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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Bernoulli Numbers</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.number_series.bernoulli_numbers"></a><a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli
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Numbers</a>
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</h3></div></div></div>
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<p>
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<a href="https://en.wikipedia.org/wiki/Bernoulli_number" target="_top">Bernoulli numbers</a>
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are a sequence of rational numbers useful for the Taylor series expansion,
|
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Euler-Maclaurin formula, and the Riemann zeta function.
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</p>
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<p>
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Bernoulli numbers are used in evaluation of some Boost.Math functions, including
|
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the <a class="link" href="../sf_gamma/tgamma.html" title="Gamma">tgamma</a>, <a class="link" href="../sf_gamma/lgamma.html" title="Log Gamma">lgamma</a>
|
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and polygamma functions.
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</p>
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<h5>
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||||
<a name="math_toolkit.number_series.bernoulli_numbers.h0"></a>
|
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<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.single_bernoulli_number"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.single_bernoulli_number">Single
|
||||
Bernoulli number</a>
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</h5>
|
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<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h1"></a>
|
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<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.synopsis"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.synopsis">Synopsis</a>
|
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</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">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
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</pre>
|
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<pre class="programlisting"><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">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">bernoulli_b2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">n</span><span class="special">);</span> <span class="comment">// Single Bernoulli number (default policy).</span>
|
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|
||||
<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>
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||||
<span class="identifier">T</span> <span class="identifier">bernoulli_b2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">n</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">// User policy for errors etc.</span>
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|
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
|
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<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h2"></a>
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<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.description"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.description">Description</a>
|
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</h5>
|
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<p>
|
||||
Both return the (2 * n)<sup>th</sup> Bernoulli number B<sub>2n</sub>.
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</p>
|
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<p>
|
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Note that since all odd numbered Bernoulli numbers are zero (apart from B<sub>1</sub> which
|
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is -½) the interface will only return the even numbered Bernoulli numbers.
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</p>
|
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<p>
|
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This function uses fast table lookup for low-indexed Bernoulli numbers, while
|
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larger values are calculated as needed and then cached. The caching mechanism
|
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requires a certain amount of thread safety code, so <code class="computeroutput"><span class="identifier">unchecked_bernoulli_b2n</span></code>
|
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may provide a better interface for performance critical code.
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</p>
|
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<p>
|
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The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
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be used to control the behaviour of the function: how it handles errors,
|
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what level of precision to use, etc.
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</p>
|
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<p>
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Refer to <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policies</a> for more details.
|
||||
</p>
|
||||
<h5>
|
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<a name="math_toolkit.number_series.bernoulli_numbers.h3"></a>
|
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<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.examples"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.examples">Examples</a>
|
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</h5>
|
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<p>
|
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A simple example computes the value of B<sub>4</sub> where the return type is <code class="computeroutput"><span class="keyword">double</span></code>, note that the argument to bernoulli_b2n
|
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is <span class="emphasis"><em>2</em></span> not <span class="emphasis"><em>4</em></span> since it computes B<sub>2N</sub>.
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</p>
|
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<pre class="programlisting"><span class="keyword">try</span>
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<span class="special">{</span> <span class="comment">// It is always wise to use try'n'catch blocks around Boost.Math functions</span>
|
||||
<span class="comment">// so that any informative error messages can be displayed in the catch block.</span>
|
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span>
|
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<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">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="number">2</span><span class="special">)</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>
|
||||
So B<sub>4</sub> == -1/30 == -0.0333333333333333
|
||||
</p>
|
||||
<p>
|
||||
If we use Boost.Multiprecision and its 50 decimal digit floating-point type
|
||||
<code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>, we can
|
||||
calculate the value of much larger numbers like B<sub>200</sub>
|
||||
and also obtain much
|
||||
higher precision.
|
||||
</p>
|
||||
<pre class="programlisting"><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="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</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">bernoulli_b2n</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">>(</span><span class="number">100</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="special">-</span><span class="number">3.6470772645191354362138308865549944904868234686191e+215</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.single_unchecked_bernoulli_numbe"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.single_unchecked_bernoulli_numbe">Single
|
||||
(unchecked) Bernoulli number</a>
|
||||
</h5>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h5"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.synopsis0"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.synopsis0">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">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">struct</span> <span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="identifier">T</span><span class="special">>;</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">inline</span> <span class="identifier">T</span> <span class="identifier">unchecked_bernoulli_b2n</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">unchecked_bernoulli_b2n</span></code> provides
|
||||
access to Bernoulli numbers <span class="bold"><strong>without any checks for
|
||||
overflow or invalid parameters</strong></span>. It is implemented as a direct
|
||||
(and very fast) table lookup, and while not recommended for general use it
|
||||
can be useful inside inner loops where the ultimate performance is required,
|
||||
and error checking is moved outside the loop.
|
||||
</p>
|
||||
<p>
|
||||
The largest value you can pass to <code class="computeroutput"><span class="identifier">unchecked_bernoulli_b2n</span><span class="special"><></span></code> is <code class="computeroutput"><span class="identifier">max_bernoulli_b2n</span><span class="special"><>::</span><span class="identifier">value</span></code>:
|
||||
passing values greater than that will result in a buffer overrun error, so
|
||||
it's clearly important to place the error handling in your own code when
|
||||
using this direct interface.
|
||||
</p>
|
||||
<p>
|
||||
The value of <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value</span></code> varies by the type T, for types
|
||||
<code class="computeroutput"><span class="keyword">float</span></code>/<code class="computeroutput"><span class="keyword">double</span></code>/<code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
|
||||
it's the largest value which doesn't overflow the target type: for example,
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span></code> is 129. However, for multiprecision
|
||||
types, it's the largest value for which the result can be represented as
|
||||
the ratio of two 64-bit integers, for example <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">value</span></code>
|
||||
is just 17. Of course larger indexes can be passed to <code class="computeroutput"><span class="identifier">bernoulli_b2n</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">n</span><span class="special">)</span></code>, but
|
||||
then you lose fast table lookup (i.e. values may need to be calculated).
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">/*For example:
|
||||
*/</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"boost::math::max_bernoulli_b2n<float>::value = "</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">value</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</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="string">"Maximum Bernoulli number using float is "</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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">value</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</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="string">"boost::math::max_bernoulli_b2n<double>::value = "</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</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="string">"Maximum Bernoulli number using double is "</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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">value</span> <span class="special">=</span> <span class="number">32</span>
|
||||
<span class="identifier">Maximum</span> <span class="identifier">Bernoulli</span> <span class="identifier">number</span> <span class="keyword">using</span> <span class="keyword">float</span> <span class="identifier">is</span> <span class="special">-</span><span class="number">2.0938e+038</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span> <span class="special">=</span> <span class="number">129</span>
|
||||
<span class="identifier">Maximum</span> <span class="identifier">Bernoulli</span> <span class="identifier">number</span> <span class="keyword">using</span> <span class="keyword">double</span> <span class="identifier">is</span> <span class="number">1.33528e+306</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h6"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.multiple_bernoulli_numbers"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.multiple_bernoulli_numbers">Multiple
|
||||
Bernoulli Numbers</a>
|
||||
</h5>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h7"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.synopsis1"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.synopsis1">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">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<pre class="programlisting"><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="comment">// Multiple Bernoulli 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">bernoulli_b2n</span><span class="special">(</span>
|
||||
<span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>
|
||||
<span class="keyword">unsigned</span> <span class="identifier">number_of_bernoullis_b2n</span><span class="special">,</span>
|
||||
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// Multiple Bernoulli 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">bernoulli_b2n</span><span class="special">(</span>
|
||||
<span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>
|
||||
<span class="keyword">unsigned</span> <span class="identifier">number_of_bernoullis_b2n</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>
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h8"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.description0"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.description0">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
Two versions of the Bernoulli number function are provided to compute multiple
|
||||
Bernoulli numbers with one call (one with default policy and the other allowing
|
||||
a user-defined policy).
|
||||
</p>
|
||||
<p>
|
||||
These return a series of Bernoulli numbers:
|
||||
</p>
|
||||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||
B<sub>2*start_index</sub>,B<sub>2*(start_index+1)</sub>,...,B<sub>2*(start_index+number_of_bernoullis_b2n-1)</sub>
|
||||
</p></blockquote></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h9"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.examples0"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.examples0">Examples</a>
|
||||
</h5>
|
||||
<p>
|
||||
We can compute and save all the float-precision Bernoulli numbers from one
|
||||
call.
|
||||
</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">bn</span><span class="special">;</span> <span class="comment">// Space for 32-bit `float` precision Bernoulli 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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>(</span><span class="number">0</span><span class="special">,</span> <span class="number">32</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">bn</span><span class="special">));</span> <span class="comment">// Fill vector with even Bernoulli 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">bn</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 Bernoulli 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="identifier">i</span><span class="special">*</span><span class="number">2</span> <span class="special"><<</span> <span class="char">' '</span>
|
||||
<span class="special"><<</span> <span class="identifier">bn</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">endl</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="number">0</span> <span class="number">1</span>
|
||||
<span class="number">2</span> <span class="number">0.166667</span>
|
||||
<span class="number">4</span> <span class="special">-</span><span class="number">0.0333333</span>
|
||||
<span class="number">6</span> <span class="number">0.0238095</span>
|
||||
<span class="number">8</span> <span class="special">-</span><span class="number">0.0333333</span>
|
||||
<span class="number">10</span> <span class="number">0.0757576</span>
|
||||
<span class="number">12</span> <span class="special">-</span><span class="number">0.253114</span>
|
||||
<span class="number">14</span> <span class="number">1.16667</span>
|
||||
<span class="number">16</span> <span class="special">-</span><span class="number">7.09216</span>
|
||||
<span class="number">18</span> <span class="number">54.9712</span>
|
||||
<span class="number">20</span> <span class="special">-</span><span class="number">529.124</span>
|
||||
<span class="number">22</span> <span class="number">6192.12</span>
|
||||
<span class="number">24</span> <span class="special">-</span><span class="number">86580.3</span>
|
||||
<span class="number">26</span> <span class="number">1.42552e+006</span>
|
||||
<span class="number">28</span> <span class="special">-</span><span class="number">2.72982e+007</span>
|
||||
<span class="number">30</span> <span class="number">6.01581e+008</span>
|
||||
<span class="number">32</span> <span class="special">-</span><span class="number">1.51163e+010</span>
|
||||
<span class="number">34</span> <span class="number">4.29615e+011</span>
|
||||
<span class="number">36</span> <span class="special">-</span><span class="number">1.37117e+013</span>
|
||||
<span class="number">38</span> <span class="number">4.88332e+014</span>
|
||||
<span class="number">40</span> <span class="special">-</span><span class="number">1.92966e+016</span>
|
||||
<span class="number">42</span> <span class="number">8.41693e+017</span>
|
||||
<span class="number">44</span> <span class="special">-</span><span class="number">4.03381e+019</span>
|
||||
<span class="number">46</span> <span class="number">2.11507e+021</span>
|
||||
<span class="number">48</span> <span class="special">-</span><span class="number">1.20866e+023</span>
|
||||
<span class="number">50</span> <span class="number">7.50087e+024</span>
|
||||
<span class="number">52</span> <span class="special">-</span><span class="number">5.03878e+026</span>
|
||||
<span class="number">54</span> <span class="number">3.65288e+028</span>
|
||||
<span class="number">56</span> <span class="special">-</span><span class="number">2.84988e+030</span>
|
||||
<span class="number">58</span> <span class="number">2.38654e+032</span>
|
||||
<span class="number">60</span> <span class="special">-</span><span class="number">2.14e+034</span>
|
||||
<span class="number">62</span> <span class="number">2.0501e+036</span>
|
||||
</pre>
|
||||
<p>
|
||||
Of course, for any floating-point type, there is a maximum Bernoulli number
|
||||
that can be computed before it overflows the exponent. By default policy,
|
||||
if we try to compute too high a Bernoulli number, an exception will be thrown.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">try</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">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">"Bernoulli number "</span> <span class="special"><<</span> <span class="number">33</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">endl</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">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>(</span><span class="number">33</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
<span class="keyword">catch</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span> <span class="identifier">ex</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="string">"Thrown Exception caught: "</span> <span class="special"><<</span> <span class="identifier">ex</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
and we will get a helpful error message (provided try'n'catch blocks are
|
||||
used).
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">Bernoulli</span> <span class="identifier">number</span> <span class="number">66</span>
|
||||
<span class="identifier">Thrown</span> <span class="identifier">Exception</span> <span class="identifier">caught</span><span class="special">:</span> <span class="identifier">Error</span> <span class="identifier">in</span> <span class="identifier">function</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>(</span><span class="identifier">n</span><span class="special">):</span>
|
||||
<span class="identifier">Overflow</span> <span class="identifier">evaluating</span> <span class="identifier">function</span> <span class="identifier">at</span> <span class="number">33</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.bernoulli_numbers.h10"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.accuracy"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
All the functions usually return values within one ULP (unit in the last
|
||||
place) for the floating-point type.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.bernoulli_numbers.h11"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.implementation"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
The implementation details are in <a href="../../../../include/boost/math/special_functions/detail/bernoulli_details.hpp" target="_top">bernoulli_details.hpp</a>
|
||||
and <a href="../../../../include/boost/math/special_functions/detail/unchecked_bernoulli.hpp" target="_top">unchecked_bernoulli.hpp</a>.
|
||||
</p>
|
||||
<p>
|
||||
For <code class="computeroutput"><span class="identifier">i</span> <span class="special"><=</span>
|
||||
<span class="identifier">max_bernoulli_index</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value</span></code> this is implemented by simple table
|
||||
lookup from a statically initialized table; for larger values of <code class="computeroutput"><span class="identifier">i</span></code>, this is implemented by the Tangent Numbers
|
||||
algorithm as described in the paper: Fast Computation of Bernoulli, Tangent
|
||||
and Secant Numbers, Richard P. Brent and David Harvey, <a href="http://arxiv.org/pdf/1108.0286v3.pdf" target="_top">http://arxiv.org/pdf/1108.0286v3.pdf</a>
|
||||
(2011).
|
||||
</p>
|
||||
<p>
|
||||
<a href="http://mathworld.wolfram.com/TangentNumber.html" target="_top">Tangent (or
|
||||
Zag) numbers</a> (an even alternating permutation number) are defined
|
||||
and their generating function is also given therein.
|
||||
</p>
|
||||
<p>
|
||||
The relation of Tangent numbers with Bernoulli numbers <span class="emphasis"><em>B<sub>i</sub></em></span>
|
||||
is given by Brent and Harvey's equation 14:
|
||||
</p>
|
||||
<p>
|
||||
   <span class="inlinemediaobject"><img src="../../../equations/tangent_numbers.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
Their relation with Bernoulli numbers <span class="emphasis"><em>B<sub>i</sub></em></span> are defined
|
||||
by
|
||||
</p>
|
||||
<p>
|
||||
if i > 0 and i is even then    <span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers.svg"></span> <br> elseif
|
||||
i == 0 then <span class="emphasis"><em>B<sub>i</sub></em></span> = 1 <br> elseif i == 1 then <span class="emphasis"><em>B<sub>i</sub></em></span>
|
||||
= -1/2 <br> elseif i < 0 or i is odd then <span class="emphasis"><em>B<sub>i</sub></em></span> =
|
||||
0
|
||||
</p>
|
||||
<p>
|
||||
Note that computed values are stored in a fixed-size table, access is thread
|
||||
safe via atomic operations (i.e. lock free programming), this imparts a much
|
||||
lower overhead on access to cached values than might otherwise be expected
|
||||
- typically for multiprecision types the cost of thread synchronisation is
|
||||
negligible, while for built in types this code is not normally executed anyway.
|
||||
For very large arguments which cannot be reasonably computed or stored in
|
||||
our cache, an asymptotic expansion <a href="http://www.luschny.de/math/primes/bernincl.html" target="_top">due
|
||||
to Luschny</a> is used:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers2.svg"></span>
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
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>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="../number_series.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../number_series.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="tangent_numbers.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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||||
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|
||||
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|
||||
</html>
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||||
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|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Prime Numbers</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
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<link rel="up" href="../number_series.html" title="Number Series">
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||||
<link rel="prev" href="tangent_numbers.html" title="Tangent Numbers">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<a accesskey="p" href="tangent_numbers.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../number_series.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="math_toolkit.number_series.primes"></a><a class="link" href="primes.html" title="Prime Numbers">Prime Numbers</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.primes.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.primes.synopsis"></a></span><a class="link" href="primes.html#math_toolkit.number_series.primes.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">prime</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<pre class="programlisting"><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">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint32_t</span> <span class="identifier">prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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="identifier">boost</span><span class="special">::</span><span class="identifier">uint32_t</span> <span class="identifier">prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">static</span> <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">max_prime</span> <span class="special">=</span> <span class="number">10000</span><span class="special">;</span>
|
||||
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.number_series.primes.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.number_series.primes.description"></a></span><a class="link" href="primes.html#math_toolkit.number_series.primes.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
The function <code class="computeroutput"><span class="identifier">prime</span></code> provides
|
||||
fast table lookup to the first 10000 prime numbers (starting from 2 as the
|
||||
zeroth prime: as 1 isn't terribly useful in practice). There are two function
|
||||
signatures one of which takes an optional <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a>
|
||||
as the second parameter to control error handling.
|
||||
</p>
|
||||
<p>
|
||||
The constant <code class="computeroutput"><span class="identifier">max_prime</span></code> is
|
||||
the largest value you can pass to <code class="computeroutput"><span class="identifier">prime</span></code>
|
||||
without incurring an error.
|
||||
</p>
|
||||
<p>
|
||||
Passing a value greater than <code class="computeroutput"><span class="identifier">max_prime</span></code>
|
||||
results in a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
||||
being raised.
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
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>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="tangent_numbers.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../number_series.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,138 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Tangent Numbers</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
||||
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
|
||||
<link rel="up" href="../number_series.html" title="Number Series">
|
||||
<link rel="prev" href="bernoulli_numbers.html" title="Bernoulli Numbers">
|
||||
<link rel="next" href="primes.html" title="Prime Numbers">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<table cellpadding="2" width="100%"><tr>
|
||||
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
|
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<td align="center"><a href="../../../../../../index.html">Home</a></td>
|
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<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
|
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<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
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<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
||||
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="bernoulli_numbers.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../number_series.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="primes.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<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>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
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>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="bernoulli_numbers.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../number_series.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="primes.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
Reference in New Issue
Block a user