ceph/src/boost/libs/math/doc/html/math_toolkit/number_series/tangent_numbers.html

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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 <p>
  46         <span class="inlinemediaobject"><img src="../../../equations/tangent_number_def.svg"></span>
  47       </p>
  48 <p>
  49         Which gives:
  50       </p>
  51 <p>
  52         <span class="inlinemediaobject"><img src="../../../equations/tangent_number_def2.svg"></span>
  53       </p>
  54 <p>
  55         Tangent numbers are used in the computation of Bernoulli numbers, but are
  56         also made available here.
  57       </p>
  58 <h5>
  59 <a name="math_toolkit.number_series.tangent_numbers.h0"></a>
  60         <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>
  61       </h5>
  62 <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>
  63 </pre>
  64 <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>
  65 <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>
  66
  67 <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>
  68 <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>
  69
  70 <span class="comment">// Multiple tangent numbers (default policy).</span>
  71 <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>
  72 <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>
  73                                     <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span>
  74                                     <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span>
  75
  76 <span class="comment">// Multiple tangent numbers (user policy).</span>
  77 <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>
  78 <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>
  79                                     <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span>
  80                                     <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span>
  81                                     <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>
  82 </pre>
  83 <h5>
  84 <a name="math_toolkit.number_series.tangent_numbers.h1"></a>
  85         <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>
  86       </h5>
  87 <p>
  88         We can compute and save a few Tangent numbers.
  89       </p>
  90 <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>
  91
  92 <span class="comment">// Start with Bernoulli number 0.</span>
  93 <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>
  94
  95 <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>
  96 <span class="special">{</span> <span class="comment">// Show vector of even Tangent numbers, showing all significant decimal digits.</span>
  97     <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>
  98         <span class="special">&lt;&lt;</span> <span class="string">" "</span>
  99         <span class="special">&lt;&lt;</span> <span class="identifier">tn</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
 100 <span class="special">}</span>
 101 <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>
 102 </pre>
 103 <p>
 104         The output is:
 105       </p>
 106 <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>
 107 </pre>
 108 <p>
 109         The source of this example is at <a href="../../../../example/bernoulli_example.cpp" target="_top">bernoulli_example.cpp</a>
 110       </p>
 111 <h5>
 112 <a name="math_toolkit.number_series.tangent_numbers.h2"></a>
 113         <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>
 114       </h5>
 115 <p>
 116         Tangent numbers are calculated as intermediates in the calculation of the
 117         <a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli numbers</a>:
 118         refer to the <a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli
 119         numbers</a> documentation for details.
 120       </p>
 121 </div>
 122 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 123 <td align="left"></td>
 124 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 125       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 126       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 127       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 128         Distributed under the Boost Software License, Version 1.0. (See accompanying
 129         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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