decab8f07b200p-1, absolute error estimate = 1.039698040693926e-02
    159. 

  159. h = 0.250000000000000, I_h = 0.577863671186883 = 0x1.27ddbf42969be00p-1, absolute error estimate = 7.970997954576120e-06
    160. 

  160. h = 0.125000000000000, I_h = 0.577863674895461 = 0x1.27ddbf6271dc000p-1, absolute error estimate = 3.708578555361441e-09
    161. 

  161. Integral = 5.778636748954611e-01, relative error estimate 6.417739540441515e-09
    162. 

  162. pi/(2e)  = 5.778636748954609e-01, difference 2.220446049250313e-16
    163. 

  163. 

  164. </pre>
    165. 

  165. <p>
    166. 

  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. 

  167.     </p>
    168. 

  168. <h6>
    169. 

  169. <a name="math_toolkit.fourier_integrals.h1"></a>
    170. 

  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. 

  171.     </h6>
    172. 

  172. <p>
    173. 

  173.       The integrator precomputes nodes and weights, and hence can be reused for many
    174. 

  174.       different frequencies with good efficiency. The integrator is pimpl'd and hence
    175. 

  175.       can be shared between threads without a <code class="computeroutput"><span class="identifier">memcpy</span></code>
    176. 

  176.       of the nodes and weights.
    177. 

  177.     </p>
    178. 

  178. <p>
    179. 

  179.       Ooura and Mori's paper identifies criteria for rapid convergence based on the
    180. 

  180.       position of the poles of the integrand in the complex plane. If these poles
    181. 

  181.       are too close to the real axis the convergence is slow. It is not trivial to
    182. 

  182.       predict the convergence rate a priori, so if you are interested in figuring
    183. 

  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. 

  184.       of printing will give you a good idea of how well this method is performing.
    185. 

  185.     </p>
    186. 

  186. <h6>
    187. 

  187. <a name="math_toolkit.fourier_integrals.h2"></a>
    188. 

  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. 

  189.       precision</a>
    190. 

  190.     </h6>
    191. 

  191. <p>
    192. 

  192.       It is simple to extend to higher precision using <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
    193. 

  193.     </p>
    194. 

  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. 

  195. <span class="comment">//auto integrator = ooura_fourier_cos&lt;Real&gt;();</span>
    196. 

  196. <span class="comment">// Decide on a (tight) tolerance.</span>
    197. 

  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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">();</span>
    198. 

  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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;(</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. 

  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. 

  201. <span class="special">{</span> <span class="comment">// More complex example function.</span>
    202. 

  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. 

  203. <span class="special">};</span>
    204. 

  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. 

  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. 

  207. </pre>
    208. 

  208. <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Integral = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">result</span> <span class="special">&lt;&lt;</span> <span class="string">", relative error estimate "</span> <span class="special">&lt;&lt;</span> <span class="identifier">relative_error</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    209. 

  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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;()</span> <span class="special">/</span> <span class="identifier">e</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;();</span> <span class="comment">// Expect integral = 1/(2e)</span>
    211. 

  211. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"pi/(2e)  = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">expected</span> <span class="special">&lt;&lt;</span> <span class="string">", difference "</span> <span class="special">&lt;&lt;</span> <span class="identifier">result</span> <span class="special">-</span> <span class="identifier">expected</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    212. 

  212. </pre>
    213. 

  213. <p>
    214. 

  214.       with output:
    215. 

  215.     </p>
    216. 

  216. <pre class="programlisting">
    217. 

  217. Integral = 0.5778636748954608589550465916563501587, relative error estimate 4.609814684522163895264277312610830278e-17
    218. 

  218. pi/(2e) = 0.5778636748954608659545328919193707407, difference -6.999486300263020581921171645255733758e-18
    219. 

  219. 

  220. </pre>
    221. 

  221. <p>
    222. 

  222.       And with diagnostics on:
    223. 

  223.     </p>
    224. 

  224. <pre class="programlisting">
    225. 

  225. ooura_fourier_cos with relative error goal 3.851859888774471706111955885169854637e-34 &amp; 15 levels.
    226. 

  226. epsilon for type = 1.925929944387235853055977942584927319e-34
    227. 

  227. h = 1.000000000000000000000000000000000, I_h = 0.588268622591776615359568690603776 = 0.5882686225917766153595686906037760, absolute error estimate = nan
    228. 

  228. h = 0.500000000000000000000000000000000, I_h = 0.577871642184837461311756940493259 = 0.5778716421848374613117569404932595, absolute error estimate = 1.039698040693915404781175011051656e-02
    229. 

  229. h = 0.250000000000000000000000000000000, I_h = 0.577863671186882539559996800783122 = 0.5778636711868825395599968007831220, absolute error estimate = 7.970997954921751760139710137450075e-06
    230. 

  230. h = 0.125000000000000000000000000000000, I_h = 0.577863674895460885593491133506723 = 0.5778636748954608855934911335067232, absolute error estimate = 3.708578346033494332723601147051768e-09
    231. 

  231. h = 0.062500000000000000000000000000000, I_h = 0.577863674895460858955046591656350 = 0.5778636748954608589550465916563502, absolute error estimate = 2.663844454185037302771663314961535e-17
    232. 

  232. h = 0.031250000000000000000000000000000, I_h = 0.577863674895460858955046591656348 = 0.5778636748954608589550465916563484, absolute error estimate = 1.733336949948512267750380148326435e-33
    233. 

  233. h = 0.015625000000000000000000000000000, I_h = 0.577863674895460858955046591656348 = 0.5778636748954608589550465916563479, absolute error estimate = 4.814824860968089632639944856462318e-34
    234. 

  234. h = 0.007812500000000000000000000000000, I_h = 0.577863674895460858955046591656347 = 0.5778636748954608589550465916563473, absolute error estimate = 6.740754805355325485695922799047246e-34
    235. 

  235. h = 0.003906250000000000000000000000000, I_h = 0.577863674895460858955046591656347 = 0.5778636748954608589550465916563475, absolute error estimate = 1.925929944387235853055977942584927e-34
    236. 

  236. Integral = 5.778636748954608589550465916563475e-01, relative error estimate 3.332844800697411177051445985473052e-34
    237. 

  237. pi/(2e)  = 5.778636748954608589550465916563481e-01, difference -6.740754805355325485695922799047246e-34
    238. 

  238. 

  239. </pre>
    240. 

  240. <p>
    241. 

  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. 

  242.     </p>
    243. 

  243. <p>
    244. 

  244.       For more examples of other functions and tests, see the full test suite at
    245. 

  245.       <a href="../../../test/ooura_fourier_integral_test.cpp" target="_top">ooura_fourier_integral_test.cpp</a>.
    246. 

  246.     </p>
    247. 

  247. <p>
    248. 

  248.       Ngyen and Nuyens make use of <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
    249. 

  249.       in their extension to multiple dimensions, showing relative errors reducing
    250. 

  250.       to &#8773; 10<sup>-2000</sup>!
    251. 

  251.     </p>
    252. 

  252. <h6>
    253. 

  253. <a name="math_toolkit.fourier_integrals.h3"></a>
    254. 

  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. 

  255.     </h6>
    256. 

  256. <p>
    257. 

  257.       This implementation is base on Ooura's 1999 paper rather than the later 2005
    258. 

  258.       paper. It does not preclude a second future implementation based on the later
    259. 

  259.       work.
    260. 

  260.     </p>
    261. 

  261. <p>
    262. 

  262.       Some of the features of the Ooura's 2005 paper that were less appealing were:
    263. 

  263.     </p>
    264. 

  264. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    265. 

  265. <li class="listitem">
    266. 

  266.           The advance of that paper is that one can compute <span class="emphasis"><em>both</em></span>
    267. 

  267.           the Fourier sine transform and Fourier cosine transform in a single shot.
    268. 

  268.           But there are examples, like sinc integral, where the Fourier sine would
    269. 

  269.           converge, but the Fourier cosine would diverge.
    270. 

  270.         </li>
    271. 

  271. <li class="listitem">
    272. 

  272.           It would force users to live in the complex plane, when many potential
    273. 

  273.           applications only need real.
    274. 

  274.         </li>
    275. 

  275. </ul></div>
    276. 

  276. <h5>
    277. 

  277. <a name="math_toolkit.fourier_integrals.h4"></a>
    278. 

  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. 

  279.     </h5>
    280. 

  280. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    281. 

  281. <li class="listitem">
    282. 

  282.           Ooura, Takuya, and Masatake Mori, <span class="emphasis"><em>A robust double exponential
    283. 

  283.           formula for Fourier-type integrals.</em></span> Journal of computational
    284. 

  284.           and applied mathematics, 112.1-2 (1999): 229-241.
    285. 

  285.         </li>
    286. 

  286. <li class="listitem">
    287. 

  287.           Ooura, Takuya, <span class="emphasis"><em>A Double Exponential Formula for the Fourier Transforms.</em></span>
    288. 

  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. 

  289.         </li>
    290. 

  290. <li class="listitem">
    291. 

  291.           Khatibi, Arezoo and Khatibi, Omid,<span class="emphasis"><em>Criteria for the Application
    292. 

  292.           of Double Exponential Transformation.</em></span> (2017) <a href="https://arxiv.org/pdf/1704.05752.pdf" target="_top">1704.05752.pdf</a>.
    293. 

  293.         </li>
    294. 

  294. <li class="listitem">
    295. 

  295.           Nguyen, Dong T.P. and Nuyens, Dirk, <span class="emphasis"><em>Multivariate integration
    296. 

  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. 

  297.         </li>
    298. 

  298. </ul></div>
    299. 

  299. </div>
    300. 

  300. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    301. 

  301. <td align="left"></td>
    302. 

  302. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    303. 

  303.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    304. 

  304.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    305. 

  305.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    306. 

  306.       Daryle Walker and Xiaogang Zhang<p>
    307. 

  307.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    308. 

  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. 

  309.       </p>
    310. 

  310. </div></td>
    311. 

  311. </tr></table>
    312. 

  312. <hr>
    313. 

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    314. 

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