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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.number_series.tangent_numbers"></a><a class="link" href="tangent_numbers.html" title="Tangent Numbers">Tangent Numbers</a> |
| </h3></div></div></div> |
| <p> |
| <a href="http://en.wikipedia.org/wiki/Tangent_numbers" target="_top">Tangent numbers</a>, |
| also called a zag function. See also <a href="http://mathworld.wolfram.com/TangentNumber.html" target="_top">Tangent |
| number</a>. |
| </p> |
| <p> |
| The first few values are 1, 2, 16, 272, 7936, 353792, 22368256, 1903757312 |
| ... (sequence <a href="http://oeis.org/A000182" target="_top">A000182 in OEIS</a>). |
| They are called tangent numbers because they appear as numerators in the |
| 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>. |
| </p> |
| <p> |
| <span class="bold"><strong>Important:</strong></span> there are two competing definitions |
| of Tangent numbers in common use (depending on whether you take the even |
| or odd numbered values as non-zero), we use: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/tangent_number_def.svg"></span> |
| </p> |
| <p> |
| Which gives: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/tangent_number_def2.svg"></span> |
| </p> |
| <p> |
| Tangent numbers are used in the computation of Bernoulli numbers, but are |
| also made available here. |
| </p> |
| <h5> |
| <a name="math_toolkit.number_series.tangent_numbers.h0"></a> |
| <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> |
| </h5> |
| <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> |
| </pre> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <span class="comment">// Multiple tangent numbers (default policy).</span> |
| <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="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> |
| <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span> |
| <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> |
| |
| <span class="comment">// Multiple tangent numbers (user policy).</span> |
| <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> |
| <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> |
| <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span> |
| <span class="identifier">OutputIterator</span> <span class="identifier">out_it</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> |
| </pre> |
| <h5> |
| <a name="math_toolkit.number_series.tangent_numbers.h1"></a> |
| <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> |
| </h5> |
| <p> |
| We can compute and save a few Tangent numbers. |
| </p> |
| <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> |
| |
| <span class="comment">// Start with Bernoulli number 0.</span> |
| <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> |
| |
| <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> |
| <span class="special">{</span> <span class="comment">// Show vector of even Tangent numbers, showing all significant decimal digits.</span> |
| <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> |
| <span class="special"><<</span> <span class="string">" "</span> |
| <span class="special"><<</span> <span class="identifier">tn</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span> |
| <span class="special">}</span> |
| <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> |
| </pre> |
| <p> |
| The output is: |
| </p> |
| <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> |
| </pre> |
| <p> |
| The source of this example is at <a href="../../../../example/bernoulli_example.cpp" target="_top">bernoulli_example.cpp</a> |
| </p> |
| <h5> |
| <a name="math_toolkit.number_series.tangent_numbers.h2"></a> |
| <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> |
| </h5> |
| <p> |
| Tangent numbers are calculated as intermediates in the calculation of the |
| <a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli numbers</a>: |
| refer to the <a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli |
| numbers</a> documentation for details. |
| </p> |
| </div> |
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| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
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