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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.number_series.tangent_numbers"></a><a class="link" href="tangent_numbers.html" title="Tangent Numbers">Tangent Numbers</a>
  28 </h3></div></div></div>
  29 <p>
  30         <a href="http://en.wikipedia.org/wiki/Tangent_numbers" target="_top">Tangent numbers</a>,
  31         also called a zag function. See also <a href="http://mathworld.wolfram.com/TangentNumber.html" target="_top">Tangent
  32         number</a>.
  33       </p>
  34 <p>
  35         The first few values are 1, 2, 16, 272, 7936, 353792, 22368256, 1903757312
  36         ... (sequence <a href="http://oeis.org/A000182" target="_top">A000182 in OEIS</a>).
  37         They are called tangent numbers because they appear as numerators in the
  38         Maclaurin series of <code class="computeroutput"><span class="identifier">tan</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code>.
  39       </p>
  40 <p>
  41         <span class="bold"><strong>Important:</strong></span> there are two competing definitions
  42         of Tangent numbers in common use (depending on whether you take the even
  43         or odd numbered values as non-zero), we use:
  44       </p>
  45 <div class="blockquote"><blockquote class="blockquote"><p>
  46           <span class="inlinemediaobject"><img src="../../../equations/tangent_number_def.svg"></span>
  47
  48         </p></blockquote></div>
  49 <p>
  50         Which gives:
  51       </p>
  52 <div class="blockquote"><blockquote class="blockquote"><p>
  53           <span class="inlinemediaobject"><img src="../../../equations/tangent_number_def2.svg"></span>
  54
  55         </p></blockquote></div>
  56 <p>
  57         Tangent numbers are used in the computation of Bernoulli numbers, but are
  58         also made available here.
  59       </p>
  60 <h5>
  61 <a name="math_toolkit.number_series.tangent_numbers.h0"></a>
  62         <span class="phrase"><a name="math_toolkit.number_series.tangent_numbers.synopsis"></a></span><a class="link" href="tangent_numbers.html#math_toolkit.number_series.tangent_numbers.synopsis">Synopsis</a>
  63       </h5>
  64 <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">special_functions</span><span class="special">/</span><span class="identifier">detail</span><span class="special">/</span><span class="identifier">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  65 </pre>
  66 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  67 <span class="identifier">T</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">);</span>  <span class="comment">// Single tangent number (default policy).</span>
  68
  69 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
  70 <span class="identifier">T</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span> <span class="special">&amp;</span><span class="identifier">pol</span><span class="special">);</span> <span class="comment">// Single tangent number (user policy).</span>
  71
  72 <span class="comment">// Multiple tangent numbers (default policy).</span>
  73 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
  74 <span class="identifier">OutputIterator</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>
  75                                     <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span>
  76                                     <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span>
  77
  78 <span class="comment">// Multiple tangent numbers (user policy).</span>
  79 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
  80 <span class="identifier">OutputIterator</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>
  81                                     <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span>
  82                                     <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span>
  83                                     <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>
  84 </pre>
  85 <h5>
  86 <a name="math_toolkit.number_series.tangent_numbers.h1"></a>
  87         <span class="phrase"><a name="math_toolkit.number_series.tangent_numbers.examples"></a></span><a class="link" href="tangent_numbers.html#math_toolkit.number_series.tangent_numbers.examples">Examples</a>
  88       </h5>
  89 <p>
  90         We can compute and save a few Tangent numbers.
  91       </p>
  92 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="identifier">tn</span><span class="special">;</span> <span class="comment">// Space for some `float` precision Tangent numbers.</span>
  93
  94 <span class="comment">// Start with Bernoulli number 0.</span>
  95 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tangent_t2n</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;(</span><span class="number">1</span><span class="special">,</span> <span class="number">6</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">back_inserter</span><span class="special">(</span><span class="identifier">tn</span><span class="special">));</span> <span class="comment">// Fill vector with even Tangent numbers.</span>
  96
  97 <span class="keyword">for</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="identifier">tn</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span> <span class="identifier">i</span><span class="special">++)</span>
  98 <span class="special">{</span> <span class="comment">// Show vector of even Tangent numbers, showing all significant decimal digits.</span>
  99     <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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="keyword">float</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">)</span>
 100         <span class="special">&lt;&lt;</span> <span class="string">" "</span>
 101         <span class="special">&lt;&lt;</span> <span class="identifier">tn</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
 102 <span class="special">}</span>
 103 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</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>
 104 </pre>
 105 <p>
 106         The output is:
 107       </p>
 108 <pre class="programlisting"><span class="number">1</span> <span class="number">2</span> <span class="number">16</span> <span class="number">272</span> <span class="number">7936</span> <span class="number">353792</span>
 109 </pre>
 110 <p>
 111         The source of this example is at <a href="../../../../example/bernoulli_example.cpp" target="_top">bernoulli_example.cpp</a>
 112       </p>
 113 <h5>
 114 <a name="math_toolkit.number_series.tangent_numbers.h2"></a>
 115         <span class="phrase"><a name="math_toolkit.number_series.tangent_numbers.implementation"></a></span><a class="link" href="tangent_numbers.html#math_toolkit.number_series.tangent_numbers.implementation">Implementation</a>
 116       </h5>
 117 <p>
 118         Tangent numbers are calculated as intermediates in the calculation of the
 119         <a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli numbers</a>:
 120         refer to the <a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli
 121         numbers</a> documentation for details.
 122       </p>
 123 </div>
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 126 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 127       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 128       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 129       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 130       Daryle Walker and Xiaogang Zhang<p>
 131         Distributed under the Boost Software License, Version 1.0. (See accompanying
 132         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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