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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
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Bernoulli number</a>
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</h5>
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<h5>
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<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>
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<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>
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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>
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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="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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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<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>
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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 20. 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 20. Policies: Controlling Precision, Error Handling etc">Policies</a> for more details.
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</p>
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<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>
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<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>
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<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>
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</pre>
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<p>
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So B<sub>4</sub> == -1/30 == -0.0333333333333333
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</p>
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<p>
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If we use Boost.Multiprecision and its 50 decimal digit floating-point type
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<code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>, we can
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calculate the value of much larger numbers like B<sub>200</sub>
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and also obtain much
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higher precision.
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</p>
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<pre class="programlisting"><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="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>
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<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>
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</pre>
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<pre class="programlisting"><span class="special">-</span><span class="number">3.6470772645191354362138308865549944904868234686191e+215</span>
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</pre>
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<h5>
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<a name="math_toolkit.number_series.bernoulli_numbers.h4"></a>
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<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
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(unchecked) Bernoulli number</a>
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</h5>
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<h5>
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<a name="math_toolkit.number_series.bernoulli_numbers.h5"></a>
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<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>
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</h5>
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<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">template</span> <span class="special"><></span>
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<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>
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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>
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<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>
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</pre>
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<p>
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<code class="computeroutput"><span class="identifier">unchecked_bernoulli_b2n</span></code> provides
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access to Bernoulli numbers <span class="bold"><strong>without any checks for
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overflow or invalid parameters</strong></span>. It is implemented as a direct
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(and very fast) table lookup, and while not recommended for general use it
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can be useful inside inner loops where the ultimate performance is required,
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and error checking is moved outside the loop.
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</p>
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<p>
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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>:
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passing values greater than that will result in a buffer overrun error, so
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it's clearly important to place the error handling in your own code when
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using this direct interface.
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</p>
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<p>
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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
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<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>
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it's the largest value which doesn't overflow the target type: for example,
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<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
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types, it's the largest value for which the result can be represented as
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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>
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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
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then you lose fast table lookup (i.e. values may need to be calculated).
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</p>
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<pre class="programlisting"><span class="comment">/*For example:
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*/</span>
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<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>
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<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>
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<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>
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<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>
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</pre>
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<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>
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<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>
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<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>
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<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>
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</pre>
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<h5>
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<a name="math_toolkit.number_series.bernoulli_numbers.h6"></a>
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<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
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Bernoulli Numbers</a>
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</h5>
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<h5>
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<a name="math_toolkit.number_series.bernoulli_numbers.h7"></a>
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<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>
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</h5>
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<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>
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<span class="comment">// Multiple Bernoulli numbers (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">OutputIterator</span><span class="special">></span>
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<span class="identifier">OutputIterator</span> <span class="identifier">bernoulli_b2n</span><span class="special">(</span>
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<span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>
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<span class="keyword">unsigned</span> <span class="identifier">number_of_bernoullis_b2n</span><span class="special">,</span>
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<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span>
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<span class="comment">// Multiple Bernoulli numbers (user 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">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>
|
|
<span class="serif_italic">[B<sub>2*start_index</sub>, B<sub>2*(start_index+1)</sub>, ..., B<sub>2*(start_index+number_of_bernoullis_b2n-1)</sub>]</span>
|
|
</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>
|
|
  
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/tangent_numbers.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers2.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
</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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>
|
|
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|
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