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24 | </div> | |
25 | <div class="section"> | |
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"><</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">></span> | |
63 | </pre> | |
64 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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">></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">&</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"><</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> | |
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"><</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">></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">&</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"><</span><span class="keyword">float</span><span class="special">></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"><</span><span class="keyword">float</span><span class="special">>(</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"><</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"><<</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"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> | |
98 | <span class="special"><<</span> <span class="string">" "</span> | |
99 | <span class="special"><<</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"><<</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 © 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 Råde, 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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